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.

翻译：本文针对大规模多智能体系统的运动规划问题，采用Cosserat杆理论对车辆编队的动态行为进行建模。该问题被表述为描述系统连续介质特性的偏微分方程最优控制问题。由于问题复杂度不受编队规模影响，该方法确保了车辆数量层面的可扩展性。通过伯恩斯坦曲面多项式实现了控制方程及问题约束的数值离散化，从而将最优控制问题转化为非线性规划问题。该非线性规划问题随后采用现成的优化软件进行求解。我们提出了若干与伯恩斯坦曲面多项式相关的性质及算法，以论证该方法选择的合理性。数值仿真结果验证了该数学框架的有效性。