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

1. Problem Statement

The paper addresses the challenge of safe multi-agent coordination in uncertain environments where agents possess asymmetric information (i.e., some agents know about obstacles or constraints that others do not).

• The Core Issue: In decentralized systems, communication is often limited, unreliable, or bandwidth-constrained. Traditional centralized planning is impractical. While Control Barrier Functions (CBFs) are widely used to ensure safety by minimally modifying control inputs, standard approaches assume agents already know all constraints.
• The Gap: Existing methods for inferring constraints (Inverse Optimal Control, Inverse Reinforcement Learning) are typically offline and require expert demonstrations. There is a lack of online, model-based methods that allow agents to infer hidden environmental constraints (like obstacle locations) simply by observing the safety-filtered actions of their teammates in real-time.

2. Methodology

The authors propose a framework where agents act as either demonstrators (who know the constraints and filter their actions) or learners (who observe the filtered actions to infer the hidden constraints).

A. Constraint Inference via Inverse Safety Filtering

The core insight is that safety filters (specifically CBF-based) modify nominal control inputs ($u_{nom}$) to satisfy safety constraints, resulting in a safe action ($u_{safe}$). The deviation $\Delta u = u_{safe} - u_{nom}$ contains information about the active constraint.

1. Single Active Constraint (Obstacle Only):

   • The authors exploit the Karush–Kuhn–Tucker (KKT) optimality conditions of the safety filter optimization problem.
   • For quadratic CBF constraints (e.g., circular obstacles), they derive a closed-form solution to recover the obstacle parameter $\theta$ (e.g., obstacle center) directly from the control deviation $\Delta u$, the current state, and the system dynamics.
   • Theorem: They prove that under conditions of sufficient actuation and constraint activity, the parameter is locally identifiable and globally unique for quadratic barriers.

2. Multiple Active Constraints (Obstacle + Formation):

   • When agents must maintain a formation while avoiding obstacles, the safety filter has multiple active constraints, making the inverse problem underdetermined.
   • The authors formulate this as a non-linear system of equations involving the obstacle parameter and Lagrange multipliers for both constraints.
   • They propose a regularized Newton solver to find the solution. They prove that this solver converges to the true constraint parameters given a sufficiently large "constraint activation distance" (the distance at which the filter activates) and proper initialization.

B. Decentralized Round-Robin Planning

To handle the dependency of learners on the future states of demonstrators (which are unknown in a fully decentralized setting), the paper introduces a Round-Robin execution framework:

• At each time step, one agent acts as the demonstrator (using its private knowledge of obstacles), while the others act as learners (using only public knowledge).
• Safety Guarantee: The demonstrator uses an inflated safety radius ($r_{demo} = r + d + \epsilon$) to ensure that even if learners are at the maximum allowed formation distance, they remain safe from the obstacle.
• Moving Obstacles: The framework is extended to handle moving obstacles (other teams) by using a Robust CBF formulation that accounts for worst-case obstacle velocities, effectively inflating the safety radius based on the obstacle's maximum speed.

3. Key Contributions

1. Provably Convergent Inference Method: A novel online method to infer constraint parameters from safety-filtered actions, providing a closed-form solution for single constraints and a convergent Newton solver for multiple constraints.
2. Generalization to Multi-Agent Formations: Extending the inference capability to scenarios where agents must simultaneously satisfy formation constraints and obstacle avoidance.
3. Decentralized Safety Framework: A round-robin planning algorithm that enables safe coordination without direct communication, with formal guarantees that agents remain in the safe set if they start in one and the formation distance is sufficiently small relative to the constraint activation distance.
4. Hardware Validation: Successful implementation on physical quadruped robots (Unitree Go2), demonstrating real-time applicability.

4. Experimental Results

The authors validated their approach through Monte Carlo simulations and hardware experiments.

• Simulation (Monte Carlo):

  • Comparison: Compared against Input Matching (IM) and non-CBF circular constraints.
  • Metrics: Collisions, "Ghosts" (false inferences), inference error, and discovery rate.
  • Results: The proposed CBF + KKT method achieved 0.1 collisions (vs. 5.6+ for non-CBF methods) and a 90% discovery rate of true constraints. It significantly outperformed the IM method in convergence region and inference accuracy.
  • Scalability: Demonstrated successful coordination for teams of 3 and 4 agents and interactions between multiple teams crossing paths.

• Hardware Experiments:

  • Setup: Two Unitree Go2 quadruped robots connected by a rope (formation constraint). One robot (rear) knew the obstacle; the other (front) had to infer it.
  • Scenarios: Navigating around a circular obstacle and through a narrow gap between walls.
  • Outcome: The front robot successfully inferred the obstacle location from the rear robot's filtered actions and navigated safely without collision, confirming the method works in real-time with physical dynamics and noise.

5. Significance

This work bridges a critical gap in multi-agent robotics by enabling implicit communication of safety constraints.

• Communication Efficiency: It eliminates the need for explicit data transmission regarding map information or obstacle locations, relying instead on the "behavior" of agents.
• Robustness: By using CBFs, the method provides formal safety guarantees even in the presence of uncertainty and dynamic environments.
• Practicality: The transition from theoretical derivation to hardware deployment on quadruped robots proves that inverse safety filtering is computationally feasible for real-world robotic systems, paving the way for more autonomous and cooperative robot teams in complex, unstructured environments.