Elementary transform sequence (ETS) models

Code author: Jesse Haviland

Elementary transforms are canonic rotations or translations about, or along, the x-, y- or z-axes. The amount of rotation or translation can be a constant or a variable. A variable amount corresponds to a joint.

Consider the example:

>>> from roboticstoolbox import ET
>>> ET.Rx(45, 'deg')
ET.Rx(eta=0.7853981633974483)
>>> ET.tz(0.75)
ET.tz(eta=0.75)
>>> e = ET.Rx(0.3) * ET.tz(0.75)
>>> print(e)
Rx(17.19°) ⊕ tz(0.75)
>>> e.fkine([])
SE3(array([[ 1.    ,  0.    ,  0.    ,  0.    ],
           [ 0.    ,  0.9553, -0.2955, -0.2216],
           [ 0.    ,  0.2955,  0.9553,  0.7165],
           [ 0.    ,  0.    ,  0.    ,  1.    ]]))

In lines 2-5 we defined two elementary transforms. Line 2 defines a rotation of 45° about the x-axis, and line 4 defines a translation of 0.75m along the z-axis. Compounding them in line 6, the result is the two elementary transforms in a sequence - an elementary transform sequence (ETS). An ETS can be arbitrarily long.

Line 8 evaluates the forward kinematics of the sequence, substituting in values, and the result is an SE(3) matrix encapsulated in an 4x4 numpy array.

The real power comes from having variable arguments to the elementary transforms as shown in this example where we define a simple two link planar manipulator.

>>> from roboticstoolbox import ET
>>> e = ET.Rz() * ET.tx(1) * ET.Rz() * ET.tx(1)
>>> print(e)
Rz(q0) ⊕ tx(1) ⊕ Rz(q1) ⊕ tx(1)
>>> len(e)
4
>>> e[1:3]
[ET.tx(eta=1.0), ET.Rz(jindex=1)]
>>> e.fkine([0, 0])
SE3(array([[1., 0., 0., 2.],
           [0., 1., 0., 0.],
           [0., 0., 1., 0.],
           [0., 0., 0., 1.]]))
>>> e.fkine([1.57, -1.57])
SE3(array([[1.    , 0.    , 0.    , 1.0008],
           [0.    , 1.    , 0.    , 1.    ],
           [0.    , 0.    , 1.    , 0.    ],
           [0.    , 0.    , 0.    , 1.    ]]))

Line 2 succinctly describes the kinematics in terms of elementary transforms: a rotation around the z-axis by the first joint angle, then a translation in the x-direction, then a rotation around the z-axis by the second joint angle, and finally a translation in the x-direction.

Line 3 creates the elementary transform sequence, with variable transforms. e is a single object that encapsulates a list of elementary transforms, and list like methods such as len as well as indexing and slicing as shown in lines 5-8.

Lines 9-18 evaluate the sequence, and substitutes elements from the passed arrays as the joint variables.

This approach is general enough to be able to describe any serial-link robot manipulator. For a branched manipulator we can use ETS to describe the connections between every parent and child link pair.

The ETS inherits list-like properties and has methods like reverse and pop.

Reference:

A simple and systematic approach to assigning Denavit-Hartenberg parameters. Peter I. Corke, IEEE Transactions on Robotics, 23(3), pp 590-594, June 2007.

ETS - 3D

class roboticstoolbox.robot.ETS.ETS(arg=None)[source]

Bases: BaseETS

This class implements an elementary transform sequence (ETS) for 3D

An instance can contain an elementary transform (ET) or an elementary transform sequence (ETS). It has list-like properties by subclassing UserList, which means we can perform indexing, slicing pop, insert, as well as using it as an iterator over its values.

ETS() an empty ETS list
ETS(et) an ETS containing a single ET
ETS([et0, et1, et2]) an ETS consisting of three ET’s

Parameters:: arg (Union[List[Union[ETS, ET]], List[ET], List[ETS], ET, ETS, None]) – Function to compute ET value

Examples

>>> from roboticstoolbox import ETS, ET
>>> e = ET.Rz(0.3) # a single ET, rotation about z
>>> ets1 = ETS(e)
>>> len(ets1)
1
>>> ets2 = ET.Rz(0.3) * ET.tx(2) # an ETS
>>> len(ets2)                    # of length 2
2
>>> ets2[1]                      # an ET sliced from the ETS
ET.tx(eta=2.0)

References

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

See also

rx(), ry(), rz(), tx(), ty(), tz()

__mul__(other)[source]

Return type:: ETS

compile()[source]

Compile an ETS

Perform constant folding for faster evaluation. Consecutive constant ETs are compounded, leading to a constant ET which is denoted by SE3 when displayed.

