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TECHNICAL PAPERS

Analysis of Fully Reversed Sequences of Rotations of a Free Rigid Body

[+] Author and Article Information
Sung K. Koh, G. K. Ananthasuresh

Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104

Christopher Croke

Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104

J. Mech. Des 126(4), 609-616 (Aug 12, 2004) (8 pages) doi:10.1115/1.1758249 History: Received June 01, 2003; Revised November 01, 2003; Online August 12, 2004
Copyright © 2004 by ASME
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References

Koh, S. K., and Ananthasuresh, G. K., 2003, “Inverse Kinematics of an Untethered Rigid Body Undergoing a Sequence of Forward and Reverse Rotations,” ASME J. Mech. Des., in press.
Li, J., Koh, S. K., Ananthasuresh, G. K., Ayyaswamy, P. S., and Ananthakrishnan, S., 2001, “A Novel Attitude Control technique for Miniature Spacecraft,” CD-ROM Proceedings of the MEMS Symposium at the 2001 ASME International Mechanical Engineering Conference and Exhibition, New York, Nov.
Koh,  S. K., Ostrowski,  J. P., and Ananthasuresh,  G. K., 2002, “Control of Micro-satellite Orientation Using Bounded-input, Fully-reversed MEMS Actuators,” Int. J. Robot. Res., 21(5–6), pp. 591–605.
Bullo,  F., Leonard,  N. E., and Lewis,  A. D., 2000, “Controllability and Motion Algorithms for Underactuated Lagrangian Systems on Lie Groups,” IEEE Trans. Autom. Control, 45(8), pp. 1437–1454.
Koh, S. K., and Ananthasuresh, G. K., 2002, “Motion-Planning for the Axis Control of Miniature Spacecraft Using Microactuators,” CD-ROM Proceedings of the ASME 2002 Biennial Mechanisms and Robotics Conference, in, Montreal Sep., Paper #DETC2002/MECH34294.
Murray, R. M., Li, Zexiang, and Sastry, S. S., 1993, A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Raton, FL.
Sastry, S., 1999, Nonlinear Systems: Analysis, Stability and Control, IAM, Springer-Verlag, New York.
Gamelin, T. W., 1999, Introduction to Topology, Second Edition, Dover Publications, New York.
Pollack, A., and Guillemin, V. W., 1974, Differential Topology, Prentice Hall, Upper Saddle River, NJ.
Marsden, J. E., and Hoffman, M. J., 1993, Elementary Classical Analysis, W. H. Freeman and Co., New York.

Figures

Grahic Jump Location
Schematic description of an FRxyzxyz sequence of a cube with θxyz=π/2. (a) faces of the cube: L (left), T (top), R (right), BT (bottom), BK (back), and F (font). (b) three orthogonal body-fixed axes x,y, and z projecting outwards from F, R, and T faces, respectively. (c) the sequence of rotations.
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ξ-plots for r=π/2 (a) type I: partially covered with the presence of two figure-8 shaped holes, one seen here and the other on the other side of the sphere (b) type II: fully covered, and (c) type III: partially covered with two exposed ends.
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The ξ plots and boundaries of holes at different radii for a type I sequence (a) ξ-plot at r=0.2π (b) ξ-plot at r=0.5π (c) ξ-plot at r=0.8π
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Intersection of the volume of the hole in ξ plot with a sphere at any radius r≤π. (a) curves overlaid on the sphere (b) curves without the sphere.
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Visualization of the hole in the ξ plot of type I sequences (a) part of the surface obtained with θy=0 condition, (b) part obtained with θy=π condition, (c) parts (a) and (b) combined, and (d) the complete volume of the hole with spherical patches added. The intersection of this volume with spheres of varying radii give the curves such as the ones shown in Fig. 4.
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The curves corresponding to the singular Jacobian condition (35b) for a type II sequence

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