Differential steering

class DiffSteer(W=1, **kwargs)

Bases: Unicycle

__init__(W=1, **kwargs)

Create differential steering kinematic model

Parameters:

• W (float) – vehicle width, defaults to 1

• kwargs (Any) – additional arguments passed to VehicleBase constructor

Model the motion of a unicycle model with equations of motion given by:

\[\begin{split}\dot{x} &= v \cos \theta \\ \dot{y} &= v \sin \theta \\ \dot{\theta} &= \omega\end{split}\]

where \(v = (v_R + v_L)/2\) is the velocity in body frame x-direction, and \(\omega = (v_R - v_L)/W\) is the turn rate.

Seealso:

f() deriv() Fx() meth:Fv Vehicle

init(x0=None, control=None, animate=True)

Initialize for simulation (superclass)

Parameters:

x0 (ndarray | Sequence[float] | None) – Initial state, defaults to value given to Vehicle constructor

Return type:

None

Performs the following initializations:

1. Clears the state history

2. Set the private random number generator to initial value

3. Sets state \(x = x_0\)

4. Sets previous velocity to zero

5. If a driver is attached, initialize it

6. If animation is enabled, initialize that

If animation is enabled by constructor and no animation object is given, use a default VehiclePolygon("car")

Seealso:

VehicleAnimationBase run()

u_limited(u)

Apply vehicle velocity and acceleration limits

Parameters:

u (ndarray | Sequence[float]) – Desired vehicle inputs \((v_L, v_R)\)

Returns:

Allowable vehicle inputs \((v_L, v_R)\)

Return type:

ndarray

Velocity and acceleration limits are applied to \(v\) and turn rate limits are applied to \(\omega\).

deriv(x, u, limits=True)

Time derivative of state

Parameters:

• x (ndarray | Sequence[float]) – vehicle state \((x, y, \theta)\)

• u (ndarray | Sequence[float]) – Desired vehicle inputs \((v_L, v_R)\)

• limits (bool) – limits are applied to input, default True

Returns:

state derivative \((\dot{x}, \dot{y}, \dot{\theta})\)

Return type:

ndarray

Returns the time derivative of state (3x1) at the state x with left and right wheel speeds u.

\[\begin{split}\dot{x} &= v \cos \theta \\ \dot{y} &= v \sin \theta \\ \dot{\theta} &= \omega\end{split}\]

where \(v = (v_R + v_L)/2\) is the velocity in body frame x-direction, and \(\omega = (v_R - v_L)/W\) is the turn rate.

If limits is True then speed and acceleration limits are applied to the wheel speeds u.

Seealso:

f()

Fv(x, odo)

Jacobian of state transition function df/dv

Parameters:

• x (ndarray | Sequence[float]) – vehicle state \((x, y, \theta)\)

• odo (ndarray | Sequence[float]) – vehicle odometry \((\delta_d, \delta_\theta)\)

Returns:

Jacobian matrix

Return type:

ndarray

Returns the Jacobian matrix \(\frac{\partial \vec{f}}{\partial \vec{v}}\) for the given state and odometry.

Seealso:

f() deriv() Fx()

Fx(x, odo)

Jacobian of state transition function df/dx

Parameters:

• x (ndarray | Sequence[float]) – vehicle state \((x, y, \theta)\)

• odo (ndarray | Sequence[float]) – vehicle odometry \((\delta_d, \delta_\theta)\)

Returns:

Jacobian matrix

Return type:

ndarray

Returns the Jacobian matrix \(\frac{\partial \vec{f}}{\partial \vec{x}}\) for the given state and odometry.

Seealso:

f() deriv() Fv()

__repr__()

Convert to string, same as __str__()

Return type:

str

Returns:

String representation of vehicle object

property accel_max: float

Get maximum acceleration of vehicle (superclass)

Returns:

maximum acceleration

Return type:

float

Set by VehicleBase() subclass constructor.

Seealso:

speed_max() steer_max u_limited()

add_driver(driver)

Add a driver agent (superclass)

Parameters:

driver (VehicleDriverBase) – a driver agent object

Return type:

None

Warning

Deprecated. Use vehicle.control = driver instead.

Seealso:

VehicleDriverBase control()

property control: Union[ndarray, Sequence[float], Callable[..., ndarray | Sequence[float]], interp1d, VehicleDriverBase, None]

Get/set vehicle control (superclass)

Getter:

Returns vehicle’s control

Setter:

Sets the vehicle’s control

Type:

2-tuple, callable, interp1d or VehicleDriver

This is the input to the vehicle during a simulation performed by run(). The control input can be:

• a constant tuple as the control inputs to the vehicle

• a function called as f(vehicle, t, x) that returns a tuple

• an interpolator called as f(t) that returns a tuple

• a driver agent, subclass of VehicleDriverBase

Example:

>>> from roboticstoolbox import Bicycle, RandomPath
>>> bike = Bicycle()
>>> bike.control = RandomPath(workspace=10)
>>> print(bike)
Bicycle: x = [ 0, 0, 0 ]
  L=1, steer_max=1.41372, speed_max=inf, accel_max=inf

Seealso:

run() eval_control() scipy.interpolate.interp1d VehicleDriverBase

property dt: float

Get sample time (superclass)

Returns:

discrete time step for simulation

Return type:

float

Set by VehicleBase() subclass constructor. Used by step() to update state.