Returns:: optimised ETS
Return type:: compile

Examples

>>> import roboticstoolbox as rtb
>>> robot = rtb.models.ETS.Panda()
>>> ets = robot.ets()
>>> ets
[ET.tz(eta=0.333), ET.Rz(jindex=0), ET.Rx(eta=-1.5707963267948966), ET.Rz(jindex=1), ET.Rx(eta=1.5707963267948966), ET.tz(eta=0.316), ET.Rz(jindex=2), ET.tx(eta=0.0825), ET.Rx(eta=1.5707963267948966), ET.Rz(jindex=3), ET.tx(eta=-0.0825), ET.Rx(eta=-1.5707963267948966), ET.tz(eta=0.384), ET.Rz(jindex=4), ET.Rx(eta=1.5707963267948966), ET.Rz(jindex=5), ET.tx(eta=0.088), ET.Rx(eta=1.5707963267948966), ET.tz(eta=0.107), ET.Rz(jindex=6), ET.tz(eta=0.10300000000000001), ET.Rz(eta=-0.7853981633974483)]
>>> ets.compile()
[ET.SE3(T=array([[1.   , 0.   , 0.   , 0.   ],
       [0.   , 1.   , 0.   , 0.   ],
       [0.   , 0.   , 1.   , 0.333],
       [0.   , 0.   , 0.   , 1.   ]])), ET.Rz(jindex=0), ET.SE3(T=array([[ 1.,  0.,  0.,  0.],
       [ 0.,  0.,  1.,  0.],
       [ 0., -1.,  0.,  0.],
       [ 0.,  0.,  0.,  1.]])), ET.Rz(jindex=1), ET.SE3(T=array([[ 1.   ,  0.   ,  0.   ,  0.   ],
       [ 0.   ,  0.   , -1.   , -0.316],
       [ 0.   ,  1.   ,  0.   ,  0.   ],
       [ 0.   ,  0.   ,  0.   ,  1.   ]])), ET.Rz(jindex=2), ET.SE3(T=array([[ 1.    ,  0.    ,  0.    ,  0.0825],
       [ 0.    ,  0.    , -1.    ,  0.    ],
       [ 0.    ,  1.    ,  0.    ,  0.    ],
       [ 0.    ,  0.    ,  0.    ,  1.    ]])), ET.Rz(jindex=3), ET.SE3(T=array([[ 1.    ,  0.    ,  0.    , -0.0825],
       [ 0.    ,  0.    ,  1.    ,  0.384 ],
       [ 0.    , -1.    ,  0.    ,  0.    ],
       [ 0.    ,  0.    ,  0.    ,  1.    ]])), ET.Rz(jindex=4), ET.SE3(T=array([[ 1.,  0.,  0.,  0.],
       [ 0.,  0., -1.,  0.],
       [ 0.,  1.,  0.,  0.],
       [ 0.,  0.,  0.,  1.]])), ET.Rz(jindex=5), ET.SE3(T=array([[ 1.   ,  0.   ,  0.   ,  0.088],
       [ 0.   ,  0.   , -1.   , -0.107],
       [ 0.   ,  1.   ,  0.   ,  0.   ],
       [ 0.   ,  0.   ,  0.   ,  1.   ]])), ET.Rz(jindex=6), ET.SE3(T=array([[ 0.7071,  0.7071,  0.    ,  0.    ],
       [-0.7071,  0.7071,  0.    ,  0.    ],
       [ 0.    ,  0.    ,  1.    ,  0.103 ],
       [ 0.    ,  0.    ,  0.    ,  1.    ]]))]

See also

isconstant()

insert(arg, i=-1)[source]

Insert value

Inserts an ET or ETS into the ET sequence. The inserted value is at position i.

