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Collaborative Altruistic Safety in Coupled Multi-Agent Systems

Brooks A. Butler, Xiao Tan, Aaron D. Ames, Magnus Egerstedt

Abstract

This paper presents a novel framework for ensuring safety in dynamically coupled multi-agent systems through collaborative control. Drawing inspiration from ecological models of altruism, we develop collaborative control barrier functions that allow agents to cooperatively enforce individual safety constraints under coupling dynamics. We introduce an altruistic safety condition based on the so-called Hamilton's rule, enabling agents to trade off their own safety to support higher-priority neighbors. By incorporating these conditions into a distributed optimization framework, we demonstrate increased feasibility and robustness in maintaining system-wide safety. The effectiveness of the proposed approach is illustrated through simulation in a simplified formation control scenario.

Collaborative Altruistic Safety in Coupled Multi-Agent Systems

Abstract

This paper presents a novel framework for ensuring safety in dynamically coupled multi-agent systems through collaborative control. Drawing inspiration from ecological models of altruism, we develop collaborative control barrier functions that allow agents to cooperatively enforce individual safety constraints under coupling dynamics. We introduce an altruistic safety condition based on the so-called Hamilton's rule, enabling agents to trade off their own safety to support higher-priority neighbors. By incorporating these conditions into a distributed optimization framework, we demonstrate increased feasibility and robustness in maintaining system-wide safety. The effectiveness of the proposed approach is illustrated through simulation in a simplified formation control scenario.

Paper Structure

This paper contains 11 sections, 4 theorems, 38 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Agent-Level Safety with Coupled Dynamics
  3. Coupled Dynamics
  4. Coupled Safety Condition Through High-Order Dynamics
  5. Distributed Optimization for Solving Safe Inputs
  6. Example Continued: Two Safety Constraints
  7. Altruistic Safety in Coupled Systems
  8. Safety-Relatedness in Coupled Systems
  9. Conditions for Altruistic Safety
  10. Example Continued: Feasibility with Altruism
  11. Conclusion

Key Result

Lemma 1

If $h_i$ is a CCBF, then, with $(u_i, u_{\mathcal{N}_i^+})$ satisfying the conditions eq:CCBF_cond and eq:CCBF_cond_2 applied, $\mathcal{C}_i$ is forward invariant. $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure B1: (Left) An example of a simple 2-agent system where $d=1$ with dynamics defined by \ref{['eq:2agent_1D_dyn']}. With no intervention by agent $1$ through $u_1$, agent $2$ will be driven by $u^f_2(x_1, x_2)$ to violate its safety constraint. (Right) A simulation of the example scenario shown on the left, with dynamics defined by \ref{['eq:2agent_1D_dyn']} and agent $1$ satisfying \ref{['eq:CCBF_cond']} for all time, where $x_2(0) = -x_1(0) = 0.3$, $r = 0.5$, $\alpha_2 = \beta_2 = 10$, $\xi=2.5$, and $\Delta=1.4$.
  • Figure B2: A simulation with the same initial conditions and parameterization as the simulation in Figure \ref{['fig:1D_example']}, but with dynamics defined by \ref{['eq:2agent_1D_dyn_2inputs']}. In (a), $k_1(x_1,x_2) = k_2(x_1,x_2) = 0$, whereas in (b), $k_i(\mathbf{x}_i)$ is defined by the "half-Sontag" feedback control law for both agents, as described in Remark \ref{['rem:k_i']}. The control inputs $u_1, u_2$ are obtained by solving \ref{['eq:prob_statement_distr']} with auxiliary variable updates defined by \ref{['eq:y_updates']}.
  • Figure C1: A simulation with the same initial conditions and parameterization as the simulation in Figure \ref{['fig:1D_example']}, but with dynamics defined by \ref{['eq:2agent_1D_dyn_2inputs']}. In (a), the control inputs $u_1, u_2$ are obtained by solving \ref{['eq:prob_statement_distr_alt']} with auxiliary variable updates defined by \ref{['eq:y_updates']}, where $\eta_1 = 1$ and $\eta_2=1000$. In (b), we compare the minimum $u_2$ that satisfies $u_2 \in \mathcal{U}_2^s(x_1, x_2, u_1)$ for two simulations: (red) $u_1, u_2$ are obtained by solving \ref{['eq:prob_statement_distr']}, and (green) $u_1, u_2$ are obtained by solving \ref{['eq:prob_statement_distr_alt']} with $\eta_1=1, \eta_2=1000$. We see that Theorem \ref{['thm:alt_feasibility']} holds, with the altruistic system that is biased towards agent 2 yielding a larger feasible set of safe inputs while both agents remain safe as they approach the boundary. The difference between the two simulations is plotted beneath in purple.

Theorems & Definitions (9)

  • Definition 1
  • Lemma 1
  • proof
  • Remark 1
  • Lemma 2: xiao2025continuous
  • Proposition 1
  • proof
  • Theorem 1
  • proof