Geometric Structures for Learning and Optimization in Robotics

This article presents an overview of geometric approaches to facilitate the acquisition and transfer of robot skills. It focuses on three complementary geometric frameworks: signed distance fields, geometric algebra, and Riemannian geometry, which provide representations facilitating learning, planning, control, and optimization problems in robotics. The first consists of representing shapes in an implicit manner through the use of a distance function, where different approaches can be used to encode and learn this function. The second, geometric algebra, is linked to Clifford algebra and allows basic geometric primitives to be treated in a unified manner, including 6D poses, planes, lines, circles, and spheres, which can represent various forms of constraints in robot applications. The third leverages the use of Riemannian manifolds to extend models and algorithms originally developed for standard Euclidean data to curved spaces. These manifolds can represent a variety of geometric objects in robotics, not only for structured objects such as spheres, matrices, and subspaces, but also for more generic smooth manifolds described by a Riemannian metric to measure distances. The article discusses the distinctions and connections between these geometric approaches and shows how they can contribute to various problems in robotics, with a focus on manipulation tasks.