Parameters:

i (int) – insert an ET or ETS into the ETS, default is at the end
arg (Union[ET, ETS]) – the elementary transform or sequence to insert

Return type:

None

Examples

>>> from roboticstoolbox import ET
>>> e = ET.Rz() * ET.tx(1) * ET.Rz() * ET.tx(1)
>>> f = ET.Ry()
>>> e.insert(f, 2)
>>> e
[ET.Rz(jindex=0), ET.tx(eta=1.0), ET.Ry(), ET.Rz(jindex=1), ET.tx(eta=1.0)]

fkine(q, base=None, tool=None, include_base=True)[source]

Forward kinematics

T = ets.fkine(q) evaluates forward kinematics for the ets 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.

q: Joint coordinates

base: A base transform applied before the ETS

tool: tool transform, optional

include_base: set to True if the base transform should be considered

Return type:: SE3
Returns:: The transformation matrix representing the pose of the end-effector

Examples

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

>>> import roboticstoolbox as rtb
>>> panda = rtb.models.Panda().ets()
>>> 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.    ]]))

Notes

A tool transform, if provided, is incorporated into the result.
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).

jacob0(q, tool=None)[source]

Manipulator geometric Jacobian in the base 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 \({}^{E}\!\nu = \mathbf{J}_m(q) \dot{q}\).

Parameters:

q (Union[ndarray, List[float], Tuple[float], Set[float]]) – Joint coordinate vector
tool (Union[ndarray, SE3, None]) – a static tool transformation matrix to apply to the end of end, defaults to None

Returns:

Manipulator Jacobian in the base frame

Return type:

J0

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().ets()
>>> 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).

jacobe(q, tool=None)[source]

Manipulator geometric Jacobian in the end-effector 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 \({}^{E}\!\nu = \mathbf{J}_m(q) \dot{q}\).

Parameters:

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

Returns:

Manipulator Jacobian in the end frame

Return type:

Je

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().ets()
>>> 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).

hessian0(q=None, J0=None, tool=None)[source]

Manipulator Hessian

The manipulator Hessian tensor maps joint acceleration to end-effector spatial acceleration, expressed in the base 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

Parameters:

q (Union[ndarray, List[float], Tuple[float], Set[float], None]) – The joint angles/configuration of the robot (Optional, if not supplied will use the stored q values).
J0 (Optional[ndarray]) – The manipulator Jacobian in the base frame
tool (Union[ndarray, SE3, None]) – a static tool transformation matrix to apply to the end of end, defaults to None

Returns:

The manipulator Hessian in the base frame

Return type:

h0

Synopsis

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

\nu    &= \mat{J}(\vec{q}) \dvec{q} \\
\alpha &= \dmat{J} \dvec{q} + \mat{J} \ddvec{q}

where .. math:

\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 .. math:

\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 .. math:

\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().ets()
>>> 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, Je=None, tool=None)[source]

Manipulator Hessian

The manipulator Hessian tensor maps joint acceleration to end-effector spatial acceleration, expressed in the end-effector 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

Parameters:

q (Union[ndarray, List[float], Tuple[float], Set[float], None]) – The joint angles/configuration of the robot (Optional, if not supplied will use the stored q values).
J0 – The manipulator Jacobian in the end-effector frame
tool (Union[ndarray, SE3, None]) – a static tool transformation matrix to apply to the end of end, defaults to None

Returns:

The manipulator Hessian in end-effector frame

Return type:

he

Synopsis

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

\nu    &= \mat{J}(\vec{q}) \dvec{q} \\
\alpha &= \dmat{J} \dvec{q} + \mat{J} \ddvec{q}

where .. math:

\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 .. math:

\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 .. math:

\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().ets()
>>> 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).

__str__(q=None)

Pretty prints the ETS

q controls how the joint variables are displayed:

None, format depends on number of joint variables
- one, display joint variable as q
- more, display joint variables as q0, q1, …
- if a joint index was provided, use this value
“”, display all joint variables as empty parentheses ()
“θ”, display all joint variables as (θ)
format string with passed joint variables (j, j+1), so “θ{0}” would display joint variables as θ0, θ1, … while “θ{1}” would display joint variables as θ1, θ2, … j is either the joint index, if provided, otherwise a sequential value.

