On a Rigid Body Subject to Point-Plane Constraints

[+] Author and Article Information
Charles W. Wampler

 General Motors R&D Center, Mail Code 480-106-359, 30500 Mound Road, Warren, MI 48090-9055charles.w.wampler@gm.com

A pure vector is a quaternion whose scalar part is zero.

A point in projective space is dehomogenized by setting one coordinate, or a linear combination of coordinates, to a constant, thereby removing the freedom to rescale.

J. Mech. Des 128(1), 151-158 (Jul 19, 2005) (8 pages) doi:10.1115/1.2120787 History: Received April 29, 2005; Revised July 19, 2005

This paper investigates the location of a rigid body such that N specified points of the body lie on N given planes in space. Variants of this problem arise in kinematics, metrology, and computer vision, including some, such as the motion of a spherical four-bar, that are not at first glance point-plane contact problems. The case N=6, the minimum number to fully constrain the body, is of special interest: We give an eigenvalue method for finding all solutions, which may number up to eight. For N7 there are, in general, no solutions, but if the constraints are compatible and not degenerate, we show how to find the unique solution by a linear least-squares method. For N5, the body is underconstrained, having in general 6N degrees of freedom; we determine the degree of the general motion for each case. We also examine the workspace of a particular three-degree-of-freedom parallel-link tripod mechanism.

Copyright © 2006 by American Society of Mechanical Engineers
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Grahic Jump Location
Figure 1

Schematic of locating six points on six planes

Grahic Jump Location
Figure 3

Viewing a feature point P with a pinhole camera

Grahic Jump Location
Figure 4

Geometric constraints of the tripod mechanism. The triangle is free to move subject to each vertex remaining on a given plane

Grahic Jump Location
Figure 5

Linear traces. Under parallel translation of the slicing line, the centroid of a witness set must follow a line. This is true for a composite set (L0) and for each of its irreducible pieces (L1,L2)

Grahic Jump Location
Figure 2

Schematic of 3-2-1 locating



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