Trajectory planning tasks for non-holonomic or collaborative systems are naturally modeled by state spaces with non-Euclidean metrics. However, existing proofs of convergence for sample-based motion planners only consider the setting of Euclidean state spaces. We resolve this issue by formulating a flexible framework and set of assumptions for which the widely-used PRM*, RRT, and RRT* algorithms remain asymptotically optimal in the non-Euclidean setting. The framework is compatible with collaborative trajectory planning: given a fleet of robotic systems that individually satisfy our assumptions, we show that the corresponding collaborative system again satisfies the assumptions and therefore has guaranteed convergence for the trajectory-finding methods. Our joint state space construction builds in a coupling parameter $1\leq p\leq \infty$, which interpolates between a preference for minimizing total energy at one extreme and a preference for minimizing the travel time at the opposite extreme. We illustrate our theory with trajectory planning for simple coupled systems, fleets of Reeds-Shepp vehicles, and a highly non-Euclidean fractal space.

翻译：非完整或协同系统的轨迹规划任务自然由具有非欧几里得度量的状态空间建模。然而，现有基于采样的运动规划器的收敛性证明仅考虑欧几里得状态空间情形。我们通过构建一个灵活框架及一组假设条件解决了这一问题，在该框架下，广泛使用的PRM*、RRT和RRT*算法在非欧几里得场景中仍保持渐近最优性。该框架与协同轨迹规划兼容：对于一组各自满足我们假设的机器人系统，我们证明相应的协同系统同样满足这些假设，从而保证轨迹搜索方法的收敛性。我们的联合状态空间构造引入了一个耦合参数$1\leq p\leq \infty$，该参数在最小化总能量与最小化旅行时间这两个极端偏好之间进行插值。我们通过简单耦合系统、里德-谢普车队以及高度非欧几里得的分形空间的轨迹规划实例验证了该理论。