Superclasses 

Returns:

trajectory

traj = robot.jtraj(q0, qf, t) is a trajectory object whose attribute traj.q is a row-wise joint-space trajectory, a quintic polynomial in time from q0 to qf. It is the method form of jtraj().

Deprecated since version 1.5.0: Passing SE3 poses for q0 or qf is deprecated and now raises TypeError. The poses were converted to joint coordinates by inverse kinematics, which is never simple: which solution is chosen depends on the solver and its starting point, and it can fail. Do that explicitly, so you control and can check it::

q0 = robot.ikine_LM(T0, q0=robot.qr).q qf = robot.ikine_LM(Tf, q0=q0).q traj = robot.jtraj(q0, qf, t)

For a Cartesian trajectory see ctraj().

Changed in version 1.5.0: The arguments are joint coordinates only, they were named T1 and T2 and could be poses. The **kwargs that were passed to the inverse kinematics method are gone.

closest_point(q, shape, inf_dist=1.0, skip=False)[source]

Find the closest point between robot and shape

Parameters:

shape (Shape) – The shape to compare distance to
inf_dist (float) – The minimum distance within which to consider the shape
skip (bool) – Skip setting all shape transforms based on q, use this option if using this method in conjunction with Swift to save time

Return type:

tuple[float | None, ndarray | None, ndarray | None]

Returns:

tuple of (distance, point on robot, point on shape)

closest_point(shape, inf_dist) returns the minimum euclidean distance between this robot and shape, provided it is less than inf_dist. It will also return the points on self and shape in the world frame which connect the line of length distance between the shapes. If the distance is negative then the shapes are collided.

isselfcollided(q=None, skip=False)[source]

Check for collisions between non-adjacent links of this robot

Parameters:

q (ndarray | Sequence[float] | None) – joint configuration, defaults to the robot’s current q
skip (bool) – use the existing world transforms instead of updating them from q, defaults to False

Return type:

Returns:

True if any non-adjacent link pair is in collision

Raises:

ValueError – the model has no collision geometry, or q is not a single finite joint configuration

Checks the collision geometry of the robot and all its grippers. Shapes attached to the same link, neighbouring links, and links in the same gripper are not tested against each other. Links without collision geometry are transparent when determining neighbours: a link’s nearest ancestor with collision geometry is its neighbour. Links in separate branches are tested, including siblings outside a shared gripper and links belonging to different grippers. Touching counts as a collision, as in iscollided().

Each gripper uses its own current joint configuration. The robot’s stored joint configuration is not changed. With skip=True, q is ignored and the caller must have updated all world transforms.

Collision checking requires the optional coal dependency. A model with only visual geometry cannot be checked; collision geometry must be supplied explicitly. Only pairs with geometry are checked, so a False result does not establish clearance for links without geometry.

Seealso:: iscollided() closest_point()

iscollided(q, shape, skip=False)[source]

Check if the robot is in collision with a shape

Parameters:

q (ndarray | Sequence[float] | None) – joint configuration, or None for the current configuration
shape (Shape | Robot) – the external shape to check, or this robot itself for self-collision checking
skip (bool) – Skip setting all shape transforms based on q, use this option if using this method in conjunction with Swift to save time

Return type:

Returns:

True if shapes have collided

robot.iscollided(q, shape) checks for collision with an external shape. robot.iscollided(q, robot) calls isselfcollided().

Seealso:: isselfcollided() closest_point()

collided(q, shape, skip=False)[source]

Check if the robot is in collision with a shape (deprecated)

Parameters:

q (ndarray | Sequence[float] | None) – joint configuration, or None for the current configuration
shape (Shape | Robot) – the external shape to check, or this robot itself for self-collision checking
skip (bool) – Skip setting all shape transforms based on q, use this option if using this method in conjunction with Swift to save time

Return type:

Returns:

True if shapes have collided

robot.collided(q, shape) issues a warning and returns the result of iscollided().

Seealso:: iscollided() isselfcollided()

joint_velocity_damper(q=None, ps=0.05, pi=0.1, n=None, gain=1.0)[source]

Compute the joint velocity damper for QP motion control

Parameters:

ps (float) – The minimum angle (in radians) in which the joint is allowed to approach to its limit
pi (float) – The influence angle (in radians) in which the velocity damper becomes active
n (int | None) – The number of joints to consider. Defaults to all joints
gain (float) – The gain for the velocity damper

Return type:

tuple[ndarray, ndarray]

Returns:

tuple of (Ain, Bin) inequality constraint matrices for an optimiser

Formulates an inequality constraint which, when optimised for will make it impossible for the robot to run into joint limits. Requires the joint limits of the robot to be specified. See examples/mmc.py for use case.

link_collision_damper(shape, q, di=0.3, ds=0.05, xi=1.0, end=None, start=None, collision_list=None)[source]

Compute a collision constrain for QP motion control

Parameters:

ds (float) – The minimum distance in which a joint is allowed to approach the collision object shape
di (float) – The influence distance in which the velocity damper becomes active
xi (float) – The gain for the velocity damper
end (Link | None) – The end link of the robot to consider
start (Link | None) – The start link of the robot to consider
collision_list (list[Shape] | None) – A list of shapes to consider for collision

Return type:

Returns:

tuple of (Ain, Bin) inequality constraint matrices for an optimiser

Formulates an inequality constraint which, when optimised for will make it impossible for the robot to run into a collision. Requires See examples/neo.py for use case.

vision_collision_damper(shape, camera=None, camera_n=0, q=None, di=0.3, ds=0.05, xi=1.0, end=None, start=None, collision_list=None)[source]

Compute a vision collision constrain for QP motion control

Parameters:

camera (Robot | SE3 | None) – The camera link, either as a robotic link or SE3 pose
camera_n (int) – Degrees of freedom of the camera link
ds (float) – The minimum distance in which a joint is allowed to approach the collision object shape
di (float) – The influence distance in which the velocity damper becomes active
xi (float) – The gain for the velocity damper
end (str | Link | Gripper | None) – The end link of the robot to consider
start (str | Link | Gripper | None) – The start link of the robot to consider
collision_list (Any) – A list of shapes to consider for collision

Return type:

tuple[ndarray | None, ndarray | None]

Returns:

tuple of (Ain, Bin) inequality constraint matrices for an optimiser

Formulates an inequality constraint which, when optimised for will make it impossible for the robot to run into a line of sight. See examples/fetch_vision.py for use case.

rne(q, qd, qdd, symbolic=False, gravity=None)[source]

Compute inverse dynamics via recursive Newton-Euler formulation

Parameters:

q (ndarray) – Joint coordinates
qd (ndarray) – Joint velocity
qdd (ndarray) – Joint acceleration
symbolic (bool) – If True, supports symbolic expressions
gravity (ndarray | Sequence[float] | None) – gravitational acceleration in the world frame, downwards gravitational force is equivalent to robot base acceleration upwards (positive); defaults to attribute of self

Return type:

Returns:

Joint force/torques

rne_dh(q, qd, qdd) where the arguments have shape (n,) where n is the number of robot joints. The result has shape (n,).

rne_dh(q, qd, qdd) where the arguments have shape (m,n) where n is the number of robot joints and where m is the number of steps in the joint trajectory. The result has shape (m,n).

rne_dh(p) where the input is a 1D array p = [q, qd, qdd] with shape (3n,), and the result has shape (n,).

rne_dh(p) where the input is a 2D array p = [q, qd, qdd] with shape (m,3n) and the result has shape (m,n).

Notes

This version supports symbolic model parameters
Verified against MATLAB code

Warning

Assumes each link’s joint is the last element of its own ETS segment (Featherstone’s spatial-vector convention – fixed geometry gets you to the joint, the joint is the last thing applied before the next link’s frame). This is guaranteed for any Robot/ERobot built normally, either from a raw ETS (Robot.__init__ splits it via ETS.split(), whose default “last” method enforces this per segment), from a URDF (each link’s ETS is built fixed-transform-then-joint, in that order), or from a PoERobot (_update_ets() appends the joint ET last too). It does not hold for a DHRobot using standard DH conventions (mdh=False), where the joint comes first, followed by d/a/alpha – calling Robot.rne(dh_instance, ...) directly (bypassing DHRobot’s own correct rne()/rne_python()) gives silently wrong answers. A DHRobot built with mdh=True is joint-last (DHLink._to_ets()’s MDH branch reorders a revolute link’s d translation to precede the joint rotation – valid since a z-rotation and a z-translation commute – so the joint ET is always last regardless of d) and works correctly through this path. See rne.md. Revisit this guard if the Robot/Link class hierarchy is ever redesigned – see https://github.com/petercorke/robotics-toolbox-python/issues/571.

property T: ndarray

Pose of the shape relative to its parent frame in the scene graph (or the world frame if it has no parent), as a 4x4 homogeneous transformation matrix. Set via the pose argument of the constructor.

This is a read/write property. The getter always returns a plain ndarray; the setter also accepts an SE3.

Warning

Because the getter returns an ndarray, in-place operators like shape.T *= delta do an elementwise multiply, not a pose composition, even when delta is an SE3 – use shape.T = shape.T * delta (or shape.T @= delta.A) instead.

Return type:: ndarray(4,4)
Setter accepts:: ndarray | SE3

__getitem__(i)

Get link

This also supports iterating over each link in the robot object, from the base to the tool.

Parameters:: i (int | str) – link number or name
Return type:: Link
Returns:: i’th link or named link

Examples

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.DH.Puma560()
>>> print(robot[1]) # print the 2nd link
RevoluteDH:   θ=q,  d=0,  a=0.4318,  ⍺=0.0
>>> print([link.a for link in robot])  # print all the a_j values
[0, 0.4318, 0.0203, 0, 0, 0]

Notes

Robot supports link lookup by name,: eg. robot['link1']

__str__()

Pretty prints the ETS Model of the robot.

Return type:: str
Returns:: Pretty print of the robot model

Notes

Constant links are shown in blue.
End-effector links are prefixed with an @
Angles in degrees
The robot base frame is denoted as BASE and is equal to the
robot’s base attribute.

accel(q, qd, torque, gravity=None)

Compute acceleration due to applied torque

Parameters:

q (Any) – Joint coordinates
qd (Any) – Joint velocity
torque (Any) – Joint torques of the robot
gravity (Any) – gravitational acceleration in the world frame, downwards gravitational force is equivalent to robot base acceleration upwards (positive); if not supplied, uses the gravity attribute of self

Returns:

Joint accelerations

Return type:

qdd = accel(q, qd, torque) calculates a vector (n) of joint accelerations that result from applying the actuator force/torque (n) to the manipulator in state q (n) and qd (n), and n is the number of robot joints.

\[\ddot{q} = \mathbf{M}^{-1} \left(\tau - \mathbf{C}(q)\dot{q} - \mathbf{g}(q)\right)\]

Trajectory operation

If q, qd, torque are matrices (m,n) then qdd is a matrix (m,n) where each row is the acceleration corresponding to the equivalent rows of q, qd, torque.

Examples

>>> import roboticstoolbox as rtb
>>> puma = rtb.models.DH.Puma560()
>>> puma.accel(puma.qz, 0.5 * np.ones(6), np.zeros(6))
array([ -7.5544, -12.22  ,  -6.4022,  -5.4303,  -4.9518,  -2.1178])

Notes

Useful for simulation of manipulator dynamics, in
conjunction with a numerical integration function.
Uses the method 1 of Walker and Orin to compute the forward
dynamics.
Featherstone’s method is more efficient for robots with large
numbers of joints.
Joint friction is considered.

References

Efficient dynamic computer simulation of robotic mechanisms,
M. W. Walker and D. E. Orin, ASME Journal of Dynamic Systems, Measurement and Control, vol. 104, no. 3, pp. 205-211, 1982.

accel_x(q, xd, wrench, gravity=None, pinv=False, representation='rpy/xyz')

Operational space acceleration due to applied wrench

Parameters:

q (Any) – Joint coordinates
xd (Any) – Operational space velocity of the end-effector
wrench (Any) – Wrench applied to the end-effector
gravity (Any) – gravitational acceleration in the world frame, downwards gravitational force is equivalent to robot base acceleration upwards (positive); if not supplied, uses the gravity attribute of self
pinv (bool) – use pseudo inverse rather than inverse
representation (Literal['rpy/xyz', 'rpy/zyx', 'eul', 'exp']) – the type of analytical Jacobian to use, default is 'rpy/xyz'

Returns:

Operational space accelerations of the end-effector

Return type:

xdd = accel_x(q, xd, wrench) is the operational space acceleration due to wrench applied to the end-effector of a robot in joint configuration q and operational space velocity xd.

\[\ddot{x} = \dot{\mathbf{J}}_a(q, \dot{q})\dot{q} + \mathbf{J}_a(q) \mathbf{M}(q)^{-1} \left( \mathbf{J}_a(q)^T w - \mathbf{C}(q)\dot{q} - \mathbf{g}(q) \right)\]

Trajectory operation

If q is a matrix (m,n), and xd and wrench are matrices (m,6), then xdd is a matrix (m,6) where each row is the acceleration corresponding to the equivalent rows of q, xd, and wrench.

Notes

Useful for simulation of manipulator dynamics, in
conjunction with a numerical integration function.
Uses the method 1 of Walker and Orin to compute the forward
dynamics.
Featherstone’s method is more efficient for robots with large
numbers of joints.
Joint friction is considered.

See also

accel()

addconfiguration(name, q)

Add a named joint configuration

Add a named configuration to the robot instance’s dictionary of named configurations.

Parameters:

name (str) – Name of the joint configuration
q (ndarray | Sequence[float]) – Joint configuration

Return type:

Examples

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.DH.Puma560()
>>> robot.addconfiguration_attr("mypos", [0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
>>> robot.configs["mypos"]
array([0.1, 0.2, 0.3, 0.4, 0.5, 0.6])

See also

addconfiguration_attr(name, q, unit='rad')

Add a named joint configuration as an attribute

Parameters:

name (str) – Name of the joint configuration
q (ndarray | Sequence[float]) – Joint configuration

Return type:

Examples

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.DH.Puma560()
>>> robot.addconfiguration_attr("mypos", [0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
>>> robot.mypos
array([0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
>>> robot.configs["mypos"]
array([0.1, 0.2, 0.3, 0.4, 0.5, 0.6])

Notes

Used in robot model init method to store the qr configuration
Dynamically adding attributes to objects can cause issues with
Python type checking.
Configuration is also added to the robot instance’s dictionary of
named configurations.

See also

attach(object)

Attach a child node

Parameters:: object (SceneNode) – the node to attach as a child of this node
Seealso:: attach_to() scene_children()
Return type:: None

attach_to(object)

Attach this node to a parent node

Parameters:: object (SceneNode) – the node to attach this node to, this node will become a child of the parent
Seealso:: attach() scene_parent()
Return type:: None

property base: SE3

Get/set robot base transform

robot.base is the robot base transform
robot.base = ... checks and sets the robot base transform

Parameters:: base – the new robot base transform
Returns:: the current robot base transform
Setter accepts:: ndarray | SE3

property base_link: Link

Get the robot base link

robot.base_link is the robot base link

Returns:: the first link in the robot tree

cinertia(q)

Deprecated, use inertia_x

Return type:: None

property comment: str

Get/set robot comment

robot.comment is the robot comment
robot.comment = ... checks and sets the robot comment

Parameters:: name – the new robot comment
Returns:: robot comment

property configs: dict[str, ndarray]

Named joint configurations

Returns:: dictionary of joint configurations, such as qr and qz, indexed by name

configurations_str(border='thin')

Return type:: Any

property control_mode: str

Get/set robot control mode

robot.control_type is the robot control mode
robot.control_type = ... checks and sets the robot control mode

Parameters:: control_mode – the new robot control mode
Returns:: the current robot control mode

copy()

Make a copy of the robot

Return type:: Any
Returns:: a deep copy, which shares no state with the original

coriolis(q, qd)

Coriolis and centripetal term

Parameters:

q (Any) – Joint coordinates
qd (Any) – Joint velocity

Returns:

Coriolis/centripetal velocity matrix

Return type:

coriolis(q, qd) calculates the Coriolis/centripetal matrix (n,n) for the robot in configuration q and velocity qd, where n is the number of joints.

The product $\mathbf{C} \dot{q}$ is the vector of joint force/torque due to velocity coupling. The diagonal elements are due to centripetal effects and the off-diagonal elements are due to Coriolis effects. This matrix is also known as the velocity coupling matrix, since it describes the disturbance forces on any joint due to velocity of all other joints.

Trajectory operation

If q and qd are matrices (m,n), each row is interpreted as a joint configuration, and the result (n,n,m) is a 3d-matrix where each plane corresponds to a row of q and qd.

Examples

>>> import roboticstoolbox as rtb
>>> puma = rtb.models.DH.Puma560()
>>> puma.coriolis(puma.qz, 0.5 * np.ones((6,)))
array([[-0.4017, -0.5513, -0.2025, -0.0007, -0.0013,  0.    ],
       [ 0.2023, -0.1937, -0.3868, -0.    , -0.002 ,  0.    ],
       [ 0.1987,  0.193 , -0.    ,  0.    , -0.0001,  0.    ],
       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
       [ 0.0007,  0.0007,  0.0001,  0.    ,  0.    ,  0.    ],
       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ]])

Notes

Joint viscous friction is also a joint force proportional to
velocity but it is eliminated in the computation of this value.
Computationally slow, involves $n^2/2$ invocations of RNE.

coriolis_x(q, qd, pinv=False, representation='rpy/xyz', J=None, Ji=None, Jd=None, C=None, Mx=None)

Operational space Coriolis and centripetal term

Parameters:

q (Any) – Joint coordinates
qd (Any) – Joint velocity
pinv (bool) – use pseudo inverse rather than inverse (Default value = False)
representation (Literal['rpy/xyz', 'rpy/zyx', 'eul', 'exp']) – the type of analytical Jacobian to use, default is 'rpy/xyz'
J (Any) – pre-computed analytical Jacobian (optional)
Ji (Any) – pre-computed inverse analytical Jacobian (optional)
Jd (Any) – pre-computed time-derivative of analytical Jacobian (optional)
C (Any) – pre-computed joint-space Coriolis matrix (optional)
Mx (Any) – pre-computed operational-space inertia matrix (optional)

Returns:

Operational space velocity matrix

Return type:

coriolis() inertia_x() hessian0()

coriolis_x(q, qd) is the Coriolis/centripetal matrix (m,m) in operational space for the robot in configuration q and velocity qd, where n is the number of joints.

\[\mathbf{C}_x = \mathbf{J}(q)^{-T} \mathbf{C}(q, \dot{q}) \mathbf{J}(q)^{-1} - \mathbf{M}_x(q) \dot{\mathbf{J}}(q, \dot{q}) \mathbf{J}(q)^{-1}\]

The product $\mathbf{C} \dot{x}$ is the operational space wrench due to joint velocity coupling. This matrix is also known as the velocity coupling matrix, since it describes the disturbance forces on any joint due to velocity of all other joints.

The transformation to operational space requires an analytical, rather than geometric, Jacobian. analytical can be one of:

Value

Rotational representation

'rpy/xyz'

RPY angular rates in XYZ order (default)

'rpy/zyx'

RPY angular rates in XYZ order

'eul'

Euler angular rates in ZYZ order

'exp'

exponential coordinate rates

Trajectory operation

If q and qd are matrices (m,n), each row is interpreted as a joint configuration, and the result is an array (m,6,6) where each matrix corresponds to a row of q and qd.

Examples

>>> import roboticstoolbox as rtb
>>> puma = rtb.models.DH.Puma560()
>>> puma.coriolis_x(puma.qn, 0.5 * np.ones((6,)))
array([[     12.3365,       4.6387,      13.7633,      -0.0409,
              0.0043,      -0.0409],
       [     -4.788 ,  -31430.2949,      -3.4675,       0.3301,
             -0.3188,       0.3301],
       [    -11.5011,      -1.7324,     -10.0545,      -0.0405,
              0.    ,      -0.0405],
       [     -0.0165, 4869725.6671,      -0.0165,      -0.1245,
              0.0278,      -0.1245],
       [     -0.1112,       0.0531,       0.1332,      -0.0251,
             -0.    ,      -0.0251],
       [     -0.0165, 4869725.6671,      -0.0165,      -0.1245,
              0.0278,      -0.1245]])

Notes

Joint viscous friction is also a joint force proportional to
velocity but it is eliminated in the computation of this value.
Computationally slow, involves $n^2/2$ invocations of RNE.
If the robot is not 6 DOF the pinv option is set True.
pinv() is around 5x slower than inv()

Warning

Assumes that the operational space has 6 DOF.

See also

property default_backend: Any

Get default graphical backend

robot.default_backend Get the default graphical backend, used when
no explicit backend is passed to Robot.plot. This is the value set for the robot, otherwise the value set for all robots by set_default_backend(), otherwise None.
robot.default_backend = ... Set the default graphical backend, used when
no explicit backend is passed to Robot.plot. The default set here will be overridden if the particular Robot subclass cannot support it.

Returns:: backend name

dfs_links(start, func=None)

A link search method

Visit all links from start in depth-first order and will apply func to each visited link

Parameters:

start (Link) – The link to start at
func (Optional[Callable[[Optional[Link]], Any]]) – An optional function to apply to each link as it is found

Return type:

list[Link]

Returns:

A list of links

dotfile(filename, etsbox=False, ets='full', jtype=False, static=True)

Write a link transform graph as a GraphViz dot file

The file can be processed using dot:

% dot -Tpng -o out.png dotfile.dot

The nodes are:

Base is shown as a grey square. This is the world frame origin, but can be changed using the base attribute of the robot.
Link frames are indicated by circles
ETS transforms are indicated by rounded boxes

The edges are:

an arrow if jtype is False or the joint is fixed
an arrow with a round head if jtype is True and the joint is revolute
an arrow with a box head if jtype is True and the joint is prismatic

Edge labels or nodes in blue have a fixed transformation to the preceding link.

Notes

If filename is a file object then the file will not be closed after the GraphViz model is written.

Parameters:

file – Name of file to write to
etsbox (bool) – Put the link ETS in a box, otherwise an edge label
ets (Literal['full', 'brief']) – Display the full ets with “full” or a brief version with “brief”
jtype (bool) – Arrowhead to node indicates revolute or prismatic type
static (bool) – Show static joints in blue and bold

Return type:

See also

showgraph() graph()

dynamics()

Pretty print the dynamic parameters

The dynamic parameters (inertial and friction) are printed in a table, with one row per link.

Examples

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.DH.Puma560()
>>> robot.dynamics()
┌───────┬───────┬─────────────────────────┬────────────────────────────────────────┬──────────┬───────────┬───────────────────┬───────┐
│   j   │   m   │            r            │                   I                    │    Jm    │     B     │        Tc         │   G   │
├───────┼───────┼─────────────────────────┼────────────────────────────────────────┼──────────┼───────────┼───────────────────┼───────┤
│ link1 │  0    │  0,  0,  0              │  0,  0.35,  0,  0,  0,  0              │  0.0002  │  0.00148  │  0.395, -0.435    │ -62.6 │
│ link2 │  17.4 │ -0.364,  0.006,  0.228  │  0.13,  0.524,  0.539,  0,  0,  0      │  0.0002  │  0.000817 │  0.126, -0.071    │  108  │
│ link3 │  4.8  │ -0.0203, -0.0141,  0.07 │  0.066,  0.086,  0.0125,  0,  0,  0    │  0.0002  │  0.00138  │  0.132, -0.105    │ -53.7 │
│ link4 │  0.82 │  0,  0.019,  0          │  0.0018,  0.0013,  0.0018,  0,  0,  0  │  3.3e-05 │  7.12e-05 │  0.0112, -0.0169  │  76   │
│ link5 │  0.34 │  0,  0,  0              │  0.0003,  0.0004,  0.0003,  0,  0,  0  │  3.3e-05 │  8.26e-05 │  0.00926, -0.0145 │  71.9 │
│ link6 │  0.09 │  0,  0,  0.032          │  0.00015,  0.00015,  4e-05,  0,  0,  0 │  3.3e-05 │  3.67e-05 │  0.00396, -0.0105 │  76.7 │
└───────┴───────┴─────────────────────────┴────────────────────────────────────────┴──────────┴───────────┴───────────────────┴───────┘

Return type:: None

dynamics_list()

Print dynamic parameters

Display the kinematic and dynamic parameters to the console in reable format

Return type:: None

dynchanged(what=None)

Dynamic parameters have changed

Called from a property setter to inform the robot that the cache of dynamic parameters is invalid.

See also

roboticstoolbox.Link._dirties_frne()

Return type:: None

property ee_links: list[Link]

ets(start=None, end=None)

Robot to ETS

robot.ets() is an ETS representing the kinematics from base to end-effector.

robot.ets(end=link) is an ETS representing the kinematics from base to the link link specified as a Link reference or a name.

robot.ets(start=l1, end=l2) is an ETS representing the kinematics from link l1 to link l2.

Parameters:

start (:param) – start of path, defaults to base_link:
end (:param) – end of path, defaults to end-effector:

Raises:

ValueError – a link does not belong to this ERobot
TypeError – a bad link argument

Return type:

ETS

Returns:

elementary transform sequence

Examples

>>> import roboticstoolbox as rtb
>>> panda = rtb.models.ETS.Panda()
>>> panda.ets()
[ET.tz(param=0.333), ET.Rz(jindex=0), ET.Rx(param=-1.5707963267948966), ET.Rz(jindex=1), ET.Rx(param=1.5707963267948966), ET.tz(param=0.316), ET.Rz(jindex=2), ET.tx(param=0.0825), ET.Rx(param=1.5707963267948966), ET.Rz(jindex=3), ET.tx(param=-0.0825), ET.Rx(param=-1.5707963267948966), ET.tz(param=0.384), ET.Rz(jindex=4), ET.Rx(param=1.5707963267948966), ET.Rz(jindex=5), ET.tx(param=0.088), ET.Rx(param=1.5707963267948966), ET.tz(param=0.107), ET.Rz(jindex=6), ET.tz(param=0.10300000000000001), ET.Rz(param=-0.7853981633974483)]

fdyn(T, q0, Q=None, Q_args=None, qd0=None, solver='RK45', solver_args=None, dt=None, progress=False)

Integrate forward dynamics

Parameters:

T (float) – integration time
q0 (ndarray | Sequence[float]) – initial joint coordinates
Q (Callable[[Any, float, ndarray, ndarray], ndarray] | None) – a function that computes generalized joint force as a function of time and/or state
Q_args (dict | None) – positional arguments passed to torque
qd0 (ndarray | Sequence[float] | None) – initial joint velocities, assumed zero if not given
solver (str) – name of SciPy integrator to use, default 'RK45'
solver_args (dict | None) – optional arguments passed to the SciPy integrator
dt (float | None) – if given, interpolate the results to this timestep
progress (bool) – show progress bar, default False

Returns:

robot trajectory

Return type:

nofriction() friction()

tg = R.fdyn(T, q) integrates the dynamics of the robot with zero input torques over the time interval 0 to T and returns the trajectory as a namedtuple with elements:

t the time vector (M,)
q the joint coordinates (M,n)
qd the joint velocities (M,n)

tg = R.fdyn(T, q, torqfun) as above but the torque applied to the joints is given by the provided function:

tau = function(robot, t, q, qd, **args)

where the inputs are:

the robot object

current time

current joint coordinates (n,)

current joint velocity (n,)

args, optional keyword arguments can be specified, these are passed in from the targs keyword argument.

The function must return a Numpy array (n,) of joint forces/torques.

Examples

To apply zero joint torque to the robot without Coulomb friction:

>>> def myfunc(robot, t, q, qd):
>>>     return np.zeros((robot.n,))

>>> tg = robot.nofriction().fdyn(5, q0, myfunc)

>>> plt.figure()
>>> plt.plot(tg.t, tg.q)
>>> plt.show()

We could also use a lambda function:

>>> tg = robot.nofriction().fdyn(
>>>     5, q0, lambda r, t, q, qd: np.zeros((r.n,)))

The robot is controlled by a PD controller. We first define a function to compute the control which has additional parameters for the setpoint and control gains (qstar, P, D):

>>> def myfunc(robot, t, q, qd, qstar, P, D):
>>>     return (qstar - q) * P + qd * D  # P, D are (6,)

>>> tg = robot.fdyn(10, q0, myfunc, torque_args=(qstar, P, D))

Many integrators have variable step length which is problematic if we want to animate the result. If dt is specified then the solver results are interpolated in time steps of dt.

Notes

This function performs poorly with non-linear joint friction,
such as Coulomb friction. The R.nofriction() method can be used to set this friction to zero.
If the function is not specified then zero force/torque is
applied to the manipulator joints.
Interpolation is performed using SciPy integrate.ode
The SciPy RK45 integrator is used by default
Interpolation is performed using SciPy interp1d

See also

accel() nofriction(), rne().

fellipse(q, opt='trans', unit='rad', centre='ee', add=True)

Create a force ellipsoid object for plotting with PyPlot.

Parameters:

q (ndarray | Sequence[float]) – the joint configuration of the robot
opt (Literal['trans', 'rot']) – 'trans' or 'rot' — plot the translational or rotational force ellipsoid
unit (Literal['rad', 'deg']) – 'rad' or 'deg'
centre (Union[Literal['ee'], ndarray, Sequence[float]]) – centre of the ellipsoid — 'ee' for the end-effector or a 3-vector [x, y, z] in the world frame
add (bool) – if True, add the ellipsoid to the active plot environment

Returns:

an EllipsePlot object

Return type:

EllipsePlot

robot.fellipse(q) creates a force ellipsoid for the robot at pose q. By default the ellipsoid is centered at the end-effector.

Notes

By default the ellipsoid related to translational motion is drawn. Use opt='rot' to draw the rotational velocity ellipsoid.
By default the ellipsoid is drawn at the end-effector. The option centre allows its origin to be set to the specified 3-vector, or the string "ee" ensures it is drawn at the end-effector position.

fkine(q, end=None, start=None, tool=None, include_base=True)

Forward kinematics

Parameters:

q (ndarray | Sequence[float]) – Joint coordinates – either global (length robot.n, addressed by each joint’s jindex, e.g. the whole robot’s current q) or, when end/start select just one branch of a branched robot, compact (length equal to the number of joints on that branch, positionally ordered from start to end – exactly what ikine_LM/ik_LM etc. return when solving for that same end)
end (str | Link | Gripper | None) – end-effector or gripper to compute forward kinematics to
start (str | Link | Gripper | None) – the link to compute forward kinematics from
tool (ndarray | SE3 | None) – tool transform, optional

Raises:

ValueError – if q’s length matches neither the compact nor the global interpretation for the selected start/end

Return type:

SE3

Returns:

The transformation matrix representing the pose of the end-effector

T = robot.fkine(q) evaluates forward kinematics for the robot at joint configuration q.

Trajectory operation: If q has multiple rows (mxn), it is considered a trajectory and the result is an SE3 instance with m values.

Examples

The following example makes a panda robot object, and solves for the forward kinematics at the listed configuration.

>>> import roboticstoolbox as rtb
>>> panda = rtb.models.Panda()
>>> panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
SE3(array([[ 0.995 , -0.    ,  0.0998,  0.484 ],
           [-0.    , -1.    ,  0.    ,  0.    ],
           [ 0.0998, -0.    , -0.995 ,  0.4126],
           [ 0.    ,  0.    ,  0.    ,  1.    ]]))

On a branched robot, end selects one branch, and q may be either the whole robot’s global q or just the solution for that branch on its own – both give the same result:

>>> import roboticstoolbox as rtb
>>> from spatialmath import SE3
>>> yumi = rtb.models.YuMi()
>>> sol = yumi.ikine_LM(SE3(0.4, 0.2, 0.3) * SE3.Rx(0.2), end="l_gripper", method="chan", k=0.1, seed=0)
>>> sol.q.shape
(7,)
>>> yumi.fkine(sol.q, end="l_gripper")
SE3(array([[ 1.    , -0.    ,  0.    ,  0.4   ],
           [ 0.    ,  0.9801, -0.1987,  0.2   ],
           [-0.    ,  0.1987,  0.9801,  0.3   ],
           [ 0.    ,  0.    ,  0.    ,  1.    ]]))

Notes

For a robot with a single end-effector there is no need to
specify end
For a robot with multiple end-effectors, the end must
be specified.
The robot’s own base transform (self.base), if set, is
always incorporated into the result.
The robot’s own tool transform (self.tool), if set, is
incorporated into the result unless the tool parameter is given explicitly, which overrides it.
Works from the end-effector link to the base

References

J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).
J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

friction(qd)

Manipulator joint friction

Parameters:: qd (ndarray) – The joint velocities of the robot
Returns:: The joint friction forces/torques for the robot
Return type:: ndarray

robot.friction(qd) is a vector of joint friction forces/torques for the robot moving with joint velocities qd.

The friction model includes:

Viscous friction which is a linear function of velocity.
Coulomb friction which is proportional to sign(qd).

\[\begin{split}\tau_j = G^2 B \dot{q}_j + |G_j| \left\{ \begin{array}{ll} \tau_{C,j}^+ & \mbox{if $\dot{q}_j > 0$} \\ \tau_{C,j}^- & \mbox{if $\dot{q}_j < 0$} \end{array} \right.\end{split}\]

Notes

The friction value should be added to the motor output torque to
determine the nett torque. It has a negative value when qd > 0.
The returned friction value is referred to the output of the
gearbox.
The friction parameters in the Link object are referred to the
motor.
Motor viscous friction is scaled up by $G^2$.
Motor Coulomb friction is scaled up by $G$.
The appropriate Coulomb friction value to use in the
non-symmetric case depends on the sign of the joint velocity, not the motor velocity.
Coulomb friction is zero for zero joint velocity, stiction is
not modeled.
The absolute value of the gear ratio is used. Negative gear
ratios are tricky: the Puma560 robot has negative gear ratio for joints 1 and 3.

See also

get_path(end=None, start=None)

Find a path from start to end

Parameters:

end (Union[Gripper, Link, str, None]) – end-effector or gripper to compute forward kinematics to
start (Union[Gripper, Link, str, None]) – name or reference to a base link, defaults to None

Raises:

ValueError – link not known or ambiguous

Return type:

tuple[list[Link], int, SE3]

Returns:

the path from start to end

Returns:

the number of joints in the path

Returns:

the tool transform present after end

graph(format='dot', filename=None, **kwargs)

Render a link transform graph as text

Parameters:

format (Literal['dot', 'mermaid', 'mermaid_fenced']) – graph output format: "dot", "mermaid" or "mermaid_fenced", defaults to "dot"
filename (Union[str, IO[str], None]) – destination path or open text stream, defaults to None
kwargs (Any) – for format="dot", forwarded to dotfile() (etsbox, ets, jtype, static); ignored otherwise

Return type:

str

Returns:

rendered graph text

Creates graph text representing the robot’s link transform graph – the same structure dotfile()/showgraph() render, in a choice of formats. If filename is provided, the rendered text is also written to it.

"dot" produces the same GraphViz output as dotfile(), just returned as a string rather than requiring a real file – etsbox/ets/jtype/static behave exactly as they do there.

"mermaid"/"mermaid_fenced" render a simplified Mermaid flowchart: one node per link (a double circle for end-effectors, a hexagon for gripper links), edges labelled with the joint type (R/P) and joint index for actuated joints. Mermaid has no equivalent of DOT’s per-edge arrowhead/colour styling, so jtype/static/etsbox only affect "dot" output. "mermaid_fenced" wraps the same text in a fenced mermaid code block, ready to paste into Markdown or render directly in a Jupyter cell:

from IPython.display import Markdown, display
display(Markdown(robot.graph(format="mermaid_fenced")))

See also

dotfile() showgraph()

property gravity: ndarray

Get/set default gravitational acceleration

robot.gravity is the default gravitational acceleration
robot.gravity = ... checks and sets default gravitational
acceleration

Parameters:: gravity – gravitational acceleration in the world frame, downwards gravitational force is equivalent to robot base acceleration upwards (positive)
Returns:: gravitational acceleration
Setter accepts:: ndarray | Sequence[float]

gravload(q=None, gravity=None)

Compute gravity load

Parameters:

q (ndarray | Sequence[float] | None) – Joint coordinates
gravity (ndarray | Sequence[float] | None) – gravitational acceleration in the world frame, downwards gravitational force is equivalent to robot base acceleration upwards (positive); if not supplied, uses the stored gravity values

Returns:

The generalised joint force/torques due to gravity

Return type:

robot.gravload(q) calculates the joint gravity loading (n) for the robot in the joint configuration q and using the default gravitational acceleration specified in the DHRobot object.

robot.gravload(q, gravity=g) as above except the gravitational acceleration is explicitly specified as g.

The return value is referenced to the load, not the motor, the gear ratio is not taken into account.

Trajectory operation

If q is a matrix (nxm) each column is interpreted as a joint configuration vector, and the result is a matrix (nxm) each column being the corresponding joint torques.

Examples

>>> import roboticstoolbox as rtb
>>> puma = rtb.models.DH.Puma560()
>>> puma.gravload(puma.qz)
array([ 0.    , 37.4837,  0.2489,  0.    ,  0.    ,  0.    ])

gravload_x(q=None, gravity=None, pinv=False, representation='rpy/xyz', Ji=None)

Operational space gravity load

Parameters:

q (Any) – Joint coordinates
gravity (Any) – gravitational acceleration in the world frame, downwards gravitational force is equivalent to robot base acceleration upwards (positive); if not supplied, uses the gravity attribute of self
pinv (bool) – use pseudo inverse rather than inverse (Default value = False)
representation (Literal['rpy/xyz', 'rpy/zyx', 'eul', 'exp']) – the type of analytical Jacobian to use, default is 'rpy/xyz'
Ji (Any) – pre-computed inverse analytical Jacobian (optional)

Returns:

The operational space gravity wrench

Return type:

ik_LM() ik_NR() ikine_GN()

robot.gravload_x(q) calculates the gravity wrench for the robot in the joint configuration q and using the default gravitational acceleration specified in the robot object.

robot.gravload_x(q, gravity=g) as above except the gravitational acceleration is explicitly specified as g.

\[\mathbf{G}_x = \mathbf{J}(q)^{-T} \mathbf{G}(q)\]

The transformation to operational space requires an analytical, rather than geometric, Jacobian. analytical can be one of:

Value

Rotational representation

'rpy/xyz'

RPY angular rates in XYZ order (default)

'rpy/zyx'

RPY angular rates in XYZ order

'eul'

Euler angular rates in ZYZ order

'exp'

exponential coordinate rates

Trajectory operation

If q is a matrix (m,n), each row is interpreted as a joint configuration vector, and the result is a matrix (m,6) with one gravity wrench per row.

Examples

>>> import roboticstoolbox as rtb
>>> puma = rtb.models.DH.Puma560()
>>> puma.gravload_x(puma.qn)
array([-30.1218,   7.5797,  53.7378,  -0.    ,  -0.0283,  -0.    ])

Notes

If the robot is not 6 DOF the pinv option is set True.
pinv() is around 5x slower than inv()

Warning

Assumes that the operational space has 6 DOF.

See also

gravload()

property grippers: list[Gripper]

Grippers attached to the robot

Returns:: A list of grippers

property hascollision: Any

Robot has collision model

Returns:: Robot has collision model
Returns:: At least one link has associated collision model.

Examples

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.DH.Puma560()
>>> robot.hascollision
False

See also

hasgeometry()

property hasdynamics: bool

Robot has dynamic parameters

Returns:: True if at least one link has a non-zero mass or motor inertia

These are the parameters that the inertial dynamics need. It is false for a model of the kinematics only, such as the default rtb.models.Panda, and the dynamics methods gravload, inertia, coriolis, accel, fdyn, itorque and rne raise ValueError for such a model, rather than return zeros or a confusing error. It is any link and not all, since the first link of the DH Puma560 has a mass of zero.

