The Computation of Reachable Surfaces for a Specified Set of Spatial Displacements

Abstract

In this chapter, we consider the geometry of configurations of points generated by arbitrary sets of positions of a rigid body. The goal is to find those points in the moving body that lie on specific algebraic surfaces. The result is a set of algebraic equations that can be remarkably complex.The problem originates with Burmester's determination (Burmester 1886[3]) of points in a body that lie on a circle for an arbitrary set of five planar positions. He used these Burmester points to design a linkage that guided the body through the specified positions. His result was a graphical solution to a set of five quadratic equations in five unknown paramters. Today, these equations are relatively easy to solve numerically, see Suh and Radcliffe (1978)[32], Sandor and Erdman (1984) [26], or McCarthy (2000) [15]. Chen and Roth (1967)[4] introduced a generalization of this problem which seeks points and lines in a moving body that take positions on surfaces associated with articulated chains used to build robot manipulators. See Craig (1989) [5] or Tsai (1999)[34] for a discussion of serial chain robots. Our focus is two-jointed chains that supports a spherical wrist. The center of this wrist traces a surface which is said to be "reachable" by the chain. Considering the various ways of assembling these articulated chains, we obtain seven reachable algebraic surfaces, and the problem reduces to computing the dimensions of these chains from a set of polynomial equations.It is interesting how quickly the complexity of the problem increases with the number of dimensional parameters and the degree of the surface. The total degree of the polynomial systems that we consider range from 32 for the simplest to over 4 million for the most complex. The equations have significant internal structure, so it is possible to use a linear product decomposition (Bernshtein 1975[2] and Morgan et al. (1995)[17]) to provide a better bound on the number of solutions, which range from 10 to over 800,000. Where possible we have used polynomial elimination to verify the number of roots (Neilsen and Roth 1995[18] and Husty 1996[6]), but in the more complex examples