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

TL;DR

The paper addresses safety-critical control for systems evolving on manifolds by developing a geometric control barrier function (CBF) framework on bundles, enabling forward-invariance guarantees beyond Euclidean spaces. It introduces a global, closed-form CBF-QP controller for control-affine systems on vector bundles and a smooth counterpart, and a backstepping-based CBF synthesis that lifts configuration constraints to the tangent bundle $TQ$ on Riemannian manifolds without resorting to higher-order tangents. Key contributions include (i) a rigorous CBF theory on bundles with safety conditions adapted to geometric settings, (ii) a constructive, globally-defined safety filter, and (iii) a backstepping method for geometric mechanical systems with an underactuated satellite on $SO(3)$ as a concrete demonstration. The approach enhances safety guarantees for robotics and aerospace applications involving non-Euclidean state spaces, offering computationally tractable controllers and scalable synthesis techniques for manifold-valued dynamics.

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).

Figures (3)

• Figure 1: Using a Riemannian formulation of backstepping, we can lift a safe configuration set $\mathop{\mathrm{\mathcal{C}}}\nolimits_0 \subseteq Q$ on the configuration manifold of a geometric mechanical system to a control barrier function on its tangent bundle.
• Figure 2: In our global geometric setting, a nonlinear control system is defined on a bundle $(\pi, \mathcal{U}, \mathcal{M})$. Fibers $\mathcal{U}_p$ are spaces of control inputs which can be applied at $p$. A controller $\kappa$ is a map taking $p \in \mathcal{M}$ to $\kappa(p) \in \mathcal{U}_p$; above, the image of $\mathcal{M}$ under a controller $\kappa$ is drawn in orange.
• Figure 3: The safe set on $SO(3)$ is visualized by projecting to the sphere via $R \in SO(3) \mapsto Re_3 \in \mathop{\mathrm{\mathbb S}}\nolimits^2$. A geometric tracking controller bullo2019geometric for an unsafe trajectory (plotted as a dotted blue line), is filtered by a CBF-QP controller. The resulting trajectory, plotted in red, maintains a safe configuration for all time. The evolution of $R \in SO(3)$ is visualized as a sequence of frames, with the red, green, and blue axes representing $Re_1, Re_2, Re_3$, respectively.

Theorems & Definitions (30)

• Definition 1
• Definition 2
• Example 1
• Definition 3
• Theorem 1
• proof
• Definition 4
• Remark 1
• Definition 5
• Example 2