Motion planning is still an open problem for many disciplines, e.g., robotics, autonomous driving, due to their need for high computational resources that hinder real-time, efficient decision-making. A class of methods striving to provide smooth solutions is gradient-based trajectory optimization. However, those methods usually suffer from bad local minima, while for many settings, they may be inapplicable due to the absence of easy-to-access gradients of the optimization objectives. In response to these issues, we introduce Motion Planning via Optimal Transport (MPOT) -- a \textit{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 the 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 cost-probing neighborhood, acting as a \textit{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 of 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 to popular motion planners, paving the way for new applications of optimal transport in motion planning.

翻译：运动规划因其对高计算资源的需求，阻碍了实时高效的决策，至今仍是机器人、自动驾驶等众多领域的开放性问题。旨在提供平滑解的一类方法是基于梯度的轨迹优化。然而，这些方法通常受困于较差的局部极小值，且在许多场景中因优化目标缺乏易获取的梯度而无法适用。针对这些问题，我们提出基于最优运输的运动规划（MPOT）——一种无梯度方法，能够在高度非线性成本下优化一批平滑轨迹，即使面对高维任务，也能通过将运动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。为促进批量轨迹优化，我们引入一种原创的零阶且高度可并行的更新规则——Sinkhorn步，它采用正则多面体族作为搜索方向。每个以轨迹航点为中心的正则多面体充当局部成本探测邻域，作为信任区域，其中Sinkhorn步将局部航点“运输”至低成本区域。我们理论上证明了Sinkhorn步引导优化参数趋向非凸目标函数的局部极小区域。随后，我们展示了MPOT在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中的高效性，证明了其相较于主流运动规划器的优越性，为最优运输在运动规划中的新应用铺平了道路。