movement_primitives.dmp.DMP#

class movement_primitives.dmp.DMP(n_dims, execution_time=1.0, dt=0.01, n_weights_per_dim=10, int_dt=0.001, p_gain=0.0, smooth_scaling=False, alpha_y=25.0, beta_y=6.25)#

Bases: WeightParametersMixin, DMPBase

Dynamical movement primitive (DMP).

Equation of transformation system (according to [1], Eq. 2.1):

\[\ddot{y} = (\alpha_y (\beta_y (g - y) - \tau \dot{y}) + f(z) + C_t) / \tau^2\]

and if smooth scaling is activated (according to [2]):

\[\ddot{y} = (\alpha_y (\beta_y (g - y) - \tau \dot{y} - \underline{\beta_y (g - y_0) z}) + f(z) + C_t) / \tau^2\]

Parameters:

n_dimsint: State space dimensions.
execution_timefloat, optional (default: 1): Execution time of the DMP: \(\tau\).
dtfloat, optional (default: 0.01): Time difference between DMP steps: \(\Delta t\).
n_weights_per_dimint, optional (default: 10): Number of weights of the function approximator per dimension.
int_dtfloat, optional (default: 0.001): Time difference for Euler integration of transformation system.
p_gainfloat, optional (default: 0): Gain for proportional controller of DMP tracking error. The domain is [0, execution_time**2/dt].
smooth_scalingbool, optional (default: False): Avoids jumps during the beginning of DMP execution when the goal is changed and the trajectory is scaled by interpolating between the old and new scaling of the trajectory.
alpha_yfloat or array-like, shape (n_dims,), optional (default: 25.0): Parameter of the transformation system.
beta_yfloat, list with length n_dims, or array with shape (n_dims,), optional (default: 6.25): Parameter of the transformation system.

Attributes:

execution_time_float: Execution time of the DMP.
dt_float: Time difference between DMP steps. This value can be changed to adapt the frequency.

Methods

`configure`([t, start_y, start_yd, start_ydd, ...])	Set meta parameters.
`get_weights`()	Get weight vector of DMP.
`imitate`(T, Y[, regularization_coefficient, ...])	Imitate demonstration.
`n_steps_open_loop`(last_y, last_yd, n_steps)	Perform 'n_steps' steps.
`open_loop`([run_t, coupling_term, step_function])	Run DMP open loop.
`reset`()	Reset DMP to initial state and time.
`set_weights`(weights)	Set weight vector of DMP.
`step`(last_y, last_yd[, coupling_term, ...])	DMP step.

get_execution_time_
set_execution_time_

References

[1]

Ijspeert, A. J., Nakanishi, J., Hoffmann, H., Pastor, P., Schaal, S. (2013). Dynamical Movement Primitives: Learning Attractor Models for Motor Behaviors. Neural Computation 25 (2), 328-373. DOI: 10.1162/NECO_a_00393, https://homes.cs.washington.edu/~todorov/courses/amath579/reading/DynamicPrimitives.pdf

[2]

Pastor, P., Hoffmann, H., Asfour, T., Schaal, S. (2009). Learning and Generalization of Motor Skills by Learning from Demonstration. In 2009 IEEE International Conference on Robotics and Automation, (pp. 763-768). DOI: 10.1109/ROBOT.2009.5152385, https://h2t.iar.kit.edu/pdf/Pastor2009.pdf

configure(t=None, start_y=None, start_yd=None, start_ydd=None, goal_y=None, goal_yd=None, goal_ydd=None)#

Set meta parameters.

Parameters:

tfloat, optional: Time at current step.
start_yarray, shape (n_dims,): Initial state.
start_ydarray, shape (n_vel_dims,): Initial velocity.
start_yddarray, shape (n_vel_dims,): Initial acceleration.
goal_yarray, shape (n_dims,): Goal state.
goal_ydarray, shape (n_vel_dims,): Goal velocity.
goal_yddarray, shape (n_vel_dims,): Goal acceleration.

Raises:

ValueError: If the length of the configured meta parameter is not correct.

get_weights()#

Get weight vector of DMP.

Returns:

weightsarray, shape (N * n_weights_per_dim,): Current weights of the DMP. N depends on the type of DMP

imitate(T, Y, regularization_coefficient=0.0, allow_final_velocity=False)#

Imitate demonstration.

Target forces of the forcing term are computed according to

\[f_{target} = \tau^2 \ddot{y}_{demo} - \alpha_y( \beta_y (g-y_{demo}) - \tau \dot{y}_{demo} - \underline{\beta_y (g-y_0) z} ),\]

where the underlined part is only used when smooth scaling is used.

Parameters:

Tarray, shape (n_steps,): Time for each step.
Yarray, shape (n_steps, n_dims): State at each step.
regularization_coefficientfloat, optional (default: 0): Regularization coefficient for regression.
allow_final_velocitybool, optional (default: False): Allow a final velocity.

n_steps_open_loop(last_y, last_yd, n_steps)#

Perform ‘n_steps’ steps.

Parameters:

last_yarray, shape (n_dims,): Last state.
last_ydarray, shape (n_dims,): Last time derivative of state (e.g., velocity).
n_stepsint: Number of steps.

Returns:

yarray, shape (n_dims,): Next state.
ydarray, shape (n_dims,): Next time derivative of state (e.g., velocity).

property n_weights#: Total number of weights configuring the forcing term.

open_loop(run_t=None, coupling_term=None, step_function='rk4-cython')#

Run DMP open loop.

Parameters:

run_tfloat, optional (default: execution_time): Run time of DMP. Can be shorter or longer than execution_time.
coupling_termobject, optional (default: None): Coupling term that will be added to velocity.
step_functionstr, optional (default: ‘rk4-cython’): DMP integration function. Possible options: ‘rk4’, ‘euler’, ‘euler-cython’, ‘rk4-cython’.

Returns:

Tarray, shape (n_steps,): Time for each step.
Yarray, shape (n_steps, n_dims): State at each step.

Raises:

ValueError: If step function is unknown.

reset()#: Reset DMP to initial state and time.

set_weights(weights)#

Set weight vector of DMP.

Parameters:

weightsarray, shape (N * n_weights_per_dim,): New weights of the DMP. N depends on the type of DMP

step(last_y, last_yd, coupling_term=None, step_function='rk4-cython')#

DMP step.

Parameters:

last_yarray, shape (n_dims,): Last state.
last_ydarray, shape (n_dims,): Last time derivative of state (e.g., velocity).
coupling_termobject, optional (default: None): Coupling term that will be added to velocity.
step_functionstr, optional (default: ‘rk4-cython’): DMP integration function. Possible options: ‘rk4’, ‘euler’, ‘euler-cython’, ‘rk4-cython’.

Returns:

yarray, shape (n_dims,): Next state.
ydarray, shape (n_dims,): Next time derivative of state (e.g., velocity).