From Bundles to Backstepping: Geometric Control Barrier Functions for Safety-Critical Control on Manifolds

TL;DR

This paper extends control barrier functions to manifolds, providing a geometric theory and synthesis techniques for safety-critical control in robotics and aerospace systems, demonstrated on a satellite example.

Contribution

It introduces a general geometric CBF theory on bundles and a constructive backstepping method on Riemannian manifolds for mechanical systems.

Findings

Unified geometric CBF framework for manifolds

Constructive CBF synthesis for mechanical systems

Application to underactuated satellite on SO(3)

Abstract

Control barrier functions (CBFs) have a well-established theory in Euclidean spaces, yet still lack general formulations and constructive synthesis tools for systems evolving on manifolds common in robotics and aerospace applications. In this paper, we develop a general theory of geometric CBFs on bundles and, for control-affine systems, recover the standard optimization-based CBF controllers and their smooth analogues. Then, by generalizing kinetic energy-based CBF backstepping to Riemannian manifolds, we provide a constructive CBF synthesis technique for geometric mechanical systems, as well as easily verifiable conditions under which it succeeds. Further, this technique utilizes mechanical structure to avoid computations on higher-order tangent bundles. We demonstrate its application to an underactuated satellite on SO(3).