Publication

An Approach for Imitation Learning on Riemannian Manifolds

Sylvain Calinon
2017
Journal Articles

Abstract

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 probabilistic learning from demonstration 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, as well as with an example of regression between two robot end-effector poses, and by extending TP-GMM and GMR to Riemannian manifolds.

Official source

https://infoscience.epfl.ch/entities/publication/32caffa6-6b67-4930-b4bc-294568dce7f9

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Sylvain Calinon
2017
Journal Articles

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/32caffa6-6b67-4930-b4bc-294568dce7f9

About this result

Related concepts (32)

Multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem.

Riemannian manifold

In differential geometry, a Riemannian manifold or Riemannian space (M, g), so called after the German mathematician Bernhard Riemann, is a real, smooth manifold M equipped with a positive-definite inner product gp on the tangent space TpM at each point p. The family gp of inner products is called a Riemannian metric (or Riemannian metric tensor). Riemannian geometry is the study of Riemannian manifolds. A common convention is to take g to be smooth, which means that for any smooth coordinate chart (U, x) on M, the n2 functions are smooth functions.

Pseudo-Riemannian manifold

In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed. Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space. A special case used in general relativity is a four-dimensional Lorentzian manifold for modeling spacetime, where tangent vectors can be classified as timelike, null, and spacelike.

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An Approach for Imitation Learning on Riemannian Manifolds

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