Parameters:: q (Optional[str]) – control how joint variables are displayed
Returns:: Pretty printed ETS
Return type:: str

Examples

>>> from roboticstoolbox import ET
>>> e = ET.Rz() * ET.tx(1) * ET.Rz()
>>> print(e[:2])
[ET.Rz(jindex=0), ET.tx(eta=1.0)]
>>> print(e)
Rz(q0) ⊕ tx(1) ⊕ Rz(q1)
>>> print(e.__str__(""))
Rz() ⊕ tx(1) ⊕ Rz()
>>> print(e.__str__("θ{0}"))  # numbering from 0
Rz(θ0) ⊕ tx(1) ⊕ Rz(θ1)
>>> print(e.__str__("θ{1}"))  # numbering from 1
Rz(θ1) ⊕ tx(1) ⊕ Rz(θ2)
>>> # explicit joint indices
>>> e = ET.Rz(jindex=3) * ET.tx(1) * ET.Rz(jindex=4)
>>> print(e)
Rz(q3) ⊕ tx(1) ⊕ Rz(q4)
>>> print(e.__str__("θ{0}"))
Rz(θ3) ⊕ tx(1) ⊕ Rz(θ4)

Angular parameters are converted to degrees, except if they are symbolic.

>>> from roboticstoolbox import ET
>>> from spatialmath.base import symbol
>>> theta, d = symbol('theta, d')
>>> e = ET.Rx(theta) * ET.tx(2) * ET.Rx(45, 'deg') * ET.Ry(0.2) * ET.ty(d)
>>> str(e)
'Rx(theta) ⊕ tx(2) ⊕ Rx(45°) ⊕ Ry(11.46°) ⊕ ty(d)'

__repr__(): Return repr(self).

__getitem__(i)

Index or slice an ETS

Parameters:: i – the index or slince
Returns:: Elementary transform
Return type:: et

Examples

>>> from roboticstoolbox import ET
>>> e = ET.Rz() * ET.tx(1) * ET.Rz() * ET.tx(1)
>>> e[0]
ET.Rz(jindex=0)
>>> e[1]
ET.tx(eta=1.0)
>>> e[1:3]
[ET.tx(eta=1.0), ET.Rz(jindex=1)]

property n: int

Number of joints

Counts the number of joints in the ETS.

Returns:: the number of joints in the ETS
Return type:: n

Examples

>>> from roboticstoolbox import ET
>>> e = ET.Rx() * ET.tx(1) * ET.tz()
>>> e.n
2

See also

joints()

property m: int

Number of transforms

Counts the number of transforms in the ETS.

Returns:: the number of transforms in the ETS
Return type:: m

Examples

>>> from roboticstoolbox import ET
>>> e = ET.Rx() * ET.tx(1) * ET.tz()
>>> e.m
3

property structure: str

Joint structure string

A string comprising the characters ‘R’ or ‘P’ which indicate the types of joints in order from left to right.

Returns:: A string indicating the joint types
Return type:: structure

Examples

>>> from roboticstoolbox import ET
>>> e = ET.tz() * ET.tx(1) * ET.Rz() * ET.tx(1)
>>> e.structure
'PR'

joints()

Get a list of the variable ETs with this ETS

Returns:: list of ETs that are joints
Return type:: joints

Examples

>>> from roboticstoolbox import ET
>>> e = ET.Rz() * ET.tx(1) * ET.Rz() * ET.tx(1)
>>> e.joints()
[ET.Rz(jindex=0), ET.Rz(jindex=1)]

split()

Split ETS into link segments

Returns:: a list of ETS, each one, apart from the last, ends with a variable ET.
Return type:: split

inv()

Inverse of ETS

The inverse of a given ETS. It is computed as the inverse of the individual ETs in the reverse order.

\[(\mathbf{E}_0, \mathbf{E}_1 \cdots \mathbf{E}_{n-1} )^{-1} = (\mathbf{E}_{n-1}^{-1}, \mathbf{E}_{n-2}^{-1} \cdots \mathbf{E}_0^{-1}{n-1} )\]

Returns:: Inverse of the ETS
Return type:: inv

Examples

>>> from roboticstoolbox import ET
>>> e = ET.Rz(jindex=2) * ET.tx(1) * ET.Rx(jindex=3,flip=True) * ET.tx(1)
>>> print(e)
Rz(q2) ⊕ tx(1) ⊕ Rx(-q3) ⊕ tx(1)
>>> print(e.inv())
tx(-1) ⊕ Rx(q3) ⊕ tx(-1) ⊕ Rz(-q2)

Notes

It is essential to use explicit joint indices to account for
the reversed order of the transforms.

