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A Hopf-Lax Type Formula for Multi-Agent Path Planning with Pattern Coordination

Christian Parkinson, Adan Baca

TL;DR

The paper addresses multi-agent path planning with prescribed formation, proposing a grid-free Hopf-Lax type representation of the Hamilton-Jacobi-Bellman equation to compute optimal coordination for nonlinear, heterogeneous agents. By decoupling per-agent Hamiltonians and employing a PDHG-based saddle-point solver, the approach yields scalable, explainable solutions without grid-based curse of dimensionality, and naturally incorporates time-varying dynamics and obstacles. Key contributions include explicit Hamiltonians for isotropic and Reeds-Shepp dynamics, a formation penalty, and a practical numerical framework that demonstrates efficient planning for formation maintenance in synthetic 2D scenarios with varying objective priorities. The work provides a robust PDE-based alternative to learning-based methods for coordinated multi-agent systems, with potential applicability to larger, more complex formations and time-dependent environments.

Abstract

We present an algorithm for a multi-agent path planning problem with pattern coordination based on dynamic programming and a Hamilton-Jacobi-Bellman equation. This falls broadly into the class of partial differential equation (PDE) based optimal path planning methods, which give a black-box-free alternative to machine learning hierarchies. Due to the high-dimensional state space of multi-agent planning problems, grid-based methods for PDE which suffer from the curse of dimensionality are infeasible, so we instead develop grid-free numerical methods based on variational Hopf-Lax type representations of solutions to Hamilton-Jacobi Equations. Our formulation is amenable to nonlinear dynamics and heterogeneous agents. We apply our method to synthetic examples wherein agents navigate around obstacles while attempting to maintain a prespecified formation, though with small changes it is likely applicable to much larger classes of problems.

A Hopf-Lax Type Formula for Multi-Agent Path Planning with Pattern Coordination

TL;DR

The paper addresses multi-agent path planning with prescribed formation, proposing a grid-free Hopf-Lax type representation of the Hamilton-Jacobi-Bellman equation to compute optimal coordination for nonlinear, heterogeneous agents. By decoupling per-agent Hamiltonians and employing a PDHG-based saddle-point solver, the approach yields scalable, explainable solutions without grid-based curse of dimensionality, and naturally incorporates time-varying dynamics and obstacles. Key contributions include explicit Hamiltonians for isotropic and Reeds-Shepp dynamics, a formation penalty, and a practical numerical framework that demonstrates efficient planning for formation maintenance in synthetic 2D scenarios with varying objective priorities. The work provides a robust PDE-based alternative to learning-based methods for coordinated multi-agent systems, with potential applicability to larger, more complex formations and time-dependent environments.

Abstract

We present an algorithm for a multi-agent path planning problem with pattern coordination based on dynamic programming and a Hamilton-Jacobi-Bellman equation. This falls broadly into the class of partial differential equation (PDE) based optimal path planning methods, which give a black-box-free alternative to machine learning hierarchies. Due to the high-dimensional state space of multi-agent planning problems, grid-based methods for PDE which suffer from the curse of dimensionality are infeasible, so we instead develop grid-free numerical methods based on variational Hopf-Lax type representations of solutions to Hamilton-Jacobi Equations. Our formulation is amenable to nonlinear dynamics and heterogeneous agents. We apply our method to synthetic examples wherein agents navigate around obstacles while attempting to maintain a prespecified formation, though with small changes it is likely applicable to much larger classes of problems.

Paper Structure

This paper contains 4 sections, 18 equations, 3 figures, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. OPTIMAL CONTROL FORMULATION
  3. NUMERICAL METHODS
  4. RESULTS AND DISCUSSION

Figures (3)

  • Figure 1: Three agents navigate around an obstacle while trying to maintain formation in an equilateral triangle. Top: the agents are heavily incentivized to minimize travel time to their final destinations. Bottom: the agents are heavily incentivized to maintain formation.
  • Figure 2: Four agents navigate around obstacles while trying to maintain formation in a square. The two agents exhibiting isotropic motion (dots) mimic the cuspidal paths that the cars take so as to keep in formation, even though their dynamics allow for smooth paths.
  • Figure 3: Three agents navigate around two moving obstacles while trying to maintain formation in an equilateral triangle. They move with local speed $v(x,y) = 1+\frac{1}{4} \sin(x)\sin(y)$ (faster in yellow regions; slower in blue). The obstacles are rotating in a circle with direction specified by the red arrow. We have chosen specifically timed snapshots to demonstrate interaction with the obstacles.