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Locating N Points of a Rigid Body on N Given Planes

[+] Author Affiliations
Charles W. Wampler

General Motors R&D Center, Warren, MI

Paper No. DETC2004-57182, pp. 513-521; 9 pages
doi:10.1115/DETC2004-57182
From:
  • ASME 2004 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
  • Volume 2: 28th Biennial Mechanisms and Robotics Conference, Parts A and B
  • Salt Lake City, Utah, USA, September 28–October 2, 2004
  • Conference Sponsors: Design Engineering Division and Computers and Information in Engineering Division
  • ISBN: 0-7918-4695-4 | eISBN: 0-7918-3742-4
  • Copyright © 2004 by ASME

abstract

This paper describes a method for finding the location of a rigid body such that N specified points of the body lie on N given planes in space. Of special interest is the case N = 6, which is the minimum number to fully constrain the body. This geometric problem arises in two seemingly disparate contexts: metrology, as a generalization of so-called “3-2-1” locating schemes; and robotics, as the forward kinematics problem for 6ES or 6SE parallel-link platform robots. For N = 6, the geometric problem can be formulated algebraically as 3 quadratic equations having, in general, eight possible solutions. We give a method for finding all eight solutions via an 8 × 8 eigenvalue problem. We also show that for N ≥ 7, the solution can be found uniquely as a linear least squares problem.

Copyright © 2004 by ASME

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