Fast Marching based Rendezvous Path Planning for a Team of Heterogeneous Vehicle

TL;DR

This work develops a deterministic, Fast Marching Method (FMM)–based framework for time-optimal rendezvous of heterogeneous vehicle teams. By computing per-agent arrival-time maps $T^i(\mathbf{x})$ under agent-specific velocity fields $V^i(\mathbf{x})$, the rendezvous point is found as $\mathbf{x}_m = \arg\min_{\mathbf{x} \in \mathcal{C}_{free}} \max_i T^i(\mathbf{x})$, and each agent’s continuous path to the rendezvous is obtained from the gradient of its arrival-time field. The methodology uses a two-pass FMM (FMS) to account for obstacle avoidance via an obstacle-distance map and to produce time grids that respect environmental constraints, followed by gradient-based path extraction. A Tampa Bay scenario with four heterogeneous vehicles demonstrates scalability, online adaptability, and a baseline comparison against RRT*-based and centroid methods, highlighting smooth, feasible trajectories and the potential for extension to dynamic obstacles and energy-aware planning. The approach offers a fast, scalable alternative for multi-agent coordination in complex, multi-domain environments with deterministic global optimality for the rendezvous task.

Abstract

This paper presents a formulation for deterministically calculating optimized paths for a multiagent system consisting of heterogeneous vehicles. The key idea is the calculation of the shortest time for each agent to reach every grid point from its known initial position. Such arrival time map is efficiently computed using the Fast Marching Method (FMM), a computational algorithm originally designed for solving boundary value problems of the Eikonal equation. By leveraging the FMM, we demonstrate that the minimal time rendezvous point and paths for all member vehicles can be uniquely determined with minimal computational overhead. The scalability and adaptability of the present method during online execution are investigated, followed by a comparison with a baseline method that highlights the effectiveness of the proposed approach. Then, the potential of the present method is showcased through a virtual rendezvous scenario involving the coordination of a ship, an underwater vehicle, an aerial vehicle, and a ground vehicle, all converging at the optimal location within the Tampa Bay area in minimal time. The results show that the developed framework can efficiently construct continuous paths of heterogeneous vehicles by accommodating operational constraints via an FMM algorithm

Figures (18)

• Figure 1: The level sets of the solution to the Eikonal equation \ref{['e:eikonal']} computed using the fast marching method, describe a surface evolving with outward normal velocity $V(\bm{x})$ = 1. The level set values are indicative of the time it takes for the initial surface (represented by the innermost blue line) to reach each grid point within the computational domain.
• Figure 2: An example of binary occupancy map. The binary image, which is on $512 \times 512$ pixel size, takes the value of 0 if the position is occupied by obstacles, and 1 otherwise.
• Figure 3: Plots of velocity functions \ref{['eqn:velocity_form']} as a function of normalized distance $d / d_\text{max}$ for different values of $\alpha$. The vertical axis $V^*(=V/\mathcal{V}_{\text{max}})$ is a normalized velocity by the maximum speed. In general, a smaller value of $\alpha$ results in a larger imposed safety distance.
• Figure 4: Comparison of velocity maps generated from the different velocity forms shown in Fig. \ref{['fig:velocityfunctioncurve']}. Sharper increase of $V^{*}$ to value $1$ results in a larger safety distance.
• Figure 5: Velocity map created using the form \ref{['eqn:velocity_form']} with $\alpha=3$. The velocity values $V^*$ are normalized with the maximum speed of agent $\mathcal{V}_{\text{max}}$.