When planning motions in a configuration space that has underlying symmetries (e.g. when manipulating one or multiple symmetric objects), the ideal planning algorithm should take advantage of those symmetries to produce shorter trajectories. However, finite symmetries lead to complicated changes to the underlying topology of configuration space, preventing the use of standard algorithms. We demonstrate how the key primitives used for sampling-based planning can be efficiently implemented in spaces with finite symmetries. A rigorous theoretical analysis, building upon a study of the geometry of the configuration space, shows improvements in the sample complexity of several standard algorithms. Furthermore, a comprehensive slate of experiments demonstrates the practical improvements in both path length and runtime.

In order for a bimanual robot to manipulate an object that is held by both hands, it must construct motion plans such that the transformation between its end effectors remains fixed. This amounts to complicated nonlinear equality constraints in the configuration space, which are difficult for trajectory optimizers. In addition, the set of feasible configurations becomes a measure zero set, which presents a challenge to sampling-based motion planners. We leverage an analytic solution to the inverse kinematics problem to parametrize the configuration space, resulting in a lower-dimensional representation where the set of valid configurations has positive measure. We describe how to use this parametrization with existing motion planning algorithms, including sampling-based approaches, trajectory optimizers, and techniques that plan through convex inner-approximations of collision-free space.

We present a new technique that enables manifold learning to accurately embed data manifolds that contain holes, without discarding any topological information. Manifold learning aims to embed high-dimensional data into a lower dimensional Euclidean space by learning a coordinate chart, but it requires that the entire manifold can be embedded in a single chart. This is impossible for manifolds with holes. In such cases, it is necessary to learn an atlas: a collection of charts that collectively cover the entire manifold. We begin with many small charts, and combine them in a bottom-up approach, where charts are only combined if doing so will not introduce problematic topological features. When it is no longer possible to combine any charts, each chart is individually embedded with standard manifold learning techniques, completing the construction of the atlas. We show the efficacy of our method by constructing atlases for challenging synthetic manifolds; learning human motion embeddings from motion capture data; and learning kinematic models of articulated objects.