ETS - 2D

class roboticstoolbox.robot.ETS.ETS2(arg=None)[source]

Bases: BaseETS

This class implements an elementary transform sequence (ETS) for 2D

Parameters:: arg (Union[List[Union[ETS2, ET2]], List[ET2], List[ETS2], ET2, ETS2, None]) – Function to compute ET value

An instance can contain an elementary transform (ET) or an elementary transform sequence (ETS). It has list-like properties by subclassing UserList, which means we can perform indexing, slicing pop, insert, as well as using it as an iterator over its values.

ETS() an empty ETS list
ET2.XY(η) is a constant elementary transform
ET2.XY(η, 'deg') as above but the angle is expressed in degrees
ET2.XY() is a joint variable, the value is left free until evaluation time
ET2.XY(j=J) as above but the joint index is explicitly given, this might correspond to the joint number of a multi-joint robot.
ET2.XY(flip=True) as above but the joint moves in the opposite sense

where XY is one of R, tx, ty.

Example

Traceback (most recent call last):
  File "<input>", line 1, in <module>
AttributeError: type object 'ETS2' has no attribute 'R'
Traceback (most recent call last):
  File "<input>", line 1, in <module>
NameError: name 'e' is not defined
Traceback (most recent call last):
  File "<input>", line 1, in <module>
NameError: name 'e' is not defined

References:

Kinematic Derivatives using the Elementary Transform Sequence, J. Haviland and P. Corke

Seealso:

r(), tx(), ty()

__str__(q=None)

Pretty prints the ETS

q controls how the joint variables are displayed:

None, format depends on number of joint variables
- one, display joint variable as q
- more, display joint variables as q0, q1, …
- if a joint index was provided, use this value
“”, display all joint variables as empty parentheses ()
“θ”, display all joint variables as (θ)
format string with passed joint variables (j, j+1), so “θ{0}” would display joint variables as θ0, θ1, … while “θ{1}” would display joint variables as θ1, θ2, … j is either the joint index, if provided, otherwise a sequential value.

Parameters:: q (Optional[str]) – control how joint variables are displayed
Returns:: Pretty printed ETS
Return type:: str

Examples

>>> from roboticstoolbox import ET
>>> e = ET.Rz() * ET.tx(1) * ET.Rz()
>>> print(e[:2])
[ET.Rz(jindex=0), ET.tx(eta=1.0)]
>>> print(e)
Rz(q0) ⊕ tx(1) ⊕ Rz(q1)
>>> print(e.__str__(""))
Rz() ⊕ tx(1) ⊕ Rz()
>>> print(e.__str__("θ{0}"))  # numbering from 0
Rz(θ0) ⊕ tx(1) ⊕ Rz(θ1)
>>> print(e.__str__("θ{1}"))  # numbering from 1
Rz(θ1) ⊕ tx(1) ⊕ Rz(θ2)
>>> # explicit joint indices
>>> e = ET.Rz(jindex=3) * ET.tx(1) * ET.Rz(jindex=4)
>>> print(e)
Rz(q3) ⊕ tx(1) ⊕ Rz(q4)
>>> print(e.__str__("θ{0}"))
Rz(θ3) ⊕ tx(1) ⊕ Rz(θ4)

Angular parameters are converted to degrees, except if they are symbolic.

>>> from roboticstoolbox import ET
>>> from spatialmath.base import symbol
>>> theta, d = symbol('theta, d')
>>> e = ET.Rx(theta) * ET.tx(2) * ET.Rx(45, 'deg') * ET.Ry(0.2) * ET.ty(d)
>>> str(e)
'Rx(theta) ⊕ tx(2) ⊕ Rx(45°) ⊕ Ry(11.46°) ⊕ ty(d)'

__repr__(): Return repr(self).

__getitem__(i)

Index or slice an ETS

Parameters:: i – the index or slince
Returns:: Elementary transform
Return type:: et

Examples

>>> from roboticstoolbox import ET
>>> e = ET.Rz() * ET.tx(1) * ET.Rz() * ET.tx(1)
>>> e[0]
ET.Rz(jindex=0)
>>> e[1]
ET.tx(eta=1.0)
>>> e[1:3]
[ET.tx(eta=1.0), ET.Rz(jindex=1)]

property n: int

Number of joints

Counts the number of joints in the ETS.

Returns:: the number of joints in the ETS
Return type:: n

Examples

>>> from roboticstoolbox import ET
>>> e = ET.Rx() * ET.tx(1) * ET.tz()
>>> e.n
2

See also

joints()

property m: int

Number of transforms

Counts the number of transforms in the ETS.

