Distributed Nash Equilibrium Seeking Algorithm in Aggregative Games for Heterogeneous Multi-Robot Systems

Abstract: This paper develops a distributed Nash Equilibrium seeking algorithm for heterogeneous multi-robot systems. The algorithm utilises distributed optimisation and output control to achieve the Nash equilibrium by leveraging information shared among neighbouring robots. Specifically, we propose a distributed optimisation algorithm that calculates the Nash equilibrium as a tailored reference for each robot and designs output control laws for heterogeneous multi-robot systems to track it in an aggregative game. We prove that our algorithm is guaranteed to converge and result in efficient outcomes. The effectiveness of our approach is demonstrated through numerical simulations and empirical testing with physical robots.

• The paper proposes a distributed Nash equilibrium seeking algorithm that uses martingale-based optimization to converge in heterogeneous multi-robot systems.
• It integrates output regulation techniques to manage diverse local dynamics and maintain robust trajectory tracking under communication disruptions.
• Simulations and real-world experiments validate the algorithm’s scalability, effectiveness, and robustness in both small-scale and large-scale robot networks.

Distributed Nash Equilibrium Seeking Algorithm in Aggregative Games for Heterogeneous Multi-Robot Systems

Introduction

The paper "Distributed Nash Equilibrium Seeking Algorithm in Aggregative Games for Heterogeneous Multi-Robot Systems" (2509.15597) presents a novel algorithm designed to tackle the challenges associated with achieving Nash equilibrium in distributed multi-agent systems. It targets heterogeneous multi-robot systems where each agent, - a robot with unique dynamics, seeks to enhance its performance through distributed optimization and cooperative strategies in aggregative games.

Problem Formulation

The central challenge addressed involves developing a distributed Nash Equilibrium Seeking (NES) algorithm for multi-agent systems in aggregative games. The agents operate under local dynamics, differing in terms of system characteristics, which are modeled as linear systems with distinct state transitions, control inputs, and output dynamics. The objective is for each agent to optimize its strategy, considering both its objectives and the aggregate strategies of others, leading to a Nash equilibrium where no agent benefits unilaterally by altering its strategy.

Algorithm Design

The proposed algorithm employs a two-stage approach:

1. Nash Equilibrium Seeking: Utilizes distributed optimization influenced by martingale theory to converge to a Nash equilibrium. It innovatively uses local estimations of aggregate strategies shared among agents to ensure convergence.
2. Tracking Control: Leverages output regulation techniques tailored for heterogeneous linear systems to ensure agents follow the generated reference trajectory accurately. This stage accounts for each system's intrinsic dynamics to ensure robustness in real-world applications.

   Figure 1: Numerical Simulation Results of Effectiveness.

Results and Validation

The effectiveness and applicability of the proposed algorithm are showcased via a series of simulations and real-world experiments:

1. Numerical Simulations:
   • Effectiveness: Demonstrated with a scenario involving six heterogeneous robots, the algorithm successfully guided each to their Nash equilibrium, as reflected in their trajectory convergence.
   • Scalability: When scaled to a network of 200 robots, the algorithm maintained robust coordination without degrading performance, evidencing its scalability.
   • Robustness: Tested under communication disruptions, the system maintained convergence, proving its robustness to dynamic network conditions.
2. Real-World Experiments:
   • Using Turtlebot robots in a GAZEBO simulation, the algorithm effectively guided robots to optimal strategies, even under constraints of limited communication.
   • In laboratory experiments, the algorithm successfully coordinated real robots to optimize positioning, showcasing practical feasibility and adaptability.

     Figure 2: Turtlebots Waffle Pi Simulation Results.

Implications and Future Work

Theoretical analyses provided in the paper confirm the convergence and stability of the proposed algorithm, making it a viable solution for diverse applications ranging from autonomous vehicular systems to complex industrial robotics where heterogeneous agent dynamics are prevalent.

Future research can explore extending the algorithm's applicability to even more complex dynamic environments, integrating nonlinear dynamics handling, and further investigating real-time adaptability and learning capabilities in rapidly changing conditions.

Figure 3: Trajectory of real robots.

Conclusion

The innovative approach of combining distributed optimization with output regulation provides a robust framework for achieving Nash equilibria in multi-agent systems with heterogeneous dynamics. The demonstrated feasibility and practical implementation potential suggest wide applicability in real-world scenarios, providing an essential tool in the advancement of autonomous and cooperative multi-robot systems.