A Linear Algebra Approach to the Analysis of Rigid Body Displacement From Initial and Final Position Data

[+] Author and Article Information
A. J. Laub

Department of Electrical Engineering—Systems, University of Southern California, Los Angeles, Calif. 90007

G. R. Shiflett

Department of Mechanical Engineering, University of Southern California, Los Angeles, Calif. 90007

J. Appl. Mech 49(1), 213-216 (Mar 01, 1982) (4 pages) doi:10.1115/1.3161978 History: Received June 01, 1980; Revised March 01, 1981; Online July 21, 2009


The location and orientation of a rigid body in space can be defined in terms of three noncollinear points in the body. As the rigid body is moved through space, the motion may be described by a series of rotations and translations. The sequence of displacements may be conveniently represented in matrix form by a series of displacement matrices that describe the motion of the body between successive positions. If the rotations and translations (and hence the displacement matrix) are known then succeeding positions of a rigid body may be easily calculated in terms of the initial position. Conversely, if successive positions of three points in the rigid body are known, it is possible to calculate the parameters of the corresponding rotation and translation. In this paper, a new solution is presented which provides explicit formulas for the rotation and translation of a rigid body in terms of the initial and final positions of three points fixed in the rigid body. The rotation matrix is determined directly whereupon appropriate rotation angles and other information can subsequently be calculated if desired.

Copyright © 1982 by ASME
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