Functional Gradient Motion Planning in Reproducing Kernel Hilbert Spaces

TL;DR

This paper presents a novel functional gradient descent algorithm for robot motion planning in Reproducing Kernel Hilbert Spaces, enabling more efficient and smooth trajectory optimization without fixed waypoint parameterizations.

Contribution

It generalizes functional gradient trajectory optimization by representing trajectories as kernel combinations, allowing larger steps and fewer iterations for optimal, smooth paths.

Findings

Effective trajectory optimization with various kernels.

Fewer iterations needed for convergence.

Improved smoothness and flexibility in planning.

Abstract

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.