  We introduce a functional gradient descent trajectory optimization algorithm
for robot motion planning in Reproducing Kernel Hilbert Spaces (RKHSs).
Functional gradient algorithms are a popular choice for motion planning in
complex many-degree-of-freedom robots, since they (in theory) work by directly
optimizing within a space of continuous trajectories to avoid obstacles while
maintaining geometric properties such as smoothness. However, in practice,
functional gradient algorithms typically commit to a fixed, finite
parameterization of trajectories, often as a list of waypoints. Such a
parameterization can lose much of the benefit of reasoning in a continuous
trajectory space: e.g., it can require taking an inconveniently small step size
and large number of iterations to maintain smoothness. Our work generalizes
functional gradient trajectory optimization by formulating it as minimization
of a cost functional in an RKHS. This generalization lets us represent
trajectories as linear combinations of kernel functions, without any need for
waypoints. As a result, we are able to take larger steps and achieve a locally
optimal trajectory in just a few iterations. Depending on the selection of
kernel, we can directly optimize in spaces of trajectories that are inherently
smooth in velocity, jerk, curvature, etc., and that have a low-dimensional,
adaptively chosen parameterization. Our experiments illustrate the
effectiveness of the planner for different kernels, including Gaussian RBFs,
Laplacian RBFs, and B-splines, as compared to the standard discretized waypoint
representation.
