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TECHNICAL PAPERS

Motion Planning for a Spherical Mobile Robot: Revisiting the Classical Ball-Plate Problem

[+] Author and Article Information
Ranjan Mukherjee

Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824-1226e-mail: mukherji@egr.msu.edu

Mark A. Minor

Department of Mechanical Engineering, University of Utah, Salt Lake City, UT 84112

Jay T. Pukrushpan

Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109

J. Dyn. Sys., Meas., Control 124(4), 502-511 (Dec 16, 2002) (10 pages) doi:10.1115/1.1513177 History: Received July 01, 2000; Revised March 01, 2002; Online December 16, 2002
Copyright © 2002 by ASME
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References

Brown,  H. B., and Xu,  Y., 1997, “A Single-Wheel Gyroscopically Stabilized Robot,” IEEE Rob. Autom. Mag., 4(3), pp. 39–44.
Koshiyama,  A., and Yamafuji,  K., 1993, “Design and Control of All-Direction Steering Type Mobile Robot,” Int. J. Robot. Res., 12(5), pp. 411–419.
Halme, A., Schonberg, T., and Wang, Y., 1996, “Motion Control of a Spherical Mobile Robot,” Proc. of 4th Int. Workshop on Advanced Motion Control (AMC), Mie Univ., Japan.
Bicchi, A., Balluchi, A., Prattichizzo, D., and Gorelli, A., 1997, “Introducing the Sphericle: An Experimental Testbed for Research and Teaching in Nonholonomy,” Proc. of IEEE Int. Conf. on Robotics and Automation, pp. 2620–2625.
Murray,  R. M., and Sastry,  S. S., 1993, “Nonholonomic Motion Planning: Steering Using Sinusoids,” IEEE Trans. Autom. Control, 38(5), pp. 700–713.
Bicchi, A., Prattichizzo, D., and Sastry, S. S., 1995, “Planning Motions of Rolling Surfaces,” Proc. of 34th IEEE Int. Conf. on Decision and Control, pp. 2812–2817.
Li,  Z., and Canny,  J., 1990, “Motion of Two Rigid Bodies with Rolling Constraint,” IEEE Trans. Rob. Autom., 6(1), pp. 62–72.
Jurdjevic,  V., 1993, “The Geometry of the Plate-Ball Problem,” Arch. Ration. Mech. Anal., 124, pp. 305–328.
Brockett, R. W., and Dai, L., 1991, “Nonholonomic Kinematics and the Role of Elliptic Functions in Constructive Controllability,” Workshop on Nonholonomic Motion Planning, IEEE Int. Conf. on Robotics and Automation.
Mukherjee,  R., Emond,  B. R., and Junkins,  J. L., 1997, “Optimal Trajectory Planning for Mobile Robots using Jacobian Elliptic Functions,” Int. J. Robot. Res., 16(6), pp. 826–839.
Levi,  M., 1993, “Geometric Phases in the Motion of Rigid Bodies,” Arch. Ration. Mech. Anal., 122(3), pp. 213–229.
Mukherjee,  R., and Pukrushpan,  J. T., 2000, “Class of Rotations Induced by Spherical Polygons,” AIAA J. Guid. Control, 23(4), pp. 746–749.
Kells, L. M., Kern, W. F., and Bland, J. R., 1940, Plane and Spherical Trigonometry, McGraw Hill, New York, NY.

Figures

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Initial and final configurations of sphere
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Motion of the sphere under control actions (A) and (B)
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Motion of the sphere under (A)-(B)-(A) sequence of control actions
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Plot of Δβ versus Δα for different values of r, shown for both counter-clockwise and clockwise paths
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Basis for complete reconfiguration of the sphere
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(a) A spherical triangle, (b) Image of the spherical triangle on the x-y plane
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Open loop reconfiguration using spherical triangles
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Reconfiguration of the sphere using the geometric motion planner
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A spherical triangle maneuver for reconfiguration

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