Returns:: the number of transforms in the ETS
Return type:: m

Examples

>>> from roboticstoolbox import ET
>>> e = ET.Rx() * ET.tx(1) * ET.tz()
>>> e.m
3

property structure: str

Joint structure string

A string comprising the characters ‘R’ or ‘P’ which indicate the types of joints in order from left to right.

Returns:: A string indicating the joint types
Return type:: structure

Examples

>>> from roboticstoolbox import ET
>>> e = ET.tz() * ET.tx(1) * ET.Rz() * ET.tx(1)
>>> e.structure
'PR'

joints()

Get a list of the variable ETs with this ETS

Returns:: list of ETs that are joints
Return type:: joints

Examples

>>> from roboticstoolbox import ET
>>> e = ET.Rz() * ET.tx(1) * ET.Rz() * ET.tx(1)
>>> e.joints()
[ET.Rz(jindex=0), ET.Rz(jindex=1)]

split()

Split ETS into link segments

Returns:: a list of ETS, each one, apart from the last, ends with a variable ET.
Return type:: split

inv()

Inverse of ETS

The inverse of a given ETS. It is computed as the inverse of the individual ETs in the reverse order.

\[(\mathbf{E}_0, \mathbf{E}_1 \cdots \mathbf{E}_{n-1} )^{-1} = (\mathbf{E}_{n-1}^{-1}, \mathbf{E}_{n-2}^{-1} \cdots \mathbf{E}_0^{-1}{n-1} )\]

Returns:: Inverse of the ETS
Return type:: inv

Examples

>>> from roboticstoolbox import ET
>>> e = ET.Rz(jindex=2) * ET.tx(1) * ET.Rx(jindex=3,flip=True) * ET.tx(1)
>>> print(e)
Rz(q2) ⊕ tx(1) ⊕ Rx(-q3) ⊕ tx(1)
>>> print(e.inv())
tx(-1) ⊕ Rx(q3) ⊕ tx(-1) ⊕ Rz(-q2)

Notes

It is essential to use explicit joint indices to account for
the reversed order of the transforms.

__mul__(other)[source]

Return type:: ETS2

compile()[source]

Compile an ETS2

Return type:: ETS2
Returns:: optimised ETS2

Perform constant folding for faster evaluation. Consecutive constant ETs are compounded, leading to a constant ET which is denoted by SE3 when displayed.

Seealso:: isconstant()

insert(arg, i=-1)[source]

Insert value

Parameters:

i (int) – insert an ET or ETS into the ETS, default is at the end
arg (Union[ET2, ETS2]) – the elementary transform or sequence to insert

Return type:

None

Inserts an ET or ETS into the ET sequence. The inserted value is at position i.

Example:

>>> from roboticstoolbox import ET2
>>> e = ET2.R() * ET2.tx(1) * ET2.R() * ET2.tx(1)
>>> f = ET2.R()
>>> e.insert(f, 2)
>>> e
[ET2.R(jindex=0), ET2.tx(eta=1.0), ET2.R(), ET2.R(jindex=1), ET2.tx(eta=1.0)]

fkine(q, base=None, tool=None, include_base=True)[source]

Forward kinematics :type q: Union[ndarray, List[float], Tuple[float], Set[float]] :param q: Joint coordinates :type q: ArrayLike :type base: Union[ndarray, SE2, None] :param base: base transform, optional :type tool: Union[ndarray, SE2, None] :param tool: tool transform, optional

Return type:: SE2
Returns:: The transformation matrix representing the pose of the end-effector

T = ets.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 SE2 instance with m values. .. note:

- The robot's base tool transform, if set, is incorporated
  into the result.
- A tool transform, if provided, is incorporated into the result.
- Works from the end-effector link to the base

References:

Kinematic Derivatives using the Elementary Transform Sequence, J. Haviland and P. Corke

jacob0(q)[source]

Return type:: ndarray

jacobe(q)[source]

Jacobian in base frame

Parameters:: q (ArrayLike) – joint coordinates
Returns:: Jacobian matrix

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

End-effector spatial velocity \(\nu = (v_x, v_y, v_z, \omega_x, \omega_y, \omega_z)^T\) is related to joint velocity by \({}^{e}\nu = {}^{e}\mathbf{J}_0(q) \dot{q}\).

Seealso:: jacob(), hessian0()