In this paper, we apply a homotopy algorithm to the problem of finding points in a moving body that lie on specific algebraic surfaces for a given set of spatial configurations of the body. This problem is a generalization of Burmester’s determination of points in a body that lie on a circle for five planar positions. We focus on seven surfaces that we term “reachable” because they correspond to serial chains with two degree-of-freedom positioning structures combined with a three degree-of-freedom spherical wrist. A homotopy algorithm based on generalized linear products is used to provide a convenient estimate of the number of solutions of these polynomial systems. A parallelized version of this algorithm was then used to numerically determine all of the solutions.
Generalized Linear Product Homotopy Algorithms and the Computation of Reachable Surfaces
Contributed by the Computer-Aided Product Development (CAPD) Committee for publication in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscript received September 2003; Revised April 2004. Associate Editor: K. Lee.
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Su, H., McCarthy, J. M., and Watson, L. T. (September 7, 2004). "Generalized Linear Product Homotopy Algorithms and the Computation of Reachable Surfaces ." ASME. J. Comput. Inf. Sci. Eng. September 2004; 4(3): 226–234. https://doi.org/10.1115/1.1760550
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