The answer is cached, and recomputed only when the mass or motor inertia of a link has been changed, so it costs nothing to ask.

Changed in version 1.5.0: This was true if any dynamic parameter had been assigned, even to zero, and was not reset when the parameters were set back to zero. A robot made from DH links with no masses reported True. The dynamics methods now raise ValueError if it is False, where they returned zeros or raised LinAlgError.

Warning

Behaviour change. Scripts that call the dynamics methods on a kinematics-only model, and accept the zeros, now get a ValueError.

Examples

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.DH.Puma560()
>>> robot.hasdynamics
True

property hasgeometry: Any

Robot has geometry model

At least one link has associated mesh to describe its shape.

Returns:: Robot has geometry model

Examples

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.DH.Puma560()
>>> robot.hasgeometry
True

See also

hascollision()

hessian0(q=None, end=None, start=None, J0=None, tool=None)

Manipulator Hessian in the start frame

Overloads:

q (ndarray | Sequence[float]) = ..., end (str | Link | Gripper | None) = None, start (str | Link | Gripper | None) = None, J0 (None) = None, tool (ndarray | SE3 | None) = None → ndarray
q (None) = None, end (str | Link | Gripper | None) = None, start (str | Link | Gripper | None) = None, J0 (ndarray) = ..., tool (ndarray | SE3 | None) = None → ndarray

Parameters:

q (Any) – The joint angles/configuration of the robot (Optional, if not supplied will use the stored q values).
end (str | Link | Gripper | None) – the final link/Gripper which the Hessian represents
start (str | Link | Gripper | None) – the first link which the Hessian represents
J0 (Any) – The manipulator Jacobian in the start frame
tool (ndarray | SE3 | None) – a static tool transformation matrix to apply to the end of end, defaults to None

Returns:

The manipulator Hessian in the start frame

The manipulator Hessian tensor maps joint acceleration to end-effector spatial acceleration, expressed in the start frame. This function calulcates this based on the ETS of the robot. One of J0 or q is required. Supply J0 if already calculated to save computation time

This method computes the manipulator Hessian in the start frame. If we take the time derivative of the differential kinematic relationship

\[\begin{split}\nu &= \mat{J}(\vec{q}) \dvec{q} \\ \alpha &= \dmat{J} \dvec{q} + \mat{J} \ddvec{q}\end{split}\]

where

\[\dmat{J} = \mat{H} \dvec{q}\]

and $\mat{H} \in \mathbb{R}^{6\times n \times n}$ is the Hessian tensor.

The elements of the Hessian are

\[\mat{H}_{i,j,k} = \frac{d^2 u_i}{d q_j d q_k}\]

where $u = \{t_x, t_y, t_z, r_x, r_y, r_z\}$ are the elements of the spatial velocity vector.

Similarly, we can write

\[\mat{J}_{i,j} = \frac{d u_i}{d q_j}\]

Examples

The following example makes a Panda robot object, and solves for the base frame Hessian at the given joint angle configuration

>>> import roboticstoolbox as rtb
>>> panda = rtb.models.Panda()
>>> panda.hessian0([0, -0.3, 0, -2.2, 0, 2, 0.7854])
array([[[-0.484 , -0.    , -0.486 , -0.    , -0.1547,  0.    , -0.    ],
        [ 0.    ,  0.0796,  0.    ,  0.2466,  0.    ,  0.2006,  0.    ],
        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
        [ 0.    , -1.    , -0.    ,  1.    ,  0.    ,  1.    ,  0.    ],
        [ 0.    , -0.    , -0.2955, -0.    ,  0.9463, -0.    ,  0.0998],
        [ 0.    ,  0.    , -0.    , -0.    ,  0.    , -0.    ,  0.    ]],

       [[-0.    , -0.484 , -0.    ,  0.4986, -0.    ,  0.1086,  0.    ],
        [ 0.0796,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
        [ 0.    , -0.0796, -0.    , -0.2466, -0.    , -0.2006, -0.    ],
        [ 0.    ,  0.    ,  0.9553,  0.    , -0.3233,  0.    , -0.995 ],
        [ 0.    ,  0.    , -0.    , -0.    ,  0.    , -0.    ,  0.    ],
        [ 0.    ,  0.    ,  0.2955,  0.    , -0.9463,  0.    , -0.0998]],

       [[-0.486 , -0.    , -0.4643, -0.    , -0.1478,  0.    ,  0.    ],
        [ 0.    ,  0.    ,  0.    ,  0.383 ,  0.    ,  0.2237,  0.    ],
        [ 0.    , -0.    , -0.1436, -0.    , -0.0457, -0.    , -0.    ],
        [ 0.    ,  0.    ,  0.    ,  0.9553,  0.    ,  0.9553,  0.    ],
        [ 0.    ,  0.    ,  0.    , -0.    ,  0.8085, -0.    , -0.1987],
        [ 0.    ,  0.    ,  0.    ,  0.2955,  0.    ,  0.2955,  0.    ]],

       [[-0.    ,  0.4986, -0.    , -0.4986,  0.    , -0.1086, -0.    ],
        [ 0.2466,  0.    ,  0.383 ,  0.    ,  0.    ,  0.    ,  0.    ],
        [ 0.    , -0.2466, -0.    ,  0.2466,  0.    ,  0.2006,  0.    ],
        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.3233,  0.    ,  0.995 ],
        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.    ],
        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.9463,  0.    ,  0.0998]],

       [[-0.1547, -0.    , -0.1478,  0.    ,  0.05  , -0.    ,  0.    ],
        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.1676, -0.    ],
        [ 0.    , -0.    , -0.0457,  0.    ,  0.1464,  0.    ,  0.    ],
        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.3233,  0.    ],
        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.    ,  0.9093],
        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.9463, -0.    ]],

       [[ 0.    ,  0.1086,  0.    , -0.1086, -0.    , -0.1086, -0.    ],
        [ 0.2006,  0.    ,  0.2237,  0.    , -0.1676,  0.    ,  0.    ],
        [ 0.    , -0.2006, -0.    ,  0.2006,  0.    ,  0.2006,  0.    ],
        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.995 ],
        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.    ],
        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.0998]],

       [[-0.    ,  0.    ,  0.    , -0.    ,  0.    , -0.    ,  0.    ],
        [ 0.    ,  0.    ,  0.    ,  0.    , -0.    ,  0.    , -0.    ],
        [ 0.    , -0.    , -0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ]]])

References

J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).
J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

hessiane(q=None, end=None, start=None, Je=None, tool=None)

Manipulator Hessian in the end-effector frame

Overloads:

q (ndarray | Sequence[float]) = ..., end (str | Link | Gripper | None) = None, start (str | Link | Gripper | None) = None, Je (None) = None, tool (ndarray | SE3 | None) = None → ndarray
q (None) = None, end (str | Link | Gripper | None) = None, start (str | Link | Gripper | None) = None, Je (ndarray) = ..., tool (ndarray | SE3 | None) = None → ndarray

Parameters:

q (Any) – The joint angles/configuration of the robot (Optional, if not supplied will use the stored q values).
end (str | Link | Gripper | None) – the final link/Gripper which the Hessian represents
start (str | Link | Gripper | None) – the first link which the Hessian represents
Je (Any) – The manipulator Jacobian in the end frame
tool (ndarray | SE3 | None) – a static tool transformation matrix to apply to the end of end, defaults to None

Returns:

The manipulator Hessian in end frame

The manipulator Hessian tensor maps joint acceleration to end-effector spatial acceleration, expressed in the end coordinate frame. This function calulcates this based on the ETS of the robot. One of J0 or q is required. Supply J0 if already calculated to save computation time

This method computes the manipulator Hessian in the end frame. If we take the time derivative of the differential kinematic relationship

\[\begin{split}\nu &= \mat{J}(\vec{q}) \dvec{q} \\ \alpha &= \dmat{J} \dvec{q} + \mat{J} \ddvec{q}\end{split}\]

where

\[\dmat{J} = \mat{H} \dvec{q}\]

and $\mat{H} \in \mathbb{R}^{6\times n \times n}$ is the Hessian tensor.

The elements of the Hessian are

\[\mat{H}_{i,j,k} = \frac{d^2 u_i}{d q_j d q_k}\]

where $u = \{t_x, t_y, t_z, r_x, r_y, r_z\}$ are the elements of the spatial velocity vector.

Similarly, we can write

\[\mat{J}_{i,j} = \frac{d u_i}{d q_j}\]

Examples

The following example makes a Panda robot object, and solves for the end-effector frame Hessian at the given joint angle configuration

>>> import roboticstoolbox as rtb
>>> panda = rtb.models.Panda()
>>> panda.hessiane([0, -0.3, 0, -2.2, 0, 2, 0.7854])
array([[[-0.4816, -0.    , -0.4835, -0.    , -0.1539, -0.    ,  0.    ],
        [ 0.    , -0.0796,  0.    , -0.2466,  0.    , -0.2006,  0.    ],
        [-0.0483, -0.    , -0.0485, -0.    , -0.0154, -0.    ,  0.    ],
        [ 0.    , -0.995 ,  0.    ,  0.995 , -0.    ,  0.995 , -0.    ],
        [ 0.    ,  0.    ,  0.2955, -0.    , -0.9463, -0.    , -0.0998],
        [ 0.    , -0.0998,  0.    ,  0.0998, -0.    ,  0.0998,  0.    ]],

       [[-0.    , -0.4896, -0.    ,  0.4715, -0.    ,  0.088 ,  0.    ],
        [-0.0796,  0.    ,  0.    , -0.    ,  0.    , -0.    ,  0.    ],
        [-0.    ,  0.0309,  0.    ,  0.2952,  0.    ,  0.2104, -0.    ],
        [ 0.    ,  0.    ,  0.9801,  0.    , -0.4161,  0.    , -1.    ],
        [ 0.    ,  0.    , -0.    , -0.    ,  0.    , -0.    ,  0.    ],
        [ 0.    ,  0.    , -0.1987, -0.    ,  0.9093, -0.    ,  0.    ]],

       [[-0.4835, -0.    , -0.4763, -0.    , -0.1516, -0.    ,  0.    ],
        [ 0.    ,  0.    ,  0.    , -0.383 ,  0.    , -0.2237,  0.    ],
        [-0.0485,  0.    ,  0.0965, -0.    ,  0.0307, -0.    ,  0.    ],
        [ 0.    ,  0.    ,  0.    ,  0.9801, -0.    ,  0.9801,  0.    ],
        [ 0.    ,  0.    ,  0.    , -0.    , -0.8085, -0.    ,  0.1987],
        [ 0.    ,  0.    ,  0.    , -0.1987, -0.    , -0.1987, -0.    ]],

       [[-0.    ,  0.4715, -0.    , -0.4715,  0.    , -0.088 ,  0.    ],
        [-0.2466, -0.    , -0.383 ,  0.    , -0.    ,  0.    , -0.    ],
        [-0.    ,  0.2952, -0.    , -0.2952, -0.    , -0.2104,  0.    ],
        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.4161,  0.    ,  1.    ],
        [ 0.    ,  0.    ,  0.    ,  0.    , -0.    ,  0.    , -0.    ],
        [ 0.    ,  0.    ,  0.    ,  0.    , -0.9093,  0.    ,  0.    ]],

       [[-0.1539, -0.    , -0.1516,  0.    ,  0.0644,  0.    , -0.    ],
        [ 0.    ,  0.    ,  0.    , -0.    , -0.    ,  0.1676, -0.    ],
        [-0.0154,  0.    ,  0.0307, -0.    , -0.1407, -0.    ,  0.    ],
        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.4161, -0.    ],
        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.9093],
        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.9093,  0.    ]],

       [[-0.    ,  0.088 , -0.    , -0.088 ,  0.    , -0.088 ,  0.    ],
        [-0.2006, -0.    , -0.2237,  0.    ,  0.1676,  0.    , -0.    ],
        [-0.    ,  0.2104, -0.    , -0.2104, -0.    , -0.2104,  0.    ],
        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  1.    ],
        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.    ],
        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ]],

       [[ 0.    ,  0.    ,  0.    ,  0.    , -0.    ,  0.    ,  0.    ],
        [ 0.    ,  0.    ,  0.    , -0.    , -0.    , -0.    , -0.    ],
        [ 0.    , -0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ]]])

References

J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).
J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

hierarchy()

Pretty print the robot link hierarchy

Return type:: None
Returns:: Pretty print of the robot model

Examples

Makes a robot and prints the hierarchy

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.URDF.Panda()
>>> robot.hierarchy()
 panda_link0
   panda_link1
     panda_link2
       panda_link3
         panda_link4
           panda_link5
             panda_link6
               panda_link7
                 panda_link8
                   panda_hand
                     panda_leftfinger
                     panda_rightfinger

ik_GN(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, pinv=True, pinv_damping=0.0, tool=None)

Fast numerical inverse kinematics by Gauss-Newton optimization

Parameters:

Tep (ndarray | SE3) – The desired end-effector pose (a single pose only, for a trajectory use ikine_GN())
end (str | Link | Gripper | None) – the link considered as the end-effector
start (str | Link | Gripper | None) – the link considered as the base frame, defaults to the robots’s base frame
q0 (ndarray | None) – initial joint configuration (default to random valid joint configuration constrained by the joint limits of the robot)
ilimit (int) – maximum number of iterations per search
slimit (int) – maximum number of search attempts
tol (float) – final error tolerance
mask (ndarray | None) – a mask vector which weights the end-effector error priority. Corresponds to translation in X, Y and Z and rotation about X, Y and Z respectively
joint_limits (bool) – constrain the solution to being within the joint limits of the robot (reject solution with invalid joint configurations and perform another search up to the slimit)
pinv (int) – Use the pseudo-inverse instead of the normal matrix inverse
pinv_damping (float) – Damping factor for the pseudo-inverse
tool (ndarray | SE3 | None) – a static tool transformation matrix to apply to the end of end; defaults to the robot’s own self.tool if not given

Returns:

an IKSolution containing joint coordinates q, success flag, iterations, searches and residual error value (reason is always empty – this fast C++ solver doesn’t produce a granular failure reason string, unlike ikine_GN())

Return type:

IKSolution

Warning

This method needs the compiled C++ extension. It raises RuntimeError if the extension is unavailable, for example in a pure-Python wheel or under Pyodide/JupyterLite, and if the sequence has a SymPy parameter, even a numeric one such as sympy.pi/2, or a joint with an offset, which the compiled code cannot use. In all of these cases use ikine_GN(), which has none of these limits.

sol = ets.ik_GN(Tep) are the joint coordinates (n) corresponding to the robot end-effector pose Tep which is an SE3 or ndarray object. This method can be used for robots with any number of degrees of freedom. This is a fast solver implemented in C++.

See the Inverse Kinematics Docs Page for more details and for a tutorial on numerical IK, see here.

Note

When using this method with redundant robots (>6 DoF), pinv must be set to True

The return value sol is an IKSolution with fields:

Field	Type	Description
`q`	ndarray(n)	joint coordinates in units of radians or metres
`success`	bool	whether a solution was found
`iterations`	int	total number of iterations
`searches`	int	total number of searches
`residual`	float	final value of cost function
`reason`	str	always empty for this C++ solver

If success == False the q values will be valid numbers, but the solution will be in error. The amount of error is indicated by the residual.

Each iteration uses the Gauss-Newton optimisation method

\[\begin{split}\vec{q}_{k+1} &= \vec{q}_k + \left( {\mat{J}(\vec{q}_k)}^\top \mat{W}_e \ {\mat{J}(\vec{q}_k)} \right)^{-1} \bf{g}_k \\ \bf{g}_k &= {\mat{J}(\vec{q}_k)}^\top \mat{W}_e \vec{e}_k\end{split}\]

where $\mat{J} = {^0\mat{J}}$ is the base-frame manipulator Jacobian. If $\mat{J}(\vec{q}_k)$ is non-singular, and $\mat{W}_e = \mat{1}_n$, then the above provides the pseudoinverse solution. However, if $\mat{J}(\vec{q}_k)$ is singular, the above can not be computed and the GN solution is infeasible.

Examples

The following example gets a panda robot object, makes a goal pose Tep, and then solves for the joint coordinates which result in the pose Tep using the ik_GN method.

>>> import roboticstoolbox as rtb
>>> panda = rtb.models.Panda()
>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
>>> panda.ik_GN(Tep)
IKSolution: q=[-1.095, -0.5422, 0.9183, -2.177, 0.472, 1.897, 0.4155], success=True, iterations=111, searches=7, residual=1.41e-10

Notes

When using the this method, the initial joint coordinates $q_0$, should correspond to a non-singular manipulator pose, since it uses the manipulator Jacobian.

References

J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).
J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

See also

ik_LM(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, k=1.0, method='chan', tool=None)

Fast Levenberg-Marquardt Numerical Inverse Kinematics Solver

Parameters:

Tep (ndarray | SE3) – The desired end-effector pose
end (str | Link | Gripper | None) – the link considered as the end-effector
start (str | Link | Gripper | None) – the link considered as the base frame, defaults to the robots’s base frame
q0 (ndarray | None) – The initial joint coordinate vector
ilimit (int) – How many iterations are allowed within a search before a new search is started
slimit (int) – How many searches are allowed before being deemed unsuccessful
tol (float) – Maximum allowed residual error E
mask (ndarray | None) – A 6 vector which assigns weights to Cartesian degrees-of-freedom error priority
joint_limits (bool) – Reject solutions with joint limit violations
k (float) – Sets the gain value for the damping matrix Wn in the next iteration
method (Literal['chan', 'wampler', 'sugihara']) – One of “chan”, “sugihara” or “wampler”. Defines which method is used to calculate the damping matrix Wn in the step method
tool (ndarray | SE3 | None) – a static tool transformation matrix to apply to the end of end; defaults to the robot’s own self.tool if not given

Returns:

an IKSolution containing joint coordinates q, success flag, iterations, searches and residual error value (reason is always empty – this fast C++ solver doesn’t produce a granular failure reason string, unlike ikine_LM())

Return type:

IKSolution

Warning

This method needs the compiled C++ extension. It raises RuntimeError if the extension is unavailable, for example in a pure-Python wheel or under Pyodide/JupyterLite, and if the sequence has a SymPy parameter, even a numeric one such as sympy.pi/2, or a joint with an offset, which the compiled code cannot use. In all of these cases use ikine_LM(), which has none of these limits.

A method which provides functionality to perform numerical inverse kinematics (IK) using the Levenberg-Marquardt method. This is a fast solver implemented in C++.

See the Inverse Kinematics Docs Page for more details and for a tutorial on numerical IK, see here.

The operation is defined by the choice of the method kwarg.

The step is defined as

\[\begin{split}\vec{q}_{k+1} &= \vec{q}_k + \left( \mat{A}_k \right)^{-1} \bf{g}_k \\ % \mat{A}_k &= {\mat{J}(\vec{q}_k)}^\top \mat{W}_e \ {\mat{J}(\vec{q}_k)} + \mat{W}_n\end{split}\]

where $\mat{W}_n = \text{diag}(\vec{w_n})(\vec{w_n} \in \mathbb{R}^n_{>0})$ is a diagonal damping matrix. The damping matrix ensures that $\mat{A}_k$ is non-singular and positive definite. The performance of the LM method largely depends on the choice of $\mat{W}_n$.

Chan’s Method

Chan proposed

\[\mat{W}_n = λ E_k \mat{1}_n\]

where λ is a constant which reportedly does not have much influence on performance. Use the kwarg k to adjust the weighting term λ.

Sugihara’s Method

Sugihara proposed

\[\mat{W}_n = E_k \mat{1}_n + \text{diag}(\hat{\vec{w}}_n)\]

where $\hat{\vec{w}}_n \in \mathbb{R}^n$, $\hat{w}_{n_i} = l^2 \sim 0.01 l^2$, and $l$ is the length of a typical link within the manipulator. We provide the variable k as a kwarg to adjust the value of $w_n$.

Wampler’s Method

Wampler proposed $\vec{w_n}$ to be a constant. This is set through the k kwarg.

Examples

The following example makes a panda robot object, makes a goal pose Tep, and then solves for the joint coordinates which result in the pose Tep using the ik_LM method.

>>> import roboticstoolbox as rtb
>>> panda = rtb.models.Panda()
>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
>>> panda.ik_LM(Tep)
IKSolution: q=[2.007, 0.5696, -2.199, -2.174, 0.5034, 1.882, 0.3916], success=True, iterations=30, searches=3, residual=2.51e-07

Notes

The value for the k kwarg will depend on the method chosen and the arm you are using. Use the following as a rough guide chan, k = 1.0 - 0.01, wampler, k = 0.01 - 0.0001, and sugihara, k = 0.1 - 0.0001

When using the this method, the initial joint coordinates $q_0$, should correspond to a non-singular manipulator pose, since it uses the manipulator Jacobian.

This class supports null-space motion to assist with maximising manipulability and avoiding joint limits. These are enabled by setting kq and km to non-zero values.

References

J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).
J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

See also

ik_NR() ik_GN() ikine_LM()

Changed in version 1.0.4: Merged the Levenberg-Marquardt IK solvers into the ik_LM method

ik_NR(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, pinv=True, pinv_damping=0.0, tool=None)

Fast numerical inverse kinematics using Newton-Raphson optimization

Parameters:

Tep (ndarray | SE3) – The desired end-effector pose (a single pose only, for a trajectory use ikine_NR())
end (str | Link | Gripper | None) – the link considered as the end-effector
start (str | Link | Gripper | None) – the link considered as the base frame, defaults to the robots’s base frame
q0 (ndarray | None) – initial joint configuration (default to random valid joint configuration constrained by the joint limits of the robot)
ilimit (int) – maximum number of iterations per search
slimit (int) – maximum number of search attempts
tol (float) – final error tolerance
mask (ndarray | None) – a mask vector which weights the end-effector error priority. Corresponds to translation in X, Y and Z and rotation about X, Y and Z respectively
joint_limits (bool) – constrain the solution to being within the joint limits of the robot (reject solution with invalid joint configurations and perform another search up to the slimit)
pinv (int) – Use the pseudo-inverse instead of the normal matrix inverse
pinv_damping (float) – Damping factor for the pseudo-inverse
tool (ndarray | SE3 | None) – a static tool transformation matrix to apply to the end of end; defaults to the robot’s own self.tool if not given

Returns:

an IKSolution containing joint coordinates q, success flag, iterations, searches and residual error value (reason is always empty – this fast C++ solver doesn’t produce a granular failure reason string, unlike ikine_NR())

Return type:

IKSolution

Warning

This method needs the compiled C++ extension. It raises RuntimeError if the extension is unavailable, for example in a pure-Python wheel or under Pyodide/JupyterLite, and if the sequence has a SymPy parameter, even a numeric one such as sympy.pi/2, or a joint with an offset, which the compiled code cannot use. In all of these cases use ikine_NR(), which has none of these limits.

sol = ets.ik_NR(Tep) are the joint coordinates (n) corresponding to the robot end-effector pose Tep which is an SE3 or ndarray object. This method can be used for robots with any number of degrees of freedom. This is a fast solver implemented in C++.

See the Inverse Kinematics Docs Page for more details and for a tutorial on numerical IK, see here.

Note

When using this method with redundant robots (>6 DoF), pinv must be set to True

The return value sol is an IKSolution with fields:

Field	Type	Description
`q`	ndarray(n)	joint coordinates in units of radians or metres
`success`	bool	whether a solution was found
`iterations`	int	total number of iterations
`searches`	int	total number of searches
`residual`	float	final value of cost function
`reason`	str	always empty for this C++ solver

If success == False the q values will be valid numbers, but the solution will be in error. The amount of error is indicated by the residual.

Each iteration uses the Newton-Raphson optimisation method

\[\vec{q}_{k+1} = \vec{q}_k + {^0\mat{J}(\vec{q}_k)}^{-1} \vec{e}_k\]

Examples

The following example gets a panda robot object, makes a goal pose Tep, and then solves for the joint coordinates which result in the pose Tep using the ik_NR method.

>>> import roboticstoolbox as rtb
>>> panda = rtb.models.Panda()
>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
>>> panda.ik_NR(Tep)
IKSolution: q=[0.753, -0.3904, -0.6602, -2.193, -0.2681, 1.968, 0.9963], success=True, iterations=11, searches=2, residual=6.32e-07

Notes

When using the this method, the initial joint coordinates $q_0$, should correspond to a non-singular manipulator pose, since it uses the manipulator Jacobian.

References

J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).
J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

See also

ik_LM() ik_GN() ikine_NR()

ikine_GN(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, pinv=False, kq=0.0, km=0.0, ps=0.0, pi=0.3, tool=None, **kwargs)

Gauss-Newton Numerical Inverse Kinematics Solver

Parameters:

Tep (ndarray | SE3) – The desired end-effector pose or pose trajectory
end (str | Link | Gripper | None) – the link considered as the end-effector
start (str | Link | Gripper | None) – the link considered as the base frame, defaults to the robots’s base frame
q0 (ndarray | Sequence[float] | None) – The initial joint coordinates, a vector (n,), or a matrix (m, n) whose rows are the starting points of the first m searches (any further searches start from random valid coordinates)
ilimit (int) – How many iterations are allowed within a search before a new search is started
slimit (int) – How many searches are allowed before being deemed unsuccessful
tol (float) – Maximum allowed residual error E
mask (ndarray | Sequence[float] | None) – A 6 vector which assigns weights to Cartesian degrees-of-freedom error priority
joint_limits (bool) – Reject solutions with joint limit violations
seed (int | None) – A seed for the private RNG used to generate random joint coordinate vectors
pinv (bool) – If True, will use the pseudoinverse in the step method instead of the normal inverse
kq (float) – The gain for joint limit avoidance. Setting to 0.0 will remove this completely from the solution
km (float) – The gain for maximisation. Setting to 0.0 will remove this completely from the solution
ps (float) – The minimum angle/distance (in radians or metres) in which the joint is allowed to approach to its limit
pi (ndarray | float) – The influence angle/distance (in radians or metres) in null space motion becomes active
tool (ndarray | SE3 | None) – a static tool transformation matrix to apply to the end of end; defaults to the robot’s own self.tool if not given

Returns:

an IKSolution containing joint coordinates q, success flag, iterations, searches, residual error value, and reason string if applicable

Return type:

If Tep is a trajectory (an SE3 with N poses, or an array with shape (N, 4, 4)) each pose is solved independently from the same q0, so consecutive solutions are not guaranteed to be on the same IK branch. q has shape (N, n), success is True only if every pose succeeded, and residual is the maximum over the poses. See solve().

Changed in version 1.5.0: For a trajectory, residual is the maximum over the poses, it was the minimum.

A method which provides functionality to perform numerical inverse kinematics (IK) using the Gauss-Newton method.

See the Inverse Kinematics Docs Page for more details and for a tutorial on numerical IK, see here.

Note

When using this method with redundant robots (>6 DoF), pinv must be set to True

Each iteration uses the Gauss-Newton optimisation method

\[\begin{split}\vec{q}_{k+1} &= \vec{q}_k + \left( {\mat{J}(\vec{q}_k)}^\top \mat{W}_e \ {\mat{J}(\vec{q}_k)} \right)^{-1} \bf{g}_k \\ \bf{g}_k &= {\mat{J}(\vec{q}_k)}^\top \mat{W}_e \vec{e}_k\end{split}\]

where $\mat{J} = {^0\mat{J}}$ is the base-frame manipulator Jacobian. If $\mat{J}(\vec{q}_k)$ is non-singular, and $\mat{W}_e = \mat{1}_n$, then the above provides the pseudoinverse solution. However, if $\mat{J}(\vec{q}_k)$ is singular, the above can not be computed and the GN solution is infeasible.

Examples

The following example gets a panda robot object, makes a goal pose Tep, and then solves for the joint coordinates which result in the pose Tep using the ikine_GN method.

>>> import roboticstoolbox as rtb
>>> panda = rtb.models.Panda()
>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
>>> panda.ikine_GN(Tep)
IKSolution: q=[1.372, 0.3554, -2.277, -1.312, -1.108, 1.806, -0.1273], success=False, reason=iteration and search limit reached, 100 numpy.LinAlgError encountered, iterations=100, searches=100, residual=1.49

Notes

When using the this method, the initial joint coordinates $q_0$, should correspond to a non-singular manipulator pose, since it uses the manipulator Jacobian.

This class supports null-space motion to assist with maximising manipulability and avoiding joint limits. These are enabled by setting kq and km to non-zero values.

References

J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).
J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

See also

IK_GN: An IK Solver class which implements the Gauss-Newton optimisation technique
ikine_LM: Implements the IK_LM class as a method within the ETS class
ikine_NR: Implements the IK_NR class as a method within the ETS class
ikine_QP: Implements the IK_QP class as a method within the ETS class
ik_GN(): The fast, C++-backed equivalent of this method (requires the compiled extension)

Changed in version 1.0.4: Added the Gauss-Newton IK solver method on the Robot class

ikine_LM(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, k=1.0, method='chan', kq=0.0, km=0.0, ps=0.0, pi=0.3, tool=None, **kwargs)

Levenberg-Marquardt Numerical Inverse Kinematics Solver

Parameters:

Tep (ndarray | SE3) – The desired end-effector pose or pose trajectory
end (str | Link | Gripper | None) – the link considered as the end-effector
start (str | Link | Gripper | None) – the link considered as the base frame, defaults to the robots’s base frame
q0 (ndarray | Sequence[float] | None) – The initial joint coordinates, a vector (n,), or a matrix (m, n) whose rows are the starting points of the first m searches (any further searches start from random valid coordinates)
ilimit (int) – How many iterations are allowed within a search before a new search is started
slimit (int) – How many searches are allowed before being deemed unsuccessful
tol (float) – Maximum allowed residual error E
mask (ndarray | Sequence[float] | None) – A 6 vector which assigns weights to Cartesian degrees-of-freedom error priority
joint_limits (bool) – Reject solutions with joint limit violations
seed (int | None) – A seed for the private RNG used to generate random joint coordinate vectors
k (float) – Sets the gain value for the damping matrix Wn in the next iteration
method (Literal['chan', 'wampler', 'sugihara']) – One of “chan”, “sugihara” or “wampler”. Defines which method is used to calculate the damping matrix Wn in the step method
kq (float) – The gain for joint limit avoidance. Setting to 0.0 will remove this completely from the solution
km (float) – The gain for maximisation. Setting to 0.0 will remove this completely from the solution
ps (float) – The minimum angle/distance (in radians or metres) in which the joint is allowed to approach to its limit
pi (ndarray | float) – The influence angle/distance (in radians or metres) in null space motion becomes active
tool (ndarray | SE3 | None) – a static tool transformation matrix to apply to the end of end; defaults to the robot’s own self.tool if not given

Returns:

an IKSolution containing joint coordinates q, success flag, iterations, searches, residual error value, and reason string if applicable

Return type:

If Tep is a trajectory (an SE3 with N poses, or an array with shape (N, 4, 4)) each pose is solved independently from the same q0, so consecutive solutions are not guaranteed to be on the same IK branch. q has shape (N, n), success is True only if every pose succeeded, and residual is the maximum over the poses. See solve().

Changed in version 1.5.0: For a trajectory, residual is the maximum over the poses, it was the minimum.

A method which provides functionality to perform numerical inverse kinematics (IK) using the Levenberg-Marquardt method.

See the Inverse Kinematics Docs Page for more details and for a tutorial on numerical IK, see here.

The operation is defined by the choice of the method kwarg.

The step is defined as

\[\begin{split}\vec{q}_{k+1} &= \vec{q}_k + \left( \mat{A}_k \right)^{-1} \bf{g}_k \\ % \mat{A}_k &= {\mat{J}(\vec{q}_k)}^\top \mat{W}_e \ {\mat{J}(\vec{q}_k)} + \mat{W}_n\end{split}\]

where $\mat{W}_n = \text{diag}(\vec{w_n})(\vec{w_n} \in \mathbb{R}^n_{>0})$ is a diagonal damping matrix. The damping matrix ensures that $\mat{A}_k$ is non-singular and positive definite. The performance of the LM method largely depends on the choice of $\mat{W}_n$.

Chan’s Method

Chan proposed

\[\mat{W}_n = λ E_k \mat{1}_n\]

where λ is a constant which reportedly does not have much influence on performance. Use the kwarg k to adjust the weighting term λ.

Sugihara’s Method

Sugihara proposed

\[\mat{W}_n = E_k \mat{1}_n + \text{diag}(\hat{\vec{w}}_n)\]

where $\hat{\vec{w}}_n \in \mathbb{R}^n$, $\hat{w}_{n_i} = l^2 \sim 0.01 l^2$, and $l$ is the length of a typical link within the manipulator. We provide the variable k as a kwarg to adjust the value of $w_n$.

Wampler’s Method

Wampler proposed $\vec{w_n}$ to be a constant. This is set through the k kwarg.

Examples

The following example makes a panda robot object, makes a goal pose Tep, and then solves for the joint coordinates which result in the pose Tep using the ikine_LM method.

>>> import roboticstoolbox as rtb
>>> panda = rtb.models.Panda()
>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
>>> panda.ikine_LM(Tep)
IKSolution: q=[-1.957, -1.617, 0.8777, -2.067, 1.339, 1.007, -0.1326], success=True, iterations=11, searches=1, residual=4.36e-11

Notes

The value for the k kwarg will depend on the method chosen and the arm you are using. Use the following as a rough guide chan, k = 1.0 - 0.01, wampler, k = 0.01 - 0.0001, and sugihara, k = 0.1 - 0.0001

When using the this method, the initial joint coordinates $q_0$, should correspond to a non-singular manipulator pose, since it uses the manipulator Jacobian.

This class supports null-space motion to assist with maximising manipulability and avoiding joint limits. These are enabled by setting kq and km to non-zero values.

References

J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).
J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

See also

IK_LM: An IK Solver class which implements the Levenberg-Marquardt optimisation technique
ikine_NR: Implements the IK_NR class as a method within the Robot class
ikine_GN: Implements the IK_GN class as a method within the Robot class
ikine_QP: Implements the IK_QP class as a method within the Robot class
ik_LM(): The fast, C++-backed equivalent of this method (requires the compiled extension)

Changed in version 1.0.4: Added the Levenberg-Marquardt IK solver method on the Robot class

ikine_NR(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, pinv=False, kq=0.0, km=0.0, ps=0.0, pi=0.3, tool=None, **kwargs)

Newton-Raphson Numerical Inverse Kinematics Solver

Parameters:

Tep (ndarray | SE3) – The desired end-effector pose or pose trajectory
end (str | Link | Gripper | None) – the link considered as the end-effector
start (str | Link | Gripper | None) – the link considered as the base frame, defaults to the robots’s base frame
q0 (ndarray | Sequence[float] | None) – The initial joint coordinates, a vector (n,), or a matrix (m, n) whose rows are the starting points of the first m searches (any further searches start from random valid coordinates)
ilimit (int) – How many iterations are allowed within a search before a new search is started
slimit (int) – How many searches are allowed before being deemed unsuccessful
tol (float) – Maximum allowed residual error E
mask (ndarray | Sequence[float] | None) – A 6 vector which assigns weights to Cartesian degrees-of-freedom error priority
joint_limits (bool) – Reject solutions with joint limit violations
seed (int | None) – A seed for the private RNG used to generate random joint coordinate vectors
pinv (bool) – If True, will use the pseudoinverse in the step method instead of the normal inverse
kq (float) – The gain for joint limit avoidance. Setting to 0.0 will remove this completely from the solution
km (float) – The gain for maximisation. Setting to 0.0 will remove this completely from the solution
ps (float) – The minimum angle/distance (in radians or metres) in which the joint is allowed to approach to its limit
pi (ndarray | float) – The influence angle/distance (in radians or metres) in null space motion becomes active
tool (ndarray | SE3 | None) – a static tool transformation matrix to apply to the end of end; defaults to the robot’s own self.tool if not given

Returns:

an IKSolution containing joint coordinates q, success flag, iterations, searches, residual error value, and reason string if applicable

Return type:

If Tep is a trajectory (an SE3 with N poses, or an array with shape (N, 4, 4)) each pose is solved independently from the same q0, so consecutive solutions are not guaranteed to be on the same IK branch. q has shape (N, n), success is True only if every pose succeeded, and residual is the maximum over the poses. See solve().

Changed in version 1.5.0: For a trajectory, residual is the maximum over the poses, it was the minimum.

A method which provides functionality to perform numerical inverse kinematics (IK) using the Newton-Raphson method.

See the Inverse Kinematics Docs Page for more details and for a tutorial on numerical IK, see here.

Note

When using this method with redundant robots (>6 DoF), pinv must be set to True

Each iteration uses the Newton-Raphson optimisation method

\[\vec{q}_{k+1} = \vec{q}_k + {^0\mat{J}(\vec{q}_k)}^{-1} \vec{e}_k\]

Examples

The following example gets a panda robot object, makes a goal pose Tep, and then solves for the joint coordinates which result in the pose Tep using the ikine_NR method.

>>> import roboticstoolbox as rtb
>>> panda = rtb.models.Panda()
>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
>>> panda.ikine_NR(Tep)
IKSolution: q=[1.949, 0.7163, 1.07, -0.1702, -2.793, 2.411, -2.799], success=False, reason=iteration and search limit reached, 100 numpy.LinAlgError encountered, iterations=100, searches=100, residual=3.23

Notes

When using the this method, the initial joint coordinates $q_0$, should correspond to a non-singular manipulator pose, since it uses the manipulator Jacobian.

This class supports null-space motion to assist with maximising manipulability and avoiding joint limits. These are enabled by setting kq and km to non-zero values.

References

J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).
J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

See also

IK_NR: An IK Solver class which implements the Newton-Raphson optimisation technique
ikine_LM: Implements the IK_LM class as a method within the ETS class
ikine_GN: Implements the IK_GN class as a method within the ETS class
ikine_QP: Implements the IK_QP class as a method within the ETS class
ik_NR(): The fast, C++-backed equivalent of this method (requires the compiled extension)

Changed in version 1.0.4: Added the Newton-Raphson IK solver method on the Robot class

ikine_QP(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, kj=1.0, ks=1.0, kq=0.0, km=0.0, ps=0.0, pi=0.3, tool=None, **kwargs)

Quadratic Programming Numerical Inverse Kinematics Solver

Parameters:

Tep (ndarray | SE3) – The desired end-effector pose or pose trajectory
end (str | Link | Gripper | None) – the link considered as the end-effector
start (str | Link | Gripper | None) – the link considered as the base frame, defaults to the robots’s base frame
q0 (ndarray | Sequence[float] | None) – The initial joint coordinates, a vector (n,), or a matrix (m, n) whose rows are the starting points of the first m searches (any further searches start from random valid coordinates)
ilimit (int) – How many iterations are allowed within a search before a new search is started
slimit (int) – How many searches are allowed before being deemed unsuccessful
tol (float) – Maximum allowed residual error E
mask (ndarray | Sequence[float] | None) – A 6 vector which assigns weights to Cartesian degrees-of-freedom error priority
joint_limits (bool) – Reject solutions with joint limit violations
seed (int | None) – A seed for the private RNG used to generate random joint coordinate vectors
kj (float) – A gain for joint velocity norm minimisation
ks (float) – A gain which adjusts the cost of slack (intentional error)
kq (float) – The gain for joint limit avoidance. Setting to 0.0 will remove this completely from the solution
km (float) – The gain for maximisation. Setting to 0.0 will remove this completely from the solution
ps (float) – The minimum angle/distance (in radians or metres) in which the joint is allowed to approach to its limit
pi (ndarray | float) – The influence angle/distance (in radians or metres) in null space motion becomes active
tool (ndarray | SE3 | None) – a static tool transformation matrix to apply to the end of end; defaults to the robot’s own self.tool if not given

Raises:

ImportError – If the package qpsolvers is not installed, see the note below

Returns:

an IKSolution containing joint coordinates q, success flag, iterations, searches, residual error value, and reason string if applicable

Return type:

If Tep is a trajectory (an SE3 with N poses, or an array with shape (N, 4, 4)) each pose is solved independently from the same q0, so consecutive solutions are not guaranteed to be on the same IK branch. q has shape (N, n), success is True only if every pose succeeded, and residual is the maximum over the poses. See solve().

