Stackelberg Game-Theoretic Trajectory Guidance for Multi-Robot Systems with Koopman Operator

TL;DR

This work tackles guided trajectory planning for a leader–follower robot pair when the follower’s decision model is unknown. It casts the interaction as a dynamic Stackelberg game and uses the Koopman operator to learn a finite-dimensional linear embedding of the follower’s feedback dynamics, enabling a tractable receding-horizon plan that approximates the Stackelberg equilibrium. Through simulations in obstacle-rich environments, the approach yields accurate multi-step follower predictions and halves the leader’s planning time compared to a model-based baseline, while maintaining successful guidance. The results underscore the practical impact of integrating data-driven Koopman learning with game-theoretic planning for fast, safe multi-robot coordination, and point to future work on formal safety guarantees and operational bounds.

Abstract

Guided trajectory planning involves a leader robot strategically directing a follower robot to collaboratively reach a designated destination. However, this task becomes notably challenging when the leader lacks complete knowledge of the follower's decision-making model. There is a need for learning-based methods to effectively design the cooperative plan. To this end, we develop a Stackelberg game-theoretic approach based on the Koopman operator to address the challenge. We first formulate the guided trajectory planning problem through the lens of a dynamic Stackelberg game. We then leverage Koopman operator theory to acquire a learning-based linear system model that approximates the follower's feedback dynamics. Based on this learned model, the leader devises a collision-free trajectory to guide the follower using receding horizon planning. We use simulations to elaborate on the effectiveness of our approach in generating learning models that accurately predict the follower's multi-step behavior when compared to alternative learning techniques. Moreover, our approach successfully accomplishes the guidance task and notably reduces the leader's planning time to nearly half when contrasted with the model-based baseline method.

Figures (4)

• Figure 1: Illustration of Stackelberg Koopman learning framework in trajectory guidance with an unknown follower robot. The guidance problem is modeled as a Stackelberg game. The leader leverages the Koopman operator to learn a linear model to predict the follower's behavior. She performs receding horizon planning to find a collision-free trajectory and guide the follower to the target destination.
• Figure 2: Prediction errors with the learned model generated by three approaches. Our Koopman-based approach provides a smaller averaged prediction error and better long-term performance compared with the other two.
• Figure 3: Interactive guidance trajectories using receding horizon planning and the models learned by different approaches. The blue and the orange represent the leader and follower trajectories, respectively. The follower start from $[0,8.5]$, $[0.5,3]$, and $[5.5,0]$ to reach the destination $[9,9]$. The leader successfully guides the follower using our approach. nn-approach (and dmd-approach) fail to guide the follower in one (and two) case.
• Figure 4: Planning details for the interactive trajectory starting from $[5.5,0]$.

Theorems & Definitions (1)

• Remark