Scalable Optimal Motion Planning for Multi-Agent Systems by Cosserat Theory of Rods

TL;DR

The paper tackles scalable motion planning for large multi-agent formations by modeling the fleet as a Cosserat rod continuum. It converts the PDE-based optimal control problem into a nonlinear program using Bernstein surface discretization, achieving complexity that is largely independent of the number of agents. Key contributions include leveraging convex-hull properties and degree-elevation techniques for safe constraint enforcement, a min-distance algorithm for obstacle avoidance, and numerical demonstrations on translational and rotational formation changes. The work demonstrates feasible, constraint-satisfying trajectories for complex 3D formations and lays groundwork for real-time planning in large-scale multi-agent systems.

Abstract

We address the motion planning problem for large multi-agent systems, utilizing Cosserat rod theory to model the dynamic behavior of vehicle formations. The problem is formulated as an optimal control problem over partial differential equations (PDEs) that describe the system as a continuum. This approach ensures scalability with respect to the number of vehicles, as the problem's complexity remains unaffected by the size of the formation. The numerical discretization of the governing equations and problem's constraints is achieved through Bernstein surface polynomials, facilitating the conversion of the optimal control problem into a nonlinear programming (NLP) problem. This NLP problem is subsequently solved using off-the-shelf optimization software. We present several properties and algorithms related to Bernstein surface polynomials to support the selection of this methodology. Numerical demonstrations underscore the efficacy of this mathematical framework.

Figures (8)

• Figure 1: Multi-agent motion.
• Figure 2: Bernstein surface of order $m=5, n=5$. The red dots represent control points, the orange dots represent points on the surface, and the blue polygon is the convex hull.
• Figure 3: Bernstein surface from Fig. \ref{['fig:convex_hull']} after being degree elevated to $m=10, n=10$.
• Figure 4: Bernstein surface from Fig. \ref{['fig:convex_hull']} being split along the red line using the de Casteljau algorithm.
• Figure 5: Case 1 - Optimal motion of agents from initial to final configurations. The black straight line depicts the initial formation of agents and the blue curve is a quarter of an ellipse for final formation, independent of number of agents. The colored curves depict the transitions of agents from their initial to final formations.

Theorems & Definitions (1)

• Remark 1