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Topological Obstructions to the Existence of Control Barrier Functions

Massimiliano de Sa, Aaron D. Ames

Abstract

In 1983, Brockett developed a topological necessary condition for the existence of continuous, asymptotically stabilizing control laws. Building upon recent work on necessary conditions for set stabilization, we develop Brockett-like necessary conditions for the existence of control barrier functions (CBFs). By leveraging the unique geometry of CBF safe sets, we provide simple and self-contained derivations of necessary conditions for the existence of CBFs and their safe, continuous controllers. We demonstrate the application of these conditions to instructive examples and kinematic nonholonomic systems, and discuss their relationship to Brockett's necessary condition.

Topological Obstructions to the Existence of Control Barrier Functions

Abstract

In 1983, Brockett developed a topological necessary condition for the existence of continuous, asymptotically stabilizing control laws. Building upon recent work on necessary conditions for set stabilization, we develop Brockett-like necessary conditions for the existence of control barrier functions (CBFs). By leveraging the unique geometry of CBF safe sets, we provide simple and self-contained derivations of necessary conditions for the existence of CBFs and their safe, continuous controllers. We demonstrate the application of these conditions to instructive examples and kinematic nonholonomic systems, and discuss their relationship to Brockett's necessary condition.
Paper Structure (19 sections, 9 theorems, 15 equations, 3 figures, 1 table)

This paper contains 19 sections, 9 theorems, 15 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Differential Geometry
  4. Nonlinear Control Systems
  5. Control Barrier Functions & Safe Set Geometry
  6. Brockett's Necessary Condition
  7. The Zeros of Vector Fields
  8. Ideas from Topology
  9. Homotopy
  10. CW Complexes
  11. Euler Characteristic
  12. Application to Zeros of Vector Fields
  13. Topological Obstructions for CBFs
  14. Non-strict CBF on the Safe Set
  15. Non-strict CBF on a Neighborhood of the Safe Set
  16. ...and 4 more sections

Key Result

Theorem 1

Let $X \in \mathfrak X^0(\mathcal{M})$ be uniquely integrable with flow $\varphi$ and $\mathcal{C}$ a regular domain in $\mathcal{M}$.

Figures (3)

  • Figure 1: For a safe set $\mathcal{C}$ of a certain geometry and a safe vector field $X$, small perturbations$Z$ (in red) of $X$ satisfy $X_p - Z_p = 0_p$ for some point $p$. This observation can be used to derive necessary conditions for safety.
  • Figure 2: We construct $\mathbb S^2$ as a finite CW complex. First, construct an equator by gluing the boundary of a 1-cell to a $0$-cell. Then, glue the boundaries of two 2-cells to the equator to form $\mathbb S^2$. This construction shows $\chi(\mathbb S^2) = 2$.
  • Figure 3: By flowing out from $\mathcal{C}$, we can inflate $\mathcal{C}$ to a new set $\tilde{\mathcal{C}}$ diffeomorphic to $\mathcal{C}$, on which a safe vector field for $\mathcal{C}$ is inward-pointing.

Theorems & Definitions (27)

  • Definition 1
  • Example 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • proof
  • Definition 4
  • Example 2
  • Proposition 1
  • Remark 1
  • ...and 17 more