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Inverse Safety Filtering: Inferring Constraints from Safety Filters for Decentralized Coordination

Minh Nguyen, Jingqi Li, Gechen Qu, Claire J. Tomlin

Abstract

Safe multi-agent coordination in uncertain environments can benefit from learning constraints from other agents. Implicitly communicating safety constraints through actions is a promising approach, allowing agents to coordinate and maintain safety without expensive communication channels. This paper introduces an online method to infer constraints from observing the safety-filtered actions of other agents. We approach the problem by using safety filters to ensure forward safety and exploit their structure to work backwards and infer constraints. We provide sufficient conditions under which we can infer these constraints and prove that our inference method converges. This constraint inference procedure is coupled with a decentralized planning method that ensures safety when the constraint activation distance is sufficiently large. We then empirically validate our method with Monte Carlo simulations and hardware experiments with quadruped robots.

Inverse Safety Filtering: Inferring Constraints from Safety Filters for Decentralized Coordination

Abstract

Safe multi-agent coordination in uncertain environments can benefit from learning constraints from other agents. Implicitly communicating safety constraints through actions is a promising approach, allowing agents to coordinate and maintain safety without expensive communication channels. This paper introduces an online method to infer constraints from observing the safety-filtered actions of other agents. We approach the problem by using safety filters to ensure forward safety and exploit their structure to work backwards and infer constraints. We provide sufficient conditions under which we can infer these constraints and prove that our inference method converges. This constraint inference procedure is coupled with a decentralized planning method that ensures safety when the constraint activation distance is sufficiently large. We then empirically validate our method with Monte Carlo simulations and hardware experiments with quadruped robots.

Paper Structure

This paper contains 25 sections, 6 theorems, 25 equations, 7 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Inferring missing information in decentralized control
  4. Safe planning methods
  5. Multi-agent planning under incomplete information
  6. Problem Formulation
  7. Parameterized Constraints
  8. Decentralized Constraint Inference Problem
  9. Inverting CBF Safety Filters
  10. KKT Optimality Conditions
  11. Closed-Form Solution
  12. Identifiability and Uniqueness
  13. Inverting Safety Filters under Formation Constraints
  14. Formation Constraints
  15. Inference with Multiple Active Constraints
  16. ...and 10 more sections

Key Result

Theorem D.1

Suppose the safety filter constraint is active, $B_s \in \mathbb{R}^{d \times d}$ is invertible, and $Q \succ 0$. Define $e \coloneqq s_t - s_{t+1}$. Define $A = \gamma\, \hat{d}^\top Q \hat{d}$, $B = -2(1{-}\gamma)(e^\top Q \hat{d})$, and $C = -(1{-}\gamma)\|e\|_Q^2 - \gamma r^2$. The obstacle posi $\blacktriangleleft$$\blacktriangleleft$

Figures (7)

  • Figure A1: Applying our decentralized constraint inference and planning method to a furniture-moving task. Agents share a goal but possess asymmetric knowledge of environmental constraints (Agent 1 initially knows of the obstacle but Agent 2 does not). Our method enables decentralized constraint inference and safe planning, with theoretical guarantees on constraint identifiability and safety without direct communication among agents.
  • Figure G1: A single trial from a Monte-Carlo rollout where we generate random obstacles and goal positions for two double integrator agents connected by a formation constraint. We compare two inference methods (KKT vs. IM) and two obstacle-avoidance formulations (CBF vs. Circle). We observe that CBF + KKT is able to avoid all collisions while the other methods have at least one collision.
  • Figure G2: Two teams of two agents cross diagonally toward opposite goals (left). The robust CBF constraint enables decentralized collision-free crossing, with the left team yielding. The distance plot (right) confirms the inter-team distance never drops below the safety threshold.
  • Figure G3: Two simulations showing 3 (left) and 4 (right) agent teams navigating through obstacles. The color of the obstacles matches with the agent that knows of it. Both scenarios are able to infer and avoid the obstacles, successfully reaching their goal positions.
  • Figure G4: Convergence regions of the regularized Newton method (left) and Input Matching (right). White denotes divergence. Newton's method converges from most initializations, whereas Input Matching requires initialization near the true obstacle.
  • ...and 2 more figures

Theorems & Definitions (12)

  • Theorem D.1: Closed-Form Solution
  • proof
  • Theorem D.2: Constraint Identifiability
  • proof
  • Corollary D.3: Global Uniqueness for Quadratic Barriers
  • proof
  • Theorem E.1: Newton Convergence
  • proof
  • Theorem F.2: Decentralized Safety Guarantee
  • proof
  • ...and 2 more