In this paper we present a novel method for approximating a finite set of n spatial locations with n spherical orientations. This is accomplished by determining a design sphere and the associated orientations on this design sphere which are nearest the n spatial locations. The design sphere and the orientations on it are optimized such that the sum of the distances between each spatial location and its approximating spherical orientation is minimized. The result is a design sphere and n spherical orientations which best approximate a set of n spatial locations. In addition, we include a modification to the method which enables the designer to require that one of the n desired spatial locations be exactly preserved. This method for approximating spatial locations with spherical orientations is directly applicable to the synthesis of spherical mechanisms for motion generation. Here we demonstrate the utility of the method for motion generation task specification in spherical mechanism design. [S1050-0472(00)00204-X]

1.
Larochelle, P. M., Dooley, A. P., Murray, A. P., and McCarthy, J. M., 1993, “Sphinx: Software for Synthesizing Spherical 4R Mechanisms,” Proceedings of the NSF Design and Manufacturing Systems Conference, 1, pp. 607–611.
2.
Larochelle
,
P. M.
, and
McCarthy
,
J. M.
,
1995
, “
Planar Motion Synthesis Using an Approximate Bi-Invariant Metric
,”
ASME J. Mech. Des.
,
117
, pp.
646
651
.
3.
Ruth, D. A., and McCarthy, J. M., 1997, “The Design of Spherical 4R Linkages for Four Specified Orientations,” Proceedings of the ASME Design Engineering Technical Conference.
4.
Osborn, S. W., and Vance, J. M., 1995, “A Virtual Reality Environment for Synthesizing Spherical Four-Bar Mechanisms,” Proceedings of the ASME Design Engineering Technical Conferences, Vol. DE-83, pp. 885–892.
5.
Larochelle, P. M., Vance, J. M., and McCarthy, J. M., 1998, “Creating a Virtual Reality Environment for Spherical Mechanism Design,” Proceedings of the NSF Design and Manufacturing Grantees Conference, pp. 83–84.
6.
Suh, C. H., and Radcliffe, C. W., 1978, Kinematics and Mechanism Design, Wiley, New York.
7.
Erdman, A., and Sandor, G. N., 1997, Advanced Mechanism Design: Analysis and Synthesis, Prentice Hall, 1, 3rd ed.
8.
Etzel, K. R., and McCarthy, J. M., 1996, “A Metric for Spatial Displacements using Biquaternions on SO(4),” Proceedings of the ASME Design Engineering Technical Conference and Computers in Engineering Conference, DETC/MECH 1164, pp. 3185–3190.
9.
Ge, Q. J., 1994, “On Matrix Algebra Realization of the Theory of Bi-quaternions,” Proceedings of the ASME Design Engineering Technical Conferences, DE-Vol. 70, pp. 425–432.
10.
Larochelle, P. M., 1994, “Design of Cooperating Robots and Spatial Mechanisms,” Ph.D. Dissertation, University of California, Irvine.
11.
McCarthy, J. M., 1990, An Introduction to Theoretical Kinematics, MIT Press.
12.
Paul, R. P., 1981, Robot Manipulators: Mathematics, Programming, and Control, MIT Press, Cambridge, Massachusetts.
13.
Larochelle, P. M., 1999, “On the Geometry of Approximate Bi-Invariant Projective Displacement Metrics,” Proceedings of the Tenth World Congress on the Theory of Machines and Mechanisms, pp. 548–553.
14.
Etzel, K. R., 1996, “Biquaternion Theory and Applications to Spatial Motion Analysis,” M.S. Thesis, University of California, Irvine.
15.
Kazerounian, K., and Rastegar, J., 1992, “Object Norms: A Class of Coordinate and Metric Independent Norms for Displacements,” Proceedings of the ASME Design Engineering Technical Conferences, DE-Vol. 47, pp. 271–275.
16.
Bobrow, J. E., and Park, F. C., 1995, “On Computing Exact Gradients for Rigid Body Guidance Using Screw Parameters,” Proceedings of the ASME Design Engineering Technical Conferences, 1, pp. 839–844.
17.
Martinez
,
J. M. R.
, and
Duffy
,
J.
,
1995
, “
On the Metrics of Rigid Body Displacements for Infinite and Finite Bodies
,”
ASME J. Mech. Des.
,
117
, pp.
41
47
.
18.
Gupta
,
K. C.
,
1997
, “
Measures of Positional Error for a Rigid Body
,”
ASME J. Mech. Des.
,
119
, pp.
346
348
.
19.
Ravani
,
R.
, and
Roth
,
B.
,
1983
, “
Motion Synthesis Using Kinematic Mapping
,”
ASME J. Mech. Trans. Aut. Des.
,
105
, pp.
460
467
.
20.
Nelder
,
J. A.
, and
Mead
,
R.
,
1965
, “
A Simplex Method for Function Minimization
,”
Comput. J. (UK)
,
7
, pp.
308
313
.
21.
Bottema, O., and Roth, B., 1979, Theoretical Kinematics, North-Holland, Amsterdam.
You do not currently have access to this content.