  We propose Parallelised Diffeomorphic Sampling-based Motion Planning (PDMP).
PDMP is a novel parallelised framework that uses bijective and differentiable
mappings, or diffeomorphisms, to transform sampling distributions of
sampling-based motion planners, in a manner akin to normalising flows. Unlike
normalising flow models which use invertible neural network structures to
represent these diffeomorphisms, we develop them from gradient information of
desired costs, and encode desirable behaviour, such as obstacle avoidance.
These transformed sampling distributions can then be used for sampling-based
motion planning. A particular example is when we wish to imbue the sampling
distribution with knowledge of the environment geometry, such that drawn
samples are less prone to be in collisions. To this end, we propose to learn a
continuous occupancy representation from environment occupancy data, such that
gradients of the representation defines a valid diffeomorphism and is amenable
to fast parallel evaluation. We use this to "morph" the sampling distribution
to draw far fewer collision-prone samples. PDMP is able to leverage gradient
information of costs, to inject specifications, in a manner similar to
optimisation-based motion planning methods, but relies on drawing from a
sampling distribution, retaining the tendency to find more global solutions,
thereby bridging the gap between trajectory optimisation and sampling-based
planning methods.
