Composable Geometric Motion Policies using Multi-Task Pullback Bundle Dynamical Systems

TL;DR

This paper introduces a geometric framework for designing reactive robot motion policies on non-Euclidean manifolds, enabling task prioritization and constraint enforcement while avoiding local minima, demonstrated on a sphere and robotic arm.

Contribution

It presents the PBDS framework that simplifies task design and composition on manifolds, with a geometric optimization approach inspired by RMPs, and provides an open-source Julia implementation.

Findings

Demonstrated real-time performance on a manipulator arm at 300-500 Hz.

Effectively avoids local minima in potential functions for task goals.

Provides a modular, geometry-aware approach for reactive motion planning.

Abstract

Despite decades of work in fast reactive planning and control, challenges remain in developing reactive motion policies on non-Euclidean manifolds and enforcing constraints while avoiding undesirable potential function local minima. This work presents a principled method for designing and fusing desired robot task behaviors into a stable robot motion policy, leveraging the geometric structure of non-Euclidean manifolds, which are prevalent in robot configuration and task spaces. Our Pullback Bundle Dynamical Systems (PBDS) framework drives desired task behaviors and prioritizes tasks using separate position-dependent and position/velocity-dependent Riemannian metrics, respectively, thus simplifying individual task design and modular composition of tasks. For enforcing constraints, we provide a class of metric-based tasks, eliminating local minima by imposing non-conflicting potential functions only for goal region attraction. We also provide a geometric optimization problem for combining tasks inspired by Riemannian Motion Policies (RMPs) that reduces to a simple least-squares problem, and we show that our approach is geometrically well-defined. We demonstrate the PBDS framework on the sphere $\mathbb S^2$ and at 300-500 Hz on a manipulator arm, and we provide task design guidance and an open-source Julia library implementation. Overall, this work presents a fast, easy-to-use framework for generating motion policies without unwanted potential function local minima on general manifolds.