Seealso:

run()

eval_control(control, x)

Evaluate vehicle control input (superclass method)

Parameters:

• control (Union[ndarray, Sequence[float], Callable[..., ndarray | Sequence[float]], interp1d, VehicleDriverBase, None]) – vehicle control

• x (ndarray | Sequence[float]) – vehicle state \((x, y, heta)\)

Raises:

ValueError – bad control

Returns:

vehicle control inputs

Return type:

ndarray

Evaluates the control for this time step and state. Control is set by the control() property to:

• a constant tuple as the control inputs to the vehicle

• a function called as f(vehicle, t, x) that returns a tuple

• an interpolator called as f(t) that returns a tuple

• a driver agent, subclass of VehicleDriverBase

Vehicle steering, speed and acceleration limits are applied to the result before being input to the vehicle model.

Seealso:

run() control() scipy.interpolate.interp1d()

f(x, odo, v=None)

State transition function

Parameters:

• x (ndarray | Sequence[float]) – vehicle state \((x, y, \theta)\)

• odo (ndarray | Sequence[float]) – vehicle odometry \((\delta_d, \delta_\theta)\)

• v (ndarray | Sequence[float] | None) – additive odometry noise, defaults to (0,0)

Returns:

predicted vehicle state

Return type:

ndarray

Predict the next state based on current state and odometry value. v is a random variable that represents additive odometry noise for simulation purposes.

\[f: \vec{x}_k, \delta_d, \delta_\theta \mapsto \vec{x}_{k+1}\]

For particle filters it is useful to apply the odometry to the states of all particles and this can be achieved if x is a 2D array with one state per row. v is ignored in this case.

Note

This is the state update equation used for EKF localization.

Seealso:

deriv() Fx() Fv()

limits_va(v, v_prev)

Apply velocity and acceleration limits (superclass)

Parameters:

• v (float) – desired velocity

• v_prev (list[float]) – previous velocity, reference to list

Returns:

allowed velocity

Return type:

float

Determine allowable velocity given desired velocity, speed and acceleration limits.

Note

This function requires previous velocity, v_prev to enable acceleration limiting. This is passed as a reference to a mutable value, a single-element list. This is reset to zero at the start of each simulation.

plot(x)

Return type:

None

plot_xy(*args, block=None, **kwargs)

Plot xy-path from history

Parameters:

• block (bool | None) – None leaves showing the figure to the environment (no plt.show call), False shows it and returns at once, True shows it and waits until it is closed; ignored by notebook backends, defaults to None

• args (Any) – positional arguments passed to plot()

• kwargs (Any) – keyword arguments passed to plot()

Return type:

None

The \((x,y)\) trajectory from the simulation history is plotted as \(x\) vs \(y\).

Seealso:

run() plot_xyt()

plot_xyt(block=None, **kwargs)

Plot configuration vs time from history

Parameters:

• block (bool | None) – None leaves showing the figure to the environment (no plt.show call), False shows it and returns at once, True shows it and waits until it is closed; ignored by notebook backends, defaults to None

• kwargs (Any) – keyword arguments passed to plot()

Return type:

None

The \((x,y, heta)\) trajectory from the simulation history is plotted against time.

Seealso:

run() plot_xy()

polygon(q)

Bounding polygon at vehicle configuration

Parameters:

q (ndarray | Sequence[float]) – vehicle configuration \((x, y, heta)\)

Returns:

bounding polygon of vehicle at configuration q

Return type:

Polygon2

Raises:

ValueError – the vehicle has no bounding polygon

The bounding polygon of the vehicle is returned for the configuration q. Can be used for collision checking using the intersects() method.

Seealso:

Polygon2

property q: ndarray

Get vehicle state/configuration (superclass)

Returns:

Vehicle state \((x, y, \theta)\)

Return type:

ndarray(3)

Seealso:

x()

property random: Generator

Get private random number generator

Returns:

NumPy random number generator

Return type:

numpy.random.Generator

Has methods including:

• integers(low, high, size, endpoint)

• random(size)

• uniform(low, high, size)

• normal(mean, std, size)

• multivariate_normal(mean, covar, size)

The generator is initialized with the seed provided at constructor time every time init() is called.

Seealso:

init() numpy.random.default_rng()

run(T=10, x0=None, control=None, animate=True)

Simulate motion of vehicle (superclass)

Parameters:

• T (float) – Simulation time in seconds, defaults to 10

• x0 (ndarray | Sequence[float] | None) – Initial state, defaults to value given to Vehicle constructor

• control (Union[ndarray, Sequence[float], Callable[..., ndarray | Sequence[float]], interp1d, VehicleDriverBase, None]) – vehicle control inputs, defaults to None

• animate (bool) – Enable graphical animation of vehicle, defaults to True

Returns:

State trajectory, each row is \((x,y,\theta)\).

Return type:

ndarray

Runs the vehicle simulation for T seconds and optionally plots an animation. The method step() performs each time step.