Changed in version 1.5.0: For a trajectory, residual is the maximum over the poses, it was the minimum.

A method that provides functionality to perform numerical inverse kinematics (IK) using a quadratic programming approach.

Note

This needs the packages qpsolvers and quadprog, which are not installed with the Toolbox by default. Install them with the qp extra: pip install roboticstoolbox-python[qp].

See the Inverse Kinematics Docs Page for more details and for a tutorial on numerical IK, see here.

Each iteration uses the following approach

\[\vec{q}_{k+1} = \vec{q}_{k} + \dot{\vec{q}}.\]

where the QP is defined as

\[\begin{split}\min_x \quad f_o(\vec{x}) &= \frac{1}{2} \vec{x}^\top \mathcal{Q} \vec{x}+ \mathcal{C}^\top \vec{x}, \\ \text{subject to} \quad \mathcal{J} \vec{x} &= \vec{\nu}, \\ \mathcal{A} \vec{x} &\leq \mathcal{B}, \\ \vec{x}^- &\leq \vec{x} \leq \vec{x}^+\end{split}\]

with

\[\begin{split}\vec{x} &= \begin{pmatrix} \dvec{q} \\ \vec{\delta} \end{pmatrix} \in \mathbb{R}^{(n+6)} \\ \mathcal{Q} &= \begin{pmatrix} \lambda_q \mat{1}_{n} & \mathbf{0}_{6 \times 6} \\ \mathbf{0}_{n \times n} & \lambda_\delta \mat{1}_{6} \end{pmatrix} \in \mathbb{R}^{(n+6) \times (n+6)} \\ \mathcal{J} &= \begin{pmatrix} \mat{J}(\vec{q}) & \mat{1}_{6} \end{pmatrix} \in \mathbb{R}^{6 \times (n+6)} \\ \mathcal{C} &= \begin{pmatrix} \mat{J}_m \\ \bf{0}_{6 \times 1} \end{pmatrix} \in \mathbb{R}^{(n + 6)} \\ \mathcal{A} &= \begin{pmatrix} \mat{1}_{n \times n + 6} \\ \end{pmatrix} \in \mathbb{R}^{(l + n) \times (n + 6)} \\ \mathcal{B} &= \eta \begin{pmatrix} \frac{\rho_0 - \rho_s} {\rho_i - \rho_s} \\ \vdots \\ \frac{\rho_n - \rho_s} {\rho_i - \rho_s} \end{pmatrix} \in \mathbb{R}^{n} \\ \vec{x}^{-, +} &= \begin{pmatrix} \dvec{q}^{-, +} \\ \vec{\delta}^{-, +} \end{pmatrix} \in \mathbb{R}^{(n+6)},\end{split}\]

where $\vec{\delta} \in \mathbb{R}^6$ is the slack vector, $\lambda_\delta \in \mathbb{R}^+$ is a gain term which adjusts the cost of the norm of the slack vector in the optimiser, $\dvec{q}^{-,+}$ are the minimum and maximum joint velocities, and $\dvec{\delta}^{-,+}$ are the minimum and maximum slack velocities.

Examples

The following example gets a panda robot object, makes a goal pose Tep, and then solves for the joint coordinates which result in the pose Tep using the ikine_QP method.

>>> import roboticstoolbox as rtb
>>> panda = rtb.models.Panda()
>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
>>> panda.ikine_QP(Tep)
!! [RUNBLOCK-ERROR] robot/Robot.py:docstring of roboticstoolbox.robot.RobotKinematics.RobotKinematicsMixin.ikine_QP:3060
    Install them using 'pip install roboticstoolbox-python[qp]'

Notes

When using the this method, the initial joint coordinates $q_0$, should correspond to a non-singular manipulator pose, since it uses the manipulator Jacobian.

This class supports null-space motion to assist with maximising manipulability and avoiding joint limits. These are enabled by setting kq and km to non-zero values.

References

J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).
J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

See also

IK_QP: An IK Solver class which implements a quadratic programming approach
ikine_LM: Implements the IK_LM class as a method within the ETS class
ikine_GN: Implements the IK_GN class as a method within the ETS class
ikine_NR: Implements the IK_NR class as a method within the ETS class

Changed in version 1.0.4: Added the Quadratic Programming IK solver method on the Robot class

inertia(q)

Manipulator inertia matrix

Parameters:: q (ndarray) – Joint coordinates
Returns:: The inertia matrix
Return type:: ndarray

inertia(q) is the symmetric joint inertia matrix (n,n) which relates joint torque to joint acceleration for the robot at joint configuration q.

Trajectory operation

If q is a matrix (m,n), each row is interpreted as a joint state vector, and the result is a 3d-matrix (nxnxk) where each plane corresponds to the inertia for the corresponding row of q.

Examples

>>> import roboticstoolbox as rtb
>>> puma = rtb.models.DH.Puma560()
>>> puma.inertia(puma.qz)
array([[ 3.9611, -0.1627, -0.1389,  0.0016, -0.0004,  0.    ],
       [-0.1627,  4.4566,  0.3727,  0.    ,  0.0019,  0.    ],
       [-0.1389,  0.3727,  0.9387,  0.    ,  0.0019,  0.    ],
       [ 0.0016,  0.    ,  0.    ,  0.1924,  0.    ,  0.    ],
       [-0.0004,  0.0019,  0.0019,  0.    ,  0.1713,  0.    ],
       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.1941]])

Notes

The diagonal elements M[j,j] are the inertia seen by joint
actuator j.
The off-diagonal elements M[j,k] are coupling inertias that
relate acceleration on joint j to force/torque on joint k.
The diagonal terms include the motor inertia reflected through
the gear ratio.

See also

cinertia()

inertia_x(q=None, pinv=False, representation='rpy/xyz', Ji=None)

Operational space inertia matrix

Parameters:

q (Any) – Joint coordinates
pinv (bool) – use pseudo inverse rather than inverse (Default value = False)
representation (Literal['rpy/xyz', 'rpy/zyx', 'eul', 'exp']) – the type of analytical Jacobian to use, default is 'rpy/xyz'
Ji (Any) – The inverse analytical Jacobian (base-frame)

Returns:

The operational space inertia matrix

Return type:

robot.inertia_x(q) is the operational space (Cartesian) inertia matrix which relates Cartesian force/torque to Cartesian acceleration at the joint configuration q.

\[\mathbf{M}_x = \mathbf{J}(q)^{-T} \mathbf{M}(q) \mathbf{J}(q)^{-1}\]

The transformation to operational space requires an analytical, rather than geometric, Jacobian. analytical can be one of:

Value

Rotational representation

'rpy/xyz'

RPY angular rates in XYZ order (default)

'rpy/zyx'

RPY angular rates in XYZ order

'eul'

Euler angular rates in ZYZ order

'exp'

exponential coordinate rates

Trajectory operation

If q is a matrix (m,n), each row is interpreted as a joint state vector, and the result is a 3d-matrix (m,n,n) where each plane corresponds to the Cartesian inertia for the corresponding row of q.

Examples

>>> import roboticstoolbox as rtb
>>> puma = rtb.models.DH.Puma560()
>>> puma.inertia_x(puma.qn)
array([[17.2954, -2.7542, -9.6233,  0.    ,  0.2795,  0.    ],
       [-2.7542, 11.6452,  1.2459,  0.0001, -0.0703,  0.0001],
       [-9.6233,  1.2459, 13.3348,  0.    ,  0.2767,  0.    ],
       [ 0.    ,  0.0001,  0.    ,  0.1941,  0.    ,  0.1941],
       [ 0.2795, -0.0703,  0.2767,  0.    ,  0.1713,  0.    ],
       [ 0.    ,  0.0001,  0.    ,  0.1941,  0.    ,  0.1941]])

Notes

If the robot is not 6 DOF the pinv option is set True.
pinv() is around 5x slower than inv()

Warning

Assumes that the operational space has 6 DOF.

See also

isprismatic() prismaticjoints()

isprismatic(j)

Check if joint is prismatic

Return type:: bool
Returns:: True if prismatic

Examples

>>> import roboticstoolbox as rtb
>>> puma = rtb.models.DH.Puma560()
>>> puma.prismaticjoints
[False, False, False, False, False, False]
>>> stanford = rtb.models.DH.Stanford()
>>> stanford.isprismatic(1)
False

See also

isrevolute(j)

Check if joint is revolute

Return type:: bool
Returns:: True if revolute

Examples

>>> import roboticstoolbox as rtb
>>> puma = rtb.models.DH.Puma560()
>>> puma.revolutejoints
[True, True, True, True, True, True]
>>> stanford = rtb.models.DH.Stanford()
>>> stanford.isrevolute(1)
True

See also

isrevolute() revolutejoints()

itorque(q, qdd)

Inertia torque

Parameters:

q (Any) – Joint coordinates
qdd (Any) – Joint acceleration

Returns:

The inertia torque vector

Return type:

itorque(q, qdd) is the inertia force/torque vector (n) at the specified joint configuration q (n) and acceleration qdd (n), and n is the number of robot joints. It is $\mathbf{I}(q) \ddot{q}$.

Trajectory operation

If q and qdd are matrices (m,n), each row is interpreted as a joint configuration, and the result is a matrix (m,n) where each row is the corresponding joint torques.

Examples

>>> import roboticstoolbox as rtb
>>> puma = rtb.models.DH.Puma560()
>>> puma.itorque(puma.qz, 0.5 * np.ones((6,)))
array([1.8304, 2.3343, 0.5872, 0.0971, 0.0873, 0.0971])

Notes

If the robot model contains non-zero motor inertia then this
will be included in the result.

See also

jacob0(q, end=None, start=None, tool=None)

Manipulator geometric Jacobian in the start frame

Parameters:

q (ndarray | Sequence[float]) – Joint coordinate vector
end (str | Link | Gripper | None) – the particular link or gripper whose velocity the Jacobian describes, defaults to the end-effector if only one is present
start (str | Link | Gripper | None) – the link considered as the base frame, defaults to the robots’s base frame
tool (ndarray | SE3 | None) – a static tool transformation matrix to apply to the end of end, defaults to None

Return type:

Returns:

Manipulator Jacobian in the start frame

robot.jacobo(q) is the manipulator Jacobian matrix which maps joint velocity to end-effector spatial velocity expressed in the base frame.

End-effector spatial velocity $\nu = (v_x, v_y, v_z, \omega_x, \omega_y, \omega_z)^T$ is related to joint velocity by ${}^{E}\!\nu = \mathbf{J}_m(q) \dot{q}$.

Examples

The following example makes a Puma560 robot object, and solves for the base-frame Jacobian at the zero joint angle configuration

>>> import roboticstoolbox as rtb
>>> puma = rtb.models.Puma560()
>>> puma.jacob0([0, 0, 0, 0, 0, 0])
array([[ 0.1295, -0.4854, -0.4854, -0.    , -0.0533,  0.    ],
       [ 0.4318,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
       [-0.    ,  0.4318,  0.    ,  0.    ,  0.    ,  0.    ],
       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
       [ 0.    , -1.    , -1.    ,  0.    , -1.    ,  0.    ],
       [ 1.    ,  0.    ,  0.    ,  1.    ,  0.    ,  1.    ]])

Notes

This is the geometric Jacobian as described in texts by
Corke, Spong etal., Siciliano etal. The end-effector velocity is described in terms of translational and angular velocity, not a velocity twist as per the text by Lynch & Park.

References

J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

jacob0_analytical(q, representation='rpy/xyz', end=None, start=None, tool=None)

Manipulator analytical Jacobian in the start frame

Parameters:

q (ndarray | Sequence[float]) – Joint coordinate vector
representation (Literal['rpy/xyz', 'rpy/zyx', 'eul', 'exp']) – angular representation
end (str | Link | Gripper | None) – the particular link or gripper whose velocity the Jacobian describes, defaults to the base link
start (str | Link | Gripper | None) – the link considered as the end-effector, defaults to the robots’s end-effector
tool (ndarray | SE3 | None) – a static tool transformation matrix to apply to the end of end, defaults to None

Return type:

Returns:

Manipulator Jacobian in the start frame

robot.jacob0_analytical(q) is the manipulator Jacobian matrix which maps joint velocity to end-effector spatial velocity expressed in the start frame.

End-effector spatial velocity $\nu = (v_x, v_y, v_z, \omega_x, \omega_y, \omega_z)^T$ is related to joint velocity by ${}^{E}\!\nu = \mathbf{J}_m(q) \dot{q}$.

`representation`	Rotational representation
`'rpy/xyz'`	RPY angular rates in XYZ order
`'rpy/zyx'`	RPY angular rates in ZYX order
`'eul'`	Euler angular rates in ZYZ order
`'exp'`	exponential coordinate rates

Examples

Makes a robot object and computes the analytic Jacobian for the given joint configuration

>>> import roboticstoolbox as rtb
>>> puma = rtb.models.ETS.Puma560()
>>> puma.jacob0_analytical([0, 0, 0, 0, 0, 0])
array([[ 0.15  ,  0.8636,  0.4318,  0.    ,  0.    ,  0.    ],
       [ 0.2203,  0.    ,  0.    ,  0.2   ,  0.    ,  0.2   ],
       [ 0.    , -0.2203, -0.2   ,  0.    , -0.2   ,  0.    ],
       [ 1.    ,  0.    ,  0.    ,  1.    ,  0.    ,  1.    ],
       [ 0.    ,  1.    ,  1.    ,  0.    ,  1.    ,  0.    ],
       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ]])

jacobe(q, end=None, start=None, tool=None)

Manipulator geometric Jacobian in the end-effector frame

Parameters:

q (ndarray | Sequence[float]) – Joint coordinate vector
end (str | Link | Gripper | None) – the particular link or gripper whose velocity the Jacobian describes, defaults to the end-effector if only one is present
start (str | Link | Gripper | None) – the link considered as the base frame, defaults to the robots’s base frame
tool (ndarray | SE3 | None) – a static tool transformation matrix to apply to the end of end, defaults to None

Return type:

Returns:

Manipulator Jacobian in the end frame

robot.jacobe(q) is the manipulator Jacobian matrix which maps joint velocity to end-effector spatial velocity expressed in the end frame.

End-effector spatial velocity $\nu = (v_x, v_y, v_z, \omega_x, \omega_y, \omega_z)^T$ is related to joint velocity by ${}^{E}\!\nu = \mathbf{J}_m(q) \dot{q}$.

Examples

The following example makes a Puma560 robot object, and solves for the end-effector frame Jacobian at the zero joint angle configuration

>>> import roboticstoolbox as rtb
>>> puma = rtb.models.Puma560()
>>> puma.jacobe([0, 0, 0, 0, 0, 0])
array([[ 0.1295, -0.4854, -0.4854, -0.    , -0.0533,  0.    ],
       [ 0.4318,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
       [-0.    ,  0.4318,  0.    ,  0.    ,  0.    ,  0.    ],
       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
       [ 0.    , -1.    , -1.    ,  0.    , -1.    ,  0.    ],
       [ 1.    ,  0.    ,  0.    ,  1.    ,  0.    ,  1.    ]])

Notes

This is the geometric Jacobian as described in texts by
Corke, Spong etal., Siciliano etal. The end-effector velocity is described in terms of translational and angular velocity, not a velocity twist as per the text by Lynch & Park.

References

J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

jointdynamics(q, qd=None)

Transfer function of joint actuator

Parameters:

q (Any) – Joint coordinates
qd (Any) – Joint velocity

Returns:

transfer function denominators

Return type:

tf = jointdynamics(qd, q) calculates a vector of n continuous-time transfer functions that represent the transfer function 1/(Js+B) for each joint based on the dynamic parameters of the robot and the configuration q (n). n is the number of robot joints. The result is a list of tuples (J, B) for each joint.

tf = jointdynamics(q, qd) as above but include the linearized effects of Coulomb friction when operating at joint velocity QD (1xN).

property keywords: list[str]

Keywords that describe the robot

Returns:: the keywords, which the model catalog can filter on

property link_dict: dict[str, Link]

linkcolormap(linkcolors='viridis')

Create a colormap for robot joints.

Parameters:: linkcolors (list[Any] | str) – list of colors or colormap name, defaults to "viridis"
Return type:: Any
Returns:: the color map

cm = robot.linkcolormap() is an n-element colormap that gives a unique color for every link. The RGBA colors for link j are cm(j).
cm = robot.linkcolormap(cmap) as above but cmap is the name of a valid matplotlib colormap. The default, example above, is the viridis colormap.
cm = robot.linkcolormap(list of colors) as above but a colormap is created from a list of n color names given as strings, tuples or hexstrings.

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.DH.Puma560()
>>> cm = robot.linkcolormap("inferno")
>>> print(cm(range(6))) # cm(i) is 3rd color in colormap
>>> cm = robot.linkcolormap(
...     ['red', 'g', (0,0.5,0), '#0f8040', 'yellow', 'cyan'])
>>> print(cm(range(6)))

Notes

Colormaps have 4-elements: red, green, blue, alpha (RGBA)
Names of supported colors and colormaps are defined in the matplotlib documentation.
- Specifying colors
- Colormaps

property links: list[Link]

Robot links

Returns:: A list of link objects

Notes

It is probably more concise to index the robot object rather than the list of links, ie. the following are equivalent: - robot.links[i] - robot[i]

property manufacturer: Any

Get/set robot manufacturer’s name

robot.manufacturer is the robot manufacturer’s name
robot.manufacturer = ... checks and sets the manufacturer’s name

Returns:: robot manufacturer’s name

property n: int

Number of joints

Returns:: Number of joints

Examples

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.DH.Puma560()
>>> robot.n
6

See also

nlinks() nbranches()

property name: str

Get/set robot name

robot.name is the robot name
robot.name = ... checks and sets the robot name

Parameters:: name – the new robot name
Returns:: the current robot name

property nbranches: int

Number of branches

Number of branches in this robot. Computed as the number of links with zero children

Returns:: number of branches in the robot’s kinematic tree

Examples

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.ETS.Panda()
>>> robot.nbranches
1

See also

n() nlinks()

property nlinks: Any

Number of links

The returned number is the total of both variable joints and static links

Returns:: Number of links

Examples

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.DH.Puma560()
>>> robot.nlinks
6

See also

n() nbranches()

nofriction(coulomb=True, viscous=False)

Remove manipulator joint friction

Parameters:

coulomb (bool) – set the Coulomb friction to 0
viscous (bool) – set the viscous friction to 0

Return type:

friction() nofriction()

Returns:

A copy of the robot with dynamic parameters perturbed

nofriction() copies the robot and returns a robot with the same link parameters except the Coulomb and/or viscous friction parameter are set to zero.

See also

partial_fkine0(q, n=3, end=None, start=None)

Manipulator Forward Kinematics nth Partial Derivative

Parameters:

q (ndarray | Sequence[float]) – The joint angles/configuration of the robot (Optional, if not supplied will use the stored q values).
n (int) – The derivative order, must be >= 3
end (str | Link | Gripper | None) – the final link/Gripper which the Hessian represents
start (str | Link | Gripper | None) – the first link which the Hessian represents

Return type:

Returns:

The nth Partial Derivative of the forward kinematics

This method computes the nth derivative of the forward kinematics where n is greater than or equal to 3. This is an extension of the differential kinematics where the Jacobian is the first partial derivative and the Hessian is the second.

Examples

The following example makes a Panda robot object, and solves for the base-effector frame 4th defivative of the forward kinematics at the given joint angle configuration

>>> import roboticstoolbox as rtb
>>> panda = rtb.models.Panda()
>>> panda.partial_fkine0([0, -0.3, 0, -2.2, 0, 2, 0.7854], n=4)
array([[[[[ 0.484 ,  0.    ,  0.486 , ...,  0.1547, -0.    ,  0.    ],
          [-0.    , -0.0796, -0.    , ..., -0.    , -0.2006,  0.    ],
          [-0.    , -0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.    ,  1.    ,  0.    , ..., -0.    , -1.    , -0.    ],
          [ 0.    ,  0.    ,  0.2955, ..., -0.9463,  0.    , -0.0998],
          [ 0.    , -0.    ,  0.    , ..., -0.    ,  0.    , -0.    ]],

         [[ 0.    ,  0.484 ,  0.    , ...,  0.    , -0.1086,  0.    ],
          [-0.    , -0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.    , -0.    , -0.    , ..., -0.    ,  0.    ,  0.    ],
          [-1.    ,  0.    , -0.9553, ...,  0.3233, -0.    ,  0.995 ],
          [-0.    ,  0.    ,  0.    , ..., -0.    ,  0.    , -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ]],

         [[ 0.4624,  0.    ,  0.4643, ...,  0.1478, -0.    ,  0.    ],
          [-0.    ,  0.067 , -0.    , ..., -0.    , -0.2237,  0.    ],
          [-0.    ,  0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
          [-0.    ,  0.9553,  0.    , ..., -0.    , -0.9553, -0.    ],
          [-0.2955, -0.    ,  0.    , ..., -0.8085,  0.    ,  0.1987],
          [-0.    , -0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ]],

         ...,

         [[-0.1565, -0.    , -0.1571, ..., -0.05  ,  0.    ,  0.    ],
          [ 0.    , -0.4323,  0.    , ..., -0.    ,  0.1676,  0.    ],
          [ 0.    , -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    , -0.3233,  0.    , ...,  0.    ,  0.3233, -0.    ],
          [ 0.9463,  0.    ,  0.8085, ...,  0.    ,  0.    , -0.9093],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ]],

         [[ 0.    , -0.484 ,  0.    , ..., -0.    ,  0.1086,  0.    ],
          [-0.    ,  0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.    ,  0.    , -0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 1.    ,  0.    ,  0.9553, ..., -0.3233,  0.    , -0.995 ],
          [-0.    , -0.    , -0.    , ..., -0.    ,  0.    ,  0.    ],
          [-0.    , -0.    , -0.    , ...,  0.    ,  0.    ,  0.    ]],

         [[-0.4816, -0.    , -0.4835, ..., -0.1539,  0.    ,  0.    ],
          [ 0.    ,  0.0309,  0.    , ...,  0.    ,  0.2104,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    , -0.995 ,  0.    , ...,  0.    ,  0.995 ,  0.    ],
          [ 0.0998,  0.    , -0.1987, ...,  0.9093, -0.    ,  0.    ],
          [ 0.    ,  0.    , -0.    , ...,  0.    , -0.    ,  0.    ]]],


        [[[ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
          [-0.    , -0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
          [-0.    , -0.0796, -0.    , ..., -0.    , -0.2006,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    , -0.    ],
          [ 0.    , -0.    ,  0.2955, ..., -0.9463,  0.    , -0.0998]],

         [[-0.    ,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
          [ 0.    , -0.0796, -0.    , ..., -0.    , -0.2006,  0.    ],
          [-0.0796, -0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
          [-0.    ,  0.    , -0.    , ...,  0.    , -0.    ,  0.    ],
          [-0.    ,  0.    ,  0.2955, ..., -0.9463,  0.    , -0.0998],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    , -0.    ]],

         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [-0.143 , -0.    , -0.1436, ..., -0.0457, -0.    ,  0.    ],
          [-0.    ,  0.    , -0.    , ..., -0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
          [-0.    , -0.2955,  0.    , ..., -0.    ,  0.2955,  0.    ],
          [-0.2955, -0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ]],

         ...,

         [[ 0.    ,  0.    , -0.    , ..., -0.    ,  0.    ,  0.    ],
          [ 0.4581,  0.    ,  0.4599, ...,  0.1464,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    , -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.9463,  0.    , ...,  0.    , -0.9463, -0.    ],
          [ 0.9463,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ]],

         [[ 0.    , -0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ],
          [-0.    ,  0.0796,  0.    , ...,  0.    ,  0.2006,  0.    ],
          [ 0.0796,  0.    , -0.    , ..., -0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    , -0.    ],
          [-0.    , -0.    , -0.2955, ...,  0.9463,  0.    ,  0.0998],
          [-0.    , -0.    , -0.    , ...,  0.    ,  0.    ,  0.    ]],

         [[-0.    ,  0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.0483,  0.    ,  0.0485, ...,  0.0154,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
          [ 0.    , -0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.0998, -0.    , ...,  0.    , -0.0998,  0.    ],
          [ 0.0998,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ]]],


        [[[ 0.4624,  0.    ,  0.4643, ...,  0.1478, -0.    ,  0.    ],
          [-0.    , -0.0761, -0.    , ..., -0.    , -0.1916,  0.    ],
          [-0.143 , -0.    , -0.1436, ..., -0.0457, -0.    ,  0.    ],
          [ 0.    ,  0.9553,  0.    , ..., -0.    , -0.9553, -0.    ],
          [ 0.    ,  0.    ,  0.2823, ..., -0.904 ,  0.    , -0.0954],
          [ 0.    , -0.2955,  0.    , ...,  0.    ,  0.2955,  0.    ]],

         [[ 0.    ,  0.4624,  0.    , ...,  0.    , -0.1037,  0.    ],
          [-0.    , -0.    , -0.1436, ..., -0.0457, -0.    ,  0.    ],
          [ 0.    ,  0.0091,  0.    , ...,  0.    ,  0.1916,  0.    ],
          [-0.9553,  0.    , -0.9127, ...,  0.3089, -0.    ,  0.9506],
          [-0.    ,  0.    ,  0.    , ..., -0.    ,  0.2955,  0.    ],
          [ 0.2955,  0.    , -0.2823, ...,  0.904 , -0.    ,  0.0954]],

         [[ 0.4418,  0.    ,  0.486 , ...,  0.1547, -0.    ,  0.    ],
          [-0.    ,  0.064 , -0.    , ..., -0.    , -0.2137,  0.    ],
          [-0.007 ,  0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
          [-0.    ,  0.9127,  0.    , ..., -0.    , -1.    , -0.    ],
          [-0.2823, -0.    ,  0.    , ..., -0.7724,  0.    ,  0.1898],
          [-0.    ,  0.2823,  0.    , ..., -0.    ,  0.    ,  0.    ]],

         ...,

         [[-0.1495, -0.    , -0.286 , ..., -0.091 ,  0.    ,  0.    ],
          [ 0.    , -0.413 ,  0.    , ..., -0.    ,  0.1601,  0.    ],
          [ 0.0223, -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    , -0.3089,  0.    , ..., -0.    ,  0.5885, -0.    ],
          [ 0.904 ,  0.    ,  0.7724, ...,  0.    ,  0.    , -0.8687],
          [-0.    , -0.904 ,  0.    , ...,  0.    , -0.    , -0.    ]],

         [[ 0.    , -0.486 ,  0.    , ..., -0.    ,  0.0444,  0.    ],
          [-0.143 ,  0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
          [-0.    ,  0.067 ,  0.    , ...,  0.    , -0.    ,  0.    ],
          [ 0.9553,  0.    ,  1.    , ..., -0.5885,  0.    , -0.9801],
          [-0.    , -0.2955, -0.    , ..., -0.    ,  0.    ,  0.    ],
          [-0.2955,  0.    , -0.    , ...,  0.    ,  0.    ,  0.    ]],

         [[-0.4601, -0.    , -0.4763, ..., -0.1516,  0.    ,  0.    ],
          [ 0.    ,  0.0295,  0.    , ...,  0.    ,  0.201 ,  0.    ],
          [ 0.0023,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    , -0.9506,  0.    , ...,  0.    ,  0.9801,  0.    ],
          [ 0.0954, -0.    , -0.1898, ...,  0.8687, -0.    ,  0.    ],
          [-0.    , -0.0954, -0.    , ...,  0.    , -0.    ,  0.    ]]],


        ...,


        [[[-0.1565, -0.    , -0.1571, ..., -0.05  ,  0.    ,  0.    ],
          [ 0.    ,  0.0257,  0.    , ...,  0.    ,  0.0648,  0.    ],
          [ 0.4581,  0.    ,  0.4599, ...,  0.1464,  0.    ,  0.    ],
          [ 0.    , -0.3233, -0.    , ...,  0.    ,  0.3233,  0.    ],
          [ 0.    , -0.    , -0.0955, ...,  0.3059, -0.    ,  0.0323],
          [ 0.    ,  0.9463, -0.    , ...,  0.    , -0.9463, -0.    ]],

         [[-0.    , -0.1565, -0.    , ..., -0.    ,  0.0351,  0.    ],
          [ 0.    , -0.    , -0.0457, ...,  0.1464,  0.    ,  0.    ],
          [-0.    ,  0.4066, -0.    , ...,  0.    , -0.0648,  0.    ],
          [ 0.3233,  0.    ,  0.3089, ..., -0.1045,  0.    , -0.3217],
          [ 0.    ,  0.    , -0.    , ...,  0.    , -0.9463, -0.    ],
          [-0.9463,  0.    , -0.713 , ..., -0.3059, -0.    , -0.0323]],

         [[-0.1495, -0.    , -0.1366, ..., -0.091 , -0.    ,  0.    ],
          [ 0.    , -0.0217,  0.    , ...,  0.    ,  0.0723,  0.    ],
          [ 0.2994,  0.    ,  0.3028, ...,  0.1251,  0.    ,  0.    ],
          [ 0.    , -0.3089,  0.    , ...,  0.    ,  0.5885,  0.    ],
          [ 0.0955,  0.    ,  0.    , ...,  0.2614, -0.    , -0.0642],
          [ 0.    ,  0.713 ,  0.    , ...,  0.    , -0.8085, -0.    ]],

         ...,

         [[ 0.0506,  0.    ,  0.0075, ...,  0.1547,  0.    ,  0.    ],
          [-0.    ,  0.1398, -0.    , ..., -0.    , -0.0542,  0.    ],
          [ 0.2945,  0.    ,  0.2885, ..., -0.    , -0.    ,  0.    ],
          [-0.    ,  0.1045, -0.    , ...,  0.    , -1.    , -0.    ],
          [-0.3059, -0.    , -0.2614, ...,  0.    ,  0.    ,  0.294 ],
          [-0.    ,  0.3059, -0.    , ...,  0.    ,  0.    ,  0.    ]],

         [[ 0.    ,  0.2318,  0.    , ...,  0.    ,  0.1547,  0.    ],
          [ 0.4581,  0.    ,  0.5056, ..., -0.    , -0.    ,  0.    ],
          [ 0.    , -0.4323,  0.    , ..., -0.    , -0.    ,  0.    ],
          [-0.3233, -0.    , -0.5885, ...,  1.    ,  0.    ,  0.4161],
          [ 0.    ,  0.9463,  0.    , ..., -0.    ,  0.    , -0.    ],
          [ 0.9463,  0.    ,  0.8085, ..., -0.    ,  0.    ,  0.    ]],

         [[ 0.1557,  0.    ,  0.1518, ...,  0.0644,  0.    ,  0.    ],
          [ 0.    , -0.01  ,  0.    , ..., -0.    , -0.068 ,  0.    ],
          [ 0.0311, -0.    ,  0.0304, ..., -0.    , -0.    ,  0.    ],
          [-0.    ,  0.3217, -0.    , ...,  0.    , -0.4161,  0.    ],
          [-0.0323,  0.    ,  0.0642, ..., -0.294 ,  0.    ,  0.    ],
          [ 0.    ,  0.0323,  0.    , ..., -0.    ,  0.    ,  0.    ]]],


        [[[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [-0.    , -0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.    ,  0.0796,  0.    , ...,  0.    ,  0.2006,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    , -0.    ],
          [ 0.    , -0.    , -0.2955, ...,  0.9463,  0.    ,  0.0998]],

         [[-0.    ,  0.    , -0.    , ...,  0.    , -0.    ,  0.    ],
          [ 0.    , -0.2006, -0.    , ...,  0.    ,  0.2006,  0.    ],
          [-0.2006, -0.    , -0.2237, ...,  0.1676,  0.    ,  0.    ],
          [-0.    ,  0.    , -0.    , ...,  0.    , -0.    ,  0.    ],
          [-0.    ,  0.    , -0.    , ..., -0.    ,  0.    ,  0.0998],
          [ 0.    ,  0.    , -0.    , ...,  0.    , -0.    ,  0.    ]],

         [[ 0.    ,  0.0593,  0.    , ...,  0.    , -0.0593,  0.    ],
          [-0.    ,  0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.    , -0.0349,  0.    , ...,  0.    ,  0.1916,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    , -0.0295],
          [-0.    ,  0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ],
          [ 0.2955,  0.    ,  0.    , ...,  0.8085, -0.    ,  0.0954]],

         ...,

         [[-0.    , -0.1898, -0.    , ...,  0.    ,  0.1898,  0.    ],
          [ 0.    , -0.    ,  0.    , ..., -0.    , -0.    ,  0.    ],
          [-0.    ,  0.3296, -0.    , ..., -0.    , -0.0648,  0.    ],
          [-0.    , -0.    , -0.    , ...,  0.    ,  0.    ,  0.0945],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ],
          [-0.9463, -0.    , -0.8085, ...,  0.    ,  0.    , -0.0323]],

         [[ 0.    , -0.    ,  0.    , ..., -0.    , -0.    ,  0.    ],
          [-0.    ,  0.2006,  0.    , ..., -0.    , -0.2006,  0.    ],
          [ 0.2006,  0.    ,  0.2237, ..., -0.1676, -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    , -0.    ],
          [-0.    ,  0.    , -0.    , ..., -0.    ,  0.    , -0.0998],
          [-0.    ,  0.    ,  0.    , ..., -0.    ,  0.    , -0.    ]],

         [[-0.    , -0.028 , -0.    , ..., -0.    ,  0.    ,  0.    ],
          [-0.0483,  0.    , -0.0485, ..., -0.0154, -0.    ,  0.    ],
          [ 0.    ,  0.0375,  0.    , ..., -0.    ,  0.    ,  0.    ],
          [ 0.    , -0.    ,  0.0295, ..., -0.0945,  0.    ,  0.    ],
          [ 0.    , -0.0998, -0.    , ...,  0.    ,  0.0998,  0.    ],
          [-0.0998, -0.    , -0.0954, ...,  0.0323,  0.    ,  0.    ]]],


        [[[-0.4816, -0.    , -0.4835, ..., -0.1539,  0.    ,  0.    ],
          [ 0.    ,  0.0792,  0.    , ...,  0.    ,  0.1996,  0.    ],
          [ 0.0483,  0.    ,  0.0485, ...,  0.0154,  0.    ,  0.    ],
          [ 0.    , -0.995 , -0.    , ...,  0.    ,  0.995 ,  0.    ],
          [ 0.    , -0.    , -0.294 , ...,  0.9416, -0.    ,  0.0993],
          [ 0.    ,  0.0998, -0.    , ...,  0.    , -0.0998,  0.    ]],

         [[-0.    , -0.4816, -0.    , ..., -0.    ,  0.108 ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [-0.    , -0.1101, -0.    , ..., -0.    , -0.41  ,  0.    ],
          [ 0.995 ,  0.    ,  0.9506, ..., -0.3217,  0.    , -0.99  ],
          [ 0.    ,  0.    , -0.    , ...,  0.    , -0.    ,  0.    ],
          [-0.0998,  0.    ,  0.4927, ..., -1.8509,  0.    , -0.0993]],

         [[-0.4601, -0.    , -0.4619, ..., -0.147 ,  0.    ,  0.    ],
          [ 0.    , -0.0666,  0.    , ...,  0.    ,  0.2226,  0.    ],
          [-0.2385, -0.    , -0.2394, ..., -0.0762,  0.    ,  0.    ],
          [ 0.    , -0.9506,  0.    , ...,  0.    ,  0.9506,  0.    ],
          [ 0.294 ,  0.    ,  0.    , ...,  0.8045, -0.    , -0.1977],
          [ 0.    , -0.4927,  0.    , ...,  0.    ,  0.4927,  0.    ]],

         ...,

         [[ 0.1557,  0.    ,  0.1563, ...,  0.0498, -0.    ,  0.    ],
          [-0.    ,  0.4302, -0.    , ...,  0.    , -0.1667,  0.    ],
          [ 0.8959,  0.    ,  0.8994, ...,  0.2863,  0.    ,  0.    ],
          [-0.    ,  0.3217, -0.    , ...,  0.    , -0.3217,  0.    ],
          [-0.9416, -0.    , -0.8045, ...,  0.    , -0.    ,  0.9048],
          [-0.    ,  1.8509, -0.    , ...,  0.    , -1.8509, -0.    ]],

         [[-0.    ,  0.4816, -0.    , ...,  0.    , -0.108 ,  0.    ],
          [ 0.    , -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.1101,  0.    , ...,  0.    ,  0.41  ,  0.    ],
          [-0.995 , -0.    , -0.9506, ...,  0.3217,  0.    ,  0.99  ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ],
          [ 0.0998, -0.    , -0.4927, ...,  1.8509,  0.    ,  0.0993]],

         [[ 0.4792,  0.    ,  0.4811, ...,  0.1532, -0.    ,  0.    ],
          [-0.    , -0.0308, -0.    , ..., -0.    , -0.2093,  0.    ],
          [ 0.0481,  0.    ,  0.0483, ...,  0.0154,  0.    ,  0.    ],
          [-0.    ,  0.99  , -0.    , ..., -0.    , -0.99  ,  0.    ],
          [-0.0993, -0.    ,  0.1977, ..., -0.9048,  0.    ,  0.    ],
          [-0.    ,  0.0993, -0.    , ...,  0.    , -0.0993,  0.    ]]]],



       [[[[ 0.    ,  0.484 ,  0.    , ...,  0.    , -0.1086,  0.    ],
          [-0.0796, -0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
          [-0.    , -0.    , -0.    , ..., -0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    , -0.9553, ...,  0.3233, -0.    ,  0.995 ],
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          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    , -0.    ]],

         [[ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
          [-0.    , -0.0796, -0.    , ..., -0.    , -0.2006,  0.    ],
          [ 0.    , -0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
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          [ 0.    ,  0.    ,  0.2955, ..., -0.9463,  0.    , -0.0998],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    , -0.    ]],

         [[ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
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          [ 0.9553,  0.    ,  0.    , ..., -0.    , -0.    ,  0.    ],
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          [-0.    , -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],

         ...,

         [[ 0.    , -0.    , -0.    , ..., -0.    ,  0.    ,  0.    ],
          [ 0.4838,  0.    ,  0.4599, ...,  0.1464,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [-0.3233, -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.9463,  0.    , ...,  0.    , -0.9463, -0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    , -0.    ,  0.    ]],

         [[-0.    , -0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ],
          [-0.    ,  0.0796,  0.    , ...,  0.    ,  0.2006,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    , -0.    ],
          [-0.    , -0.    , -0.2955, ...,  0.9463,  0.    ,  0.0998],
          [-0.    , -0.    , -0.    , ...,  0.    ,  0.    ,  0.    ]],

         [[-0.    ,  0.    , -0.    , ..., -0.    ,  0.    ,  0.    ],
          [ 0.1276,  0.    ,  0.0485, ...,  0.0154,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [-0.995 , -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.0998, -0.    , ...,  0.    , -0.0998,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ]]],


