Transformation Matrices in Kinematics

John J. Uicker; Bahram Ravani; Pradip N. Sheth

doi:10.1017/CBO9781139032339.004

3 - Transformation Matrices in Kinematics

Published online by Cambridge University Press: 05 April 2013

John J. Uicker ,

Bahram Ravani and

Pradip N. Sheth

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John J. Uicker: Affiliation:
University of Wisconsin, Madison
Bahram Ravani: Affiliation:
University of California, Davis
Pradip N. Sheth: Affiliation:
University of Virginia

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Summary

Introduction

Before formulating a numeric method for design analysis of mechanisms and multibody systems, let us first consider the essential characteristics of the problem being addressed. What are the chief difficulties encountered in the design analysis of a mechanism or multibody system? It is not the laws of mechanics as such that cause difficulty. It is the fact that, once a problem has been formulated, it is often too formidable algebraically to be easily solved. This complexity does not arise from static and dynamic force relationships, but from the kinematics – the changing geometry. The basic constraint equations that govern the motions within a machine or multibody system come from the fact that rigid bodies hold their respective joint elements in constant spatial relationships to one another. This type of constraint invariably leads to a set of highly nonlinear simultaneous algebraic equations.

Because the difficulties in an analytic approach to mechanism and multibody system analysis stem from the geometry, it is wise to choose a mathematical formulation suited to this type of problem. One such formulation is based on the use of matrices to represent the transformation equations between strategically located coordinate systems fixed in successive bodies. This approach has been developed into an extremely general and powerful technique for mechanism and multibody system analysis, and the next several chapters are devoted to its presentation. Before this can be presented effectively, however, we must become familiar with a number of basic operations that render matrix algebra so useful in performing coordinate transformations. The purpose of this chapter, therefore, is to develop this foundation.

Type: Chapter
Information: Matrix Methods in the Design Analysis of Mechanisms and Multibody Systems , pp. 42 - 79

DOI: https://doi.org/10.1017/CBO9781139032339.004 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2013

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References

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