This paper shows that in the use of Lie groups for the study of the relative motion of rigid bodies some assumptions are not explicitly stated. A commutation diagram is shown which points out the “reference problem” and its simplification to the usual Lie group approach under certain conditions which are made explicit.

Issue Section:
Technical Papers
Keywords:
Lie groups, classical mechanics
Topics:
Kinematics, Space, Theorems (Mathematics), Classical mechanics, Screws, Lie algebras
1.
Ball, R. S., 1900, A Treatise on the Theory of Screws, Cambridge University Press, Cambridge, UK.
2.
Lipkin, H., 1985, “Geometry and Mappings of Screws With Applications to the Hybrid Control of Robotic Manipulators,” Ph.D. thesis, University of Florida.
3.
Lipkin
,
H.
, and
Duffy
,
J.
,
1985
, “
The Elliptic Polarity of Screws
,”
ASME J. Mech., Transm., Autom. Des.
,
107
, pp.
377
–
387
.
4.
Lipkin
,
H.
, and
Duffy
,
J.
,
1988
, “
Hybrid Twist and Wrench Control for a Robotic Manipulator
,”
ASME J. Mech. Des.
,
110
, pp.
138
–
144
.
5.
Olver, P. J., 1993, Applications of Lie Groups to Differential Equations, Vol. 107 (Graduate Texts in Mathematics) 2nd Ed., Springer-Verlag, New York.
6.
Gilmore, R., 1974, Lie Groups, Lie Algebras, and Some of Their Applications, John Wiley and Sons, New York.
7.
Murray, R. M., Li, Z., and Shankar Sastry, S., 1994, A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Raton, FL.
8.
Stramigioli, S., Maschke, B., and Bidard, C. 2000, “Hamiltonian Formulation of the Dynamics of Spatial Mechanisms Using Lie Groups and Screw Theory,” Proceedings of the Symposium Commemorating the Legacy, Works, and Life of Sir Robert Stawell Ball H. Lipkin, ed., Cambridge University Press, Cambridge, UK.
9.
Selig, J. M., 1996, Geometric Methods in Robotics (Monographs in Computer Sciences), Springer-Verlag, New York.
10.
Maschke, B. M. J., 1996, Modeling and Control of Mechanisms and Robots, World Scientific, Singapore, pp. 1–38.
11.
Stramigioli, S., 2001, Modeling and Interactive Mechanical Systems: A Coordinate Free Approach, Springer, Berlin.
12.
Karger, A., and Novak, J., 1978, Space Kinematics and Lie Groups, Gordon and Breach, New York.
13.
Herve
,
J. M.
,
1999
, “
The Lie Group of Rigid Body Displacements, a Fundamental Tool for Mechanism Design
,”
Mech. Mach. Theory
,
34
, pp.
719
–
730
.
14.
Abraham, R., and Marsden, J. E., 1994, Foundations of Mechanics, 2nd Ed., Addison-Wesley, Reading, MA.
15.
Samuel, P., 1988, Projective Geometry (Undergraduate Texts in Mathematics, Readings in Mathematics), Springer-Verlag, New York.
Copyright © 2001
 by ASME
You do not currently have access to this content.