        [[[ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
          [-0.    , -0.0796, -0.    , ..., -0.    , -0.2006,  0.    ],
          [-0.    , -0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.    ,  0.    , -0.    , ...,  0.    , -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.2955, ..., -0.9463,  0.    , -0.0998],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    , -0.    ]],

         [[ 0.    ,  0.484 ,  0.    , ...,  0.    , -0.1086,  0.    ],
          [-0.    , -0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.    ,  0.0796,  0.    , ...,  0.    ,  0.2006,  0.    ],
          [ 0.    ,  0.    , -0.9553, ...,  0.3233, -0.    ,  0.995 ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    , -0.    ],
          [ 0.    ,  0.    , -0.2955, ...,  0.9463, -0.    ,  0.0998]],

         [[ 0.4154,  0.    ,  0.4643, ...,  0.1478, -0.    ,  0.    ],
          [-0.    ,  0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.2952,  0.    ,  0.1436, ...,  0.0457,  0.    ,  0.    ],
          [ 0.    ,  0.9553,  0.    , ..., -0.    , -0.9553, -0.    ],
          [-0.2955, -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [-0.    ,  0.2955,  0.    , ...,  0.    , -0.2955, -0.    ]],

         ...,

         [[-0.0058, -0.    , -0.1571, ..., -0.05  ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [-0.5095, -0.    , -0.4599, ..., -0.1464, -0.    ,  0.    ],
          [-0.    , -0.3233,  0.    , ...,  0.    ,  0.3233, -0.    ],
          [ 0.9463,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
          [ 0.    , -0.9463, -0.    , ...,  0.    ,  0.9463,  0.    ]],

         [[ 0.    , -0.484 ,  0.    , ..., -0.    ,  0.1086,  0.    ],
          [-0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    , -0.0796, -0.    , ..., -0.    , -0.2006,  0.    ],
          [ 0.    ,  0.    ,  0.9553, ..., -0.3233,  0.    , -0.995 ],
          [-0.    , -0.    , -0.    , ...,  0.    ,  0.    ,  0.    ],
          [-0.    ,  0.    ,  0.2955, ..., -0.9463,  0.    , -0.0998]],

         [[-0.4657, -0.    , -0.4835, ..., -0.1539,  0.    ,  0.    ],
          [ 0.    , -0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
          [-0.2068, -0.    , -0.0485, ..., -0.0154, -0.    ,  0.    ],
          [-0.    , -0.995 ,  0.    , ...,  0.    ,  0.995 ,  0.    ],
          [ 0.0998,  0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.    , -0.0998,  0.    , ..., -0.    ,  0.0998,  0.    ]]],


        [[[ 0.    ,  0.4624,  0.    , ...,  0.    , -0.1037,  0.    ],
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          [ 0.    ,  0.0761,  0.    , ...,  0.    ,  0.1916,  0.    ],
          [ 0.    ,  0.    , -0.9127, ...,  0.3089, -0.    ,  0.9506],
          [ 0.    , -0.2955,  0.    , ..., -0.    ,  0.2955,  0.    ],
          [ 0.    , -0.    , -0.2823, ...,  0.904 , -0.    ,  0.0954]],

         [[-0.0235,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
          [ 0.    , -0.067 ,  0.    , ...,  0.    ,  0.2237,  0.    ],
          [ 0.0761,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [-0.    ,  0.    , -0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.2955,  0.    ,  0.    , ...,  0.8085, -0.    , -0.1987],
          [ 0.    ,  0.    , -0.    , ...,  0.    ,  0.    , -0.    ]],

         [[-0.    ,  0.    ,  0.    , ...,  0.    , -0.0661,  0.    ],
          [-0.2232,  0.    ,  0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.2137,  0.    ],
          [ 0.9127,  0.    ,  0.    , ..., -0.2389, -0.    ,  0.0587],
          [-0.    , -0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ],
          [ 0.2823,  0.    ,  0.    , ...,  0.7724, -0.    , -0.1898]],

         ...,

         [[ 0.    , -0.0644, -0.    , ..., -0.    ,  0.0495,  0.    ],
          [ 0.1154, -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.3914, -0.    , ...,  0.    , -0.1601,  0.    ],
          [-0.3089, -0.    ,  0.2389, ...,  0.    ,  0.    , -0.2687],
          [ 0.    , -0.8085,  0.    , ...,  0.    ,  0.    , -0.    ],
          [-0.904 , -0.    , -0.7724, ..., -0.    , -0.    ,  0.8687]],

         [[ 0.    , -0.    ,  0.    , ..., -0.    , -0.    ,  0.    ],
          [-0.    ,  0.    , -0.    , ...,  0.    , -0.    ,  0.    ],
          [ 0.    , -0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    , -0.    ],
          [-0.2955,  0.    , -0.    , ..., -0.    ,  0.    , -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ]],

         [[-0.    ,  0.0158, -0.    , ..., -0.    ,  0.0622,  0.    ],
          [ 0.2227, -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [-0.    , -0.0962, -0.    , ..., -0.    , -0.201 ,  0.    ],
          [-0.9506, -0.    , -0.0587, ...,  0.2687,  0.    ,  0.    ],
          [-0.    ,  0.1987, -0.    , ...,  0.    ,  0.    ,  0.    ],
          [-0.0954,  0.    ,  0.1898, ..., -0.8687,  0.    ,  0.    ]]],


        ...,


        [[[ 0.    , -0.1565, -0.    , ..., -0.    ,  0.0351,  0.    ],
          [ 0.4838,  0.    ,  0.4599, ...,  0.1464,  0.    ,  0.    ],
          [ 0.    , -0.0257,  0.    , ...,  0.    , -0.0648,  0.    ],
          [ 0.    , -0.    ,  0.3089, ..., -0.1045,  0.    , -0.3217],
          [ 0.    ,  0.9463, -0.    , ...,  0.    , -0.9463, -0.    ],
          [ 0.    ,  0.    ,  0.0955, ..., -0.3059, -0.    , -0.0323]],

         [[ 0.0754, -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.4323,  0.    , ...,  0.    , -0.1676,  0.    ],
          [-0.0257,  0.    ,  0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    , -0.    ],
          [-0.9463,  0.    , -0.8085, ..., -0.    , -0.    ,  0.9093],
          [-0.    ,  0.    , -0.    , ..., -0.    ,  0.    ,  0.    ]],

         [[-0.    , -0.    , -0.    , ..., -0.    ,  0.0495,  0.    ],
          [ 0.3925,  0.    ,  0.3929, ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.1601,  0.    ],
          [-0.3089, -0.    ,  0.    , ...,  0.    ,  0.    , -0.2687],
          [ 0.    ,  0.8085,  0.    , ...,  0.    ,  0.    , -0.    ],
          [-0.0955,  0.    ,  0.    , ..., -0.    , -0.    ,  0.8687]],

         ...,

         [[ 0.    ,  0.    ,  0.    , ...,  0.    , -0.1586,  0.    ],
          [ 0.0668,  0.    ,  0.1251, ...,  0.    , -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.0542,  0.    ],
          [ 0.1045,  0.    , -0.    , ...,  0.    , -0.    ,  0.8605],
          [-0.    ,  0.    , -0.    , ...,  0.    , -0.    ,  0.    ],
          [ 0.3059,  0.    ,  0.    , ...,  0.    , -0.    , -0.294 ]],

         [[-0.    ,  0.    , -0.    , ..., -0.    ,  0.    ,  0.    ],
          [ 0.    , -0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
          [ 0.    , -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [-0.    ,  0.    , -0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.9463,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],

         [[ 0.    , -0.0724,  0.    , ...,  0.    , -0.1991,  0.    ],
          [-0.4578, -0.    , -0.4726, ...,  0.    , -0.    ,  0.    ],
          [-0.    ,  0.4401, -0.    , ...,  0.    ,  0.068 ,  0.    ],
          [ 0.3217,  0.    ,  0.2687, ..., -0.8605, -0.    ,  0.    ],
          [ 0.    , -0.9093,  0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.0323, -0.    , -0.8687, ...,  0.294 , -0.    , -0.    ]]],


        [[[-0.    ,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
          [-0.    ,  0.0796,  0.    , ...,  0.    ,  0.2006,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    , -0.    , ...,  0.    , -0.    ,  0.    ],
          [ 0.    , -0.    , -0.2955, ...,  0.9463,  0.    ,  0.0998],
          [ 0.    , -0.    , -0.    , ...,  0.    , -0.    ,  0.    ]],

         [[-0.    , -0.484 , -0.    , ..., -0.    ,  0.1086,  0.    ],
          [ 0.    , -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    , -0.0796, -0.    , ..., -0.    , -0.2006,  0.    ],
          [-0.    ,  0.    ,  0.9553, ..., -0.3233,  0.    , -0.995 ],
          [ 0.    ,  0.    , -0.    , ...,  0.    , -0.    , -0.    ],
          [ 0.    ,  0.    ,  0.2955, ..., -0.9463,  0.    , -0.0998]],

         [[-0.5217, -0.    , -0.5304, ..., -0.0983, -0.    ,  0.    ],
          [ 0.    ,  0.0897,  0.    , ..., -0.    ,  0.2237,  0.    ],
          [ 0.0486,  0.    ,  0.0701, ..., -0.2058,  0.    ,  0.    ],
          [ 0.    , -0.9553,  0.    , ...,  0.    ,  0.9553,  0.    ],
          [ 0.2955,  0.    ,  0.    , ...,  1.617 , -0.    , -0.1987],
          [ 0.    , -0.2955,  0.    , ...,  0.    ,  0.2955,  0.    ]],

         ...,

         [[ 0.3463,  0.    ,  0.3688, ..., -0.1086,  0.    ,  0.    ],
          [-0.    ,  0.697 , -0.    , ...,  0.    , -0.1676,  0.    ],
          [ 0.3932,  0.    ,  0.3875, ...,  0.2006,  0.    ,  0.    ],
          [-0.    ,  0.3233, -0.    , ...,  0.    , -0.3233,  0.    ],
          [-0.9463, -0.    , -1.617 , ...,  0.    , -0.    ,  0.9093],
          [-0.    ,  0.9463, -0.    , ...,  0.    , -0.9463, -0.    ]],

         [[-0.    ,  0.484 , -0.    , ...,  0.    , -0.1086,  0.    ],
          [-0.    ,  0.    ,  0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.    ,  0.0796,  0.    , ..., -0.    ,  0.2006,  0.    ],
          [ 0.    , -0.    , -0.9553, ...,  0.3233,  0.    ,  0.995 ],
          [-0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ],
          [ 0.    , -0.    , -0.2955, ...,  0.9463,  0.    ,  0.0998]],

         [[ 0.4937,  0.    ,  0.5059, ...,  0.1372, -0.    ,  0.    ],
          [-0.    , -0.2413, -0.    , ...,  0.    ,  0.    ,  0.    ],
          [-0.072 , -0.    , -0.1741, ...,  0.1822,  0.    ,  0.    ],
          [-0.    ,  0.995 , -0.    , ..., -0.    , -0.995 ,  0.    ],
          [-0.0998,  0.    ,  0.1987, ..., -0.9093,  0.    ,  0.    ],
          [-0.    ,  0.0998, -0.    , ...,  0.    , -0.0998,  0.    ]]],


        [[[-0.    , -0.4816, -0.    , ..., -0.    ,  0.108 ,  0.    ],
          [ 0.1276,  0.    ,  0.0485, ...,  0.0154,  0.    ,  0.    ],
          [-0.    , -0.0792, -0.    , ..., -0.    , -0.1996,  0.    ],
          [ 0.    , -0.    ,  0.9506, ..., -0.3217,  0.    , -0.99  ],
          [ 0.    ,  0.0998, -0.    , ...,  0.    , -0.0998,  0.    ],
          [ 0.    ,  0.    ,  0.294 , ..., -0.9416,  0.    , -0.0993]],

         [[ 0.0079, -0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ],
          [-0.    , -0.0309, -0.    , ..., -0.    , -0.2104,  0.    ],
          [-0.0792, -0.    ,  0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    , -0.    ],
          [-0.0998,  0.    ,  0.1987, ..., -0.9093,  0.    ,  0.    ],
          [-0.    ,  0.    ,  0.    , ..., -0.    ,  0.    , -0.    ]],

         [[-0.    , -0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
          [-0.0181, -0.    , -0.0965, ..., -0.0307,  0.    ,  0.    ],
          [-0.    , -0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
          [-0.9506, -0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ],
          [ 0.    , -0.1987,  0.    , ...,  0.    ,  0.1987, -0.    ],
          [-0.294 , -0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ]],

         ...,

         [[ 0.    , -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.407 ,  0.    ,  0.4419, ...,  0.1407, -0.    ,  0.    ],
          [ 0.    , -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.3217,  0.    ,  0.    , ...,  0.    , -0.    , -0.    ],
          [-0.    ,  0.9093, -0.    , ...,  0.    , -0.9093,  0.    ],
          [ 0.9416,  0.    ,  0.    , ...,  0.    , -0.    , -0.    ]],

         [[-0.0079,  0.    , -0.    , ...,  0.    , -0.    ,  0.    ],
          [ 0.    ,  0.0309,  0.    , ...,  0.    ,  0.2104,  0.    ],
          [ 0.0792,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [-0.    ,  0.    , -0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.0998, -0.    , -0.1987, ...,  0.9093,  0.    ,  0.    ],
          [-0.    ,  0.    , -0.    , ...,  0.    ,  0.    ,  0.    ]],

         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [-0.0796, -0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.99  ,  0.    , -0.    , ...,  0.    , -0.    ,  0.    ],
          [-0.    ,  0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ],
          [ 0.0993,  0.    , -0.    , ...,  0.    , -0.    ,  0.    ]]]],



       [[[[ 0.486 ,  0.    ,  0.4643, ...,  0.1478, -0.    ,  0.    ],
          [-0.    , -0.    , -0.    , ..., -0.    , -0.2237,  0.    ],
          [-0.    , -0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    , -0.9553, -0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.8085,  0.    ,  0.1987],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ]],

         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [-0.    , -0.    , -0.1436, ..., -0.0457, -0.    ,  0.    ],
          [-0.    , -0.    , -0.    , ..., -0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    , -0.    , -0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.2955,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],

         [[ 0.4643,  0.    ,  0.486 , ...,  0.1547, -0.    ,  0.    ],
          [-0.    , -0.    , -0.    , ..., -0.    , -0.2137,  0.    ],
          [-0.    , -0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    , -1.    , -0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.7724,  0.    ,  0.1898],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ]],

         ...,

         [[-0.1571, -0.    , -0.286 , ..., -0.091 ,  0.    ,  0.    ],
          [-0.    , -0.3914,  0.    , ..., -0.    ,  0.1601,  0.    ],
          [ 0.    , -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.5885, -0.    ],
          [ 0.8085,  0.    ,  0.7724, ...,  0.    ,  0.    , -0.8687],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ]],

         [[ 0.    , -0.486 ,  0.    , ..., -0.    ,  0.0444,  0.    ],
          [-0.143 ,  0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
          [-0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.9553,  0.    ,  1.    , ..., -0.5885, -0.    , -0.9801],
          [-0.    , -0.2955, -0.    , ..., -0.    ,  0.    ,  0.    ],
          [-0.    , -0.    , -0.    , ...,  0.    ,  0.    ,  0.    ]],

         [[-0.4835, -0.    , -0.4763, ..., -0.1516,  0.    ,  0.    ],
          [ 0.    ,  0.0962,  0.    , ...,  0.    ,  0.201 ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.9801,  0.    ],
          [-0.1987, -0.    , -0.1898, ...,  0.8687, -0.    ,  0.    ],
          [-0.    ,  0.    , -0.    , ...,  0.    , -0.    ,  0.    ]]],


        [[[ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
          [-0.    , -0.    , -0.1436, ..., -0.0457, -0.    ,  0.    ],
          [-0.    , -0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    , -0.    , -0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.2955,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ]],

         [[ 0.486 ,  0.    ,  0.4643, ...,  0.1478, -0.    ,  0.    ],
          [ 0.    , -0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
          [-0.    ,  0.    ,  0.1436, ...,  0.0457,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    , -0.9553, -0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.2955, -0.    ]],

         [[ 0.    ,  0.    ,  0.    , ...,  0.    , -0.0661,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.2137,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.2389, -0.    ,  0.0587],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.7724, -0.    , -0.1898]],

         ...,

         [[ 0.    , -0.0644, -0.    , ..., -0.    ,  0.0495,  0.    ],
          [ 0.    , -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.3914, -0.    , ...,  0.    , -0.1601,  0.    ],
          [ 0.    ,  0.    ,  0.2389, ...,  0.    ,  0.    , -0.2687],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    , -0.    ],
          [ 0.    , -0.    , -0.7724, ...,  0.    , -0.    ,  0.8687]],

         [[-0.0471, -0.    ,  0.    , ..., -0.    , -0.    ,  0.    ],
          [-0.    ,  0.    , -0.    , ...,  0.    , -0.    ,  0.    ],
          [ 0.2191, -0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.9553,  0.    , ..., -0.    ,  0.    , -0.    ],
          [-0.2955, -0.    , -0.    , ..., -0.    ,  0.    , -0.    ],
          [-0.    ,  0.2955,  0.    , ...,  0.    ,  0.    , -0.    ]],

         [[-0.    ,  0.0158, -0.    , ..., -0.    ,  0.0622,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [-0.    , -0.0962, -0.    , ..., -0.    , -0.201 ,  0.    ],
          [ 0.    ,  0.    , -0.0587, ...,  0.2687,  0.    ,  0.    ],
          [-0.    , -0.    , -0.    , ...,  0.    ,  0.    , -0.    ],
          [-0.    ,  0.    ,  0.1898, ..., -0.8687,  0.    ,  0.    ]]],


        [[[ 0.4643,  0.    ,  0.486 , ...,  0.1547, -0.    ,  0.    ],
          [-0.    , -0.    , -0.    , ..., -0.    , -0.2137,  0.    ],
          [-0.    , -0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    , -1.    , -0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.7724,  0.    ,  0.1898],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ]],

         [[ 0.    ,  0.    ,  0.    , ...,  0.    , -0.0661,  0.    ],
          [-0.    , -0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.2137,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.2389, -0.    ,  0.0587],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.7724, -0.    , -0.1898]],

         [[ 0.4435,  0.    ,  0.4643, ...,  0.1478, -0.    ,  0.    ],
          [-0.    , -0.    , -0.    , ..., -0.    , -0.2237,  0.    ],
          [ 0.1372,  0.    ,  0.1436, ...,  0.0457,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    , -0.9553, -0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.8085,  0.    ,  0.1987],
          [ 0.    ,  0.    ,  0.    , ..., -0.    , -0.2955, -0.    ]],

         ...,

         [[-0.0344, -0.    , -0.2732, ..., -0.087 ,  0.    ,  0.    ],
          [-0.    , -0.3929,  0.    , ..., -0.    ,  0.1676,  0.    ],
          [-0.0274, -0.    , -0.0845, ..., -0.0269,  0.    ,  0.    ],
          [ 0.    ,  0.2389,  0.    , ...,  0.    ,  0.5622, -0.    ],
          [ 0.7724,  0.    ,  0.8085, ...,  0.    ,  0.    , -0.9093],
          [ 0.    , -0.7724,  0.    , ...,  0.    ,  0.1739, -0.    ]],

         [[ 0.    , -0.484 ,  0.    , ..., -0.    ,  0.0425,  0.    ],
          [-0.2872,  0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
          [-0.    , -0.0796,  0.    , ..., -0.    ,  0.0131,  0.    ],
          [ 1.    ,  0.    ,  0.9553, ..., -0.5622,  0.    , -0.9363],
          [-0.    , -0.    , -0.    , ..., -0.    ,  0.    ,  0.    ],
          [-0.    ,  0.    ,  0.2955, ..., -0.1739,  0.    , -0.2896]],

         [[-0.4904, -0.    , -0.455 , ..., -0.1448,  0.    ,  0.    ],
          [ 0.    ,  0.0965,  0.    , ...,  0.    ,  0.2104,  0.    ],
          [-0.1476, -0.    , -0.1407, ..., -0.0448,  0.    ,  0.    ],
          [ 0.    , -0.0587,  0.    , ...,  0.    ,  0.9363,  0.    ],
          [-0.1898, -0.    , -0.1987, ...,  0.9093, -0.    ,  0.    ],
          [-0.    ,  0.1898,  0.    , ...,  0.    ,  0.2896,  0.    ]]],


        ...,


        [[[-0.1571, -0.    , -0.286 , ..., -0.091 , -0.    ,  0.    ],
          [-0.    ,  0.    ,  0.    , ...,  0.    ,  0.0723,  0.    ],
          [ 0.3914,  0.    ,  0.3929, ...,  0.1251,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.5885,  0.    ],
          [ 0.    , -0.    ,  0.    , ...,  0.2614, -0.    , -0.0642],
          [ 0.    ,  0.8085, -0.    , ...,  0.    , -0.8085, -0.    ]],

         [[-0.    , -0.0644, -0.    , ..., -0.    ,  0.0495,  0.    ],
          [ 0.    , -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.3914, -0.    , ...,  0.    , -0.1601,  0.    ],
          [-0.    ,  0.    ,  0.2389, ...,  0.    ,  0.    , -0.2687],
          [ 0.    ,  0.    , -0.    , ...,  0.    , -0.    , -0.    ],
          [-0.8085,  0.    , -0.7724, ...,  0.    , -0.    ,  0.8687]],

         [[-0.2657, -0.    , -0.3893, ..., -0.1239, -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.1316,  0.    ],
          [ 0.3274,  0.    ,  0.2908, ...,  0.0926,  0.    ,  0.    ],
          [-0.    , -0.2389,  0.    , ...,  0.    ,  0.8011,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.4758, -0.    , -0.1169],
          [ 0.    ,  0.7724,  0.    , ...,  0.    , -0.5985, -0.    ]],

         ...,

         [[ 0.0508,  0.    ,  0.0555, ...,  0.1478,  0.    ,  0.    ],
          [ 0.    ,  0.1874, -0.    , ..., -0.    , -0.0986,  0.    ],
          [ 0.2737,  0.    ,  0.3901, ...,  0.0457,  0.    ,  0.    ],
          [-0.    , -0.    , -0.    , ...,  0.    , -0.9553, -0.    ],
          [-0.2614, -0.    , -0.4758, ...,  0.    ,  0.    ,  0.5351],
          [-0.    , -0.    , -0.    , ...,  0.    , -0.2955, -0.    ]],

         [[ 0.    ,  0.3492,  0.    , ...,  0.    ,  0.1478,  0.    ],
          [ 0.4141,  0.    ,  0.3929, ..., -0.    , -0.    ,  0.    ],
          [ 0.    , -0.3445,  0.    , ...,  0.    ,  0.0457,  0.    ],
          [-0.5885, -0.    , -0.8011, ...,  0.9553,  0.    ,  0.3976],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ],
          [ 0.8085,  0.    ,  0.5985, ...,  0.2955,  0.    ,  0.123 ]],

         [[ 0.2864,  0.    ,  0.2936, ...,  0.0615,  0.    ,  0.    ],
          [ 0.    , -0.0461,  0.    , ..., -0.    , -0.1238,  0.    ],
          [ 0.0063, -0.    , -0.0008, ...,  0.019 ,  0.    ,  0.    ],
          [-0.    ,  0.2687, -0.    , ...,  0.    , -0.3976,  0.    ],
          [ 0.0642,  0.    ,  0.1169, ..., -0.5351, -0.    ,  0.    ],
          [ 0.    , -0.8687,  0.    , ...,  0.    , -0.123 ,  0.    ]]],


        [[[ 0.    , -0.0235,  0.    , ...,  0.    , -0.0593,  0.    ],
          [-0.143 ,  0.    ,  0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.    ,  0.0761,  0.    , ...,  0.    ,  0.1916,  0.    ],
          [ 0.    ,  0.    ,  0.0873, ..., -0.2797, -0.    , -0.0295],
          [ 0.    , -0.2955, -0.    , ..., -0.    ,  0.    ,  0.    ],
          [ 0.    , -0.    , -0.2823, ...,  0.904 , -0.    ,  0.0954]],

         [[-0.486 , -0.    , -0.4643, ..., -0.1478, -0.    ,  0.    ],
          [ 0.    , -0.067 ,  0.    , ...,  0.    ,  0.2237,  0.    ],
          [ 0.    , -0.    , -0.1436, ..., -0.0457,  0.    ,  0.    ],
          [-0.    ,  0.    , -0.    , ...,  0.    ,  0.9553,  0.    ],
          [ 0.2955,  0.    ,  0.    , ...,  0.8085, -0.    , -0.1987],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.2955,  0.    ]],

         [[ 0.    ,  0.0198,  0.    , ...,  0.    , -0.0661,  0.    ],
          [-0.1436,  0.    ,  0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.    , -0.064 ,  0.    , ...,  0.    ,  0.2137,  0.    ],
          [-0.0873,  0.    ,  0.    , ..., -0.2389, -0.    ,  0.0587],
          [ 0.    , -0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ],
          [ 0.2823, -0.    ,  0.    , ...,  0.7724, -0.    , -0.1898]],

         ...,

         [[ 0.    , -0.2899, -0.    , ...,  0.    ,  0.2117,  0.    ],
          [ 0.0685, -0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
          [-0.    ,  0.3252, -0.    , ..., -0.    , -0.0723,  0.    ],
          [ 0.2797, -0.    ,  0.2389, ...,  0.    ,  0.    , -0.188 ],
          [ 0.    , -0.8085,  0.    , ...,  0.    ,  0.    , -0.    ],
          [-0.904 , -0.    , -0.7724, ...,  0.    ,  0.    ,  0.0642]],

         [[-0.0357, -0.    , -0.0661, ...,  0.0495, -0.    ,  0.    ],
          [-0.    ,  0.2237,  0.    , ..., -0.    , -0.2237,  0.    ],
          [ 0.0486,  0.    ,  0.2137, ..., -0.1601,  0.    ,  0.    ],
          [ 0.    , -0.9553,  0.    , ..., -0.    ,  0.    , -0.    ],
          [-0.    ,  0.    , -0.    , ..., -0.    ,  0.    ,  0.1987],
          [ 0.    , -0.2955,  0.    , ..., -0.    ,  0.    , -0.    ]],

         [[-0.    ,  0.049 , -0.    , ..., -0.    ,  0.    ,  0.    ],
          [ 0.0962,  0.    ,  0.0965, ...,  0.0307, -0.    ,  0.    ],
          [-0.    , -0.008 ,  0.    , ..., -0.    ,  0.    ,  0.    ],
          [ 0.0295, -0.    , -0.0587, ...,  0.188 ,  0.    ,  0.    ],
          [-0.    ,  0.1987, -0.    , ...,  0.    , -0.1987,  0.    ],
          [-0.0954, -0.    ,  0.1898, ..., -0.0642,  0.    ,  0.    ]]],


        [[[-0.4835, -0.    , -0.4763, ..., -0.1516, -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.2226,  0.    ],
          [-0.0962,  0.    , -0.0965, ..., -0.0307,  0.    ,  0.    ],
          [ 0.    , -0.    ,  0.    , ...,  0.    ,  0.9801,  0.    ],
          [ 0.    , -0.    ,  0.    , ...,  0.8045, -0.    , -0.1977],
          [ 0.    , -0.1987, -0.    , ...,  0.    ,  0.1987,  0.    ]],

         [[-0.    ,  0.0158, -0.    , ..., -0.    ,  0.0622,  0.    ],
          [ 0.    , -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [-0.    , -0.0962, -0.    , ..., -0.    , -0.201 ,  0.    ],
          [ 0.    ,  0.    , -0.0587, ...,  0.2687,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    , -0.    ],
          [ 0.1987,  0.    ,  0.1898, ..., -0.8687,  0.    ,  0.    ]],

         [[-0.4335, -0.    , -0.4265, ..., -0.1358, -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.2192,  0.    ],
          [-0.2348, -0.    , -0.233 , ..., -0.0742,  0.    ,  0.    ],
          [-0.    ,  0.0587,  0.    , ...,  0.    ,  0.8776,  0.    ],
          [ 0.    , -0.    ,  0.    , ...,  0.7924, -0.    , -0.1947],
          [ 0.    , -0.1898,  0.    , ...,  0.    ,  0.4794,  0.    ]],

         ...,

         [[ 0.0262,  0.    ,  0.0234, ...,  0.0074, -0.    ,  0.    ],
          [-0.    ,  0.3958, -0.    , ...,  0.    , -0.1642,  0.    ],
          [ 0.8781,  0.    ,  0.8728, ...,  0.2779,  0.    ,  0.    ],
          [-0.    , -0.2687, -0.    , ...,  0.    , -0.0481,  0.    ],
          [-0.8045, -0.    , -0.7924, ...,  0.    , -0.    ,  0.8912],
          [-0.    ,  0.8687, -0.    , ...,  0.    , -1.7961, -0.    ]],

         [[-0.    ,  0.4586, -0.    , ...,  0.    , -0.1686,  0.    ],
          [ 0.    , -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.1742,  0.    , ...,  0.    ,  0.3976,  0.    ],
          [-0.9801, -0.    , -0.8776, ...,  0.0481,  0.    ,  0.9752],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ],
          [-0.1987, -0.    , -0.4794, ...,  1.7961,  0.    ,  0.0978]],

         [[ 0.4811,  0.    ,  0.4739, ...,  0.1509, -0.    ,  0.    ],
          [-0.    , -0.0973, -0.    , ..., -0.    , -0.2062,  0.    ],
          [ 0.0483,  0.    ,  0.0475, ...,  0.0151,  0.    ,  0.    ],
          [-0.    , -0.    , -0.    , ..., -0.    , -0.9752,  0.    ],
          [ 0.1977,  0.    ,  0.1947, ..., -0.8912,  0.    ,  0.    ],
          [-0.    , -0.    , -0.    , ...,  0.    , -0.0978,  0.    ]]]],



       ...,



       [[[[ 0.1547,  0.    ,  0.1478, ..., -0.05  ,  0.    ,  0.    ],
          [-0.    , -0.    , -0.    , ..., -0.    ,  0.1676,  0.    ],
          [-0.    , -0.    , -0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.3233, -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.9093],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    , -0.    ]],

         [[ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ],
          [-0.    , -0.    , -0.0457, ...,  0.1464,  0.    ,  0.    ],
          [-0.    , -0.    , -0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.9463, -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ]],

         [[ 0.1478,  0.    ,  0.1547, ..., -0.091 ,  0.    ,  0.    ],
          [-0.    , -0.    , -0.    , ..., -0.    ,  0.1601,  0.    ],
          [-0.    , -0.    , -0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.5885, -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.8687],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    , -0.    ]],

         ...,

         [[-0.05  , -0.    , -0.091 , ...,  0.1547,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    , -0.0542,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -1.    , -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.294 ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],

         [[ 0.    ,  0.2318,  0.    , ...,  0.    ,  0.1547,  0.    ],
          [ 0.4581,  0.    ,  0.5056, ..., -0.    , -0.    ,  0.    ],
          [ 0.    , -0.    ,  0.    , ..., -0.    , -0.    ,  0.    ],
          [-0.3233, -0.    , -0.5885, ...,  1.    ,  0.    ,  0.4161],
          [-0.    ,  0.9463, -0.    , ..., -0.    ,  0.    , -0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ]],

         [[-0.1539, -0.    , -0.1516, ...,  0.0644,  0.    ,  0.    ],
          [ 0.    , -0.4401,  0.    , ..., -0.    , -0.068 ,  0.    ],
          [ 0.    , -0.    , -0.    , ...,  0.    , -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.4161,  0.    ],
          [ 0.9093,  0.    ,  0.8687, ..., -0.294 ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ]]],


        [[[ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ],
          [-0.    , -0.    , -0.0457, ...,  0.1464,  0.    ,  0.    ],
          [-0.    , -0.    , -0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.9463, -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    , -0.    ]],

         [[ 0.1547,  0.    ,  0.1478, ..., -0.05  ,  0.    ,  0.    ],
          [ 0.    , -0.    , -0.    , ...,  0.    ,  0.    ,  0.    ],
          [-0.    ,  0.    ,  0.0457, ..., -0.1464, -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.3233, -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    , -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.9463,  0.    ]],

         [[ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.0495,  0.    ],
          [ 0.    ,  0.    , -0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.1601,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.2687],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    ,  0.8687]],

         ...,

         [[-0.    , -0.    , -0.    , ...,  0.    , -0.1586,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    , -0.    ,  0.    ],
          [-0.    , -0.    , -0.    , ...,  0.    ,  0.0542,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    ,  0.8605],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    , -0.294 ]],

         [[-0.2358, -0.    , -0.3049, ..., -0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [-0.4838, -0.    , -0.5056, ...,  0.    ,  0.    ,  0.    ],
          [-0.    , -0.3233, -0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.9463,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
          [ 0.    , -0.9463,  0.    , ...,  0.    ,  0.    ,  0.    ]],

         [[ 0.    , -0.0724, -0.    , ...,  0.    , -0.1991,  0.    ],
          [ 0.    , -0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
          [-0.    ,  0.4401, -0.    , ...,  0.    ,  0.068 ,  0.    ],
          [ 0.    ,  0.    ,  0.2687, ..., -0.8605, -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.    , -0.    , -0.8687, ...,  0.294 , -0.    , -0.    ]]],


        [[[ 0.1478,  0.    ,  0.1547, ..., -0.091 ,  0.    ,  0.    ],
          [-0.    , -0.    , -0.    , ..., -0.    ,  0.1601,  0.    ],
          [-0.    , -0.    , -0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.5885, -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.8687],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    , -0.    ]],

         [[ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.0495,  0.    ],
          [-0.    , -0.    , -0.    , ..., -0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.1601,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.2687],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    ,  0.8687]],

         [[ 0.1412,  0.    ,  0.1478, ..., -0.087 ,  0.    ,  0.    ],
          [-0.    , -0.    , -0.    , ..., -0.    ,  0.1676,  0.    ],
          [ 0.0437,  0.    ,  0.0457, ..., -0.0269,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.5622, -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.9093],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.1739, -0.    ]],

         ...,

         [[-0.0478, -0.    , -0.087 , ...,  0.1478,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    , -0.0986,  0.    ],
          [-0.0148, -0.    , -0.0269, ...,  0.0457,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.9553, -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.5351],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.2955, -0.    ]],

         [[ 0.    ,  0.2849,  0.    , ...,  0.    ,  0.1478,  0.    ],
          [ 0.4141,  0.    ,  0.3929, ..., -0.    , -0.    ,  0.    ],
          [ 0.    ,  0.0469,  0.    , ...,  0.    ,  0.0457,  0.    ],
          [-0.5885, -0.    , -0.5622, ...,  0.9553,  0.    ,  0.3976],
          [-0.    , -0.    , -0.    , ..., -0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    , -0.1739, ...,  0.2955,  0.    ,  0.123 ]],

         [[-0.017 , -0.    , -0.1448, ...,  0.0615,  0.    ,  0.    ],
          [ 0.    , -0.4419,  0.    , ..., -0.    , -0.1238,  0.    ],
          [-0.0241, -0.    , -0.0448, ...,  0.019 ,  0.    ,  0.    ],
          [ 0.    ,  0.2687,  0.    , ...,  0.    , -0.3976,  0.    ],
          [ 0.8687,  0.    ,  0.9093, ..., -0.5351, -0.    ,  0.    ],
          [ 0.    , -0.8687,  0.    , ...,  0.    , -0.123 ,  0.    ]]],


        ...,


        [[[-0.05  , -0.    , -0.091 , ...,  0.1547,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    , -0.0542,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -1.    , -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.294 ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],

         [[-0.    , -0.    , -0.    , ...,  0.    , -0.1586,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    , -0.    ,  0.    ],
          [-0.    , -0.    , -0.    , ...,  0.    ,  0.0542,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    ,  0.8605],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    , -0.294 ]],

         [[-0.0478, -0.    , -0.087 , ...,  0.1478,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    , -0.0986,  0.    ],
          [-0.0148, -0.    , -0.0269, ...,  0.0457,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.9553, -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.5351],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.2955, -0.    ]],

         ...,

         [[ 0.0162,  0.    ,  0.0294, ..., -0.05  ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.1676,  0.    ],
          [ 0.0473,  0.    ,  0.0861, ..., -0.1464, -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.3233, -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.9093],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.9463,  0.    ]],

         [[ 0.    , -0.484 ,  0.    , ..., -0.    , -0.05  ,  0.    ],
          [ 0.2928,  0.    ,  0.2501, ...,  0.    ,  0.    ,  0.    ],
          [-0.    , -0.0796, -0.    , ..., -0.    , -0.1464,  0.    ],
          [ 1.    ,  0.    ,  0.9553, ..., -0.3233,  0.    , -0.1345],
          [-0.    , -0.    , -0.    , ..., -0.    ,  0.    , -0.    ],
          [-0.    ,  0.    ,  0.2955, ..., -0.9463,  0.    , -0.3938]],

         [[-0.3667, -0.    , -0.4107, ..., -0.0208,  0.    ,  0.    ],
          [ 0.    ,  0.2108,  0.    , ..., -0.    ,  0.2104,  0.    ],
          [ 0.4286,  0.    ,  0.4207, ..., -0.0609, -0.    ,  0.    ],
          [ 0.    , -0.8605,  0.    , ...,  0.    ,  0.1345,  0.    ],
          [-0.294 , -0.    , -0.5351, ...,  0.9093,  0.    ,  0.    ],
          [-0.    ,  0.294 ,  0.    , ..., -0.    ,  0.3938,  0.    ]]],


        [[[ 0.    ,  0.0754,  0.    , ...,  0.    ,  0.1898,  0.    ],
          [ 0.4581,  0.    ,  0.5056, ..., -0.    , -0.    ,  0.    ],
          [-0.    , -0.0257, -0.    , ..., -0.    , -0.0648,  0.    ],
          [ 0.    , -0.    , -0.2797, ...,  0.8955,  0.    ,  0.0945],
          [ 0.    ,  0.9463,  0.    , ..., -0.    ,  0.    , -0.    ],
          [ 0.    , -0.    ,  0.0955, ..., -0.3059,  0.    , -0.0323]],

         [[-0.1547, -0.    , -0.1478, ...,  0.05  ,  0.    ,  0.    ],
          [ 0.    ,  0.4323,  0.    , ...,  0.    , -0.1676,  0.    ],
          [ 0.    , -0.    , -0.0457, ...,  0.1464,  0.    ,  0.    ],
          [ 0.    ,  0.    , -0.    , ...,  0.    , -0.3233,  0.    ],
          [-0.9463,  0.    , -0.8085, ..., -0.    , -0.    ,  0.9093],
          [ 0.    ,  0.    ,  0.    , ..., -0.    , -0.9463, -0.    ]],

         [[ 0.    , -0.0634,  0.    , ...,  0.    ,  0.2117,  0.    ],
          [ 0.3456,  0.    ,  0.3929, ..., -0.    , -0.    ,  0.    ],
          [-0.    ,  0.0217, -0.    , ..., -0.    , -0.0723,  0.    ],
          [ 0.2797,  0.    ,  0.    , ...,  0.7651,  0.    , -0.188 ],
          [-0.    ,  0.8085,  0.    , ..., -0.    ,  0.    , -0.    ],
          [-0.0955, -0.    ,  0.    , ..., -0.2614,  0.    ,  0.0642]],

         ...,

         [[-0.    ,  0.4091, -0.    , ..., -0.    , -0.1586,  0.    ],
          [ 0.1464, -0.    ,  0.1251, ...,  0.    ,  0.    ,  0.    ],
          [ 0.    , -0.1398,  0.    , ..., -0.    ,  0.0542,  0.    ],
          [-0.8955, -0.    , -0.7651, ...,  0.    ,  0.    ,  0.8605],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ],
          [ 0.3059,  0.    ,  0.2614, ...,  0.    ,  0.    , -0.294 ]],