The control inputs are provided by control which can be:

• a constant tuple as the control inputs to the vehicle

• a function called as f(vehicle, t, x) that returns a tuple

• an interpolator called as f(t) that returns a tuple, see SciPy interp1d

• a driver agent, subclass of VehicleDriverBase

This is evaluated by eval_control() which applies velocity, acceleration and steering limits.

The simulation can be stopped prematurely by the control function calling stopsim().

Note

• the simulation is fixed-time step with the step given by the dt attribute set by the constructor.

• integration uses rectangular integration.

Seealso:

init() step() control() run_animation()

run_animation(T=10, x0=None, control=None, format=None, file=None)

Animate motion of vehicle (superclass)

Parameters:

• T (float) – Simulation time in seconds, defaults to 10

• x0 (ndarray | Sequence[float] | None) – Initial state, defaults to value given to Vehicle constructor

• control (Union[ndarray, Sequence[float], Callable[..., ndarray | Sequence[float]], interp1d, VehicleDriverBase, None]) – vehicle control inputs, defaults to None

• format (str | None) – Output format

• file (str | None) – File name

Raises:

ValueError – file is None for format "gif" or "mp4", or format is not recognised

Returns:

Matplotlib animation object

Return type:

FuncAnimation | str | None

Runs the vehicle simulation for T seconds and returns an animation in various formats:

``format``    ``file``   description
============  =========  ============================
``"html"``    str, None  return HTML5 video
``"jshtml"``  str, None  return JS+HTML video
``"gif"``     str        return animated GIF
``"mp4"``     str        return MP4/H264 video
``None``                 return a ``FuncAnimation`` object

The allowables types for file are given in the second column. A str value is the file name. If None is an option then return the video as a string.

For the last case, a reference to the animation object must be held if used for animation in a Jupyter cell:

anim = robot.run_animation(T=20)

The control inputs are provided by control which can be:

• a constant tuple as the control inputs to the vehicle

• a function called as f(vehicle, t, x) that returns a tuple

• an interpolator called as f(t) that returns a tuple, see SciPy interp1d

• a driver agent, subclass of VehicleDriverBase

This is evaluated by eval_control() which applies velocity, acceleration and steering limits.

The simulation can be stopped prematurely by the control function calling stopsim().

Note

• the simulation is fixed-time step with the step given by the dt attribute set by the constructor.

• integration uses rectangular integration.

Seealso:

init() step() control() run()

property speed_max: float

Get maximum speed of vehicle (superclass)

Returns:

maximum speed

Return type:

float

Set by VehicleBase() subclass constructor.

Seealso:

accel_max() steer_max u_limited()

step(u=None, animate=True, pause=True)

Step simulator by one time step (superclass)

Returns:

odometry \((\delta_d, \delta_\theta)\)

Return type:

ndarray

1. Obtain the vehicle control inputs

2. Integrate the vehicle state forward one timestep

3. Updates the stored state and state history

4. Update animation if enabled.

5. Returns the odometry

• veh.step((vel, steer)) for a Bicycle vehicle model

• veh.step((vel, turn_rate)) for a Unicycle vehicle model

• veh.step() as above but control is taken from the control attribute which might be a function or driver agent.

Example:

>>> from roboticstoolbox import Bicycle
>>> bike = Bicycle()  # default bicycle model
>>> bike.step((1, 0.2))  # one step: v=1, γ=0.2
array([0.1   , 0.0203])
>>> bike.x
array([0.1   , 0.    , 0.0203])
>>> bike.step((1, 0.2))  # another step: v=1, γ=0.2
array([0.1   , 0.0203])
>>> bike.x
array([0.2   , 0.002 , 0.0405])

Note

Vehicle control input limits are applied.

Seealso:

control(), update, run()

stopsim()

Stop the simulation (superclass)

A control function can stop the simulation initiated by the run method.

Seealso:

run()

Return type:

None

property verbose: bool

Get verbosity (superclass)

Returns:

verbosity level

Return type:

bool

Set by VehicleBase subclass constructor.

property workspace: ndarray | None

Size of robot workspace (superclass)

Returns:

workspace bounds [xmin, xmax, ymin, ymax]

Return type:

ndarray(4)

Returns the bounds of the workspace as specified by constructor option workspace

Setter accepts:

Union[float, ndarray, Sequence[float], HasWorkspace, None]

property x: ndarray

Get vehicle state/configuration (superclass)

Returns:

Vehicle state \((x, y, \theta)\)

Return type:

ndarray(3)

Seealso:

q()

property x0: ndarray

Get vehicle initial state/configuration (superclass)

Returns:

Vehicle state \((x, y, \theta)\)

Return type:

ndarray(3)

Set by VehicleBase subclass constructor. The state is set to this value at the beginning of each simulation run.

Seealso:

run()

Setter accepts:

ndarray | Sequence[float]

property x_hist: ndarray

Get vehicle state/configuration history (superclass)

Returns:

Vehicle state history

Return type:

ndarray(n,3)

The state \((x, y, heta)\) at each time step resulting from a simulation run, one row per timestep.

Seealso:

run()