In this article we examine the problem of designing a mechanism whose tool frame comes closest to reaching a set of desired goal frames. The basic mathematical question we address is characterizing the set of distance metrics in SE(3), the Euclidean group of rigid-body motions. Using Lie theory, we show that no bi-invariant distance metric (i.e., one that is invariant under both left and right translations) exists in SE(3), and that because physical space does not have a natural length scale, any distance metric in SE(3) will ultimately depend on a choice of length scale. We show how to construct left- and right-invariant distance metrics in SE(3), and suggest a particular left-invariant distance metric parametrized by length scale that is useful for kinematic applications. Ways of including engineering considerations into the choice of length scale are suggested, and applications of this distance metric to the design and positioning of certain planar and spherical mechanisms are given.

Bodduluri, R. M. C., 1990, “Design and Planned Movement of Multi-Degree-of-Freedom Spatial Mechanisms,” Ph.D. Thesis, Department of Mechanical and Aerospace Engineering, University of California, Irvine.
Boothby, W., 1975, An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, New York.
Cheeger, J., and Ebin, D. G., 1975, Comparison Theorems in Riemannian Geometry, North-Holland, Amsterdam.
Chevalley, C., 1946, Theory of Lie Groups, Princeton University Press, Princeton.
Gallot, S., Hulin, D., and Lafontaine, J., 1990, Riemannian Geometry, Springer-Verlag, Berlin.
Kazerounian, K., and Rastegar, J., 1992, “Object Norms: A Class of Coordinate and Metric Independent Norms for Displacements,” Flexible Mechanisms, Dynamics and Analysis. ASME DE-Vol. 47, G. Kinzel et al., eds., New York.
Loncaric, J., 1985, “Geometric Analysis of Compliant Mechanisms in Robotics,” Ph.D. Thesis, Harvard University.
Milnor, J., 1969, Morse Theory, Princeton University Press, Princeton.
Ravani, B., 1982, “Kinematic Mapping as Applied to Motion Approximation and Mechanism Synthesis,” Ph.D. Thesis, Department of Mechanical Engineering, Stanford University.
This content is only available via PDF.
You do not currently have access to this content.