         [[ 0.501 ,  0.    ,  0.5166, ..., -0.1586,  0.    ,  0.    ],
          [ 0.    , -0.1676,  0.    , ...,  0.    ,  0.1676,  0.    ],
          [ 0.3932,  0.    ,  0.4333, ...,  0.0542, -0.    ,  0.    ],
          [-0.    ,  0.3233, -0.    , ..., -0.    ,  0.    , -0.    ],
          [-0.    ,  0.    , -0.    , ..., -0.    ,  0.    , -0.9093],
          [-0.    ,  0.9463, -0.    , ..., -0.    ,  0.    ,  0.    ]],

         [[-0.    , -0.2116, -0.    , ..., -0.    ,  0.    ,  0.    ],
          [-0.4401, -0.    , -0.4419, ..., -0.1407,  0.    ,  0.    ],
          [ 0.    , -0.0887,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [-0.0945, -0.    ,  0.188 , ..., -0.8605,  0.    ,  0.    ],
          [ 0.    , -0.9093,  0.    , ...,  0.    ,  0.9093, -0.    ],
          [ 0.0323,  0.    , -0.0642, ...,  0.294 , -0.    , -0.    ]]],


        [[[-0.1539, -0.    , -0.1516, ...,  0.0644,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.1667,  0.    ],
          [ 0.4401,  0.    ,  0.4419, ...,  0.1407,  0.    ,  0.    ],
          [ 0.    ,  0.    , -0.    , ...,  0.    , -0.4161,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    ,  0.9048],
          [ 0.    ,  0.9093, -0.    , ..., -0.    , -0.9093, -0.    ]],

         [[-0.    , -0.0724, -0.    , ..., -0.    , -0.1991,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
          [ 0.    ,  0.4401,  0.    , ...,  0.    ,  0.068 ,  0.    ],
          [-0.    ,  0.    ,  0.2687, ..., -0.8605, -0.    ,  0.    ],
          [-0.    ,  0.    , -0.    , ...,  0.    , -0.    ,  0.    ],
          [-0.9093,  0.    , -0.8687, ...,  0.294 , -0.    , -0.    ]],

         [[-0.2771, -0.    , -0.2754, ...,  0.0199,  0.    ,  0.    ],
          [ 0.    , -0.    ,  0.    , ...,  0.    , -0.1642,  0.    ],
          [ 0.375 ,  0.    ,  0.3773, ...,  0.1534,  0.    ,  0.    ],
          [ 0.    , -0.2687,  0.    , ...,  0.    , -0.1288,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    ,  0.8912],
          [ 0.    ,  0.8687,  0.    , ..., -0.    , -0.9917, -0.    ]],

         ...,

         [[ 0.4663,  0.    ,  0.4672, ...,  0.1123,  0.    ,  0.    ],
          [-0.    ,  0.    , -0.    , ..., -0.    ,  0.0697,  0.    ],
          [ 0.0034, -0.    ,  0.0006, ..., -0.1064, -0.    ,  0.    ],
          [-0.    ,  0.8605, -0.    , ...,  0.    , -0.7259, -0.    ],
          [ 0.    , -0.    ,  0.    , ...,  0.    ,  0.    , -0.3784],
          [ 0.    , -0.294 ,  0.    , ...,  0.    ,  0.6878,  0.    ]],

         [[ 0.    , -0.129 ,  0.    , ...,  0.    ,  0.2443,  0.    ],
          [-0.    , -0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.    , -0.4733,  0.    , ..., -0.    , -0.1515,  0.    ],
          [ 0.4161,  0.    ,  0.1288, ...,  0.7259,  0.    , -0.4141],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ],
          [ 0.9093,  0.    ,  0.9917, ..., -0.6878,  0.    , -0.0415]],

         [[ 0.1532,  0.    ,  0.1509, ..., -0.0641,  0.    ,  0.    ],
          [-0.    ,  0.4452, -0.    , ...,  0.    ,  0.0876,  0.    ],
          [ 0.0154, -0.    ,  0.0151, ..., -0.0064, -0.    ,  0.    ],
          [-0.    , -0.    , -0.    , ...,  0.    ,  0.4141,  0.    ],
          [-0.9048, -0.    , -0.8912, ...,  0.3784, -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.0415,  0.    ]]]],



       [[[[-0.    , -0.1086, -0.    , ...,  0.    ,  0.1086,  0.    ],
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          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],

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          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],

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          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],

         ...,

         [[ 0.    , -0.1547,  0.    , ...,  0.    ,  0.1547,  0.    ],
          [ 0.0648,  0.    ,  0.0723, ..., -0.0542, -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.4161],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],

         [[ 0.    ,  0.    ,  0.    , ..., -0.    , -0.    ,  0.    ],
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          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.0998],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ]],

         [[-0.    , -0.4016,  0.    , ..., -0.    ,  0.    ,  0.    ],
          [ 0.1512,  0.    ,  0.1741, ..., -0.1822, -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.995 ,  0.    ,  0.9801, ..., -0.4161,  0.    ,  0.    ],
          [-0.    , -0.0998, -0.    , ...,  0.    ,  0.0998,  0.    ],
          [-0.    , -0.    , -0.    , ...,  0.    ,  0.    ,  0.    ]]],


        [[[-0.    , -0.    , -0.    , ...,  0.    ,  0.    ,  0.    ],
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          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ],
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          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],

         [[-0.    , -0.1086, -0.    , ...,  0.    ,  0.1086,  0.    ],
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          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.995 ],
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          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.0998]],

         [[-0.0593, -0.    , -0.0661, ...,  0.0495, -0.    ,  0.    ],
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          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ]],

         ...,

         [[ 0.1898,  0.    ,  0.2117, ..., -0.1586,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ],
          [-0.0648, -0.    , -0.0723, ...,  0.0542,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],

         [[-0.    ,  0.1086, -0.    , ..., -0.    , -0.1086,  0.    ],
          [-0.    ,  0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
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          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.995 ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.0998]],

         [[ 0.4937,  0.    ,  0.5059, ...,  0.1372, -0.    ,  0.    ],
          [-0.    , -0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ],
          [-0.072 , -0.    , -0.1741, ...,  0.1822,  0.    ,  0.    ],
          [ 0.    ,  0.995 ,  0.    , ..., -0.    , -0.995 ,  0.    ],
          [-0.0998, -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [-0.    ,  0.0998,  0.    , ..., -0.    , -0.0998,  0.    ]]],


        [[[-0.    , -0.0444, -0.    , ...,  0.    ,  0.0444,  0.    ],
          [-0.1916, -0.    , -0.2137, ...,  0.1601, -0.    ,  0.    ],
          [-0.    ,  0.    , -0.    , ...,  0.    , -0.    ,  0.    ],
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          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],

         [[-0.0593, -0.    , -0.0661, ...,  0.0495, -0.    ,  0.    ],
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          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ]],

         [[-0.    , -0.0425, -0.    , ...,  0.    ,  0.0425,  0.    ],
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          [ 0.    , -0.0131,  0.    , ...,  0.    ,  0.0131,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.9363],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.2896]],

         ...,

         [[ 0.    , -0.1478,  0.    , ...,  0.    ,  0.1478,  0.    ],
          [ 0.118 ,  0.    ,  0.1316, ..., -0.0986, -0.    ,  0.    ],
          [-0.    , -0.0457, -0.    , ...,  0.    ,  0.0457,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.3976],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.123 ]],

         [[-0.    ,  0.    , -0.    , ...,  0.    , -0.    ,  0.    ],
          [-0.    ,  0.2237, -0.    , ..., -0.    , -0.2237,  0.    ],
          [ 0.    , -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.1987],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ]],

         [[-0.    , -0.3903,  0.    , ..., -0.    ,  0.    ,  0.    ],
          [ 0.1481,  0.    ,  0.3158, ..., -0.1335, -0.    ,  0.    ],
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          [ 0.9801,  0.    ,  0.9363, ..., -0.3976,  0.    ,  0.    ],
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          [-0.    ,  0.    ,  0.2896, ..., -0.123 ,  0.    ,  0.    ]]],


        ...,


        [[[ 0.    , -0.1547,  0.    , ...,  0.    ,  0.1547,  0.    ],
          [ 0.0648,  0.    ,  0.0723, ..., -0.0542, -0.    ,  0.    ],
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          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ]],

         [[ 0.1898,  0.    ,  0.2117, ..., -0.1586,  0.    ,  0.    ],
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          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],

         [[ 0.    , -0.1478,  0.    , ...,  0.    ,  0.1478,  0.    ],
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          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.123 ]],

         ...,

         [[-0.    ,  0.05  , -0.    , ...,  0.    , -0.05  ,  0.    ],
          [-0.2006, -0.    , -0.2237, ...,  0.1676,  0.    ,  0.    ],
          [ 0.    ,  0.1464,  0.    , ..., -0.    , -0.1464,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.1345],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.3938]],

         [[ 0.    , -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    , -0.1676,  0.    , ...,  0.    ,  0.1676,  0.    ],
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          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.9093],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],

         [[-0.    ,  0.173 , -0.    , ..., -0.    ,  0.    ,  0.    ],
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          [ 0.    , -0.0501,  0.    , ...,  0.    ,  0.    ,  0.    ],
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          [ 0.    ,  0.    , -0.    , ...,  0.    ,  0.9093, -0.    ],
          [ 0.    , -0.    , -0.123 , ...,  0.3938, -0.    , -0.    ]]],


        [[[-0.    ,  0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
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          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ]],

         [[-0.    ,  0.1086, -0.    , ..., -0.    , -0.1086,  0.    ],
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          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.0998]],

         [[-0.    ,  0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
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          [ 0.    , -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.1987],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ]],

         ...,

         [[ 0.    , -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    , -0.1676,  0.    , ...,  0.    ,  0.1676,  0.    ],
          [-0.    ,  0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.9093],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],

         [[ 0.    , -0.1086,  0.    , ...,  0.    ,  0.1086,  0.    ],
          [-0.    ,  0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
          [-0.    ,  0.2006, -0.    , ..., -0.    , -0.2006,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.995 ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.0998]],

         [[-0.5217, -0.    , -0.5282, ..., -0.1205,  0.    ,  0.    ],
          [ 0.    ,  0.2413,  0.    , ..., -0.    , -0.    ,  0.    ],
          [ 0.3508,  0.    ,  0.3966, ..., -0.3489, -0.    ,  0.    ],
          [ 0.    , -0.995 ,  0.    , ...,  0.    ,  0.995 ,  0.    ],
          [ 0.0998,  0.    , -0.1987, ...,  0.9093, -0.    ,  0.    ],
          [ 0.    , -0.0998,  0.    , ..., -0.    ,  0.0998,  0.    ]]],


        [[[-0.    ,  0.0801, -0.    , ..., -0.    , -0.108 ,  0.    ],
          [ 0.1512,  0.    ,  0.1741, ..., -0.1822, -0.    ,  0.    ],
          [ 0.    ,  0.0792,  0.    , ...,  0.    ,  0.1996,  0.    ],
          [ 0.    ,  0.    ,  0.0295, ..., -0.0945,  0.    ,  0.99  ],
          [ 0.    , -0.0998, -0.    , ...,  0.    ,  0.0998, -0.    ],
          [ 0.    , -0.    , -0.294 , ...,  0.9416, -0.    ,  0.0993]],

         [[ 0.02  ,  0.    ,  0.0223, ..., -0.0167, -0.    ,  0.    ],
          [ 0.    ,  0.0309,  0.    , ...,  0.    ,  0.2104,  0.    ],
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          [-0.    ,  0.    , -0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.0998,  0.    , -0.1987, ...,  0.9093, -0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],

         [[-0.    ,  0.0908,  0.    , ..., -0.    , -0.1064,  0.    ],
          [ 0.2927,  0.    ,  0.3158, ..., -0.1335, -0.    ,  0.    ],
          [ 0.    , -0.0406,  0.    , ...,  0.    ,  0.1966,  0.    ],
          [-0.0295,  0.    ,  0.    , ..., -0.0807, -0.    ,  0.9752],
          [ 0.    ,  0.1987,  0.    , ..., -0.    , -0.1987, -0.    ],
          [ 0.294 , -0.    ,  0.    , ...,  0.8045,  0.    ,  0.0978]],

         ...,

         [[-0.    , -0.0716, -0.    , ..., -0.    ,  0.0452,  0.    ],
          [-0.5236, -0.    , -0.535 , ..., -0.0709, -0.    ,  0.    ],
          [ 0.    ,  0.3469,  0.    , ...,  0.    , -0.0835,  0.    ],
          [ 0.0945, -0.    ,  0.0807, ...,  0.    ,  0.    , -0.4141],
          [-0.    , -0.9093,  0.    , ...,  0.    ,  0.9093,  0.    ],
          [-0.9416, -0.    , -0.8045, ...,  0.    , -0.    , -0.0415]],

         [[-0.02  , -0.    , -0.0223, ...,  0.0167, -0.    ,  0.    ],
          [-0.    , -0.0309, -0.    , ..., -0.    , -0.2104,  0.    ],
          [ 0.1996,  0.    ,  0.2226, ..., -0.1667, -0.    ,  0.    ],
          [-0.    ,  0.    ,  0.    , ..., -0.    ,  0.    , -0.    ],
          [-0.0998,  0.    ,  0.1987, ..., -0.9093,  0.    ,  0.    ],
          [ 0.    ,  0.    , -0.    , ...,  0.    ,  0.    , -0.    ]],

         [[-0.    ,  0.3996, -0.    , ...,  0.    , -0.    ,  0.    ],
          [-0.2006, -0.    , -0.2237, ...,  0.1676,  0.    ,  0.    ],
          [ 0.    ,  0.0401,  0.    , ..., -0.    , -0.    ,  0.    ],
          [-0.99  , -0.    , -0.9752, ...,  0.4141,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ],
          [-0.0993, -0.    , -0.0978, ...,  0.0415,  0.    ,  0.    ]]]],



       [[[[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
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         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
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         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
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          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]]]]],
      shape=(7, 7, 7, 6, 7))

References

J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).
J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

payload(m, p=None)

Add a payload to the end-effector

Parameters:

m (float) – mass (kg)
p (ndarray | Sequence[float] | None) – position in end-effector frame

Return type:

payload(m, p) adds a payload with point mass m at position p in the end-effector coordinate frame.

payload(m) adds a payload with point mass m at the origin of the end-effector coordinate frame.

payload(0) removes added payload.

The point mass is combined with the mass, centre of mass and inertia of the end-effector link, and p is expressed in that link’s frame. The link’s own parameters are kept, so each call replaces the payload added by the previous one, and a payload of zero mass leaves the dynamics unchanged.

Note

The end-effector link is ee_links[0] when the robot has a single end-effector. A robot with several end-effectors has no single end-effector link, and the payload is added to links[n - 1] as in earlier releases.

payload_capacity(q, tau_m, w, frame='base')

Compute the payload capacity of the manipulator

Parameters:

q (ndarray | Sequence[float]) – Joint coordinates for trajectory
tau_m (ndarray | Sequence[float]) – actuator torque limits
w (ndarray | Sequence[float]) – External wrench applied to the end-effector
frame (Literal['base', 'ee']) – the frame in which w is expressed, "base" for the base frame (default) and "ee" for the end-effector frame, which includes the tool transform

Returns:

Payload capacity of the manipulator

Return type:

For each configuration, row of q, compute the multiple of the applied external wrench w that can be applied to the end-effector without exceeding the actuator torque limits tau_m. If w is the weight of a 1 kg mass then the multiple is the capacity in kg. A load such as a weight is fixed in the world, so w is in the base frame by default; use frame="ee" for a wrench that is fixed to the end-effector, such as a force along the tool axis. This multiple is computed for each joint and can be

a positive real number
inf the joint has no role in determining the payload capacity
nan the gravity load already exceeds the joint torque limit

The joint with the lowest payload capacity is the one that determines the overall payload capacity for that configuration.

The result is a matrix of payload capacities for each joint and configuration.

The joint force/torque limits tau_m are given as an array of shape (n,) or (1,n) of maximum values, in which case the minimum is the negative of the maximum, or (2,n), in which case the first row is the minimum and the second row is the maximum.

Examples

In this example we compute the payload capacity of the Puma560 robot along a straight line trajectory in the x-direction, moving from tightly folded to almost straight. The end-effector is oriented with the tool frame pointing in the x-direction, and a tool offset is added so that the load causes a moment on the wrist. The external wrench is a downward force of 9.81 N, which is equivalent to a 1 kg mass. The torque limits are the motor fuse limits from Table 2.17 of “Visual Control of Robots”, Corke (1996). The payload capacity, as well as the limiting joint, is plotted as a function of the x-position of the end-effector. The minimum payload capacity is also plotted, along with the index of the joint that determines the payload capacity.

>>> from roboticstoolbox import models
>>> import matplotlib.pyplot as plt
>>> import numpy as np
>>> from spatialmath import SE3
>>> puma = models.DH.Puma560()
>>> puma.tool = SE3.Tz(0.05)
>>> T = SE3.Tx(np.arange(0.35, 0.85, 0.01)) * SE3.Tz(puma.links[0].d) * SE3.Ry(90, "deg")
>>> sol = puma.ikine_a(T)
>>> w = [0,0,-9.81,0,0,0]
>>> cap = puma.payload_capacity(sol.q, [56, 97, 52,10,10,10], w)
>>> plt.plot(T.x, [np.min(c) for c in cap])
>>> plt.plot(T.x, [np.argmin(c) for c in cap])

Derived from the MATLAB functions pay and paycap originally contributed to the MATLAB Robotics Toolbox by Bryan Moutrie (@au-phiware). Ported to Python and adapted for robotics-toolbox-python with permission from the original author.

Added in version 1.5.0.

Seealso:: gravload() payload()

perturb(p=0.1)

Perturb robot parameters

Parameters:: p (float) – The percent (+/-) to be perturbed. Default 10%
Return type:: Any
Returns:: A copy of the robot with dynamic parameters perturbed

rp = perturb(p) is a new robot object in which the dynamic parameters (link mass and inertia) have been perturbed. The perturbation is multiplicative so that values are multiplied by random numbers in the interval (1-p) to (1+p). The name string of the perturbed robot is prefixed by ‘P/’.

Useful for investigating the robustness of various model-based control schemes. For example to vary parameters in the range +/- 10 percent is: r2 = puma.perturb(0.1)

plot(q, backend=None, block=False, dt=0.05, limits=None, vellipse=False, fellipse=False, fig=None, movie=None, loop=False, notebook=None, ax=None, env=None, view=None, **kwargs)

Graphical display and animation

robot.plot(q, 'pyplot') displays a graphical view of a robot based on the kinematic model and the joint configuration q. This is a stick figure polyline which joins the origins of the link coordinate frames. The plot will autoscale with an aspect ratio of 1.

If q (m,n) representing a joint-space trajectory it will create an animation with a pause of dt seconds between each frame.

Parameters:

q (ndarray | Sequence[float]) – The joint configuration of the robot.
backend (Optional[Literal['swift', 'pyplot', 'pyplot2']]) – The graphical backend to use, currently ‘swift’ and ‘pyplot’ are implemented. Defaults to ‘swift’ of a Robot and ‘pyplot` for a DHRobot
block (bool) – True waits until the window is closed; the PyPlot backend has already shown the figure, so False and None return at once, defaults to False
dt (float) – if q is a trajectory, this describes the delay in seconds between frames
limits (ndarray | Sequence[float] | None) – Custom view limits for the plot. If not supplied will autoscale, [x1, x2, y1, y2, z1, z2] (this option is for ‘pyplot’ only)
vellipse (bool) – (Plot Option) Plot the velocity ellipse at the end-effector (this option is for ‘pyplot’ only)
fellipse (bool) – (Plot Option) Plot the force ellipse at the end-effector (this option is for ‘pyplot’ only)
fig (str | None) – (Plot Option) The figure label to plot in (this option is for ‘pyplot’ only)
ax (Any) – (Plot Option) Existing Matplotlib 3D axes to draw the scene on, instead of making a figure (this option is for ‘pyplot’ only)
env (Connector | None) – The scene returned by an earlier plot, to add this robot to, instead of making a new scene; see Several configurations below. The backend of the scene is used, so backend is ignored, and the options that are set when a scene is made (limits, fig, ax, render_mode, inline_*, notebook and browser) cannot be given
color – (Plot Option) The color of the robot’s line, any Matplotlib color (this option is for ‘pyplot’ only)
view (str | None) – (Plot Option) "xy", "xz" or "yz" to start with a flat view of that plane, the first axis to the right and the second up, or None (default) for the usual three-dimensional view. The view is not locked, the plot can be rotated as usual. With env it changes the view of that scene (this option is for ‘pyplot’ only)
movie (str | None) – (Plot Option) The filename to save the movie to (this option is for ‘pyplot’ only)
loop (bool) – (Plot Option) Loop the movie (this option is for ‘pyplot’ only)
notebook (bool | None) – (Plot Option, Swift only) True shows Swift in a frame of the current notebook cell, False opens it in a browser window, and None (default) chooses the frame in a notebook, unless a browser is named, and a window otherwise, so the same call works in a script and in a notebook. The browser showing the notebook must be on the machine running the kernel, since the frame points at localhost
browser – (Plot Option, Swift only) the browser for the window, or None (default) for the system’s. This is a name known to Python’s webbrowser module, which depends on the operating system, for example "chrome", "firefox" or "safari" on macOS and "google-chrome" or "chromium" on Linux, or a command line with %s where the URL goes, for any other browser; webbrowser.get(name) tells you whether a name works. Naming a browser means a window, not the notebook frame. browser="notebook" is deprecated, use notebook=True. See swift.Swift.Swift.launch()
realtime – (Plot Option) Pace the steps of a trajectory in real time, one each dt, defaults to True for a trajectory (this option is for ‘swift’ only)
jointaxes – (Plot Option) Plot an arrow indicating the axes in which the joint revolves around(revolute joint) or translates along (prosmatic joint) (this option is for ‘pyplot’ only)
eeframe – (Plot Option) Plot the end-effector coordinate frame at the location of the end-effector. Uses three arrows, red, green and blue to indicate the x, y, and z-axes. (this option is for ‘pyplot’ only)
shadow – (Plot Option) Plot a shadow of the robot in the x-y plane. (this option is for ‘pyplot’ only)
name – (Plot Option) Plot the name of the robot near its base (this option is for ‘pyplot’ only)
render_mode – (Plot Option) Rendering mode for matplotlib backends: 'window', 'notebook-widget', or 'notebook-inline'. If omitted, a mode is chosen from the environment, so the same call works for any %matplotlib setting: for the interactive ipympl backend a single pose is a widget that can be rotated, but a trajectory uses 'notebook-inline', since a widget cannot update while the cell is running.
inline_every_n – (Plot Option) In notebook-inline mode, push one rendered frame every N simulation steps. Larger N reduces output load.
inline_format – (Plot Option) In notebook-inline mode, frame format: 'svg' (default) or 'png'.
inline_dpi – (Plot Option) DPI for PNG inline frames only; ignored when inline_format='svg'.

Return type:

Returns:

A reference to the environment object which controls the figure

Note

Python’s webbrowser module cannot tell whether a browser is installed, so a browser that cannot be opened is not reported as such. On macOS the names "chrome", "firefox" and "safari" are always accepted, whether or not the app is installed. If it is not, no window opens and there is no error: Swift waits about 10 seconds for the page to connect and then reports Could not connect to the Swift simulator. If you see that, check that the browser named by browser is installed, or leave it as None to use the system’s default browser. See swift.Swift.Swift.launch().

Several configurations

plot returns the scene it made, and passing that as env to another plot adds a robot to it instead of making a new scene. The robot is the same model, and each plot of it holds the pose it was given, so the robot is shown in two configurations at once, here the initial and the final. Give each a color, and turn off what would be drawn twice.

import roboticstoolbox as rtb

puma = rtb.models.DH.Puma560()
env = puma.plot(puma.qr, color="gray", block=False)
puma.plot(puma.qn, env=env, color="red", name=False, block=False)

The scene can be given a pose to draw directly, handle = env.add(puma, q=q), and handle.q = q moves only that drawing. A robot that is added without a q follows puma.q, as it always has. A trajectory can be plotted on top of an earlier plot in the same way; the earlier plot stays where it was. In a notebook, an animation needs the scene to have been made with render_mode="notebook-inline".

Notes

By default this method will block until the figure is dismissed.
To avoid this set block=False.
For PyPlot, the polyline joins the origins of the link frames,
but for some Denavit-Hartenberg models those frames may not actually be on the robot, ie. the lines to not necessarily represent the links of the robot.

See also

teach()

plot_ellipse(ellipse, block=True, limits=None, jointaxes=True, eeframe=True, shadow=True, name=True)

Plot an ellipsoid.

Parameters:

ellipse (EllipsePlot) – the ellipsoid to plot
block (bool) – True waits until the window is closed; the PyPlot backend has already shown the figure, so False and None return at once, defaults to True
limits (ndarray | Sequence[float] | None) – custom view limits [x1, x2, y1, y2, z1, z2]; autoscales if not supplied
jointaxes (bool) – plot an arrow indicating the joint axis
eeframe (bool) – plot the end-effector coordinate frame
shadow (bool) – plot a shadow of the robot in the x-y plane
name (bool) – plot the name of the robot near its base

Returns:

a reference to the PyPlot object controlling the matplotlib figure

Return type:

robot.plot_ellipse(ellipsoid) displays the ellipsoid.

plot_fellipse(q, block=True, fellipse=None, limits=None, opt='trans', centre='ee', jointaxes=True, eeframe=True, shadow=True, name=True)

Plot the force ellipsoid for a manipulator.

Parameters:

q (ndarray | Sequence[float] | None) – the joint configuration of the robot
block (bool) – True waits until the window is closed; the PyPlot backend has already shown the figure, so False and None return at once, defaults to True
fellipse (EllipsePlot | None) – a pre-built force ellipsoid to plot
limits (ndarray | Sequence[float] | None) – custom view limits [x1, x2, y1, y2, z1, z2]; autoscales if not supplied
opt (Literal['trans', 'rot']) – 'trans' or 'rot' — plot the translational or rotational force ellipsoid
centre (Union[Literal['ee'], ndarray, Sequence[float]]) – coordinates to plot the ellipse — [x, y, z] or "ee"
jointaxes (bool) – plot an arrow indicating the joint axis
eeframe (bool) – plot the end-effector coordinate frame
shadow (bool) – plot a shadow of the robot in the x-y plane
name (bool) – plot the name of the robot near its base

Raises:

ValueError – if neither q nor fellipse is supplied

Returns:

a reference to the PyPlot object controlling the matplotlib figure

Return type:

robot.plot_fellipse(q) displays the force ellipsoid for the robot at pose q. The plot will autoscale with an aspect ratio of 1.

robot.plot_fellipse(vellipse=ell) uses a pre-built ellipse object.

Notes

By default the ellipsoid related to translational motion is drawn. Use opt='rot' to draw the rotational velocity ellipsoid.
By default the ellipsoid is drawn at the origin. Use centre to specify a 3-vector, or "ee" to draw at the end-effector.

plot_vellipse(q, block=True, vellipse=None, limits=None, opt='trans', centre='ee', jointaxes=True, eeframe=True, shadow=True, name=True)

Plot the velocity ellipsoid for a manipulator.

Parameters:

q (ndarray | Sequence[float] | None) – the joint configuration of the robot
block (bool) – True waits until the window is closed; the PyPlot backend has already shown the figure, so False and None return at once, defaults to True
vellipse (EllipsePlot | None) – a pre-built velocity ellipsoid to plot
limits (ndarray | Sequence[float] | None) – custom view limits [x1, x2, y1, y2, z1, z2]; autoscales if not supplied
opt (Literal['trans', 'rot']) – 'trans' or 'rot' — plot the translational or rotational velocity ellipsoid
centre (Union[Literal['ee'], ndarray, Sequence[float]]) – coordinates to plot the ellipse — [x, y, z] or "ee"
jointaxes (bool) – plot an arrow indicating the joint axis
eeframe (bool) – plot the end-effector coordinate frame
shadow (bool) – plot a shadow of the robot in the x-y plane
name (bool) – plot the name of the robot near its base

Raises:

ValueError – if neither q nor vellipse is supplied

Returns:

a reference to the PyPlot object controlling the matplotlib figure

Return type:

isprismatic() revolutejoints()

robot.plot_vellipse(q) displays the velocity ellipsoid for the robot at pose q. The plot will autoscale with an aspect ratio of 1.

robot.plot_vellipse(vellipse=ell) uses a pre-built ellipse object.

Notes

By default the ellipsoid related to translational motion is drawn. Use opt='rot' to draw the rotational velocity ellipsoid.
By default the ellipsoid is drawn at the origin. Use centre to specify a 3-vector, or "ee" to draw at the end-effector.

property prismaticjoints: list[bool]

Prismatic joints as bool array

Returns:: array of joint type, True if prismatic

Examples

>>> import roboticstoolbox as rtb
>>> puma = rtb.models.DH.Puma560()
>>> puma.prismaticjoints
[False, False, False, False, False, False]
>>> stanford = rtb.models.DH.Stanford()
>>> stanford.prismaticjoints
[False, False, True, False, False, False]

Notes

Fixed joints, that maintain a constant link relative pose, are not included.

See also

property q: ndarray

Get/set robot joint configuration

robot.q is the robot joint configuration
robot.q = ... checks and sets the joint configuration

Parameters:: q – the new robot joint configuration
Returns:: robot joint configuration
Setter accepts:: ndarray | Sequence[float]

property qd: ndarray

Get/set robot joint velocity

robot.qd is the robot joint velocity
robot.qd = ... checks and sets the joint velocity

Returns:: robot joint velocity
Setter accepts:: ndarray | Sequence[float]

property qdd: ndarray

Get/set robot joint acceleration

robot.qdd is the robot joint acceleration
robot.qdd = ... checks and sets the robot joint acceleration

Returns:: robot joint acceleration
Setter accepts:: ndarray | Sequence[float]

property qlim: ndarray

Joint limits

Limits are extracted from the link objects. If joints limits are not set for:

a revolute joint [-𝜋. 𝜋] is returned
a prismatic joint an exception is raised

Parameters:: qlim – An array of joints limits (2, n)
Raises:: ValueError – unset limits for a prismatic joint
Returns:: Array of joint limit values

Examples

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.DH.Puma560()
>>> robot.qlim
array([[-2.7925, -0.7854, -3.927 , -1.9199, -1.7453, -4.6426],
       [ 2.7925,  3.927 ,  0.7854,  2.9671,  1.7453,  4.6426]])

Setter accepts:: ndarray | Sequence[float]

random_q()

Return a random joint configuration

The value for each joint is uniform randomly distributed between the limits set for the robot.

Notes

The joint limit for all joints must be set.

Return type:: Any
Returns:: Random joint configuration :rtype: ndarray(n)

See also

qlim() qlim()

property revolutejoints: list[bool]

Revolute joints as bool array

Returns:: array of joint type, True if revolute

Examples

>>> import roboticstoolbox as rtb
>>> puma = rtb.models.DH.Puma560()
>>> puma.revolutejoints
[True, True, True, True, True, True]
>>> stanford = rtb.models.DH.Stanford()
>>> stanford.revolutejoints
[True, True, False, True, True, True]

Notes

Fixed joints, that maintain a constant link relative pose, are not included.

See also

isrevolute() prismaticjoints()

property scene_children: list[SceneNode]

Return the child nodes of this object in the scene graph.

Setting a new list of children updates each child’s scene_parent to this object, but does not remove this object from any previous parent’s scene_children.

This is a read/write property.

Return type:: list(SceneNode)
Seealso:: scene_parent() attach() attach_to()

property scene_parent: SceneNode | None

Return the parent node of this object in the scene graph.

Setting a new parent adds this object to the new parent’s scene_children.

This is a read/write property.

Return type:: SceneNode | None
Seealso:: scene_children() attach() attach_to()
Setter accepts:: SceneNode

segments()

Segments of branched robot

For a single-chain robot with structure:

L1 - L2 - L3

the return is [[None, L1, L2, L3]]

For a robot with structure:

L1 - L2 +-  L3 - L4
        +- L5 - L6

the return is [[None, L1, L2], [L2, L3, L4], [L2, L5, L6]]

Return type:: list[list[Optional[Link]]]
Returns:: Segment list

Notes

the length of the list is the number of segments in the robot
the first segment always starts with None which represents
the base transform (since there is no base link)
the last link of one segment is also the first link of subsequent
segments

classmethod set_default_backend(be)

Set the default graphical backend for all robots

Parameters:: be (Any) – backend name, for example "swift", or None to clear
Return type:: None

Robot.set_default_backend("swift") sets the backend used by plot() for every robot, existing or created later, that has not been given one of its own by robot.default_backend = .... The robot’s own value takes precedence.

See also

default_backend

showgraph(display_graph=True, **kwargs)

Display a link transform graph in browser

robot.showgraph() displays a graph of the robot’s link frames and the ETS between them. It uses GraphViz dot.

The nodes are:

Base is shown as a grey square. This is the world frame origin, but can be changed using the base attribute of the robot.
Link frames are indicated by circles
ETS transforms are indicated by rounded boxes

The edges are:

an arrow if jtype is False or the joint is fixed
an arrow with a round head if jtype is True and the joint is revolute
an arrow with a box head if jtype is True and the joint is prismatic

Edge labels or nodes in blue have a fixed transformation to the preceding link.

Parameters:

display_graph (bool) – Open the graph in a browser if True. Otherwise will return the file path
etsbox – Put the link ETS in a box, otherwise an edge label
jtype – Arrowhead to node indicates revolute or prismatic type
static – Show static joints in blue and bold

Return type:

str | None

Examples

>>> import roboticstoolbox as rtb
>>> panda = rtb.models.URDF.Panda()
>>> panda.showgraph()

See also

dotfile() graph()

property structure: str

Return the joint structure string

A string with one letter per joint: R for a revolute joint, and P for a prismatic joint.

Returns:: joint configuration string

Examples

>>> import roboticstoolbox as rtb
>>> puma = rtb.models.DH.Puma560()
>>> puma.structure
'RRRRRR'
>>> stanford = rtb.models.DH.Stanford()
>>> stanford.structure
'RRPRRR'

Notes

Fixed joints, that maintain a constant link relative pose, are not included. len(self.structure) == self.n.

property symbolic: bool

teach(q, block=True, limits=None, vellipse=False, fellipse=False, backend=None, view=None)

Graphical teach pendant

robot.teach(q) opens a graphical view (PyPlot or Swift – see backend) which allows the user to “drive” a graphical robot using a graphical slider panel. The robot’s initial joint configuration is q.

robot.teach() as above except the robot’s stored value of q is used.

Parameters:

q (ndarray | Sequence[float] | None) – The joint configuration of the robot (Optional, if not supplied will use the stored q values).
block (bool) – True waits until the window is closed; the PyPlot backend has already shown the figure, so False and None return at once, defaults to True
limits (ndarray | Sequence[float] | None) – Custom view limits for the plot. If not supplied will autoscale, [x1, x2, y1, y2, z1, z2] (this option is for ‘pyplot’/’pyplot2’ only)
vellipse (bool) – (Plot Option) Plot the velocity ellipse at the end-effector (this option is for ‘pyplot’ only)
fellipse (bool) – (Plot Option) Plot the force ellipse at the end-effector (this option is for ‘pyplot’ only)
backend (Optional[Literal['swift', 'pyplot', 'pyplot2']]) – The graphical backend to use – ‘swift’, ‘pyplot’, or ‘pyplot2’. Defaults to whatever plot() would pick for this robot (see its own backend for the resolution rule).
view (str | None) – "xy", "xz" or "yz" to start with a flat view of that plane, the first axis to the right and the second up, or None (default) for the usual three-dimensional view. The view is not locked, the plot can be rotated as usual (this option is for ‘pyplot’ only)

Return type:

Returns:

A reference to the environment object which controls the figure/view

Notes

Program execution is blocked until the teach window is dismissed. If block=False the method is non-blocking but you need to poll the window manager (PyPlot) or keep calling env.step() yourself (Swift) to ensure the window/panel remains responsive.
Once teach() returns (block=True), robot.q holds the final pose the sliders were left at – true for every backend, even though Swift’s own live joint state during the session lives on the returned environment’s handle (env.swift_objects[0].q), not robot.q itself, to keep the robot model plain and shareable while teaching is in progress (see swift’s AssemblyHandle).
The slider limits are derived from the joint limit properties. If not set then:
- For revolute joints they are assumed to be [-pi, +pi]
- For prismatic joint they are assumed unknown and an error occurs.

todegrees(q)

Convert joint angles to degrees

Parameters:: q (Any) – The joint configuration of the robot
Return type:: ndarray
Returns:: a vector of joint coordinates in degrees and metres
Returns:: robot.todegrees(q) converts joint coordinates q to degrees
Returns:: taking into account whether elements of q correspond to revolute
Returns:: or prismatic joints, ie. prismatic joint values are not converted.
Returns:: If q is a matrix, with one column per joint, the conversion is
Returns:: performed columnwise.

Examples

>>> import roboticstoolbox as rtb
>>> from math import pi
>>> stanford = rtb.models.DH.Stanford()
>>> stanford.todegrees([pi/4, pi/8, 2, -pi/4, pi/6, pi/3])
array([ 45. ,  22.5,   2. , -45. ,  30. ,  60. ])

property tool: SE3

Get/set robot tool transform

robot.tool is the robot tool transform as an SE3 object
robot._tool is the robot tool transform as a numpy array
robot.tool = ... checks and sets the robot tool transform

Parameters:: tool – the new robot tool transform (as an SE(3))
Returns:: robot tool transform
Setter accepts:: SE3 | ndarray

toradians(q)

Convert joint angles to radians

robot.toradians(q) converts joint coordinates q to radians taking into account whether elements of q correspond to revolute or prismatic joints, ie. prismatic joint values are not converted.

If q is a matrix, with one column per joint, the conversion is performed columnwise.

Parameters:: q (Any) – The joint configuration of the robot
Return type:: ndarray
Returns:: a vector of joint coordinates in radians and metres

Examples

>>> import roboticstoolbox as rtb
>>> stanford = rtb.models.DH.Stanford()
>>> stanford.toradians([10, 20, 2, 30, 40, 50])
array([0.1745, 0.3491, 2.    , 0.5236, 0.6981, 0.8727])

tree()

Render the whole tree this node lives in as indented text – walks up to the root first (the same root-finding as update()), then renders down from there, with this node marked with a trailing <== so its position in the tree is visible.

Return type:: str

tree_children()

Render this node’s own subtree as indented text (one repr() per line), not walking through parents – the same “not through parents” scope as _propagate_scene_children().

Nodes have no .name in this package, so each line is that node’s own repr() (type, constructor params, color, pose) – usually enough to tell siblings apart, since it’s rare for two distinct nodes to share an identical pose as well as everything else.

Return type:: str

update()

Recompute the world transform of every node in the scene graph this node belongs to, starting from the root and working down.

Call this after changing any node’s T/pose (or its scene_parent) – nothing propagates automatically. It doesn’t matter which node in the graph you call it on: this always walks up to the root first, then pushes fresh world transforms down through the whole tree, not just this node’s own subtree.

Return type:: None

property urdf_filepath: str

property urdf_string: str

vellipse(q, opt='trans', unit='rad', centre='ee', scale=0.1, add=True)

Create a velocity ellipsoid object for plotting with PyPlot.

