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Deconvolution on the Euclidean motion group and planar robotic manipulator design

Published online by Cambridge University Press:  15 January 2009

Peter T. Kim ,
Yan Liu ,
Zhi-Ming Luo  and
Yunfeng Wang
Peter T. Kim*
Affiliation:
Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario N1G 2W1Canada.
Yan Liu
Affiliation:
Google New York, 76 9th Ave. 4th Floor New York, NY 10011.
Zhi-Ming Luo
Affiliation:
Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario N1G 2W1Canada.
Yunfeng Wang
Affiliation:
Department of Mechanical Engineering, The College of New Jersey, Ewing, NJ 08628-0718, USA.
*
*Corresponding author. E-mail: pkim@uoguelph.ca
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Summary

Several problems of practical interest in robotics can be modelled as the convolution of functions on the Euclidean motion group. These include the evaluation of reachable positions and orientations at the distal end of a robot manipulator arm. A natural inverse problem arises when one wishes to design rather than to model manipulators. Namely, by considering a serial-chain robot arm as a concatenation of segments, we examine how statistics of known segments can be used to select, or design, the remainder of the structure so as to attain the desired statistical properties of the whole structure. This is then a deconvolution density estimation problem for the Euclidean motion group. We prove several results about the convergence of these deconvolution estimators to the true underlying density under certain smoothness assumptions. A practical implementation to the design of planar robot arms is demonstrated.

Keywords

Degenerate diffusionFourier analysisGaussian distributionInverse problemIrreducible representationsKinematicsManipulator arm
Type
Article
Information
Robotica , Volume 27 , Issue 6 , October 2009 , pp. 861 - 872
Copyright
Copyright © Cambridge University Press 2009

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