Motion planning is still an open problem for many disciplines, e.g., robotics, autonomous driving, due to issues like high planning times that hinder real-time, efficient decision-making. A class of methods striving to provide smooth solutions is gradient-based trajectory optimization. However, those methods might suffer from bad local minima, while for many settings, they may be inapplicable due to the absence of easy access to objectives-gradients. In response to these issues, we introduce Motion Planning via Optimal Transport (MPOT) - a gradient-free method that optimizes a batch of smooth trajectories over highly nonlinear costs, even for high-dimensional tasks, while imposing smoothness through a Gaussian Process dynamics prior via planning-as-inference perspective. To facilitate batch trajectory optimization, we introduce an original zero-order and highly-parallelizable update rule -- the Sinkhorn Step, which uses the regular polytope family for its search directions; each regular polytope, centered on trajectory waypoints, serves as a local neighborhood, effectively acting as a trust region, where the Sinkhorn Step "transports" local waypoints toward low-cost regions. We theoretically show that Sinkhorn Step guides the optimizing parameters toward local minima regions on non-convex objective functions. We then show the efficiency of MPOT in a range of problems from low-dimensional point-mass navigation to high-dimensional whole-body robot motion planning, evincing its superiority compared with popular motion planners and paving the way for new applications of optimal transport in motion planning.

翻译：运动规划仍是许多学科（如机器人学、自动驾驶）中的开放性问题，其面临的挑战包括规划时间过长，阻碍了实时高效的决策制定。一类旨在提供平滑解的方法是基于梯度的轨迹优化。然而，这类方法可能陷入不良局部极小值，并且在许多场景中由于难以直接获取目标函数的梯度而难以应用。针对这些问题，我们提出基于最优传输的运动规划（MPOT）——一种无梯度方法，能够优化一批具有高度非线性代价的平滑轨迹，即使面对高维任务也能适用，同时通过将规划视为推理的视角，利用高斯过程动力学先验施加平滑性约束。为促进批量轨迹优化，我们引入了一种原创的零阶且高度可并行的更新规则——Sinkhorn步长，该规则利用正则多面体族作为搜索方向；每个以轨迹路点为中心的正则多面体作为局部邻域，有效充当信任区域，其中Sinkhorn步长将局部路点“传输”至低代价区域。我们从理论上证明，Sinkhorn步长能够引导优化参数在非凸目标函数上趋向局部极小值区域。随后，我们通过从低维质点导航到高维全身机器人运动规划等一系列问题展示了MPOT的高效性，证明了其相较于主流运动规划器的优越性，并为最优传输在运动规划中的新应用铺平了道路。