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RESEARCH PAPERS

# 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

## Abstract

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 $N⩾7$ 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 $N⩽5$, the body is underconstrained, having in general $6−N$ 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.

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## Figures

Figure 1

Schematic of locating six points on six planes

Figure 3

Viewing a feature point P with a pinhole camera

Figure 4

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

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)

Figure 2

Schematic of 3-2-1 locating

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