Learning Task-Space Synergies using Riemannian Geometry

 In the context of robotic control, synergies can form elementary units of behavior. By specifying taskdependent coordination behaviors at a low control level, one can achieve task-specific disturbance rejection. In this work we present an approach to learn the parameters of such lowlevel controllers by demonstration. We identify a synergy by extracting covariance information from demonstration data. The extracted synergy is used to derive a time-invariant state feedback controller through optimal control. To cope with the non-Euclidean nature of robot poses, we utilize Riemannian geometry, where both estimation of the covariance and the associated controller take into account the geometry of the pose manifold. We demonstrate the efficacy of the approach experimentally in a bimanual manipulation task.

 

An Approach for Imitation Learning on Riemannian Manifolds

In imitation learning, multivariate Gaussians are widely used to encode robot behaviors. Such approaches do not provide the ability to properly represent end-effector orientation, as the distance metric in the space of orientations is not Euclidean.

In this work we present an extension of common imitation learning techniques to Riemannian manifolds. This generalization enables the encoding of joint distributions that include the robot pose. We show that Gaussian conditioning, Gaussian product and nonlinear regression can be achieved with this representation. The proposed approach is illustrated with examples on a 2-dimensional sphere, with an example of regression between two robot end-effector poses, as well as an extension of Task-Parameterized Gaussian Mixture Model (TP-GMM) and Gaussian Mixture Regression (GMR) to Riemannian manifolds. 

The work is accompanied with source code that can be downloaded here.