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Multi-agent path-planning in a moving medium via Wasserstein Hamiltonian Flow

Christina Frederick, Haomin Zhou

TL;DR

A finite dimensional variational model for multi-agent path-planning in which a group of agents traverses from initial positions to a target distribution in a moving medium using the Wasserstein Hamiltonian flows that transport between probability distributions while optimizing a running cost is presented.

Abstract

We present a finite dimensional variational model for multi-agent path-planning in which a group of agents traverses from initial positions to a target distribution in a moving medium. The model is derived using the agent-based formulation of the Wasserstein Hamiltonian flows that transport between probability distributions while optimizing a running cost. The objective is the mismatch between their final positions and the target distribution. The constraints are a system of Hamiltonian equations that provide the trajectories of the agents. The free variables on which the optimization is defined form a finite vector of the initial velocities for the agents. The model is solved numerically by the L-BFGS method in conjunction with a shooting strategy. Several simulation examples, including a time-dependent moving medium, are presented to illustrate the performance of the model.

Multi-agent path-planning in a moving medium via Wasserstein Hamiltonian Flow

TL;DR

A finite dimensional variational model for multi-agent path-planning in which a group of agents traverses from initial positions to a target distribution in a moving medium using the Wasserstein Hamiltonian flows that transport between probability distributions while optimizing a running cost is presented.

Abstract

We present a finite dimensional variational model for multi-agent path-planning in which a group of agents traverses from initial positions to a target distribution in a moving medium. The model is derived using the agent-based formulation of the Wasserstein Hamiltonian flows that transport between probability distributions while optimizing a running cost. The objective is the mismatch between their final positions and the target distribution. The constraints are a system of Hamiltonian equations that provide the trajectories of the agents. The free variables on which the optimization is defined form a finite vector of the initial velocities for the agents. The model is solved numerically by the L-BFGS method in conjunction with a shooting strategy. Several simulation examples, including a time-dependent moving medium, are presented to illustrate the performance of the model.
Paper Structure (17 sections, 1 theorem, 37 equations, 7 figures, 1 table, 1 algorithm)

This paper contains 17 sections, 1 theorem, 37 equations, 7 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Relevant literature
  4. Contributions and organization
  5. Wasserstein-Hamiltonian flows in a moving medium
  6. Agent-based formulation
  7. Optimization Strategy for the Initial Value Problem
  8. Linear Example
  9. Challenges in Optimization
  10. Numerical Implementation: Agent Discretization and Optimization
  11. Agent Dynamics and Density Estimation
  12. Boundary Regularization
  13. Numerical Experiments
  14. Single-agent experiments (Gaussian target)
  15. Multi-agent ensembles
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\mu, \nu$ be probability measures on $\mathbb{R}^d$ with bounded second-order moment and let $w(t, x)$ be a $C^1$ vector field. The critical points $X(t)$ and $q(t)$ minimizing the control energy functional are characterized by the Hamiltonian system: where $\boldsymbol{H}(x, q, t) = \frac{1}{2}|q|^2 + q\cdot w(t, x)$ is the control Hamiltonian derived via Pontryagin’s minimum principle. F

Figures (7)

  • Figure 1: Single-agent path planning in representative flow fields. We consider several steady and time-dependent background flows: (a) Circle: $w(x,y) = (-y,\,x)^\top$, (b) Point Attractor: $w(x,y) = (-x+2y,\,-y-x)^\top$, (c) Point Repeller: $w(x,y) = (x+y,\,-x+y)^\top$, (d) Vertical: $w(x,y) = (0,\,5y)^\top$, (e) Stagnation Point: $w(x,y) = (x-2y,\,-x-y)^\top$, and (f) Gyre: $w(t,x,y) =\left(-2\pi\sin(\pi f),\, 2\pi\cos(\pi f)\,\partial_x f \right)^\top$, where $f(t,x) = a(t)x^2 + b(t)x$, $a(t) = \epsilon\sin(\omega t)$, and $b(t) = 1 - 2\epsilon\sin(\omega t)$, with $\epsilon = 0.1$, $\omega = 2\pi$.
  • Figure 2: Left: Single-agent path-planning in a time-dependent 'gyre' background flow: $w(t,x,y) =\left(-2\pi\sin(\pi f),\, 2\pi\cos(\pi f)\,\partial_x f \right)^\top$, where $f(t,x) = a(t)x^2 + b(t)x$, $a(t) = \epsilon\sin(\omega t)$, and $b(t) = 1 - 2\epsilon\sin(\omega t)$, with $\epsilon = 0.1$, $\omega = 2\pi$. Right: Instantaneous control effort $\|\dot{X}(t) - w(t, X(t))\|^2,$.
  • Figure 3: Multi-agent trajectory optimization and target formation in a rotational "circle" background flow for varying ensemble sizes. The agents navigate from a compact initial cluster at the origin to an equilateral distribution along the annular target manifold. The results demonstrate that the Wasserstein-Hamiltonian Flow (WHF) maintains high-fidelity density matching via KL-divergence minimization while scaling effectively with agent count $N$.
  • Figure 4: The savings in energy are proportional to the number of agents, as demonstrated by these plots.
  • Figure 5: Plots generated by minimizers of \ref{['eq:objective']} for the target distributions $\nu_{\mathrm{ring}}$ (top) and $\nu_{\mathrm{heart}}$ (bottom) and $N=25$ agents. The final objective values are similar, however they have different control energy values, each corresponding to local minimizers. The plots show energy expenditure in increasing order from left to right.
  • ...and 2 more figures

Theorems & Definitions (1)

  • Theorem 1: Wasserstein-Hamiltonian Flow in a Moving Medium