Parameters:

q (ndarray | Sequence[float]) – the joint configuration of the robot
opt (Literal['trans', 'rot']) – 'trans' or 'rot' — plot the translational or rotational velocity ellipsoid
unit (Literal['rad', 'deg']) – 'rad' or 'deg'
centre (Union[Literal['ee'], ndarray, Sequence[float]]) – centre of the ellipsoid — 'ee' for the end-effector or a 3-vector [x, y, z] in the world frame
scale (float) – scale factor for the ellipsoid
add (bool) – if True, add the ellipsoid to the active plot environment

Returns:

an EllipsePlot object

Return type:

EllipsePlot

robot.vellipse(q) creates a velocity ellipsoid for the robot at pose q. By default the ellipsoid is centered at the end-effector.

Notes

By default the ellipsoid related to translational motion is drawn. Use opt='rot' to draw the rotational velocity ellipsoid.
By default the ellipsoid is drawn at the end-effector. The option centre allows its origin to be set to the specified 3-vector, or the string "ee" ensures it is drawn at the end-effector position.

class Robot2(arg, **kwargs)[source]

Bases: BaseRobot[Link2]

property T: ndarray

Pose of the shape relative to its parent frame in the scene graph (or the world frame if it has no parent), as a 4x4 homogeneous transformation matrix. Set via the pose argument of the constructor.

This is a read/write property. The getter always returns a plain ndarray; the setter also accepts an SE3.

Warning

Because the getter returns an ndarray, in-place operators like shape.T *= delta do an elementwise multiply, not a pose composition, even when delta is an SE3 – use shape.T = shape.T * delta (or shape.T @= delta.A) instead.

Return type:: ndarray(4,4)
Setter accepts:: ndarray | SE3

__getitem__(i)

Get link

This also supports iterating over each link in the robot object, from the base to the tool.

Parameters:: i (int | str) – link number or name
Return type:: Link2
Returns:: i’th link or named link

Examples

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.DH.Puma560()
>>> print(robot[1]) # print the 2nd link
RevoluteDH:   θ=q,  d=0,  a=0.4318,  ⍺=0.0
>>> print([link.a for link in robot])  # print all the a_j values
[0, 0.4318, 0.0203, 0, 0, 0]

Notes

Robot supports link lookup by name,: eg. robot['link1']

__str__()

Pretty prints the ETS Model of the robot.

Return type:: str
Returns:: Pretty print of the robot model

Notes

Constant links are shown in blue.
End-effector links are prefixed with an @
Angles in degrees
The robot base frame is denoted as BASE and is equal to the
robot’s base attribute.

accel(q, qd, torque, gravity=None)

Compute acceleration due to applied torque

Parameters:

q (Any) – Joint coordinates
qd (Any) – Joint velocity
torque (Any) – Joint torques of the robot
gravity (Any) – gravitational acceleration in the world frame, downwards gravitational force is equivalent to robot base acceleration upwards (positive); if not supplied, uses the gravity attribute of self

Returns:

Joint accelerations

Return type:

qdd = accel(q, qd, torque) calculates a vector (n) of joint accelerations that result from applying the actuator force/torque (n) to the manipulator in state q (n) and qd (n), and n is the number of robot joints.

\[\ddot{q} = \mathbf{M}^{-1} \left(\tau - \mathbf{C}(q)\dot{q} - \mathbf{g}(q)\right)\]

Trajectory operation

If q, qd, torque are matrices (m,n) then qdd is a matrix (m,n) where each row is the acceleration corresponding to the equivalent rows of q, qd, torque.

Examples

>>> import roboticstoolbox as rtb
>>> puma = rtb.models.DH.Puma560()
>>> puma.accel(puma.qz, 0.5 * np.ones(6), np.zeros(6))
array([ -7.5544, -12.22  ,  -6.4022,  -5.4303,  -4.9518,  -2.1178])

Notes

Useful for simulation of manipulator dynamics, in
conjunction with a numerical integration function.
Uses the method 1 of Walker and Orin to compute the forward
dynamics.
Featherstone’s method is more efficient for robots with large
numbers of joints.
Joint friction is considered.

References

Efficient dynamic computer simulation of robotic mechanisms,
M. W. Walker and D. E. Orin, ASME Journal of Dynamic Systems, Measurement and Control, vol. 104, no. 3, pp. 205-211, 1982.

accel_x(q, xd, wrench, gravity=None, pinv=False, representation='rpy/xyz')

Operational space acceleration due to applied wrench

Parameters:

q (Any) – Joint coordinates
xd (Any) – Operational space velocity of the end-effector
wrench (Any) – Wrench applied to the end-effector
gravity (Any) – gravitational acceleration in the world frame, downwards gravitational force is equivalent to robot base acceleration upwards (positive); if not supplied, uses the gravity attribute of self
pinv (bool) – use pseudo inverse rather than inverse
representation (Literal['rpy/xyz', 'rpy/zyx', 'eul', 'exp']) – the type of analytical Jacobian to use, default is 'rpy/xyz'

Returns:

Operational space accelerations of the end-effector

Return type:

xdd = accel_x(q, xd, wrench) is the operational space acceleration due to wrench applied to the end-effector of a robot in joint configuration q and operational space velocity xd.

\[\ddot{x} = \dot{\mathbf{J}}_a(q, \dot{q})\dot{q} + \mathbf{J}_a(q) \mathbf{M}(q)^{-1} \left( \mathbf{J}_a(q)^T w - \mathbf{C}(q)\dot{q} - \mathbf{g}(q) \right)\]

Trajectory operation

If q is a matrix (m,n), and xd and wrench are matrices (m,6), then xdd is a matrix (m,6) where each row is the acceleration corresponding to the equivalent rows of q, xd, and wrench.

Notes

Useful for simulation of manipulator dynamics, in
conjunction with a numerical integration function.
Uses the method 1 of Walker and Orin to compute the forward
dynamics.
Featherstone’s method is more efficient for robots with large
numbers of joints.
Joint friction is considered.

See also

accel()

addconfiguration(name, q)

Add a named joint configuration

Add a named configuration to the robot instance’s dictionary of named configurations.

Parameters:

name (str) – Name of the joint configuration
q (ndarray | Sequence[float]) – Joint configuration

Return type:

Examples

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.DH.Puma560()
>>> robot.addconfiguration_attr("mypos", [0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
>>> robot.configs["mypos"]
array([0.1, 0.2, 0.3, 0.4, 0.5, 0.6])

See also

addconfiguration_attr(name, q, unit='rad')

Add a named joint configuration as an attribute

Parameters:

name (str) – Name of the joint configuration
q (ndarray | Sequence[float]) – Joint configuration

Return type:

Examples

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.DH.Puma560()
>>> robot.addconfiguration_attr("mypos", [0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
>>> robot.mypos
array([0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
>>> robot.configs["mypos"]
array([0.1, 0.2, 0.3, 0.4, 0.5, 0.6])

Notes

Used in robot model init method to store the qr configuration
Dynamically adding attributes to objects can cause issues with
Python type checking.
Configuration is also added to the robot instance’s dictionary of
named configurations.

See also

attach(object)

Attach a child node

Parameters:: object (SceneNode) – the node to attach as a child of this node
Seealso:: attach_to() scene_children()
Return type:: None

attach_to(object)

Attach this node to a parent node

Parameters:: object (SceneNode) – the node to attach this node to, this node will become a child of the parent
Seealso:: attach() scene_parent()
Return type:: None

property base_link: Link2

Get the robot base link

robot.base_link is the robot base link

Returns:: the first link in the robot tree

cinertia(q)

Deprecated, use inertia_x

Return type:: None

property comment: str

Get/set robot comment

robot.comment is the robot comment
robot.comment = ... checks and sets the robot comment

Parameters:: name – the new robot comment
Returns:: robot comment

property configs: dict[str, ndarray]

Named joint configurations

Returns:: dictionary of joint configurations, such as qr and qz, indexed by name

configurations_str(border='thin')

Return type:: Any

property control_mode: str

Get/set robot control mode

robot.control_type is the robot control mode
robot.control_type = ... checks and sets the robot control mode

Parameters:: control_mode – the new robot control mode
Returns:: the current robot control mode

copy()

Make a copy of the robot

Return type:: Any
Returns:: a deep copy, which shares no state with the original

coriolis(q, qd)

Coriolis and centripetal term

Parameters:

q (Any) – Joint coordinates
qd (Any) – Joint velocity

Returns:

Coriolis/centripetal velocity matrix

Return type:

coriolis(q, qd) calculates the Coriolis/centripetal matrix (n,n) for the robot in configuration q and velocity qd, where n is the number of joints.

The product $\mathbf{C} \dot{q}$ is the vector of joint force/torque due to velocity coupling. The diagonal elements are due to centripetal effects and the off-diagonal elements are due to Coriolis effects. This matrix is also known as the velocity coupling matrix, since it describes the disturbance forces on any joint due to velocity of all other joints.

Trajectory operation

If q and qd are matrices (m,n), each row is interpreted as a joint configuration, and the result (n,n,m) is a 3d-matrix where each plane corresponds to a row of q and qd.

Examples

>>> import roboticstoolbox as rtb
>>> puma = rtb.models.DH.Puma560()
>>> puma.coriolis(puma.qz, 0.5 * np.ones((6,)))
array([[-0.4017, -0.5513, -0.2025, -0.0007, -0.0013,  0.    ],
       [ 0.2023, -0.1937, -0.3868, -0.    , -0.002 ,  0.    ],
       [ 0.1987,  0.193 , -0.    ,  0.    , -0.0001,  0.    ],
       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
       [ 0.0007,  0.0007,  0.0001,  0.    ,  0.    ,  0.    ],
       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ]])

Notes

Joint viscous friction is also a joint force proportional to
velocity but it is eliminated in the computation of this value.
Computationally slow, involves $n^2/2$ invocations of RNE.

coriolis_x(q, qd, pinv=False, representation='rpy/xyz', J=None, Ji=None, Jd=None, C=None, Mx=None)

Operational space Coriolis and centripetal term

Parameters:

q (Any) – Joint coordinates
qd (Any) – Joint velocity
pinv (bool) – use pseudo inverse rather than inverse (Default value = False)
representation (Literal['rpy/xyz', 'rpy/zyx', 'eul', 'exp']) – the type of analytical Jacobian to use, default is 'rpy/xyz'
J (Any) – pre-computed analytical Jacobian (optional)
Ji (Any) – pre-computed inverse analytical Jacobian (optional)
Jd (Any) – pre-computed time-derivative of analytical Jacobian (optional)
C (Any) – pre-computed joint-space Coriolis matrix (optional)
Mx (Any) – pre-computed operational-space inertia matrix (optional)

Returns:

Operational space velocity matrix

Return type:

coriolis() inertia_x() hessian0()

coriolis_x(q, qd) is the Coriolis/centripetal matrix (m,m) in operational space for the robot in configuration q and velocity qd, where n is the number of joints.

\[\mathbf{C}_x = \mathbf{J}(q)^{-T} \mathbf{C}(q, \dot{q}) \mathbf{J}(q)^{-1} - \mathbf{M}_x(q) \dot{\mathbf{J}}(q, \dot{q}) \mathbf{J}(q)^{-1}\]

The product $\mathbf{C} \dot{x}$ is the operational space wrench due to joint velocity coupling. This matrix is also known as the velocity coupling matrix, since it describes the disturbance forces on any joint due to velocity of all other joints.

The transformation to operational space requires an analytical, rather than geometric, Jacobian. analytical can be one of:

Value

Rotational representation

'rpy/xyz'

RPY angular rates in XYZ order (default)

'rpy/zyx'

RPY angular rates in XYZ order

'eul'

Euler angular rates in ZYZ order

'exp'

exponential coordinate rates

Trajectory operation

If q and qd are matrices (m,n), each row is interpreted as a joint configuration, and the result is an array (m,6,6) where each matrix corresponds to a row of q and qd.

Examples

>>> import roboticstoolbox as rtb
>>> puma = rtb.models.DH.Puma560()
>>> puma.coriolis_x(puma.qn, 0.5 * np.ones((6,)))
array([[     12.3365,       4.6387,      13.7633,      -0.0409,
              0.0043,      -0.0409],
       [     -4.788 ,  -31430.2949,      -3.4675,       0.3301,
             -0.3188,       0.3301],
       [    -11.5011,      -1.7324,     -10.0545,      -0.0405,
              0.    ,      -0.0405],
       [     -0.0165, 4869725.6671,      -0.0165,      -0.1245,
              0.0278,      -0.1245],
       [     -0.1112,       0.0531,       0.1332,      -0.0251,
             -0.    ,      -0.0251],
       [     -0.0165, 4869725.6671,      -0.0165,      -0.1245,
              0.0278,      -0.1245]])

Notes

Joint viscous friction is also a joint force proportional to
velocity but it is eliminated in the computation of this value.
Computationally slow, involves $n^2/2$ invocations of RNE.
If the robot is not 6 DOF the pinv option is set True.
pinv() is around 5x slower than inv()

Warning

Assumes that the operational space has 6 DOF.

See also

property default_backend: Any

Get default graphical backend

robot.default_backend Get the default graphical backend, used when
no explicit backend is passed to Robot.plot. This is the value set for the robot, otherwise the value set for all robots by set_default_backend(), otherwise None.
robot.default_backend = ... Set the default graphical backend, used when
no explicit backend is passed to Robot.plot. The default set here will be overridden if the particular Robot subclass cannot support it.

Returns:: backend name

dfs_links(start, func=None)

A link search method

Visit all links from start in depth-first order and will apply func to each visited link

Parameters:

start (Link2) – The link to start at
func (Optional[Callable[[Optional[Link2]], Any]]) – An optional function to apply to each link as it is found

Return type:

list[Link2]

Returns:

A list of links

dotfile(filename, etsbox=False, ets='full', jtype=False, static=True)

Write a link transform graph as a GraphViz dot file

The file can be processed using dot:

% dot -Tpng -o out.png dotfile.dot

The nodes are:

Base is shown as a grey square. This is the world frame origin, but can be changed using the base attribute of the robot.
Link frames are indicated by circles
ETS transforms are indicated by rounded boxes

The edges are:

an arrow if jtype is False or the joint is fixed
an arrow with a round head if jtype is True and the joint is revolute
an arrow with a box head if jtype is True and the joint is prismatic

Edge labels or nodes in blue have a fixed transformation to the preceding link.

Notes

If filename is a file object then the file will not be closed after the GraphViz model is written.

Parameters:

file – Name of file to write to
etsbox (bool) – Put the link ETS in a box, otherwise an edge label
ets (Literal['full', 'brief']) – Display the full ets with “full” or a brief version with “brief”
jtype (bool) – Arrowhead to node indicates revolute or prismatic type
static (bool) – Show static joints in blue and bold

Return type:

See also

showgraph() graph()

dynamics()

Pretty print the dynamic parameters

The dynamic parameters (inertial and friction) are printed in a table, with one row per link.

Examples

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.DH.Puma560()
>>> robot.dynamics()
┌───────┬───────┬─────────────────────────┬────────────────────────────────────────┬──────────┬───────────┬───────────────────┬───────┐
│   j   │   m   │            r            │                   I                    │    Jm    │     B     │        Tc         │   G   │
├───────┼───────┼─────────────────────────┼────────────────────────────────────────┼──────────┼───────────┼───────────────────┼───────┤
│ link1 │  0    │  0,  0,  0              │  0,  0.35,  0,  0,  0,  0              │  0.0002  │  0.00148  │  0.395, -0.435    │ -62.6 │
│ link2 │  17.4 │ -0.364,  0.006,  0.228  │  0.13,  0.524,  0.539,  0,  0,  0      │  0.0002  │  0.000817 │  0.126, -0.071    │  108  │
│ link3 │  4.8  │ -0.0203, -0.0141,  0.07 │  0.066,  0.086,  0.0125,  0,  0,  0    │  0.0002  │  0.00138  │  0.132, -0.105    │ -53.7 │
│ link4 │  0.82 │  0,  0.019,  0          │  0.0018,  0.0013,  0.0018,  0,  0,  0  │  3.3e-05 │  7.12e-05 │  0.0112, -0.0169  │  76   │
│ link5 │  0.34 │  0,  0,  0              │  0.0003,  0.0004,  0.0003,  0,  0,  0  │  3.3e-05 │  8.26e-05 │  0.00926, -0.0145 │  71.9 │
│ link6 │  0.09 │  0,  0,  0.032          │  0.00015,  0.00015,  4e-05,  0,  0,  0 │  3.3e-05 │  3.67e-05 │  0.00396, -0.0105 │  76.7 │
└───────┴───────┴─────────────────────────┴────────────────────────────────────────┴──────────┴───────────┴───────────────────┴───────┘

Return type:: None

dynamics_list()

Print dynamic parameters

Display the kinematic and dynamic parameters to the console in reable format

Return type:: None

dynchanged(what=None)

Dynamic parameters have changed

Called from a property setter to inform the robot that the cache of dynamic parameters is invalid.

See also

roboticstoolbox.Link._dirties_frne()

Return type:: None

property ee_links: list[Link2]

ets(start=None, end=None)

Robot to ETS

robot.ets() is an ETS representing the kinematics from base to end-effector.

robot.ets(end=link) is an ETS representing the kinematics from base to the link link specified as a Link reference or a name.

robot.ets(start=l1, end=l2) is an ETS representing the kinematics from link l1 to link l2.

Parameters:

start (:param) – start of path, defaults to base_link:
end (:param) – end of path, defaults to end-effector:

Raises:

ValueError – a link does not belong to this ERobot
TypeError – a bad link argument

Return type:

ETS

Returns:

elementary transform sequence

Examples

>>> import roboticstoolbox as rtb
>>> panda = rtb.models.ETS.Panda()
>>> panda.ets()
[ET.tz(param=0.333), ET.Rz(jindex=0), ET.Rx(param=-1.5707963267948966), ET.Rz(jindex=1), ET.Rx(param=1.5707963267948966), ET.tz(param=0.316), ET.Rz(jindex=2), ET.tx(param=0.0825), ET.Rx(param=1.5707963267948966), ET.Rz(jindex=3), ET.tx(param=-0.0825), ET.Rx(param=-1.5707963267948966), ET.tz(param=0.384), ET.Rz(jindex=4), ET.Rx(param=1.5707963267948966), ET.Rz(jindex=5), ET.tx(param=0.088), ET.Rx(param=1.5707963267948966), ET.tz(param=0.107), ET.Rz(jindex=6), ET.tz(param=0.10300000000000001), ET.Rz(param=-0.7853981633974483)]

fdyn(T, q0, Q=None, Q_args=None, qd0=None, solver='RK45', solver_args=None, dt=None, progress=False)

Integrate forward dynamics

Parameters:

T (float) – integration time
q0 (ndarray | Sequence[float]) – initial joint coordinates
Q (Callable[[Any, float, ndarray, ndarray], ndarray] | None) – a function that computes generalized joint force as a function of time and/or state
Q_args (dict | None) – positional arguments passed to torque
qd0 (ndarray | Sequence[float] | None) – initial joint velocities, assumed zero if not given
solver (str) – name of SciPy integrator to use, default 'RK45'
solver_args (dict | None) – optional arguments passed to the SciPy integrator
dt (float | None) – if given, interpolate the results to this timestep
progress (bool) – show progress bar, default False

Returns:

robot trajectory

Return type:

nofriction() friction()

tg = R.fdyn(T, q) integrates the dynamics of the robot with zero input torques over the time interval 0 to T and returns the trajectory as a namedtuple with elements:

t the time vector (M,)
q the joint coordinates (M,n)
qd the joint velocities (M,n)

tg = R.fdyn(T, q, torqfun) as above but the torque applied to the joints is given by the provided function:

tau = function(robot, t, q, qd, **args)

where the inputs are:

the robot object

current time

current joint coordinates (n,)

current joint velocity (n,)

args, optional keyword arguments can be specified, these are passed in from the targs keyword argument.

The function must return a Numpy array (n,) of joint forces/torques.

Examples

To apply zero joint torque to the robot without Coulomb friction:

>>> def myfunc(robot, t, q, qd):
>>>     return np.zeros((robot.n,))

>>> tg = robot.nofriction().fdyn(5, q0, myfunc)

>>> plt.figure()
>>> plt.plot(tg.t, tg.q)
>>> plt.show()

We could also use a lambda function:

>>> tg = robot.nofriction().fdyn(
>>>     5, q0, lambda r, t, q, qd: np.zeros((r.n,)))

The robot is controlled by a PD controller. We first define a function to compute the control which has additional parameters for the setpoint and control gains (qstar, P, D):

>>> def myfunc(robot, t, q, qd, qstar, P, D):
>>>     return (qstar - q) * P + qd * D  # P, D are (6,)

>>> tg = robot.fdyn(10, q0, myfunc, torque_args=(qstar, P, D))

Many integrators have variable step length which is problematic if we want to animate the result. If dt is specified then the solver results are interpolated in time steps of dt.

Notes

This function performs poorly with non-linear joint friction,
such as Coulomb friction. The R.nofriction() method can be used to set this friction to zero.
If the function is not specified then zero force/torque is
applied to the manipulator joints.
Interpolation is performed using SciPy integrate.ode
The SciPy RK45 integrator is used by default
Interpolation is performed using SciPy interp1d

See also

accel() nofriction(), rne().

fellipse(q, opt='trans', unit='rad', centre='ee', add=True)

Create a force ellipsoid object for plotting with PyPlot.

Parameters:

q (ndarray | Sequence[float]) – the joint configuration of the robot
opt (Literal['trans', 'rot']) – 'trans' or 'rot' — plot the translational or rotational force ellipsoid
unit (Literal['rad', 'deg']) – 'rad' or 'deg'
centre (Union[Literal['ee'], ndarray, Sequence[float]]) – centre of the ellipsoid — 'ee' for the end-effector or a 3-vector [x, y, z] in the world frame
add (bool) – if True, add the ellipsoid to the active plot environment

Returns:

an EllipsePlot object

Return type:

EllipsePlot

robot.fellipse(q) creates a force ellipsoid for the robot at pose q. By default the ellipsoid is centered at the end-effector.

Notes

By default the ellipsoid related to translational motion is drawn. Use opt='rot' to draw the rotational velocity ellipsoid.
By default the ellipsoid is drawn at the end-effector. The option centre allows its origin to be set to the specified 3-vector, or the string "ee" ensures it is drawn at the end-effector position.

friction(qd)

Manipulator joint friction

Parameters:: qd (ndarray) – The joint velocities of the robot
Returns:: The joint friction forces/torques for the robot
Return type:: ndarray

robot.friction(qd) is a vector of joint friction forces/torques for the robot moving with joint velocities qd.

The friction model includes:

Viscous friction which is a linear function of velocity.
Coulomb friction which is proportional to sign(qd).

\[\begin{split}\tau_j = G^2 B \dot{q}_j + |G_j| \left\{ \begin{array}{ll} \tau_{C,j}^+ & \mbox{if $\dot{q}_j > 0$} \\ \tau_{C,j}^- & \mbox{if $\dot{q}_j < 0$} \end{array} \right.\end{split}\]

Notes

The friction value should be added to the motor output torque to
determine the nett torque. It has a negative value when qd > 0.
The returned friction value is referred to the output of the
gearbox.
The friction parameters in the Link object are referred to the
motor.
Motor viscous friction is scaled up by $G^2$.
Motor Coulomb friction is scaled up by $G$.
The appropriate Coulomb friction value to use in the
non-symmetric case depends on the sign of the joint velocity, not the motor velocity.
Coulomb friction is zero for zero joint velocity, stiction is
not modeled.
The absolute value of the gear ratio is used. Negative gear
ratios are tricky: the Puma560 robot has negative gear ratio for joints 1 and 3.

See also

get_path(end=None, start=None)

Find a path from start to end

Parameters:

end (Union[Gripper, Link2, str, None]) – end-effector or gripper to compute forward kinematics to
start (Union[Gripper, Link2, str, None]) – name or reference to a base link, defaults to None

Raises:

ValueError – link not known or ambiguous

Return type:

tuple[list[Link2], int, SE3]

Returns:

the path from start to end

Returns:

the number of joints in the path

Returns:

the tool transform present after end

graph(format='dot', filename=None, **kwargs)

Render a link transform graph as text

Parameters:

format (Literal['dot', 'mermaid', 'mermaid_fenced']) – graph output format: "dot", "mermaid" or "mermaid_fenced", defaults to "dot"
filename (Union[str, IO[str], None]) – destination path or open text stream, defaults to None
kwargs (Any) – for format="dot", forwarded to dotfile() (etsbox, ets, jtype, static); ignored otherwise

Return type:

str

Returns:

rendered graph text

Creates graph text representing the robot’s link transform graph – the same structure dotfile()/showgraph() render, in a choice of formats. If filename is provided, the rendered text is also written to it.

"dot" produces the same GraphViz output as dotfile(), just returned as a string rather than requiring a real file – etsbox/ets/jtype/static behave exactly as they do there.

"mermaid"/"mermaid_fenced" render a simplified Mermaid flowchart: one node per link (a double circle for end-effectors, a hexagon for gripper links), edges labelled with the joint type (R/P) and joint index for actuated joints. Mermaid has no equivalent of DOT’s per-edge arrowhead/colour styling, so jtype/static/etsbox only affect "dot" output. "mermaid_fenced" wraps the same text in a fenced mermaid code block, ready to paste into Markdown or render directly in a Jupyter cell:

from IPython.display import Markdown, display
display(Markdown(robot.graph(format="mermaid_fenced")))

See also

dotfile() showgraph()

property gravity: ndarray

Get/set default gravitational acceleration

robot.gravity is the default gravitational acceleration
robot.gravity = ... checks and sets default gravitational
acceleration

Parameters:: gravity – gravitational acceleration in the world frame, downwards gravitational force is equivalent to robot base acceleration upwards (positive)
Returns:: gravitational acceleration
Setter accepts:: ndarray | Sequence[float]

gravload(q=None, gravity=None)

Compute gravity load

Parameters:

q (ndarray | Sequence[float] | None) – Joint coordinates
gravity (ndarray | Sequence[float] | None) – gravitational acceleration in the world frame, downwards gravitational force is equivalent to robot base acceleration upwards (positive); if not supplied, uses the stored gravity values

Returns:

The generalised joint force/torques due to gravity

Return type:

robot.gravload(q) calculates the joint gravity loading (n) for the robot in the joint configuration q and using the default gravitational acceleration specified in the DHRobot object.

robot.gravload(q, gravity=g) as above except the gravitational acceleration is explicitly specified as g.

The return value is referenced to the load, not the motor, the gear ratio is not taken into account.

Trajectory operation

If q is a matrix (nxm) each column is interpreted as a joint configuration vector, and the result is a matrix (nxm) each column being the corresponding joint torques.

Examples

>>> import roboticstoolbox as rtb
>>> puma = rtb.models.DH.Puma560()
>>> puma.gravload(puma.qz)
array([ 0.    , 37.4837,  0.2489,  0.    ,  0.    ,  0.    ])

gravload_x(q=None, gravity=None, pinv=False, representation='rpy/xyz', Ji=None)

Operational space gravity load

Parameters:

q (Any) – Joint coordinates
gravity (Any) – gravitational acceleration in the world frame, downwards gravitational force is equivalent to robot base acceleration upwards (positive); if not supplied, uses the gravity attribute of self
pinv (bool) – use pseudo inverse rather than inverse (Default value = False)
representation (Literal['rpy/xyz', 'rpy/zyx', 'eul', 'exp']) – the type of analytical Jacobian to use, default is 'rpy/xyz'
Ji (Any) – pre-computed inverse analytical Jacobian (optional)

Returns:

The operational space gravity wrench

Return type:

robot.gravload_x(q) calculates the gravity wrench for the robot in the joint configuration q and using the default gravitational acceleration specified in the robot object.

robot.gravload_x(q, gravity=g) as above except the gravitational acceleration is explicitly specified as g.

\[\mathbf{G}_x = \mathbf{J}(q)^{-T} \mathbf{G}(q)\]

The transformation to operational space requires an analytical, rather than geometric, Jacobian. analytical can be one of:

Value

Rotational representation

'rpy/xyz'

RPY angular rates in XYZ order (default)

'rpy/zyx'

RPY angular rates in XYZ order

'eul'

Euler angular rates in ZYZ order

'exp'

exponential coordinate rates

Trajectory operation

If q is a matrix (m,n), each row is interpreted as a joint configuration vector, and the result is a matrix (m,6) with one gravity wrench per row.

Examples

>>> import roboticstoolbox as rtb
>>> puma = rtb.models.DH.Puma560()
>>> puma.gravload_x(puma.qn)
array([-30.1218,   7.5797,  53.7378,  -0.    ,  -0.0283,  -0.    ])

Notes

If the robot is not 6 DOF the pinv option is set True.
pinv() is around 5x slower than inv()

Warning

Assumes that the operational space has 6 DOF.

See also

gravload()

property grippers: list[Gripper]

Grippers attached to the robot

Returns:: A list of grippers

property hascollision: Any

Robot has collision model

Returns:: Robot has collision model
Returns:: At least one link has associated collision model.

Examples

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.DH.Puma560()
>>> robot.hascollision
False

See also

hasgeometry()

property hasdynamics: bool

Robot has dynamic parameters

Returns:: True if at least one link has a non-zero mass or motor inertia

These are the parameters that the inertial dynamics need. It is false for a model of the kinematics only, such as the default rtb.models.Panda, and the dynamics methods gravload, inertia, coriolis, accel, fdyn, itorque and rne raise ValueError for such a model, rather than return zeros or a confusing error. It is any link and not all, since the first link of the DH Puma560 has a mass of zero.

The answer is cached, and recomputed only when the mass or motor inertia of a link has been changed, so it costs nothing to ask.

Changed in version 1.5.0: This was true if any dynamic parameter had been assigned, even to zero, and was not reset when the parameters were set back to zero. A robot made from DH links with no masses reported True. The dynamics methods now raise ValueError if it is False, where they returned zeros or raised LinAlgError.

Warning

Behaviour change. Scripts that call the dynamics methods on a kinematics-only model, and accept the zeros, now get a ValueError.

Examples

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.DH.Puma560()
>>> robot.hasdynamics
True

property hasgeometry: Any

Robot has geometry model

At least one link has associated mesh to describe its shape.

Returns:: Robot has geometry model

Examples

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.DH.Puma560()
>>> robot.hasgeometry
True

See also

hascollision()

hierarchy()

Pretty print the robot link hierarchy

Return type:: None
Returns:: Pretty print of the robot model

Examples

Makes a robot and prints the hierarchy

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.URDF.Panda()
>>> robot.hierarchy()
 panda_link0
   panda_link1
     panda_link2
       panda_link3
         panda_link4
           panda_link5
             panda_link6
               panda_link7
                 panda_link8
                   panda_hand
                     panda_leftfinger
                     panda_rightfinger

inertia(q)

Manipulator inertia matrix

Parameters:: q (ndarray) – Joint coordinates
Returns:: The inertia matrix
Return type:: ndarray

inertia(q) is the symmetric joint inertia matrix (n,n) which relates joint torque to joint acceleration for the robot at joint configuration q.

Trajectory operation

If q is a matrix (m,n), each row is interpreted as a joint state vector, and the result is a 3d-matrix (nxnxk) where each plane corresponds to the inertia for the corresponding row of q.

Examples

>>> import roboticstoolbox as rtb
>>> puma = rtb.models.DH.Puma560()
>>> puma.inertia(puma.qz)
array([[ 3.9611, -0.1627, -0.1389,  0.0016, -0.0004,  0.    ],
       [-0.1627,  4.4566,  0.3727,  0.    ,  0.0019,  0.    ],
       [-0.1389,  0.3727,  0.9387,  0.    ,  0.0019,  0.    ],
       [ 0.0016,  0.    ,  0.    ,  0.1924,  0.    ,  0.    ],
       [-0.0004,  0.0019,  0.0019,  0.    ,  0.1713,  0.    ],
       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.1941]])

Notes

The diagonal elements M[j,j] are the inertia seen by joint
actuator j.
The off-diagonal elements M[j,k] are coupling inertias that
relate acceleration on joint j to force/torque on joint k.
The diagonal terms include the motor inertia reflected through
the gear ratio.

See also

cinertia()

inertia_x(q=None, pinv=False, representation='rpy/xyz', Ji=None)

Operational space inertia matrix

Parameters:

q (Any) – Joint coordinates
pinv (bool) – use pseudo inverse rather than inverse (Default value = False)
representation (Literal['rpy/xyz', 'rpy/zyx', 'eul', 'exp']) – the type of analytical Jacobian to use, default is 'rpy/xyz'
Ji (Any) – The inverse analytical Jacobian (base-frame)

Returns:

The operational space inertia matrix

Return type:

robot.inertia_x(q) is the operational space (Cartesian) inertia matrix which relates Cartesian force/torque to Cartesian acceleration at the joint configuration q.

\[\mathbf{M}_x = \mathbf{J}(q)^{-T} \mathbf{M}(q) \mathbf{J}(q)^{-1}\]

The transformation to operational space requires an analytical, rather than geometric, Jacobian. analytical can be one of:

Value

Rotational representation

'rpy/xyz'

RPY angular rates in XYZ order (default)

'rpy/zyx'

RPY angular rates in XYZ order

'eul'

Euler angular rates in ZYZ order

'exp'

exponential coordinate rates

Trajectory operation

If q is a matrix (m,n), each row is interpreted as a joint state vector, and the result is a 3d-matrix (m,n,n) where each plane corresponds to the Cartesian inertia for the corresponding row of q.

Examples

>>> import roboticstoolbox as rtb
>>> puma = rtb.models.DH.Puma560()
>>> puma.inertia_x(puma.qn)
array([[17.2954, -2.7542, -9.6233,  0.    ,  0.2795,  0.    ],
       [-2.7542, 11.6452,  1.2459,  0.0001, -0.0703,  0.0001],
       [-9.6233,  1.2459, 13.3348,  0.    ,  0.2767,  0.    ],
       [ 0.    ,  0.0001,  0.    ,  0.1941,  0.    ,  0.1941],
       [ 0.2795, -0.0703,  0.2767,  0.    ,  0.1713,  0.    ],
       [ 0.    ,  0.0001,  0.    ,  0.1941,  0.    ,  0.1941]])

Notes

If the robot is not 6 DOF the pinv option is set True.
pinv() is around 5x slower than inv()

Warning

Assumes that the operational space has 6 DOF.

See also

isprismatic() prismaticjoints()

isprismatic(j)

Check if joint is prismatic

Return type:: bool
Returns:: True if prismatic

Examples

>>> import roboticstoolbox as rtb
>>> puma = rtb.models.DH.Puma560()
>>> puma.prismaticjoints
[False, False, False, False, False, False]
>>> stanford = rtb.models.DH.Stanford()
>>> stanford.isprismatic(1)
False

See also

isrevolute(j)

Check if joint is revolute

Return type:: bool
Returns:: True if revolute

Examples

>>> import roboticstoolbox as rtb
>>> puma = rtb.models.DH.Puma560()
>>> puma.revolutejoints
[True, True, True, True, True, True]
>>> stanford = rtb.models.DH.Stanford()
>>> stanford.isrevolute(1)
True

See also

isrevolute() revolutejoints()

itorque(q, qdd)

Inertia torque

Parameters:

q (Any) – Joint coordinates
qdd (Any) – Joint acceleration

Returns:

The inertia torque vector

Return type:

itorque(q, qdd) is the inertia force/torque vector (n) at the specified joint configuration q (n) and acceleration qdd (n), and n is the number of robot joints. It is $\mathbf{I}(q) \ddot{q}$.

Trajectory operation

If q and qdd are matrices (m,n), each row is interpreted as a joint configuration, and the result is a matrix (m,n) where each row is the corresponding joint torques.

Examples

>>> import roboticstoolbox as rtb
>>> puma = rtb.models.DH.Puma560()
>>> puma.itorque(puma.qz, 0.5 * np.ones((6,)))
array([1.8304, 2.3343, 0.5872, 0.0971, 0.0873, 0.0971])

Notes

If the robot model contains non-zero motor inertia then this
will be included in the result.

See also

jointdynamics(q, qd=None)

Transfer function of joint actuator

Parameters:

q (Any) – Joint coordinates
qd (Any) – Joint velocity

Returns:

transfer function denominators

Return type:

tf = jointdynamics(qd, q) calculates a vector of n continuous-time transfer functions that represent the transfer function 1/(Js+B) for each joint based on the dynamic parameters of the robot and the configuration q (n). n is the number of robot joints. The result is a list of tuples (J, B) for each joint.

tf = jointdynamics(q, qd) as above but include the linearized effects of Coulomb friction when operating at joint velocity QD (1xN).

property keywords: list[str]

Keywords that describe the robot

Returns:: the keywords, which the model catalog can filter on

property link_dict: dict[str, Link2]

linkcolormap(linkcolors='viridis')

Create a colormap for robot joints.

Parameters:: linkcolors (list[Any] | str) – list of colors or colormap name, defaults to "viridis"
Return type:: Any
Returns:: the color map

cm = robot.linkcolormap() is an n-element colormap that gives a unique color for every link. The RGBA colors for link j are cm(j).
cm = robot.linkcolormap(cmap) as above but cmap is the name of a valid matplotlib colormap. The default, example above, is the viridis colormap.
cm = robot.linkcolormap(list of colors) as above but a colormap is created from a list of n color names given as strings, tuples or hexstrings.

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.DH.Puma560()
>>> cm = robot.linkcolormap("inferno")
>>> print(cm(range(6))) # cm(i) is 3rd color in colormap
>>> cm = robot.linkcolormap(
...     ['red', 'g', (0,0.5,0), '#0f8040', 'yellow', 'cyan'])
>>> print(cm(range(6)))

Notes

Colormaps have 4-elements: red, green, blue, alpha (RGBA)
Names of supported colors and colormaps are defined in the matplotlib documentation.
- Specifying colors
- Colormaps

property links: list[Link2]

Robot links

Returns:: A list of link objects

Notes

It is probably more concise to index the robot object rather than the list of links, ie. the following are equivalent: - robot.links[i] - robot[i]

property manufacturer: Any

Get/set robot manufacturer’s name

robot.manufacturer is the robot manufacturer’s name
robot.manufacturer = ... checks and sets the manufacturer’s name

Returns:: robot manufacturer’s name

property n: int

Number of joints

Returns:: Number of joints

Examples

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.DH.Puma560()
>>> robot.n
6

See also

nlinks() nbranches()

property name: str

Get/set robot name

robot.name is the robot name
robot.name = ... checks and sets the robot name

Parameters:: name – the new robot name
Returns:: the current robot name

property nbranches: int

Number of branches

Number of branches in this robot. Computed as the number of links with zero children

Returns:: number of branches in the robot’s kinematic tree

Examples

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.ETS.Panda()
>>> robot.nbranches
1

See also

n() nlinks()

property nlinks: Any

Number of links

The returned number is the total of both variable joints and static links

Returns:: Number of links

Examples

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.DH.Puma560()
>>> robot.nlinks
6

See also

n() nbranches()

nofriction(coulomb=True, viscous=False)

Remove manipulator joint friction

Parameters:

coulomb (bool) – set the Coulomb friction to 0
viscous (bool) – set the viscous friction to 0

Return type:

friction() nofriction()

Returns:

A copy of the robot with dynamic parameters perturbed

nofriction() copies the robot and returns a robot with the same link parameters except the Coulomb and/or viscous friction parameter are set to zero.

See also

payload(m, p=None)

Add a payload to the end-effector

Parameters:

m (float) – mass (kg)
p (ndarray | Sequence[float] | None) – position in end-effector frame

Return type:

payload(m, p) adds a payload with point mass m at position p in the end-effector coordinate frame.

payload(m) adds a payload with point mass m at the origin of the end-effector coordinate frame.

payload(0) removes added payload.

