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Steering with Contingencies: Combinatorial Stabilization and Reach-Avoid Filters

Yana Lishkova, Pio Ong, Sander Tonkens, Sylvia Herbert, Aaron D. Ames

Abstract

In applications such as autonomous landing and navigation, it is often desirable to steer toward a target while retaining the ability to divert to at least $r$ (out of $p$) alternative sites if conditions change. In this work, we formalize this combinatorial contingency requirement and develop tractable control filters for enforcement. Combinatorial stabilization requires asymptotic stability of a selected equilibrium while ensuring the trajectory remains within the safe region of attraction of at least $r$-out-of-$p$ candidates. To enforce this requirement, we use control Lyapunov functions (CLFs) to construct regions of attraction, which are combined combinatorially within an optimization-based filter. Combinatorial targeting extends this framework to finite-horizon problems using Hamilton-Jacobi backward reach-avoid sets, accommodating shrinking reachable regions due to finite horizons or resource depletion. In both formulations, the resulting combinatorial stability filter and combinatorial reach-avoid filter require only $p+1$ constraints, preventing combinatorial blow-up and enabling safe real-time switching between targets. The framework is demonstrated on two examples where the filters ensure steering with contingency and enable safe diversion.

Steering with Contingencies: Combinatorial Stabilization and Reach-Avoid Filters

Abstract

In applications such as autonomous landing and navigation, it is often desirable to steer toward a target while retaining the ability to divert to at least (out of ) alternative sites if conditions change. In this work, we formalize this combinatorial contingency requirement and develop tractable control filters for enforcement. Combinatorial stabilization requires asymptotic stability of a selected equilibrium while ensuring the trajectory remains within the safe region of attraction of at least -out-of- candidates. To enforce this requirement, we use control Lyapunov functions (CLFs) to construct regions of attraction, which are combined combinatorially within an optimization-based filter. Combinatorial targeting extends this framework to finite-horizon problems using Hamilton-Jacobi backward reach-avoid sets, accommodating shrinking reachable regions due to finite horizons or resource depletion. In both formulations, the resulting combinatorial stability filter and combinatorial reach-avoid filter require only constraints, preventing combinatorial blow-up and enabling safe real-time switching between targets. The framework is demonstrated on two examples where the filters ensure steering with contingency and enable safe diversion.

Paper Structure

This paper contains 16 sections, 7 theorems, 34 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Control Lyapunov functions
  4. Control barrier functions
  5. Combinatorial control barrier functions
  6. Combinatorial Stabilization
  7. Combinatorial Stabilization via CLFs
  8. Combinatorially stabilizing controllers
  9. Discussion on results
  10. Combinatorial Backward Reachability
  11. Combinatorial reach-avoid value functions
  12. Combinatorial targeting controllers
  13. Simulations
  14. Example 1: Linear System (CLF-based)
  15. Example 2: Airplane Landing (HJR-based)
  16. ...and 1 more sections

Key Result

Lemma 1

Let $V$ be a local CLF for system sys:ctrl_affine, and let $\mathbf{k}:\mathbb{R}^n \rightarrow \mathbb{R}^m$ be a continuous feedback controller such that and $\mathbf{k}(\mathbf{x}^\star)\!=\!\mathbf{u}^\star$ for all $\mathbf{x}\!\in\! B_r(\mathbf{x}^\star)\!\setminus\!\{\mathbf{x}^\star\}$. Then the controller $\mathbf{k}$ asymptotically stabilizes the equilibrium $\mathbf{x}^\star$. $\blacks

Figures (2)

  • Figure D1: Example 1: Linear system with $p=3$ targets. The simulation starts with $j^\dagger=1$ and $r=2$, switches to $j^\dagger=2$ at $t=0.5\,\mathrm{s}$, and later raises $r$ to $3$. The filtered trajectory remains within $r$-out-of-$p$ safe sets throughout, while the nominal trajectory violates obstacle constraints.
  • Figure E1: Example 2: Simplified aircraft with $p=6$ runway targets navigating between obstacles. Dashed line shows the unfiltered trajectory, while solid lines show the filtered trajectories colored by a currently active contingency target. Four cases are presented with varying $r$ and horizon times $\tau_1(0), \tau_{2}(0)$. Reach-avoid $0$-superlevel sets are plotted at the state (and time) indicated by the arrow (and dotted line in inset). Insets in (a) show the evolution of the steering reach–avoid 0-superlevel set for the green runway for decreasing horizon $\tau_1$.

Theorems & Definitions (20)

  • Definition 1: Local control Lyapunov function
  • Lemma 1
  • Lemma 2
  • Definition 2: Control Barrier Function Ames19_CBFsOng25_combinatorialCBF
  • Lemma 3
  • Lemma 4
  • Definition 3
  • Definition 4
  • Theorem 1
  • proof
  • ...and 10 more