  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.
