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SafeSpace: Aggregating Safe Sets from Backup Control Barrier Functions under Input Constraints

Pio Ong, David E. J. van Wijk, Massimiliano de Sa, Joel W. Burdick, Aaron D. Ames

Abstract

Control barrier functions (CBFs) provide a principled framework for enforcing safety in control systems -- yet the certified safe operating region in practice is often conservative, especially under input bounds. In many applications, multiple smaller safe sets can be certified independently, e.g., around distinct equilibria with different stabilizing controllers. This paper proposes a framework for uniting such regions into a single certified safe set using \emph{combinatorial CBFs}. We refine the combinatorial CBF framework by introducing an auxiliary variable that enables logical compositions of individual CBFs. In the proposed framework, we show that such compositions yield a \emph{generalized combinatorial CBF} under a condition termed \emph{conjunctive compatibility}. Building on this result, we extend the framework to enable the aggregation of multiple implicit safe sets generated by the backup CBF framework. We show that the resulting CBF-based quadratic program yields a continuous safety filter over the aggregated safe region. The approach is demonstrated on two spacecraft safety problems, safe attitude control and safe station keeping, where multiple certified safe regions are combined to expand the operational envelope.

SafeSpace: Aggregating Safe Sets from Backup Control Barrier Functions under Input Constraints

Abstract

Control barrier functions (CBFs) provide a principled framework for enforcing safety in control systems -- yet the certified safe operating region in practice is often conservative, especially under input bounds. In many applications, multiple smaller safe sets can be certified independently, e.g., around distinct equilibria with different stabilizing controllers. This paper proposes a framework for uniting such regions into a single certified safe set using \emph{combinatorial CBFs}. We refine the combinatorial CBF framework by introducing an auxiliary variable that enables logical compositions of individual CBFs. In the proposed framework, we show that such compositions yield a \emph{generalized combinatorial CBF} under a condition termed \emph{conjunctive compatibility}. Building on this result, we extend the framework to enable the aggregation of multiple implicit safe sets generated by the backup CBF framework. We show that the resulting CBF-based quadratic program yields a continuous safety filter over the aggregated safe region. The approach is demonstrated on two spacecraft safety problems, safe attitude control and safe station keeping, where multiple certified safe regions are combined to expand the operational envelope.

Paper Structure

This paper contains 11 sections, 5 theorems, 38 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Safety via Control Barrier Functions
  4. Safety Filter Framework
  5. Set Expansion via Backup CBF Framework
  6. Uniting Multiple Safe Sets
  7. Uniting Multiple Implicit Safe Sets
  8. Applications to Space Systems
  9. Spacecraft attitude
  10. Safe Spacecraft Station Keeping
  11. Conclusion

Key Result

Lemma 1

Suppose that a function $h_j:\mathbb{R}^n \rightarrow \mathbb{R}$ is a CBF for sys:ctrl_affine under a given input constraint $\mathcal{U}\subseteq\mathbb{R}^m$. Then, for any function $h:\mathbb{R}^n \rightarrow \mathbb{R}$ and any positive definite function $\rho:\mathbb{R} \rightarrow \mathbb{R}_ holds for any $\mathbf{x}\in\mathcal{C}$ in the zero-superlevel set of $h$ in eq:safe_set. $\blackt

Figures (2)

  • Figure 1: Simulation results for the satellite are depicted on the sphere under the projection $\mathbf{R} \mapsto \mathbf{R}\mathbf{e}_3$. The satellite is commanded to track a trajectory circling around the border of the safe set. Five backup sets are used, with the level set of each $h_j$ plotted as a solid circle and its implicit safe set as a dotted line. Using the proposed combinatorial bCBF, plotted in green, both safety and convergence to the boundary are achieved. As evidenced by the red trajectory, convergence is not possible with a single backup set.
  • Figure 2: Simulation results for safe station keeping comparing the standard CBF (a), the standard bCBF (b), the proposed combinatorial CBF (c) and the proposed combinatorial bCBF (d). While all four approaches guarantee the safety of the spacecraft by adhering to the keep-in and keep-out constraints (a-d), and obeying the input bounds (e,$\mkern2mu$f) they vary in mission performance. By using a CBF without expansion (a,$\mkern2mu$b), the spacecraft cannot approach the desired orbit (dashed black), though using multiple CBFs without expansion allows a closer approach (b). Expanding a single CBF with the backup method allows for improved orbit tracking (c), but expanding with multiple backup controllers and CBFs using the proposed approach yields the largest control invariant safe set, and thus achieves superior orbit tracking (d).

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • proof
  • Definition 4
  • Definition 5
  • Proposition 1
  • proof
  • Remark 1
  • ...and 7 more