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Allocating Corrective Control to Mitigate Multi-agent Safety Violations Under Private Preferences

Johnathan Corbin, Sarah H. Q. Li, Jonathan Rogers

TL;DR

The paper addresses safety in multi-agent systems where agents have private preferences by pairing high-order control barrier functions ($HOCBF$) with a privacy-preserving progressive second-price ($PSP$) auction to allocate the corrective control burden as avoidance credits. It maps the safety requirement into a divisible resource and uses PSP (VCG-based payments) to compute a socially optimal, incentive-compatible allocation without revealing private valuations. After auction convergence, agents implement their assigned burden via a minimum-norm control adjustment, ensuring forward invariance of the safety set. Hardware experiments on the Robotarium with four unicycle-like robots demonstrate that the auction-based allocation preserves safety while distributing burden according to evolving private valuations, improving fairness without increasing total effort.

Abstract

We propose a novel framework that computes the corrective control efforts to ensure joint safety in multi-agent dynamical systems. This framework efficiently distributes the required corrective effort without revealing individual agents' private preferences. Our framework integrates high-order control barrier functions (HOCBFs), which enforce safety constraints with formal guarantees of safety for complex dynamical systems, with a privacy-preserving resource allocation mechanism based on the progressive second price (PSP) auction. When a joint safety constraint is violated, agents iteratively bid on new corrective efforts via 'avoidance credits' rather than explicitly solving for feasible corrective efforts that remove the safety violation. The resulting correction, determined via a second price payment rule, coincides with the socially optimal safe distribution of corrective actions. Critically, the bidding process achieves this optimal allocation efficiently and without revealing private preferences of individual agents. We demonstrate this method through multi-robot hardware experiments on the Robotarium platform.

Allocating Corrective Control to Mitigate Multi-agent Safety Violations Under Private Preferences

TL;DR

The paper addresses safety in multi-agent systems where agents have private preferences by pairing high-order control barrier functions () with a privacy-preserving progressive second-price () auction to allocate the corrective control burden as avoidance credits. It maps the safety requirement into a divisible resource and uses PSP (VCG-based payments) to compute a socially optimal, incentive-compatible allocation without revealing private valuations. After auction convergence, agents implement their assigned burden via a minimum-norm control adjustment, ensuring forward invariance of the safety set. Hardware experiments on the Robotarium with four unicycle-like robots demonstrate that the auction-based allocation preserves safety while distributing burden according to evolving private valuations, improving fairness without increasing total effort.

Abstract

We propose a novel framework that computes the corrective control efforts to ensure joint safety in multi-agent dynamical systems. This framework efficiently distributes the required corrective effort without revealing individual agents' private preferences. Our framework integrates high-order control barrier functions (HOCBFs), which enforce safety constraints with formal guarantees of safety for complex dynamical systems, with a privacy-preserving resource allocation mechanism based on the progressive second price (PSP) auction. When a joint safety constraint is violated, agents iteratively bid on new corrective efforts via 'avoidance credits' rather than explicitly solving for feasible corrective efforts that remove the safety violation. The resulting correction, determined via a second price payment rule, coincides with the socially optimal safe distribution of corrective actions. Critically, the bidding process achieves this optimal allocation efficiently and without revealing private preferences of individual agents. We demonstrate this method through multi-robot hardware experiments on the Robotarium platform.
Paper Structure (20 sections, 1 theorem, 29 equations, 3 figures, 2 tables)

This paper contains 20 sections, 1 theorem, 29 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Research
  3. Preliminaries and Problem Formulation
  4. Notation
  5. Agent Dynamics and Nominal Control
  6. Multi-agent Safety and Problem Formulation
  7. Safety Enforcement via HOCBFs
  8. Safety Allocation Mechanism
  9. Safety Deficit and Responsibility Allocation
  10. An Auction Game for Optimal Allocation
  11. Control Synthesis from Optimal Allocation
  12. Experimental Demonstration
  13. Experimental Model
  14. HOCBF Constraint Derivation
  15. Valuation Function Design
  16. ...and 5 more sections

Key Result

Lemma 1

Let $\tilde{H}(\mathbf{x}, t)$ be the LSE approximation eq:lse_approx of $H(\mathbf{x}, t)$eq:safe_set with relative degree $m$. The safe set $\mathcal{C}(t)$eq:safe_set is rendered forward invariant $\forall t \geq 0$ if the joint control $\mathbf{u}(t)$ satisfies the affine constraint where $A \coloneqq L_G L_f^{m-1}\tilde{H}$ and $b \coloneqq -L_f^m\tilde{H}-\frac{\partial^m \tilde{H}}{\partia

Figures (3)

  • Figure 1: Agent trajectories for the 4-agent collision scenario with arrows indicating direction of travel. (a) With the baseline QP-HOCBF controller, Agent 1 (blue) is forced to deviate significantly in all three collision events. (b) With our proposed Auction-HOCBF controller, Agent 1 deviates significantly in the first collision, but its dynamically increasing valuation function causes other agents to bear more of the burden in subsequent collisions.
  • Figure 2: Snapshot of the hardware experiment on the Robotarium platform for the system utilizing the Auction-HOCBF controller during the third collision event.
  • Figure 3: Stacked area plot of safety correction ($\Delta_i$) over time. The combined height of the stacks corresponds to the instantaneous safety deficit $S(t)$ (black dashed line). (a) The baseline QP-HOCBF divides the effort in a similar way for each collision. (b) Our Auction-CBF dynamically shifts the effort allocation in later collisions as Agent 1's valuation increases.

Theorems & Definitions (6)

  • Definition 1: Private Valuation Function
  • Definition 2: Relative Degree
  • Lemma 1: Sufficient Condition for Forward Invariance
  • proof
  • Definition 3: Avoidance Credit Auction Game
  • Definition 4: Nash Equilibrium