This paper provides a new perspective to the problem of reconfiguration of a rolling sphere. It is shown that the motion of a rolling sphere can be characterized by evolute-involute geometry. This characterization, which is a manifestation of our specific selection of Euler angle coordinates and choice of angular velocities in a rotating coordinate frame, allows us to recast the three-dimensional kinematics problem as a problem in planar geometry. This, in turn, allows a variety of optimization problems to be defined and admits infinite solution trajectories. It is shown that logarithmic spirals form a class of solution trajectories and they result in exponential convergence of the configuration variables.
Issue Section:
Technical Papers
1.
Hammersley
, J.
, 1983, “Oxford Commemoration Ball
,” London Mathematical Society Lecture Notes
, 79
, pp. 112
–142
.2.
Li
, Z.
, and Canny
, J.
, 1990, “Motion of Two Rigid Bodies with Rolling Constraints
,” IEEE Trans. Rob. Autom.
1042-296X, 6
(1
), pp. 62
–72
.3.
Jurdjevic
, V.
, 1993, “The Geometry of the Plate-Ball Problem
,” Arch. Ration. Mech. Anal.
0003-9527, 124
, pp. 305
–328
.4.
Mukherjee
, R.
, Minor
, M.
, and Pukrushpan
, J. T.
, 2002, “Motion Planning for a Spherical Mobile Robot: Revisiting the Classical Ball-Plate Problem
,” ASME J. Dyn. Syst., Meas., Control
0022-0434, 124
(4
), pp. 502
–511
.5.
Krener
, A. J.
, and Nikitin
, S.
, 1997, “Generalized Isoperimetric Problem
,” Journal of Mathematical Systems, Estimation, and Control
, 7
(3
), pp. 1
–15
.6.
Das
, T.
, and Mukherjee
, R.
, 2004, “Exponential Stabilization of the Rolling Sphere
,” Automatica
0005-1098, 40
, pp. 1877
–1889
.7.
Courant
, R.
, and Hilbert
, D.
, 2003, Methods of Mathematical Physics
, Wiley Interscience
, New York
, pp. 135
–136
.Copyright © 2006
by American Society of Mechanical Engineers
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