The point mass is combined with the mass, centre of mass and inertia of the end-effector link, and p is expressed in that link’s frame. The link’s own parameters are kept, so each call replaces the payload added by the previous one, and a payload of zero mass leaves the dynamics unchanged.

Note

The end-effector link is ee_links[0] when the robot has a single end-effector. A robot with several end-effectors has no single end-effector link, and the payload is added to links[n - 1] as in earlier releases.

payload_capacity(q, tau_m, w, frame='base')

Compute the payload capacity of the manipulator

Parameters:

q (ndarray | Sequence[float]) – Joint coordinates for trajectory
tau_m (ndarray | Sequence[float]) – actuator torque limits
w (ndarray | Sequence[float]) – External wrench applied to the end-effector
frame (Literal['base', 'ee']) – the frame in which w is expressed, "base" for the base frame (default) and "ee" for the end-effector frame, which includes the tool transform

Returns:

Payload capacity of the manipulator

Return type:

For each configuration, row of q, compute the multiple of the applied external wrench w that can be applied to the end-effector without exceeding the actuator torque limits tau_m. If w is the weight of a 1 kg mass then the multiple is the capacity in kg. A load such as a weight is fixed in the world, so w is in the base frame by default; use frame="ee" for a wrench that is fixed to the end-effector, such as a force along the tool axis. This multiple is computed for each joint and can be

a positive real number
inf the joint has no role in determining the payload capacity
nan the gravity load already exceeds the joint torque limit

The joint with the lowest payload capacity is the one that determines the overall payload capacity for that configuration.

The result is a matrix of payload capacities for each joint and configuration.

The joint force/torque limits tau_m are given as an array of shape (n,) or (1,n) of maximum values, in which case the minimum is the negative of the maximum, or (2,n), in which case the first row is the minimum and the second row is the maximum.

Examples

In this example we compute the payload capacity of the Puma560 robot along a straight line trajectory in the x-direction, moving from tightly folded to almost straight. The end-effector is oriented with the tool frame pointing in the x-direction, and a tool offset is added so that the load causes a moment on the wrist. The external wrench is a downward force of 9.81 N, which is equivalent to a 1 kg mass. The torque limits are the motor fuse limits from Table 2.17 of “Visual Control of Robots”, Corke (1996). The payload capacity, as well as the limiting joint, is plotted as a function of the x-position of the end-effector. The minimum payload capacity is also plotted, along with the index of the joint that determines the payload capacity.

>>> from roboticstoolbox import models
>>> import matplotlib.pyplot as plt
>>> import numpy as np
>>> from spatialmath import SE3
>>> puma = models.DH.Puma560()
>>> puma.tool = SE3.Tz(0.05)
>>> T = SE3.Tx(np.arange(0.35, 0.85, 0.01)) * SE3.Tz(puma.links[0].d) * SE3.Ry(90, "deg")
>>> sol = puma.ikine_a(T)
>>> w = [0,0,-9.81,0,0,0]
>>> cap = puma.payload_capacity(sol.q, [56, 97, 52,10,10,10], w)
>>> plt.plot(T.x, [np.min(c) for c in cap])
>>> plt.plot(T.x, [np.argmin(c) for c in cap])

Derived from the MATLAB functions pay and paycap originally contributed to the MATLAB Robotics Toolbox by Bryan Moutrie (@au-phiware). Ported to Python and adapted for robotics-toolbox-python with permission from the original author.

Added in version 1.5.0.

Seealso:: gravload() payload()

perturb(p=0.1)

Perturb robot parameters

Parameters:: p (float) – The percent (+/-) to be perturbed. Default 10%
Return type:: Any
Returns:: A copy of the robot with dynamic parameters perturbed

rp = perturb(p) is a new robot object in which the dynamic parameters (link mass and inertia) have been perturbed. The perturbation is multiplicative so that values are multiplied by random numbers in the interval (1-p) to (1+p). The name string of the perturbed robot is prefixed by ‘P/’.

Useful for investigating the robustness of various model-based control schemes. For example to vary parameters in the range +/- 10 percent is: r2 = puma.perturb(0.1)

plot(q, backend=None, block=False, dt=0.05, limits=None, vellipse=False, fellipse=False, fig=None, movie=None, loop=False, notebook=None, ax=None, env=None, view=None, **kwargs)

Graphical display and animation

robot.plot(q, 'pyplot') displays a graphical view of a robot based on the kinematic model and the joint configuration q. This is a stick figure polyline which joins the origins of the link coordinate frames. The plot will autoscale with an aspect ratio of 1.

If q (m,n) representing a joint-space trajectory it will create an animation with a pause of dt seconds between each frame.

Parameters:

q (ndarray | Sequence[float]) – The joint configuration of the robot.
backend (Optional[Literal['swift', 'pyplot', 'pyplot2']]) – The graphical backend to use, currently ‘swift’ and ‘pyplot’ are implemented. Defaults to ‘swift’ of a Robot and ‘pyplot` for a DHRobot
block (bool) – True waits until the window is closed; the PyPlot backend has already shown the figure, so False and None return at once, defaults to False
dt (float) – if q is a trajectory, this describes the delay in seconds between frames
limits (ndarray | Sequence[float] | None) – Custom view limits for the plot. If not supplied will autoscale, [x1, x2, y1, y2, z1, z2] (this option is for ‘pyplot’ only)
vellipse (bool) – (Plot Option) Plot the velocity ellipse at the end-effector (this option is for ‘pyplot’ only)
fellipse (bool) – (Plot Option) Plot the force ellipse at the end-effector (this option is for ‘pyplot’ only)
fig (str | None) – (Plot Option) The figure label to plot in (this option is for ‘pyplot’ only)
ax (Any) – (Plot Option) Existing Matplotlib 3D axes to draw the scene on, instead of making a figure (this option is for ‘pyplot’ only)
env (Connector | None) – The scene returned by an earlier plot, to add this robot to, instead of making a new scene; see Several configurations below. The backend of the scene is used, so backend is ignored, and the options that are set when a scene is made (limits, fig, ax, render_mode, inline_*, notebook and browser) cannot be given
color – (Plot Option) The color of the robot’s line, any Matplotlib color (this option is for ‘pyplot’ only)
view (str | None) – (Plot Option) "xy", "xz" or "yz" to start with a flat view of that plane, the first axis to the right and the second up, or None (default) for the usual three-dimensional view. The view is not locked, the plot can be rotated as usual. With env it changes the view of that scene (this option is for ‘pyplot’ only)
movie (str | None) – (Plot Option) The filename to save the movie to (this option is for ‘pyplot’ only)
loop (bool) – (Plot Option) Loop the movie (this option is for ‘pyplot’ only)
notebook (bool | None) – (Plot Option, Swift only) True shows Swift in a frame of the current notebook cell, False opens it in a browser window, and None (default) chooses the frame in a notebook, unless a browser is named, and a window otherwise, so the same call works in a script and in a notebook. The browser showing the notebook must be on the machine running the kernel, since the frame points at localhost
browser – (Plot Option, Swift only) the browser for the window, or None (default) for the system’s. This is a name known to Python’s webbrowser module, which depends on the operating system, for example "chrome", "firefox" or "safari" on macOS and "google-chrome" or "chromium" on Linux, or a command line with %s where the URL goes, for any other browser; webbrowser.get(name) tells you whether a name works. Naming a browser means a window, not the notebook frame. browser="notebook" is deprecated, use notebook=True. See swift.Swift.Swift.launch()
realtime – (Plot Option) Pace the steps of a trajectory in real time, one each dt, defaults to True for a trajectory (this option is for ‘swift’ only)
jointaxes – (Plot Option) Plot an arrow indicating the axes in which the joint revolves around(revolute joint) or translates along (prosmatic joint) (this option is for ‘pyplot’ only)
eeframe – (Plot Option) Plot the end-effector coordinate frame at the location of the end-effector. Uses three arrows, red, green and blue to indicate the x, y, and z-axes. (this option is for ‘pyplot’ only)
shadow – (Plot Option) Plot a shadow of the robot in the x-y plane. (this option is for ‘pyplot’ only)
name – (Plot Option) Plot the name of the robot near its base (this option is for ‘pyplot’ only)
render_mode – (Plot Option) Rendering mode for matplotlib backends: 'window', 'notebook-widget', or 'notebook-inline'. If omitted, a mode is chosen from the environment, so the same call works for any %matplotlib setting: for the interactive ipympl backend a single pose is a widget that can be rotated, but a trajectory uses 'notebook-inline', since a widget cannot update while the cell is running.
inline_every_n – (Plot Option) In notebook-inline mode, push one rendered frame every N simulation steps. Larger N reduces output load.
inline_format – (Plot Option) In notebook-inline mode, frame format: 'svg' (default) or 'png'.
inline_dpi – (Plot Option) DPI for PNG inline frames only; ignored when inline_format='svg'.

Return type:

Returns:

A reference to the environment object which controls the figure

Note

Python’s webbrowser module cannot tell whether a browser is installed, so a browser that cannot be opened is not reported as such. On macOS the names "chrome", "firefox" and "safari" are always accepted, whether or not the app is installed. If it is not, no window opens and there is no error: Swift waits about 10 seconds for the page to connect and then reports Could not connect to the Swift simulator. If you see that, check that the browser named by browser is installed, or leave it as None to use the system’s default browser. See swift.Swift.Swift.launch().

Several configurations

plot returns the scene it made, and passing that as env to another plot adds a robot to it instead of making a new scene. The robot is the same model, and each plot of it holds the pose it was given, so the robot is shown in two configurations at once, here the initial and the final. Give each a color, and turn off what would be drawn twice.

import roboticstoolbox as rtb

puma = rtb.models.DH.Puma560()
env = puma.plot(puma.qr, color="gray", block=False)
puma.plot(puma.qn, env=env, color="red", name=False, block=False)

The scene can be given a pose to draw directly, handle = env.add(puma, q=q), and handle.q = q moves only that drawing. A robot that is added without a q follows puma.q, as it always has. A trajectory can be plotted on top of an earlier plot in the same way; the earlier plot stays where it was. In a notebook, an animation needs the scene to have been made with render_mode="notebook-inline".

Notes

By default this method will block until the figure is dismissed.
To avoid this set block=False.
For PyPlot, the polyline joins the origins of the link frames,
but for some Denavit-Hartenberg models those frames may not actually be on the robot, ie. the lines to not necessarily represent the links of the robot.

See also

teach()

plot_ellipse(ellipse, block=True, limits=None, jointaxes=True, eeframe=True, shadow=True, name=True)

Plot an ellipsoid.

Parameters:

ellipse (EllipsePlot) – the ellipsoid to plot
block (bool) – True waits until the window is closed; the PyPlot backend has already shown the figure, so False and None return at once, defaults to True
limits (ndarray | Sequence[float] | None) – custom view limits [x1, x2, y1, y2, z1, z2]; autoscales if not supplied
jointaxes (bool) – plot an arrow indicating the joint axis
eeframe (bool) – plot the end-effector coordinate frame
shadow (bool) – plot a shadow of the robot in the x-y plane
name (bool) – plot the name of the robot near its base

Returns:

a reference to the PyPlot object controlling the matplotlib figure

Return type:

robot.plot_ellipse(ellipsoid) displays the ellipsoid.

plot_fellipse(q, block=True, fellipse=None, limits=None, opt='trans', centre='ee', jointaxes=True, eeframe=True, shadow=True, name=True)

Plot the force ellipsoid for a manipulator.

Parameters:

q (ndarray | Sequence[float] | None) – the joint configuration of the robot
block (bool) – True waits until the window is closed; the PyPlot backend has already shown the figure, so False and None return at once, defaults to True
fellipse (EllipsePlot | None) – a pre-built force ellipsoid to plot
limits (ndarray | Sequence[float] | None) – custom view limits [x1, x2, y1, y2, z1, z2]; autoscales if not supplied
opt (Literal['trans', 'rot']) – 'trans' or 'rot' — plot the translational or rotational force ellipsoid
centre (Union[Literal['ee'], ndarray, Sequence[float]]) – coordinates to plot the ellipse — [x, y, z] or "ee"
jointaxes (bool) – plot an arrow indicating the joint axis
eeframe (bool) – plot the end-effector coordinate frame
shadow (bool) – plot a shadow of the robot in the x-y plane
name (bool) – plot the name of the robot near its base

Raises:

ValueError – if neither q nor fellipse is supplied

Returns:

a reference to the PyPlot object controlling the matplotlib figure

Return type:

robot.plot_fellipse(q) displays the force ellipsoid for the robot at pose q. The plot will autoscale with an aspect ratio of 1.

robot.plot_fellipse(vellipse=ell) uses a pre-built ellipse object.

Notes

By default the ellipsoid related to translational motion is drawn. Use opt='rot' to draw the rotational velocity ellipsoid.
By default the ellipsoid is drawn at the origin. Use centre to specify a 3-vector, or "ee" to draw at the end-effector.

plot_vellipse(q, block=True, vellipse=None, limits=None, opt='trans', centre='ee', jointaxes=True, eeframe=True, shadow=True, name=True)

Plot the velocity ellipsoid for a manipulator.

Parameters:

q (ndarray | Sequence[float] | None) – the joint configuration of the robot
block (bool) – True waits until the window is closed; the PyPlot backend has already shown the figure, so False and None return at once, defaults to True
vellipse (EllipsePlot | None) – a pre-built velocity ellipsoid to plot
limits (ndarray | Sequence[float] | None) – custom view limits [x1, x2, y1, y2, z1, z2]; autoscales if not supplied
opt (Literal['trans', 'rot']) – 'trans' or 'rot' — plot the translational or rotational velocity ellipsoid
centre (Union[Literal['ee'], ndarray, Sequence[float]]) – coordinates to plot the ellipse — [x, y, z] or "ee"
jointaxes (bool) – plot an arrow indicating the joint axis
eeframe (bool) – plot the end-effector coordinate frame
shadow (bool) – plot a shadow of the robot in the x-y plane
name (bool) – plot the name of the robot near its base

Raises:

ValueError – if neither q nor vellipse is supplied

Returns:

a reference to the PyPlot object controlling the matplotlib figure

Return type:

isprismatic() revolutejoints()

robot.plot_vellipse(q) displays the velocity ellipsoid for the robot at pose q. The plot will autoscale with an aspect ratio of 1.

robot.plot_vellipse(vellipse=ell) uses a pre-built ellipse object.

Notes

By default the ellipsoid related to translational motion is drawn. Use opt='rot' to draw the rotational velocity ellipsoid.
By default the ellipsoid is drawn at the origin. Use centre to specify a 3-vector, or "ee" to draw at the end-effector.

property prismaticjoints: list[bool]

Prismatic joints as bool array

Returns:: array of joint type, True if prismatic

Examples

>>> import roboticstoolbox as rtb
>>> puma = rtb.models.DH.Puma560()
>>> puma.prismaticjoints
[False, False, False, False, False, False]
>>> stanford = rtb.models.DH.Stanford()
>>> stanford.prismaticjoints
[False, False, True, False, False, False]

Notes

Fixed joints, that maintain a constant link relative pose, are not included.

See also

property q: ndarray

Get/set robot joint configuration

robot.q is the robot joint configuration
robot.q = ... checks and sets the joint configuration

Parameters:: q – the new robot joint configuration
Returns:: robot joint configuration
Setter accepts:: ndarray | Sequence[float]

property qd: ndarray

Get/set robot joint velocity

robot.qd is the robot joint velocity
robot.qd = ... checks and sets the joint velocity

Returns:: robot joint velocity
Setter accepts:: ndarray | Sequence[float]

property qdd: ndarray

Get/set robot joint acceleration

robot.qdd is the robot joint acceleration
robot.qdd = ... checks and sets the robot joint acceleration

Returns:: robot joint acceleration
Setter accepts:: ndarray | Sequence[float]

property qlim: ndarray

Joint limits

Limits are extracted from the link objects. If joints limits are not set for:

a revolute joint [-𝜋. 𝜋] is returned
a prismatic joint an exception is raised

Parameters:: qlim – An array of joints limits (2, n)
Raises:: ValueError – unset limits for a prismatic joint
Returns:: Array of joint limit values

Examples

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.DH.Puma560()
>>> robot.qlim
array([[-2.7925, -0.7854, -3.927 , -1.9199, -1.7453, -4.6426],
       [ 2.7925,  3.927 ,  0.7854,  2.9671,  1.7453,  4.6426]])

Setter accepts:: ndarray | Sequence[float]

random_q()

Return a random joint configuration

The value for each joint is uniform randomly distributed between the limits set for the robot.

Notes

The joint limit for all joints must be set.

Return type:: Any
Returns:: Random joint configuration :rtype: ndarray(n)

See also

qlim() qlim()

property revolutejoints: list[bool]

Revolute joints as bool array

Returns:: array of joint type, True if revolute

Examples

>>> import roboticstoolbox as rtb
>>> puma = rtb.models.DH.Puma560()
>>> puma.revolutejoints
[True, True, True, True, True, True]
>>> stanford = rtb.models.DH.Stanford()
>>> stanford.revolutejoints
[True, True, False, True, True, True]

Notes

Fixed joints, that maintain a constant link relative pose, are not included.

See also

isrevolute() prismaticjoints()

property scene_children: list[SceneNode]

Return the child nodes of this object in the scene graph.

Setting a new list of children updates each child’s scene_parent to this object, but does not remove this object from any previous parent’s scene_children.

This is a read/write property.

Return type:: list(SceneNode)
Seealso:: scene_parent() attach() attach_to()

property scene_parent: SceneNode | None

Return the parent node of this object in the scene graph.

Setting a new parent adds this object to the new parent’s scene_children.

This is a read/write property.

Return type:: SceneNode | None
Seealso:: scene_children() attach() attach_to()
Setter accepts:: SceneNode

segments()

Segments of branched robot

For a single-chain robot with structure:

L1 - L2 - L3

the return is [[None, L1, L2, L3]]

For a robot with structure:

L1 - L2 +-  L3 - L4
        +- L5 - L6

the return is [[None, L1, L2], [L2, L3, L4], [L2, L5, L6]]

Return type:: list[list[Optional[Link2]]]
Returns:: Segment list

Notes

the length of the list is the number of segments in the robot
the first segment always starts with None which represents
the base transform (since there is no base link)
the last link of one segment is also the first link of subsequent
segments

classmethod set_default_backend(be)

Set the default graphical backend for all robots

Parameters:: be (Any) – backend name, for example "swift", or None to clear
Return type:: None

Robot.set_default_backend("swift") sets the backend used by plot() for every robot, existing or created later, that has not been given one of its own by robot.default_backend = .... The robot’s own value takes precedence.

See also

default_backend

showgraph(display_graph=True, **kwargs)

Display a link transform graph in browser

robot.showgraph() displays a graph of the robot’s link frames and the ETS between them. It uses GraphViz dot.

The nodes are:

Base is shown as a grey square. This is the world frame origin, but can be changed using the base attribute of the robot.
Link frames are indicated by circles
ETS transforms are indicated by rounded boxes

The edges are:

an arrow if jtype is False or the joint is fixed
an arrow with a round head if jtype is True and the joint is revolute
an arrow with a box head if jtype is True and the joint is prismatic

Edge labels or nodes in blue have a fixed transformation to the preceding link.

Parameters:

display_graph (bool) – Open the graph in a browser if True. Otherwise will return the file path
etsbox – Put the link ETS in a box, otherwise an edge label
jtype – Arrowhead to node indicates revolute or prismatic type
static – Show static joints in blue and bold

Return type:

str | None

Examples

>>> import roboticstoolbox as rtb
>>> panda = rtb.models.URDF.Panda()
>>> panda.showgraph()

See also

dotfile() graph()

property structure: str

Return the joint structure string

A string with one letter per joint: R for a revolute joint, and P for a prismatic joint.

Returns:: joint configuration string

Examples

>>> import roboticstoolbox as rtb
>>> puma = rtb.models.DH.Puma560()
>>> puma.structure
'RRRRRR'
>>> stanford = rtb.models.DH.Stanford()
>>> stanford.structure
'RRPRRR'

Notes

Fixed joints, that maintain a constant link relative pose, are not included. len(self.structure) == self.n.

property symbolic: bool

teach(q, block=True, limits=None, vellipse=False, fellipse=False, backend=None, view=None)

Graphical teach pendant

robot.teach(q) opens a graphical view (PyPlot or Swift – see backend) which allows the user to “drive” a graphical robot using a graphical slider panel. The robot’s initial joint configuration is q.

robot.teach() as above except the robot’s stored value of q is used.

Parameters:

q (ndarray | Sequence[float] | None) – The joint configuration of the robot (Optional, if not supplied will use the stored q values).
block (bool) – True waits until the window is closed; the PyPlot backend has already shown the figure, so False and None return at once, defaults to True
limits (ndarray | Sequence[float] | None) – Custom view limits for the plot. If not supplied will autoscale, [x1, x2, y1, y2, z1, z2] (this option is for ‘pyplot’/’pyplot2’ only)
vellipse (bool) – (Plot Option) Plot the velocity ellipse at the end-effector (this option is for ‘pyplot’ only)
fellipse (bool) – (Plot Option) Plot the force ellipse at the end-effector (this option is for ‘pyplot’ only)
backend (Optional[Literal['swift', 'pyplot', 'pyplot2']]) – The graphical backend to use – ‘swift’, ‘pyplot’, or ‘pyplot2’. Defaults to whatever plot() would pick for this robot (see its own backend for the resolution rule).
view (str | None) – "xy", "xz" or "yz" to start with a flat view of that plane, the first axis to the right and the second up, or None (default) for the usual three-dimensional view. The view is not locked, the plot can be rotated as usual (this option is for ‘pyplot’ only)

Return type:

Returns:

A reference to the environment object which controls the figure/view

Notes

Program execution is blocked until the teach window is dismissed. If block=False the method is non-blocking but you need to poll the window manager (PyPlot) or keep calling env.step() yourself (Swift) to ensure the window/panel remains responsive.
Once teach() returns (block=True), robot.q holds the final pose the sliders were left at – true for every backend, even though Swift’s own live joint state during the session lives on the returned environment’s handle (env.swift_objects[0].q), not robot.q itself, to keep the robot model plain and shareable while teaching is in progress (see swift’s AssemblyHandle).
The slider limits are derived from the joint limit properties. If not set then:
- For revolute joints they are assumed to be [-pi, +pi]
- For prismatic joint they are assumed unknown and an error occurs.

todegrees(q)

Convert joint angles to degrees

Parameters:: q (Any) – The joint configuration of the robot
Return type:: ndarray
Returns:: a vector of joint coordinates in degrees and metres
Returns:: robot.todegrees(q) converts joint coordinates q to degrees
Returns:: taking into account whether elements of q correspond to revolute
Returns:: or prismatic joints, ie. prismatic joint values are not converted.
Returns:: If q is a matrix, with one column per joint, the conversion is
Returns:: performed columnwise.

Examples

>>> import roboticstoolbox as rtb
>>> from math import pi
>>> stanford = rtb.models.DH.Stanford()
>>> stanford.todegrees([pi/4, pi/8, 2, -pi/4, pi/6, pi/3])
array([ 45. ,  22.5,   2. , -45. ,  30. ,  60. ])

property tool: SE3

Get/set robot tool transform

robot.tool is the robot tool transform as an SE3 object
robot._tool is the robot tool transform as a numpy array
robot.tool = ... checks and sets the robot tool transform

Parameters:: tool – the new robot tool transform (as an SE(3))
Returns:: robot tool transform
Setter accepts:: SE3 | ndarray

toradians(q)

Convert joint angles to radians

robot.toradians(q) converts joint coordinates q to radians taking into account whether elements of q correspond to revolute or prismatic joints, ie. prismatic joint values are not converted.

If q is a matrix, with one column per joint, the conversion is performed columnwise.

Parameters:: q (Any) – The joint configuration of the robot
Return type:: ndarray
Returns:: a vector of joint coordinates in radians and metres

Examples

>>> import roboticstoolbox as rtb
>>> stanford = rtb.models.DH.Stanford()
>>> stanford.toradians([10, 20, 2, 30, 40, 50])
array([0.1745, 0.3491, 2.    , 0.5236, 0.6981, 0.8727])

tree()

Render the whole tree this node lives in as indented text – walks up to the root first (the same root-finding as update()), then renders down from there, with this node marked with a trailing <== so its position in the tree is visible.

Return type:: str

tree_children()

Render this node’s own subtree as indented text (one repr() per line), not walking through parents – the same “not through parents” scope as _propagate_scene_children().

Nodes have no .name in this package, so each line is that node’s own repr() (type, constructor params, color, pose) – usually enough to tell siblings apart, since it’s rare for two distinct nodes to share an identical pose as well as everything else.

Return type:: str

update()

Recompute the world transform of every node in the scene graph this node belongs to, starting from the root and working down.

Call this after changing any node’s T/pose (or its scene_parent) – nothing propagates automatically. It doesn’t matter which node in the graph you call it on: this always walks up to the root first, then pushes fresh world transforms down through the whole tree, not just this node’s own subtree.

Return type:: None

property urdf_filepath: str

property urdf_string: str

vellipse(q, opt='trans', unit='rad', centre='ee', scale=0.1, add=True)

Create a velocity ellipsoid object for plotting with PyPlot.

Parameters:

q (ndarray | Sequence[float]) – the joint configuration of the robot
opt (Literal['trans', 'rot']) – 'trans' or 'rot' — plot the translational or rotational velocity ellipsoid
unit (Literal['rad', 'deg']) – 'rad' or 'deg'
centre (Union[Literal['ee'], ndarray, Sequence[float]]) – centre of the ellipsoid — 'ee' for the end-effector or a 3-vector [x, y, z] in the world frame
scale (float) – scale factor for the ellipsoid
add (bool) – if True, add the ellipsoid to the active plot environment

Returns:

an EllipsePlot object

Return type:

EllipsePlot

robot.vellipse(q) creates a velocity ellipsoid for the robot at pose q. By default the ellipsoid is centered at the end-effector.

Notes

By default the ellipsoid related to translational motion is drawn. Use opt='rot' to draw the rotational velocity ellipsoid.
By default the ellipsoid is drawn at the end-effector. The option centre allows its origin to be set to the specified 3-vector, or the string "ee" ensures it is drawn at the end-effector position.

property base: SE2

Get/set robot base transform

Returns:: robot tool transform

robot.base is the robot base transform

robot.base = ... checks and sets the robot base transform

Notes

The private attribute _base will be None in the case of
no base transform, but this property will return SE3() which is an identity matrix.

Setter accepts:: Any

jacob0(q, start=None, end=None)[source]

Return type:: Any

jacobe(q, start=None, end=None)[source]

Return type:: Any

fkine(q, end=None, start=None)[source]

Return type:: Any

property reach: float

Reach of the robot

Returns:: Maximum reach of the robot
Return type:: float

A conservative estimate of the reach of the robot. It is computed as the sum of the translational ETs that define the link transform.

Computed on the first access. If kinematic parameters subsequently change this will not be reflected.

Notes

Probably an overestimate of reach
Used by numerical inverse kinematics to scale translational error.
For a prismatic joint, uses qlim if it is set

fkine_all(q)[source]

Compute the pose of every link frame

Parameters:: q (ndarray | Sequence[float]) – The joint configuration
Return type:: SE2
Returns:: Pose of all links

T = robot.fkine_all(q) is an SE3 instance with robot.nlinks + 1 values:

T[0] is the base transform
T[i] is the pose of link whose number is i

References

J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

Link

The ERobot is defined by a tree of ELink subclass objects.

class BaseLink(ets=[], name=None, parent=None, joint_name=None, m=None, r=None, I=None, Jm=None, B=None, Tc=None, G=None, qlim=None, qdlim=None, tlim=None, geometry=[], collision=[], **kwargs)[source]

Bases: SceneNode, ABC

An abstract link superclass for all link types.

Parameters:

ets (ETS | ETS2 | ET | ET2) – kinematic - The elementary transforms which make up the link
name (Any) – name of the link
parent (Union[Self, str, None]) – a reference to the parent link in the kinematic chain
joint_name (str | None) – the name of the joint variable
m (float | None) – dynamic - link mass
r (ndarray | Sequence[float] | None) – dynamic - position of COM with respect to link frame
I (ndarray | Sequence[float] | None) – dynamic - inertia of link with respect to COM
Jm (float | None) – dynamic - motor inertia
B (float | None) – dynamic - motor viscous friction
Tc (ndarray | Sequence[float] | None) – dynamic - motor Coulomb friction [Tc⁺, Tc⁻]
G (float | None) – dynamic - gear ratio, default 1 (motor directly coupled to the joint)
qlim (ndarray | Sequence[float] | None) – joint variable limits [min, max]
geometry (list[Shape]) – the visual geometry which represents the link. This is used to display the link in Swift
collision (list[Shape]) – the collision geometry which represents the link in collision checkers

It holds metadata related to:

a robot link, such as rigid-body inertial parameters defined in the link frame, and link name
a robot joint, that connects this link to its parent, such as joint limits, direction of motion, motor and transmission parameters.

Notes

For a more sophisticated actuator model use the actuator
attribute which is not initialized or used by this Toolbox.
There is no ability to name a joint as supported by URDF

property Ts: ndarray | None

Constant part of link ETS

Returns:: constant part of link transform
Return type:: ndarray(4,4) or None

The ETS for each Link comprises a constant part (possible the identity) followed by an optional joint variable transform. This property returns the constant part. If no constant part is given, this returns an identity matrix.

Examples

>>> from roboticstoolbox import Link, ET
>>> link = Link( ET.tz(0.333) * ET.Rx(90, 'deg') * ET.Rz() )
>>> link.Ts
array([[ 1.   ,  0.   ,  0.   ,  0.   ],
       [ 0.   ,  0.   , -1.   ,  0.   ],
       [ 0.   ,  1.   ,  0.   ,  0.333],
       [ 0.   ,  0.   ,  0.   ,  1.   ]])
>>> link = Link( ET.Rz() )
>>> link.Ts

property ets: ETS

Get/set link ets

link.ets is the link ets
link.ets = ... checks and sets the link ets

Parameters:: ets – the new link ets
Returns:: the current link ets
Setter accepts:: ETS | ETS2

__str__()[source]

Pretty prints the ETS Model of the link

Return type:: str
Returns:: pretty print of the robot link

Will output angles in degrees

copy()[source]

Copy of link object

Return type:: Self
Returns:: copy of link object

link.copy() is a new Link subclass instance with a copy of all the parameters.

property v: BaseET | None

Variable part of link ETS

Returns:: joint variable transform

The ETS for each Link comprises a constant part (possible the identity) followed by an optional joint variable transform. This property returns the latter.

Examples

>>> from roboticstoolbox import Link, ET, ETS
>>> link = Link( ET.tz(0.333) * ET.Rx(90, 'deg') * ET.Rz() )
>>> print(link.v)
Rz(q0)

property name: str

Get/set link name

link.name is the link name
link.name = ... checks and sets the link name

Returns:: link name

property robot: BaseRobot | None

Get forward reference to the robot which owns this link

link.robot is the robot reference
link.robot = ... checks and sets the robot reference

Returns:: The robot object
Setter accepts:: BaseRobot

property qlim: ndarray | None

Get/set joint limits

link.qlim is the joint limits
link.qlim = ... checks and sets the joint limits

Returns:: joint limits
Return type:: ndarray(2,) or None

Notes

The limits are not widely enforced within the toolbox.
If no joint limits are specified the value is None

See also

Setter accepts:: ndarray | Sequence[float]

property qdlim: float | None

Get/set joint velocity limits

link.qdlim is the joint velocity limits
link.qdlim = ... checks and sets the joint velocity limits

Returns:: joint velocity limits

Notes

The limits are not widely enforced within the toolbox.
If no joint velocity limits are specified the value is None

See also

Setter accepts:: float

property tlim: float | None

Get/set joint torque limits

link.tlim is the joint torque/force limits
link.tlim = ... checks and sets the joint torque limits

Returns:: joint torque limits

Notes

The limits are not widely enforced within the toolbox.
If no joint torque limits are specified the value is None

See also

Setter accepts:: float

property hasdynamics: bool

Link has dynamic parameters (Link superclass)

Returns:: True if the link has a non-zero mass or motor inertia

These are the parameters that the inertial dynamics need. The value is kept up to date by the setters of m and Jm, so it follows a mass that is set, and set back to zero.

Changed in version 1.5.0: This was true if any dynamic parameter, or even a kinematic parameter of a DH link, had been assigned, even to zero. A link with only friction parameters, or one made with m=0, now reports False.

Examples

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.DH.Puma560()
>>> robot[1].hasdynamics
True

property isflip: bool

Get/set joint flip

Returns:: joint flip

link.flip is the joint flip status
link.flip = ... checks and sets the joint flip status

Joint flip defines the direction of motion of the joint.

flip = False is conventional motion direction:

revolute motion is a positive rotation about the z-axis

prismatic motion is a positive translation along the z-axis

flip = True is the opposite motion direction:

revolute motion is a negative rotation about the z-axis

prismatic motion is a negative translation along the z-axis

property m: float

Get/set link mass

link.m is the link mass
link.m = ... checks and sets the link mass

Returns:: link mass

property r: ndarray

Get/set link centre of mass

Returns:: link centre of mass
Return type:: ndarray(3,)

The link centre of mass is a 3-vector defined with respect to the link frame.

link.r is the link centre of mass
link.r = ... checks and sets the link centre of mass

Warning

The array is read-only. To change the centre of mass assign a new value, link.r = [0, 0.1, 0], since editing the array in place, link.r[1] = 0.1, would bypass the setter and so the dynamics of the robot would not know that it had changed.

Changed in version 1.5.0: The array is read-only, and link.r[1] = 0.1 raises ValueError. It was writeable, and an edit was silently not seen by the dynamics of the robot.

Setter accepts:: ndarray | Sequence[float]

property I: ndarray

Get/set link inertia

Returns:: link inertia
Return type:: ndarray(3,3)

Link inertia is a symmetric 3x3 matrix describing the inertia with respect to a frame with its origin at the centre of mass, and with axes parallel to those of the link frame.

link.I is the link inertia
link.I = ... checks and sets the link inertia

The inertia matrix is

$\begin{bmatrix} I_{xx} & I_{xy} & I_{xz} \\ I_{xy} & I_{yy} & I_{yz} \\I_{xz} & I_{yz} & I_{zz} \end{bmatrix}$

and can be specified as either:

a 3 ⨉ 3 symmetric matrix
a 3-vector $(I_{xx}, I_{yy}, I_{zz})$
a 6-vector $(I_{xx}, I_{yy}, I_{zz}, I_{xy}, I_{yz}, I_{xz})$

Notes

Referred to the link side of the gearbox.

Warning

The array is read-only. To change the inertia assign a new value, link.I = ..., since editing the array in place, link.I[0, 0] = 1, would bypass the setter and so the dynamics of the robot would not know that it had changed.

Changed in version 1.5.0: The array is read-only, and link.I[0, 0] = 1 raises ValueError. It was writeable, and an edit was silently not seen by the dynamics of the robot.

Setter accepts:: ndarray | Sequence[float]

property Jm: float

Get/set motor inertia

link.Jm is the motor inertia
link.Jm = ... checks and sets the motor inertia

Returns:: motor inertia

Notes

Referred to the motor side of the gearbox.

property B: float

Get/set motor viscous friction

link.B is the motor viscous friction
link.B = ... checks and sets the motor viscous friction

Returns:: motor viscous friction

Notes

Referred to the motor side of the gearbox.
Viscous friction is the same for positive and negative motion.

property Tc: ndarray

Get/set motor Coulomb friction

Returns:: motor Coulomb friction
Return type:: ndarray(2,)

link.Tc is the motor Coulomb friction
link.Tc = ... checks and sets the motor Coulomb friction. If a scalar is given the value is set to [T, -T], if a 2-vector it is assumed to be in the order [Tc⁺, Tc⁻]

Coulomb friction is a non-linear friction effect defined by two parameters such that

\[\begin{split}\tau = \left\{ \begin{array}{ll} \tau_C^+ & \mbox{if $\dot{q} > 0$} \\ \tau_C^- & \mbox{if $\dot{q} < 0$} \end{array} \right.\end{split}\]

Notes

Referred to the motor side of the gearbox.
$\tau_C^+$ must be $> 0$, and $\tau_C^-$ must
be $< 0$.

Warning

The array is read-only. To change the Coulomb friction assign a new value, link.Tc = ..., since editing the array in place would bypass the setter and so the dynamics of the robot would not know that it had changed.

Changed in version 1.5.0: The array is read-only, and an edit in place raises ValueError. It was writeable, and an edit was silently not seen by the dynamics of the robot.

Setter accepts:: ndarray | Sequence[float]

property G: float

Get/set gear ratio

link.G is the transmission gear ratio
link.G = ... checks and sets the gear ratio

Returns:: gear ratio

Notes

The ratio of motor motion : link motion
The gear ratio can be negative, see also the flip attribute.
The default is 1, the motor is coupled directly to the joint.

Changed in version 1.5.0: The default was 0, which silently discarded the motor inertia and friction of a link that did not set G.

See also

flip

property geometry: SceneGroup

Get/set joint visual geometry

link.geometry is the list of the visual geometries which
represent the shape of the link :return: the visual geometries :rtype: list of Shape
link.geometry = ... checks and sets the geometry
link.geometry.append(...) add geometry

Setter accepts:: SceneGroup | list[Shape] | Shape

property collision: SceneGroup

Get/set joint collision geometry

The collision geometries are what is used to check for collisions.

link.collision is the list of the collision geometries which
represent the collidable shape of the link. :return: the collision geometries :rtype: list of Shape
link.collision = ... checks and sets the collision geometry
link.collision.append(...) add collision geometry

Setter accepts:: SceneGroup | list[Shape] | Shape

property isjoint: bool

Test if link has joint

Returns:: test if link has a joint

The ETS for each Link comprises a constant part (possible the identity) followed by an optional joint variable transform. This property returns the whether the Link contains the variable transform.

Examples

>>> from roboticstoolbox import models
>>> robot = models.URDF.Panda()
>>> robot[1].isjoint  # link with joint
True
>>> robot[8].isjoint  # static link
False

property jindex: int | None

Get/set joint index

link.jindex is the joint index
link.jindex = ... checks and sets the joint index

Returns:: joint index

For a serial-link manipulator the joints are numbered starting at zero and increasing sequentially toward the end-effector. For branched mechanisms this is not so straightforward. The link’s jindex property specifies the index of its joint variable within a vector of joint coordinates.

Notes

jindex values must be a sequence of integers starting
at zero.

Setter accepts:: int

property T: ndarray

Pose of the shape relative to its parent frame in the scene graph (or the world frame if it has no parent), as a 4x4 homogeneous transformation matrix. Set via the pose argument of the constructor.

This is a read/write property. The getter always returns a plain ndarray; the setter also accepts an SE3.

Warning

Because the getter returns an ndarray, in-place operators like shape.T *= delta do an elementwise multiply, not a pose composition, even when delta is an SE3 – use shape.T = shape.T * delta (or shape.T @= delta.A) instead.

Return type:: ndarray(4,4)
Setter accepts:: ndarray | SE3

attach(object)

Attach a child node

Parameters:: object (SceneNode) – the node to attach as a child of this node
Seealso:: attach_to() scene_children()
Return type:: None

attach_to(object)

Attach this node to a parent node

Parameters:: object (SceneNode) – the node to attach this node to, this node will become a child of the parent
Seealso:: attach() scene_parent()
Return type:: None

property isprismatic: bool

Checks if the joint is of prismatic type

Returns:: True if is prismatic

property scene_children: list[SceneNode]

Return the child nodes of this object in the scene graph.

Setting a new list of children updates each child’s scene_parent to this object, but does not remove this object from any previous parent’s scene_children.

