Carnegie Mellon University
The
Functional Gradient Motion Planning in Reproducing Kernel Hilbert Spaces

Zita Alexandra Magalhaes Marinho , Anca Dragan, Arunkumar Byravan, Byron Boots, Geoffrey Gordon, and Siddhartha Srinivasa
Proceedings of Robotics: Science and Systems (RSS-2016), July, 2016.


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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, implementations such as CHOMP and TrajOpt typically commit to a fixed, finite parametrization of trajectories, often as a sequence 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. 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 with independent and coupled interactions among robot joints, Laplacian RBFs, and B-splines, as compared to the standard discretized waypoint representation.

Keywords
RKHSs, Functional gradient, trajectory optimization

Notes
Grant ID: Carnegie Mellon Portugal Program under Grant SFRH/BD/52015/2012
Associated Center(s) / Consortia: Quality of Life Technology Center, National Robotics Engineering Center, and Center for the Foundations of Robotics
Associated Lab(s) / Group(s): Personal Robotics

Text Reference
Zita Alexandra Magalhaes Marinho , Anca Dragan, Arunkumar Byravan, Byron Boots, Geoffrey Gordon, and Siddhartha Srinivasa, "Functional Gradient Motion Planning in Reproducing Kernel Hilbert Spaces," Proceedings of Robotics: Science and Systems (RSS-2016), July, 2016.

BibTeX Reference
@inproceedings{Magalhaes_Marinho__2016_8148,
   author = "Zita Alexandra {Magalhaes Marinho } and Anca Dragan and Arunkumar Byravan and Byron Boots and Geoffrey Gordon and Siddhartha Srinivasa",
   title = "Functional Gradient Motion Planning in Reproducing Kernel Hilbert Spaces",
   booktitle = "Proceedings of Robotics: Science and Systems (RSS-2016)",
   month = "July",
   year = "2016",
}