Adaptation of Motor Primitives to the Environment Through Learning and Statistical Generalization

Abstract

In this paper we propose a method of adapting motion to the environment based on force feedback. Our method combines two approaches of motor primitive adaptation. Starting from a single demonstration of motion, we use iterative learning control to adapt the motion to different conditions of the environment, for example, the height of the table. The adaptation is realized through coupling terms at the velocity level of a dynamic movement primitive, and acts as a feedforward component, predetermined for the given external condition. As adaptation to each condition takes several iterations, we combine this method with statistical generalization, employing Gaussian process regression. By generating a small database of coupling terms through iterative learning, we adapt to the environment by generalizing between the coupling terms in the database, thus either already achieving an appropriate coupling term for our demonstration trajectory or providing an initial estimate for the adaptation. Consequently, the learning doesn’t need to be executed for every condition of the environment, but only for a small set. In the paper we provide the details of the method and evaluate it in a simulated setting for the use case of placing a glass on a table.

Appendix

Dynamic movement primitives have been extensively studied in robotics. For the completeness of the paper we provide a short summary based on [8]. A nonlinear system of differential equations defines DMP for discrete movements

$$\tau \dot{z} = \alpha_{z} (\beta_{z} (g - y) - z) + f(x)$$

(13)

$$\tau \dot{y} = z .$$

(14)

f(x) is defined as a linear combination of nonlinear radial basis functions

$$f\left( x \right) = \frac{{\mathop \sum \nolimits_{i = 1}^{N} w_{i}\Psi _{i} (x) }}{{\Psi _{i} (x)}}x ,$$

(15)

$$\Psi _{i} \left( x \right) = \exp ( - h_{i} \left( {x - c_{i} } \right)^{2} ) ,$$

(16)

where c _i are the centres of radial basis functions distributed along the trajectory and h_i > 0 their widths. If $\alpha_{\text{z}} ,\,\upbeta_{\text{z}} ,\,\,\uptau > 0,\upalpha_{\text{z}} = 4\upbeta_{\text{z}}$ the linear part of the system (13) and (14) is critically damped and has a unique attractor point at y = g, z = 0. A phase variable x is used in (15) and (16). It is utilized to avoid direct dependency of f(x) on time. Its dynamics is defined by

$$\tau \dot{x} = - \alpha_{x} x ,$$

(17)

with initial value at x(0) = 1. α _x is a positive constant. The weight vector w composed of w _i defines the shape of the encoded trajectory. The learning of the weight vector is described in [8]. Multiple DOFs are realized by maintaining separate sets of (13)–(16), while a single canonical system given by (17) is used for synchronization.