This is a read/write property.

Return type:: list(SceneNode)
Seealso:: scene_parent() attach() attach_to()

property scene_parent: SceneNode | None

Return the parent node of this object in the scene graph.

Setting a new parent adds this object to the new parent’s scene_children.

This is a read/write property.

Return type:: SceneNode | None
Seealso:: scene_children() attach() attach_to()
Setter accepts:: SceneNode

tree()

Render the whole tree this node lives in as indented text – walks up to the root first (the same root-finding as update()), then renders down from there, with this node marked with a trailing <== so its position in the tree is visible.

Return type:: str

tree_children()

Render this node’s own subtree as indented text (one repr() per line), not walking through parents – the same “not through parents” scope as _propagate_scene_children().

Nodes have no .name in this package, so each line is that node’s own repr() (type, constructor params, color, pose) – usually enough to tell siblings apart, since it’s rare for two distinct nodes to share an identical pose as well as everything else.

Return type:: str

update()

Recompute the world transform of every node in the scene graph this node belongs to, starting from the root and working down.

Call this after changing any node’s T/pose (or its scene_parent) – nothing propagates automatically. It doesn’t matter which node in the graph you call it on: this always walks up to the root first, then pushes fresh world transforms down through the whole tree, not just this node’s own subtree.

Return type:: None

property isrevolute: bool

Checks if the joint is of revolute type

Returns:: True if is revolute

property parent: Optional[Self]

Parent link

Returns:: Link’s parent

This is a reference to the links parent in the kinematic chain

Examples

>>> from roboticstoolbox import models
>>> robot = models.URDF.Panda()
>>> robot[0].parent  # base link has no parent
>>> robot[1].parent  # second link's parent
Link(name = "panda_link0")

property parent_name: str | None

Parent link name

Returns:: Link’s parent name

property children: list[Link] | None

List of child links

Returns:: child links

The list will be empty for a end-effector link

property nchildren: int

Number of child links

Returns:: number of child links

Will be zero for an end-effector link

closest_point(shape, inf_dist=1.0, skip=False)[source]

Finds the closest point to a shape

Parameters:

shape (Shape) – The shape to compare distance to
inf_dist (float) – The minimum distance within which to consider the shape
skip (bool) – Skip setting all shape transforms

Return type:

tuple[float | None, ndarray | None, ndarray | None]

Returns:

(d, p1, p2) where d is the distance between the shapes, p1 is the point in the world frame on the link [x, y, z], and p2 is the point in the world frame on the shape [x, y, z].

closest_point(shape, inf_dist) returns the minimum euclidean distance between this link and shape, provided it is less than inf_dist. It will also return the points on self and shape in the world frame which connect the line of length distance between the shapes. If the distance is negative then the shapes are collided.

iscollided(shape, skip=False)[source]

Checks for collision with a shape

Parameters:

shape (Shape) – The shape to compare distance to
skip (bool) – Skip setting all shape transforms

Return type:

Returns:

True if shapes have collided

iscollided(shape) checks if this link and shape have collided

collided(shape, skip=False)[source]

Checks for collision with a shape

Parameters:

shape (Shape) – The shape to compare distance to
skip (bool) – Skip setting all shape transforms

Return type:

Returns:

True if shapes have collided

iscollided(shape) checks if this link and shape have collided

dyn(indent=0)[source]

Inertial properties of link as a string

Parameters:: indent (int) – indent each line by this many spaces
Return type:: Any

link.dyn() is a string representation the inertial properties of the link object in a multi-line format. The properties shown are mass, centre of mass, inertia, friction, gear ratio and motor properties.

Examples

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.DH.Puma560()
>>> print(robot.links[2])        # kinematic parameters
RevoluteDH:   θ=q,  d=0.15005,  a=0.0203,  ⍺=-1.5707963267948966
>>> print(robot.links[2].dyn())  # dynamic parameters
m     =       4.8 
r     =     -0.02   -0.014     0.07 
        |    0.066        0        0 | 
I     = |        0    0.086        0 | 
        |        0        0    0.013 | 
Jm    =    0.0002 
B     =    0.0014 
Tc    =      0.13(+)     -0.1(-) 
G     =       -54 
qlim  =      -3.9 to     0.79

See also

dyntable

islimit(q)[source]

Checks if joint exceeds limit

Parameters:: q (float) – joint coordinate
Return type:: Any
Returns:: True if joint is exceeded

link.islimit(q) is True if q exceeds the joint limits defined by link.

Notes

If no limits are set always return False.

See also

qlim()

nofriction(coulomb=True, viscous=False)[source]

Clone link without friction

Parameters:

coulomb (bool) – if True, will set the Coulomb friction to 0
viscous (bool) – if True, will set the viscous friction to 0

Return type:

link.nofriction() is a copy of the link instance with the same parameters except, the Coulomb and/or viscous friction parameters are set to zero.

Notes

For simulation it can be useful to remove Coulomb friction
which can cause problems for numerical integration.

friction(qd, coulomb=True)[source]

Compute joint friction

Parameters:

qd (float) – The joint velocity
coulomb (bool) – include Coulomb friction

Return type:

Returns:

the friction force/torque

friction(qd) is the joint friction force/torque for joint velocity qd. The friction model includes:

Viscous friction which is a linear function of velocity.
Coulomb friction which is proportional to sign(qd).

\[\begin{split}\tau = G^2 B \dot{q} + |G| \left\{ \begin{array}{ll} \tau_C^+ & \mbox{if $\dot{q} > 0$} \\ \tau_C^- & \mbox{if $\dot{q} < 0$} \end{array} \right.\end{split}\]

Notes

The friction value should be added to the motor output torque to
determine the nett torque. It has a negative value when qd > 0.
The returned friction value is referred to the output of the
gearbox.
The friction parameters in the Link object are referred to the
motor.
Motor viscous friction is scaled up by $G^2$.
Motor Coulomb friction is scaled up by math:G.
The appropriate Coulomb friction value to use in the
non-symmetric case depends on the sign of the joint velocity, not the motor velocity.
Coulomb friction is zero for zero joint velocity, stiction is
not modeled.
The absolute value of the gear ratio is used. Negative gear
ratios are tricky: the Puma560 robot has negative gear ratio for joints 1 and 3.

class Link(ets=[], jindex=None, **kwargs)[source]

Bases: BaseLink

ETS link class

Parameters:

ets (ETS | ET) – kinematic - The elementary transforms which make up the link
jindex (int | None) – the joint variable index
name – name of the link
parent – a reference to the parent link in the kinematic chain
joint_name – the name of the joint variable
m – dynamic - link mass
r – dynamic - position of COM with respect to link frame
I – dynamic - inertia of link with respect to COM
Jm – dynamic - motor inertia
B – dynamic - motor viscous friction
Tc – dynamic - motor Coulomb friction [Tc⁺, Tc⁻]
G – dynamic - gear ratio
qlim – joint variable limits [min, max]
geometry – the visual geometry which represents the link. This is used to display the link in Swift
collision – the collision geometry which represents the link in collision checkers

The Link object holds all information related to a robot link and can form a serial-connected chain or a rigid-body tree. It inherits from the Link class which provides common functionality such as joint and link such as kinematics parameters, The transform to the next link is given as an ETS with the joint variable, if present, as the last term. This is preprocessed and the object stores:

Ts the constant part as a NumPy array, or None
v a pointer to an ETS object representing the joint variable, or None

See also

Link2, DHLink

property B: float

Get/set motor viscous friction

link.B is the motor viscous friction
link.B = ... checks and sets the motor viscous friction

Returns:: motor viscous friction

Notes

Referred to the motor side of the gearbox.
Viscous friction is the same for positive and negative motion.

property G: float

Get/set gear ratio

link.G is the transmission gear ratio
link.G = ... checks and sets the gear ratio

Returns:: gear ratio

Notes

The ratio of motor motion : link motion
The gear ratio can be negative, see also the flip attribute.
The default is 1, the motor is coupled directly to the joint.

Changed in version 1.5.0: The default was 0, which silently discarded the motor inertia and friction of a link that did not set G.

See also

flip

property I: ndarray

Get/set link inertia

Returns:: link inertia
Return type:: ndarray(3,3)

Link inertia is a symmetric 3x3 matrix describing the inertia with respect to a frame with its origin at the centre of mass, and with axes parallel to those of the link frame.

link.I is the link inertia
link.I = ... checks and sets the link inertia

The inertia matrix is

$\begin{bmatrix} I_{xx} & I_{xy} & I_{xz} \\ I_{xy} & I_{yy} & I_{yz} \\I_{xz} & I_{yz} & I_{zz} \end{bmatrix}$

and can be specified as either:

a 3 ⨉ 3 symmetric matrix
a 3-vector $(I_{xx}, I_{yy}, I_{zz})$
a 6-vector $(I_{xx}, I_{yy}, I_{zz}, I_{xy}, I_{yz}, I_{xz})$

Notes

Referred to the link side of the gearbox.

Warning

The array is read-only. To change the inertia assign a new value, link.I = ..., since editing the array in place, link.I[0, 0] = 1, would bypass the setter and so the dynamics of the robot would not know that it had changed.

Changed in version 1.5.0: The array is read-only, and link.I[0, 0] = 1 raises ValueError. It was writeable, and an edit was silently not seen by the dynamics of the robot.

Setter accepts:: ndarray | Sequence[float]

property Jm: float

Get/set motor inertia

link.Jm is the motor inertia
link.Jm = ... checks and sets the motor inertia

Returns:: motor inertia

Notes

Referred to the motor side of the gearbox.

property T: ndarray

Pose of the shape relative to its parent frame in the scene graph (or the world frame if it has no parent), as a 4x4 homogeneous transformation matrix. Set via the pose argument of the constructor.

This is a read/write property. The getter always returns a plain ndarray; the setter also accepts an SE3.

Warning

Because the getter returns an ndarray, in-place operators like shape.T *= delta do an elementwise multiply, not a pose composition, even when delta is an SE3 – use shape.T = shape.T * delta (or shape.T @= delta.A) instead.

Return type:: ndarray(4,4)
Setter accepts:: ndarray | SE3

property Tc: ndarray

Get/set motor Coulomb friction

Returns:: motor Coulomb friction
Return type:: ndarray(2,)

link.Tc is the motor Coulomb friction
link.Tc = ... checks and sets the motor Coulomb friction. If a scalar is given the value is set to [T, -T], if a 2-vector it is assumed to be in the order [Tc⁺, Tc⁻]

Coulomb friction is a non-linear friction effect defined by two parameters such that

\[\begin{split}\tau = \left\{ \begin{array}{ll} \tau_C^+ & \mbox{if $\dot{q} > 0$} \\ \tau_C^- & \mbox{if $\dot{q} < 0$} \end{array} \right.\end{split}\]

Notes

Referred to the motor side of the gearbox.
$\tau_C^+$ must be $> 0$, and $\tau_C^-$ must
be $< 0$.

Warning

The array is read-only. To change the Coulomb friction assign a new value, link.Tc = ..., since editing the array in place would bypass the setter and so the dynamics of the robot would not know that it had changed.

Changed in version 1.5.0: The array is read-only, and an edit in place raises ValueError. It was writeable, and an edit was silently not seen by the dynamics of the robot.

Setter accepts:: ndarray | Sequence[float]

property Ts: ndarray | None

Constant part of link ETS

Returns:: constant part of link transform
Return type:: ndarray(4,4) or None

The ETS for each Link comprises a constant part (possible the identity) followed by an optional joint variable transform. This property returns the constant part. If no constant part is given, this returns an identity matrix.

Examples

>>> from roboticstoolbox import Link, ET
>>> link = Link( ET.tz(0.333) * ET.Rx(90, 'deg') * ET.Rz() )
>>> link.Ts
array([[ 1.   ,  0.   ,  0.   ,  0.   ],
       [ 0.   ,  0.   , -1.   ,  0.   ],
       [ 0.   ,  1.   ,  0.   ,  0.333],
       [ 0.   ,  0.   ,  0.   ,  1.   ]])
>>> link = Link( ET.Rz() )
>>> link.Ts

__str__()

Pretty prints the ETS Model of the link

Return type:: str
Returns:: pretty print of the robot link

Will output angles in degrees

attach(object)

Attach a child node

Parameters:: object (SceneNode) – the node to attach as a child of this node
Seealso:: attach_to() scene_children()
Return type:: None

attach_to(object)

Attach this node to a parent node

Parameters:: object (SceneNode) – the node to attach this node to, this node will become a child of the parent
Seealso:: attach() scene_parent()
Return type:: None

property children: list[Link] | None

List of child links

Returns:: child links

The list will be empty for a end-effector link

closest_point(shape, inf_dist=1.0, skip=False)

Finds the closest point to a shape

Parameters:

shape (Shape) – The shape to compare distance to
inf_dist (float) – The minimum distance within which to consider the shape
skip (bool) – Skip setting all shape transforms

Return type:

tuple[float | None, ndarray | None, ndarray | None]

Returns:

(d, p1, p2) where d is the distance between the shapes, p1 is the point in the world frame on the link [x, y, z], and p2 is the point in the world frame on the shape [x, y, z].

closest_point(shape, inf_dist) returns the minimum euclidean distance between this link and shape, provided it is less than inf_dist. It will also return the points on self and shape in the world frame which connect the line of length distance between the shapes. If the distance is negative then the shapes are collided.

collided(shape, skip=False)

Checks for collision with a shape

Parameters:

shape (Shape) – The shape to compare distance to
skip (bool) – Skip setting all shape transforms

Return type:

Returns:

True if shapes have collided

iscollided(shape) checks if this link and shape have collided

property collision: SceneGroup

Get/set joint collision geometry

The collision geometries are what is used to check for collisions.

link.collision is the list of the collision geometries which
represent the collidable shape of the link. :return: the collision geometries :rtype: list of Shape
link.collision = ... checks and sets the collision geometry
link.collision.append(...) add collision geometry

Setter accepts:: SceneGroup | list[Shape] | Shape

copy()

Copy of link object

Return type:: Self
Returns:: copy of link object

link.copy() is a new Link subclass instance with a copy of all the parameters.

dyn(indent=0)

Inertial properties of link as a string

Parameters:: indent (int) – indent each line by this many spaces
Return type:: Any

link.dyn() is a string representation the inertial properties of the link object in a multi-line format. The properties shown are mass, centre of mass, inertia, friction, gear ratio and motor properties.

Examples

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.DH.Puma560()
>>> print(robot.links[2])        # kinematic parameters
RevoluteDH:   θ=q,  d=0.15005,  a=0.0203,  ⍺=-1.5707963267948966
>>> print(robot.links[2].dyn())  # dynamic parameters
m     =       4.8 
r     =     -0.02   -0.014     0.07 
        |    0.066        0        0 | 
I     = |        0    0.086        0 | 
        |        0        0    0.013 | 
Jm    =    0.0002 
B     =    0.0014 
Tc    =      0.13(+)     -0.1(-) 
G     =       -54 
qlim  =      -3.9 to     0.79

See also

dyntable

property ets: ETS

Get/set link ets

link.ets is the link ets
link.ets = ... checks and sets the link ets

Parameters:: ets – the new link ets
Returns:: the current link ets
Setter accepts:: ETS | ETS2

friction(qd, coulomb=True)

Compute joint friction

Parameters:

qd (float) – The joint velocity
coulomb (bool) – include Coulomb friction

Return type:

Returns:

the friction force/torque

friction(qd) is the joint friction force/torque for joint velocity qd. The friction model includes:

Viscous friction which is a linear function of velocity.
Coulomb friction which is proportional to sign(qd).

\[\begin{split}\tau = G^2 B \dot{q} + |G| \left\{ \begin{array}{ll} \tau_C^+ & \mbox{if $\dot{q} > 0$} \\ \tau_C^- & \mbox{if $\dot{q} < 0$} \end{array} \right.\end{split}\]

Notes

The friction value should be added to the motor output torque to
determine the nett torque. It has a negative value when qd > 0.
The returned friction value is referred to the output of the
gearbox.
The friction parameters in the Link object are referred to the
motor.
Motor viscous friction is scaled up by $G^2$.
Motor Coulomb friction is scaled up by math:G.
The appropriate Coulomb friction value to use in the
non-symmetric case depends on the sign of the joint velocity, not the motor velocity.
Coulomb friction is zero for zero joint velocity, stiction is
not modeled.
The absolute value of the gear ratio is used. Negative gear
ratios are tricky: the Puma560 robot has negative gear ratio for joints 1 and 3.

property geometry: SceneGroup

Get/set joint visual geometry

link.geometry is the list of the visual geometries which
represent the shape of the link :return: the visual geometries :rtype: list of Shape
link.geometry = ... checks and sets the geometry
link.geometry.append(...) add geometry

Setter accepts:: SceneGroup | list[Shape] | Shape

property hasdynamics: bool

Link has dynamic parameters (Link superclass)

Returns:: True if the link has a non-zero mass or motor inertia

These are the parameters that the inertial dynamics need. The value is kept up to date by the setters of m and Jm, so it follows a mass that is set, and set back to zero.

Changed in version 1.5.0: This was true if any dynamic parameter, or even a kinematic parameter of a DH link, had been assigned, even to zero. A link with only friction parameters, or one made with m=0, now reports False.

Examples

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.DH.Puma560()
>>> robot[1].hasdynamics
True

iscollided(shape, skip=False)

Checks for collision with a shape

Parameters:

shape (Shape) – The shape to compare distance to
skip (bool) – Skip setting all shape transforms

Return type:

Returns:

True if shapes have collided

iscollided(shape) checks if this link and shape have collided

property isflip: bool

Get/set joint flip

Returns:: joint flip

link.flip is the joint flip status
link.flip = ... checks and sets the joint flip status

Joint flip defines the direction of motion of the joint.

flip = False is conventional motion direction:

revolute motion is a positive rotation about the z-axis

prismatic motion is a positive translation along the z-axis

flip = True is the opposite motion direction:

revolute motion is a negative rotation about the z-axis

prismatic motion is a negative translation along the z-axis

property isjoint: bool

Test if link has joint

Returns:: test if link has a joint

The ETS for each Link comprises a constant part (possible the identity) followed by an optional joint variable transform. This property returns the whether the Link contains the variable transform.

Examples

>>> from roboticstoolbox import models
>>> robot = models.URDF.Panda()
>>> robot[1].isjoint  # link with joint
True
>>> robot[8].isjoint  # static link
False

islimit(q)

Checks if joint exceeds limit

Parameters:: q (float) – joint coordinate
Return type:: Any
Returns:: True if joint is exceeded

link.islimit(q) is True if q exceeds the joint limits defined by link.

Notes

If no limits are set always return False.

See also

qlim()

property isprismatic: bool

Checks if the joint is of prismatic type

Returns:: True if is prismatic

property isrevolute: bool

Checks if the joint is of revolute type

Returns:: True if is revolute

property jindex: int | None

Get/set joint index

link.jindex is the joint index
link.jindex = ... checks and sets the joint index

Returns:: joint index

For a serial-link manipulator the joints are numbered starting at zero and increasing sequentially toward the end-effector. For branched mechanisms this is not so straightforward. The link’s jindex property specifies the index of its joint variable within a vector of joint coordinates.

Notes

jindex values must be a sequence of integers starting
at zero.

Setter accepts:: int

property m: float

Get/set link mass

link.m is the link mass
link.m = ... checks and sets the link mass

Returns:: link mass

property name: str

Get/set link name

link.name is the link name
link.name = ... checks and sets the link name

Returns:: link name

property nchildren: int

Number of child links

Returns:: number of child links

Will be zero for an end-effector link

nofriction(coulomb=True, viscous=False)

Clone link without friction

Parameters:

coulomb (bool) – if True, will set the Coulomb friction to 0
viscous (bool) – if True, will set the viscous friction to 0

Return type:

link.nofriction() is a copy of the link instance with the same parameters except, the Coulomb and/or viscous friction parameters are set to zero.

Notes

For simulation it can be useful to remove Coulomb friction
which can cause problems for numerical integration.

property parent: Optional[Self]

Parent link

Returns:: Link’s parent

This is a reference to the links parent in the kinematic chain

Examples

>>> from roboticstoolbox import models
>>> robot = models.URDF.Panda()
>>> robot[0].parent  # base link has no parent
>>> robot[1].parent  # second link's parent
Link(name = "panda_link0")

property parent_name: str | None

Parent link name

Returns:: Link’s parent name

property qdlim: float | None

Get/set joint velocity limits

link.qdlim is the joint velocity limits
link.qdlim = ... checks and sets the joint velocity limits

Returns:: joint velocity limits

Notes

The limits are not widely enforced within the toolbox.
If no joint velocity limits are specified the value is None

See also

Setter accepts:: float

property qlim: ndarray | None

Get/set joint limits

link.qlim is the joint limits
link.qlim = ... checks and sets the joint limits

Returns:: joint limits
Return type:: ndarray(2,) or None

Notes

The limits are not widely enforced within the toolbox.
If no joint limits are specified the value is None

See also

Setter accepts:: ndarray | Sequence[float]

property r: ndarray

Get/set link centre of mass

Returns:: link centre of mass
Return type:: ndarray(3,)

The link centre of mass is a 3-vector defined with respect to the link frame.

link.r is the link centre of mass
link.r = ... checks and sets the link centre of mass

Warning

The array is read-only. To change the centre of mass assign a new value, link.r = [0, 0.1, 0], since editing the array in place, link.r[1] = 0.1, would bypass the setter and so the dynamics of the robot would not know that it had changed.

Changed in version 1.5.0: The array is read-only, and link.r[1] = 0.1 raises ValueError. It was writeable, and an edit was silently not seen by the dynamics of the robot.

Setter accepts:: ndarray | Sequence[float]

property robot: BaseRobot | None

Get forward reference to the robot which owns this link

link.robot is the robot reference
link.robot = ... checks and sets the robot reference

Returns:: The robot object
Setter accepts:: BaseRobot

property scene_children: list[SceneNode]

Return the child nodes of this object in the scene graph.

Setting a new list of children updates each child’s scene_parent to this object, but does not remove this object from any previous parent’s scene_children.

This is a read/write property.

Return type:: list(SceneNode)
Seealso:: scene_parent() attach() attach_to()

property scene_parent: SceneNode | None

Return the parent node of this object in the scene graph.

Setting a new parent adds this object to the new parent’s scene_children.

This is a read/write property.

Return type:: SceneNode | None
Seealso:: scene_children() attach() attach_to()
Setter accepts:: SceneNode

property tlim: float | None

Get/set joint torque limits

link.tlim is the joint torque/force limits
link.tlim = ... checks and sets the joint torque limits

Returns:: joint torque limits

Notes

The limits are not widely enforced within the toolbox.
If no joint torque limits are specified the value is None

See also

Setter accepts:: float

tree()

Render the whole tree this node lives in as indented text – walks up to the root first (the same root-finding as update()), then renders down from there, with this node marked with a trailing <== so its position in the tree is visible.

Return type:: str

tree_children()

Render this node’s own subtree as indented text (one repr() per line), not walking through parents – the same “not through parents” scope as _propagate_scene_children().

Nodes have no .name in this package, so each line is that node’s own repr() (type, constructor params, color, pose) – usually enough to tell siblings apart, since it’s rare for two distinct nodes to share an identical pose as well as everything else.

Return type:: str

update()

Recompute the world transform of every node in the scene graph this node belongs to, starting from the root and working down.

Call this after changing any node’s T/pose (or its scene_parent) – nothing propagates automatically. It doesn’t matter which node in the graph you call it on: this always walks up to the root first, then pushes fresh world transforms down through the whole tree, not just this node’s own subtree.

Return type:: None

property v: BaseET | None

Variable part of link ETS

Returns:: joint variable transform

The ETS for each Link comprises a constant part (possible the identity) followed by an optional joint variable transform. This property returns the latter.

Examples

>>> from roboticstoolbox import Link, ET, ETS
>>> link = Link( ET.tz(0.333) * ET.Rx(90, 'deg') * ET.Rz() )
>>> print(link.v)
Rz(q0)

A(q=0.0)[source]

Link transform matrix

Parameters:: q (float) – Joint coordinate (radians or metres). Not required for links with no variable
Return type:: SE3
Returns:: link frame transformation matrix

link.A(q) is an SE(3) matrix that describes the rigid-body transformation from the previous to the current link frame to the next, which depends on the joint coordinate q.

class Link2(ets=[], jindex=None, **kwargs)[source]

Bases: BaseLink

property B: float

Get/set motor viscous friction

link.B is the motor viscous friction
link.B = ... checks and sets the motor viscous friction

Returns:: motor viscous friction

Notes

Referred to the motor side of the gearbox.
Viscous friction is the same for positive and negative motion.

property G: float

Get/set gear ratio

link.G is the transmission gear ratio
link.G = ... checks and sets the gear ratio

Returns:: gear ratio

Notes

The ratio of motor motion : link motion
The gear ratio can be negative, see also the flip attribute.
The default is 1, the motor is coupled directly to the joint.

Changed in version 1.5.0: The default was 0, which silently discarded the motor inertia and friction of a link that did not set G.

See also

flip

property I: ndarray

Get/set link inertia

Returns:: link inertia
Return type:: ndarray(3,3)

Link inertia is a symmetric 3x3 matrix describing the inertia with respect to a frame with its origin at the centre of mass, and with axes parallel to those of the link frame.

link.I is the link inertia
link.I = ... checks and sets the link inertia

The inertia matrix is

$\begin{bmatrix} I_{xx} & I_{xy} & I_{xz} \\ I_{xy} & I_{yy} & I_{yz} \\I_{xz} & I_{yz} & I_{zz} \end{bmatrix}$

and can be specified as either:

a 3 ⨉ 3 symmetric matrix
a 3-vector $(I_{xx}, I_{yy}, I_{zz})$
a 6-vector $(I_{xx}, I_{yy}, I_{zz}, I_{xy}, I_{yz}, I_{xz})$

Notes

Referred to the link side of the gearbox.

Warning

The array is read-only. To change the inertia assign a new value, link.I = ..., since editing the array in place, link.I[0, 0] = 1, would bypass the setter and so the dynamics of the robot would not know that it had changed.

Changed in version 1.5.0: The array is read-only, and link.I[0, 0] = 1 raises ValueError. It was writeable, and an edit was silently not seen by the dynamics of the robot.

Setter accepts:: ndarray | Sequence[float]

property Jm: float

Get/set motor inertia

link.Jm is the motor inertia
link.Jm = ... checks and sets the motor inertia

Returns:: motor inertia

Notes

Referred to the motor side of the gearbox.

property T: ndarray

Pose of the shape relative to its parent frame in the scene graph (or the world frame if it has no parent), as a 4x4 homogeneous transformation matrix. Set via the pose argument of the constructor.

This is a read/write property. The getter always returns a plain ndarray; the setter also accepts an SE3.

Warning

Because the getter returns an ndarray, in-place operators like shape.T *= delta do an elementwise multiply, not a pose composition, even when delta is an SE3 – use shape.T = shape.T * delta (or shape.T @= delta.A) instead.

Return type:: ndarray(4,4)
Setter accepts:: ndarray | SE3

property Tc: ndarray

Get/set motor Coulomb friction

Returns:: motor Coulomb friction
Return type:: ndarray(2,)

link.Tc is the motor Coulomb friction
link.Tc = ... checks and sets the motor Coulomb friction. If a scalar is given the value is set to [T, -T], if a 2-vector it is assumed to be in the order [Tc⁺, Tc⁻]

Coulomb friction is a non-linear friction effect defined by two parameters such that

\[\begin{split}\tau = \left\{ \begin{array}{ll} \tau_C^+ & \mbox{if $\dot{q} > 0$} \\ \tau_C^- & \mbox{if $\dot{q} < 0$} \end{array} \right.\end{split}\]

Notes

Referred to the motor side of the gearbox.
$\tau_C^+$ must be $> 0$, and $\tau_C^-$ must
be $< 0$.

Warning

The array is read-only. To change the Coulomb friction assign a new value, link.Tc = ..., since editing the array in place would bypass the setter and so the dynamics of the robot would not know that it had changed.

Changed in version 1.5.0: The array is read-only, and an edit in place raises ValueError. It was writeable, and an edit was silently not seen by the dynamics of the robot.

Setter accepts:: ndarray | Sequence[float]

property Ts: ndarray | None

Constant part of link ETS

Returns:: constant part of link transform
Return type:: ndarray(4,4) or None

The ETS for each Link comprises a constant part (possible the identity) followed by an optional joint variable transform. This property returns the constant part. If no constant part is given, this returns an identity matrix.

Examples

>>> from roboticstoolbox import Link, ET
>>> link = Link( ET.tz(0.333) * ET.Rx(90, 'deg') * ET.Rz() )
>>> link.Ts
array([[ 1.   ,  0.   ,  0.   ,  0.   ],
       [ 0.   ,  0.   , -1.   ,  0.   ],
       [ 0.   ,  1.   ,  0.   ,  0.333],
       [ 0.   ,  0.   ,  0.   ,  1.   ]])
>>> link = Link( ET.Rz() )
>>> link.Ts

__str__()

Pretty prints the ETS Model of the link

Return type:: str
Returns:: pretty print of the robot link

Will output angles in degrees

attach(object)

Attach a child node

Parameters:: object (SceneNode) – the node to attach as a child of this node
Seealso:: attach_to() scene_children()
Return type:: None

attach_to(object)

Attach this node to a parent node

Parameters:: object (SceneNode) – the node to attach this node to, this node will become a child of the parent
Seealso:: attach() scene_parent()
Return type:: None

property children: list[Link] | None

List of child links

Returns:: child links

The list will be empty for a end-effector link

closest_point(shape, inf_dist=1.0, skip=False)

Finds the closest point to a shape

Parameters:

shape (Shape) – The shape to compare distance to
inf_dist (float) – The minimum distance within which to consider the shape
skip (bool) – Skip setting all shape transforms

Return type:

tuple[float | None, ndarray | None, ndarray | None]

Returns:

(d, p1, p2) where d is the distance between the shapes, p1 is the point in the world frame on the link [x, y, z], and p2 is the point in the world frame on the shape [x, y, z].

closest_point(shape, inf_dist) returns the minimum euclidean distance between this link and shape, provided it is less than inf_dist. It will also return the points on self and shape in the world frame which connect the line of length distance between the shapes. If the distance is negative then the shapes are collided.

collided(shape, skip=False)

Checks for collision with a shape

Parameters:

shape (Shape) – The shape to compare distance to
skip (bool) – Skip setting all shape transforms

Return type:

Returns:

True if shapes have collided

iscollided(shape) checks if this link and shape have collided

property collision: SceneGroup

Get/set joint collision geometry

The collision geometries are what is used to check for collisions.

link.collision is the list of the collision geometries which
represent the collidable shape of the link. :return: the collision geometries :rtype: list of Shape
link.collision = ... checks and sets the collision geometry
link.collision.append(...) add collision geometry

Setter accepts:: SceneGroup | list[Shape] | Shape

copy()

Copy of link object

Return type:: Self
Returns:: copy of link object

link.copy() is a new Link subclass instance with a copy of all the parameters.

dyn(indent=0)

Inertial properties of link as a string

Parameters:: indent (int) – indent each line by this many spaces
Return type:: Any

link.dyn() is a string representation the inertial properties of the link object in a multi-line format. The properties shown are mass, centre of mass, inertia, friction, gear ratio and motor properties.

Examples

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.DH.Puma560()
>>> print(robot.links[2])        # kinematic parameters
RevoluteDH:   θ=q,  d=0.15005,  a=0.0203,  ⍺=-1.5707963267948966
>>> print(robot.links[2].dyn())  # dynamic parameters
m     =       4.8 
r     =     -0.02   -0.014     0.07 
        |    0.066        0        0 | 
I     = |        0    0.086        0 | 
        |        0        0    0.013 | 
Jm    =    0.0002 
B     =    0.0014 
Tc    =      0.13(+)     -0.1(-) 
G     =       -54 
qlim  =      -3.9 to     0.79

See also

dyntable

property ets: ETS

Get/set link ets

link.ets is the link ets
link.ets = ... checks and sets the link ets

Parameters:: ets – the new link ets
Returns:: the current link ets
Setter accepts:: ETS | ETS2

friction(qd, coulomb=True)

Compute joint friction

Parameters:

qd (float) – The joint velocity
coulomb (bool) – include Coulomb friction

Return type:

Returns:

the friction force/torque

friction(qd) is the joint friction force/torque for joint velocity qd. The friction model includes:

Viscous friction which is a linear function of velocity.
Coulomb friction which is proportional to sign(qd).

\[\begin{split}\tau = G^2 B \dot{q} + |G| \left\{ \begin{array}{ll} \tau_C^+ & \mbox{if $\dot{q} > 0$} \\ \tau_C^- & \mbox{if $\dot{q} < 0$} \end{array} \right.\end{split}\]

Notes

The friction value should be added to the motor output torque to
determine the nett torque. It has a negative value when qd > 0.
The returned friction value is referred to the output of the
gearbox.
The friction parameters in the Link object are referred to the
motor.
Motor viscous friction is scaled up by $G^2$.
Motor Coulomb friction is scaled up by math:G.
The appropriate Coulomb friction value to use in the
non-symmetric case depends on the sign of the joint velocity, not the motor velocity.
Coulomb friction is zero for zero joint velocity, stiction is
not modeled.
The absolute value of the gear ratio is used. Negative gear
ratios are tricky: the Puma560 robot has negative gear ratio for joints 1 and 3.

property geometry: SceneGroup

Get/set joint visual geometry

link.geometry is the list of the visual geometries which
represent the shape of the link :return: the visual geometries :rtype: list of Shape
link.geometry = ... checks and sets the geometry
link.geometry.append(...) add geometry

Setter accepts:: SceneGroup | list[Shape] | Shape

property hasdynamics: bool

Link has dynamic parameters (Link superclass)

Returns:: True if the link has a non-zero mass or motor inertia

These are the parameters that the inertial dynamics need. The value is kept up to date by the setters of m and Jm, so it follows a mass that is set, and set back to zero.

Changed in version 1.5.0: This was true if any dynamic parameter, or even a kinematic parameter of a DH link, had been assigned, even to zero. A link with only friction parameters, or one made with m=0, now reports False.

Examples

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.DH.Puma560()
>>> robot[1].hasdynamics
True

iscollided(shape, skip=False)

Checks for collision with a shape

Parameters:

shape (Shape) – The shape to compare distance to
skip (bool) – Skip setting all shape transforms

Return type:

Returns:

True if shapes have collided

iscollided(shape) checks if this link and shape have collided

property isflip: bool

Get/set joint flip

Returns:: joint flip

link.flip is the joint flip status
link.flip = ... checks and sets the joint flip status

Joint flip defines the direction of motion of the joint.

flip = False is conventional motion direction:

revolute motion is a positive rotation about the z-axis

prismatic motion is a positive translation along the z-axis

flip = True is the opposite motion direction:

revolute motion is a negative rotation about the z-axis

prismatic motion is a negative translation along the z-axis

property isjoint: bool

Test if link has joint

Returns:: test if link has a joint

The ETS for each Link comprises a constant part (possible the identity) followed by an optional joint variable transform. This property returns the whether the Link contains the variable transform.

Examples

>>> from roboticstoolbox import models
>>> robot = models.URDF.Panda()
>>> robot[1].isjoint  # link with joint
True
>>> robot[8].isjoint  # static link
False

islimit(q)

Checks if joint exceeds limit

Parameters:: q (float) – joint coordinate
Return type:: Any
Returns:: True if joint is exceeded

link.islimit(q) is True if q exceeds the joint limits defined by link.

Notes

If no limits are set always return False.

See also

qlim()

property isprismatic: bool

Checks if the joint is of prismatic type

Returns:: True if is prismatic

property isrevolute: bool

Checks if the joint is of revolute type

Returns:: True if is revolute

property jindex: int | None

Get/set joint index

link.jindex is the joint index
link.jindex = ... checks and sets the joint index

Returns:: joint index

For a serial-link manipulator the joints are numbered starting at zero and increasing sequentially toward the end-effector. For branched mechanisms this is not so straightforward. The link’s jindex property specifies the index of its joint variable within a vector of joint coordinates.

Notes

jindex values must be a sequence of integers starting
at zero.

Setter accepts:: int

property m: float

Get/set link mass

link.m is the link mass
link.m = ... checks and sets the link mass

Returns:: link mass

property name: str

Get/set link name

link.name is the link name
link.name = ... checks and sets the link name

Returns:: link name

property nchildren: int

Number of child links

Returns:: number of child links

Will be zero for an end-effector link

nofriction(coulomb=True, viscous=False)

Clone link without friction

Parameters:

coulomb (bool) – if True, will set the Coulomb friction to 0
viscous (bool) – if True, will set the viscous friction to 0

Return type:

link.nofriction() is a copy of the link instance with the same parameters except, the Coulomb and/or viscous friction parameters are set to zero.

Notes

For simulation it can be useful to remove Coulomb friction
which can cause problems for numerical integration.

property parent: Optional[Self]

Parent link

Returns:: Link’s parent

This is a reference to the links parent in the kinematic chain

Examples

>>> from roboticstoolbox import models
>>> robot = models.URDF.Panda()
>>> robot[0].parent  # base link has no parent
>>> robot[1].parent  # second link's parent
Link(name = "panda_link0")

property parent_name: str | None

Parent link name

Returns:: Link’s parent name

property qdlim: float | None

Get/set joint velocity limits

link.qdlim is the joint velocity limits
link.qdlim = ... checks and sets the joint velocity limits

Returns:: joint velocity limits

Notes

The limits are not widely enforced within the toolbox.
If no joint velocity limits are specified the value is None

See also

Setter accepts:: float

property qlim: ndarray | None

Get/set joint limits

link.qlim is the joint limits
link.qlim = ... checks and sets the joint limits

Returns:: joint limits
Return type:: ndarray(2,) or None

Notes

The limits are not widely enforced within the toolbox.
If no joint limits are specified the value is None

See also

Setter accepts:: ndarray | Sequence[float]

property r: ndarray

Get/set link centre of mass

Returns:: link centre of mass
Return type:: ndarray(3,)

The link centre of mass is a 3-vector defined with respect to the link frame.

link.r is the link centre of mass
link.r = ... checks and sets the link centre of mass

Warning

The array is read-only. To change the centre of mass assign a new value, link.r = [0, 0.1, 0], since editing the array in place, link.r[1] = 0.1, would bypass the setter and so the dynamics of the robot would not know that it had changed.

Changed in version 1.5.0: The array is read-only, and link.r[1] = 0.1 raises ValueError. It was writeable, and an edit was silently not seen by the dynamics of the robot.

Setter accepts:: ndarray | Sequence[float]

property robot: BaseRobot | None

Get forward reference to the robot which owns this link

link.robot is the robot reference
link.robot = ... checks and sets the robot reference

Returns:: The robot object
Setter accepts:: BaseRobot

property scene_children: list[SceneNode]

Return the child nodes of this object in the scene graph.

Setting a new list of children updates each child’s scene_parent to this object, but does not remove this object from any previous parent’s scene_children.

This is a read/write property.

Return type:: list(SceneNode)
Seealso:: scene_parent() attach() attach_to()

property scene_parent: SceneNode | None

Return the parent node of this object in the scene graph.

Setting a new parent adds this object to the new parent’s scene_children.

This is a read/write property.

Return type:: SceneNode | None
Seealso:: scene_children() attach() attach_to()
Setter accepts:: SceneNode

property tlim: float | None

Get/set joint torque limits

link.tlim is the joint torque/force limits
link.tlim = ... checks and sets the joint torque limits

Returns:: joint torque limits

Notes

The limits are not widely enforced within the toolbox.
If no joint torque limits are specified the value is None

See also