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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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References

1.Pieper, D. L., The Kinematics of Manipulators under Computer Control Ph.D. Dissertation (Stanford, CA: Stanford University, Oct. 1968).Google Scholar
2.Roth, B., Rastegar, J. and Scheinman, V., “On the Design of Computer Controlled Manipulators,” First CISM-IFTMM Symposium on Theory and Practice of Robots and Manipulators (1973), pp. 93–113.Google Scholar
3.Koliskor, A., “The l-Coordinate Approach to the Industrial Robots Design,” Information Control Problems in Manufacturing Technology 1986. Proceedings of the 5th IFAC/IFIP/IMACS/IFORS Conference, Suzdal, USSR, (1987) pp. 225–232 (Preprint).Google Scholar
4.Chirikjian, G. S., “Kinematic synthesis of mechanisms and robotic manipulators with binary actuators,” ASME J. Mech. Des. 117, 573–580 (Dec. 1995).CrossRefGoogle Scholar
5.Sen, D. and Mruthyunjaya, T. S., “A discrete state perspective of manipulator workspaces,” Mech. Mach. Theory 29 (4), 591–605 (1994).CrossRefGoogle Scholar
6.Basavaraj, U. and Duffy, J., “End-effector motion capabilities of serial manipulators,” Int. J. Rob. Res. 12 (2), 132–145 (Apr. 1993).CrossRefGoogle Scholar
7.Korein, J. U., A Geometric Investigation of Reach (MIT Press, Cambridge, MA, 1985).Google Scholar
8.Wang, Y. F. and Chirikjian, G. S., “Workspace generation of hyper-redundant manipulators as a diffusion process on SE(N),” IEEE Trans. Rob. Autom. 20 (3), 399–408 (Jun. 2004).CrossRefGoogle Scholar
9.Liu, Y., Probability Density Estimation on Rotation and Motion Groups Ph.D. Dissertation (Baltimore, MD: Department of Mechanical Engineering, Johns Hopkins University, Apr. 2007).Google Scholar
10.Chirikjian, G. S., “Fredholm integral equations on the Euclidean motion group,” Inverse Probl. 12, 579–599 (Oct. 1996).CrossRefGoogle Scholar
11.Chirikjian, G. S. and Ebert-Uphoff, I., “Numerical convolution on the Euclidean group with applications to workspace generation,” IEEE Trans. Rob. Autom. 14 (1), 123–136 (Feb. 1998).CrossRefGoogle Scholar
12.Ebert-Uphoff, I. and Chirikjian, G. S., “Efficient workspace generation for binary manipulators with many actuators,” J. Rob. Syst. 12 (6), 383–400 (Jun. 1995).CrossRefGoogle Scholar
13.Ebert-Uphoff, I. and Chirikjian, G. S., “Inverse Kinematics of Discretely Actuated Hyper-Redundant Manipulators Using Workspace Densities,” Proceedings of the IEEE International Conference on Robotics and Automation ICRA, Minneapolis, MN (Apr. 1996) pp. 139–145.Google Scholar
14.Hörmander, L., “Hypoelliptic second order differential equations,” Acta Math. 119, 147–171 (1967).CrossRefGoogle Scholar
15.Chirikjian, G. S., and Kyatkin, A. B., Engineering Applications of Noncommutative Harmonic Analysis (CRC Press, Boca Raton, FL, 2001).Google Scholar
16.Chirikjian, G. S. and Kyatkin, A. B., “An operational calculus for the Euclidean motion group with applications in robotics and polymer science,” J. Fourier Anal. Appl. 6 (6), 583–606 (Dec. 2000).CrossRefGoogle Scholar
17.Wang, Y., Zhou, Y., Maslen, D. K. and Chirikjian, G. S., “Solving the phase-noise Fokker–Planck equation using the motion-group Fourier transform,” IEEE Trans. Commun. 54 (5), 868–877 (May 2006).CrossRefGoogle Scholar
18.Zhou, Y. and Chirikjian, G. S., “Probabilistic Models of Dead-Reckoning Error in Nonholonomic Mobile Robots,” Proceedings of the IEEE International Conference on Robotics and Automation ICRA, Taipei, Taiwan (Sep. 2003).Google Scholar
19.Higham, D. J., “An algorithmic introduction to numerical simulation of stochastic differential equations,” SIAM Rev. 43 (3), 525–546 (2001).CrossRefGoogle Scholar