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3 - Transformation Matrices in Kinematics

Published online by Cambridge University Press:  05 April 2013

John J. Uicker ,
Bahram Ravani  and
Pradip N. Sheth
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
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Matrix Methods in the Design Analysis of Mechanisms and Multibody Systems , pp. 42 - 79
Publisher: Cambridge University Press
Print publication year: 2013

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References

Cardano, G., Opus nouum de proportionibus numerorum, motuum, ponderum, sonorum aliarumque rerum mensurandarum. Item de aliza regula. Basel, 1570. A schematic drawing of the Cardan joint is pictured in this manuscript. However, there is no evidence that Cardan ever constructed such a device.
Ceccarelli, M., “Screw axis defined by Giulio Mozzi in 1763 and early studies on helicoidal motion,” Mechanism and Machine Theory, vol. 35, 2000, pp. 761–70.CrossRefGoogle Scholar
Chasles, M., “Note sur les propriétés générales du système de deux corps semblables entr'eux et placés d'une manière quelconque dans l'espace; et sur le déplacement fini ou infiniment petit d'un corps solide libre, [Notes on general properties of a system of two identical bodies arbitrarily located in space; and on the finite or infinitesimal motion of a free solid body],” Bulletin des Sciences Mathématiques, Astronomiques, Physiques et Chimiques de Ferrussac [Bulletin of the Sciences of Mathematics, Astronomy, Physics, and Chemistry by Ferrussac], vol. 14, Paris, 1830, pp. 321–26.
Coolidge, J. L., A History of Geometrical Methods, Dover Publications, Inc., New York, 1963. Includes an extensive bibliography.
Descartes, R., Discours de la méthode pour bien conduire sa raison et chercher la vérité dans les sciences, [Discourse on methods for reasoning and seeking truth in science], Leiden: Jan Maire, 1637. Cartesian coordinates were introduced in the third appendix that is titled La Géométrie, which focuses on the connections between geometry and algebra.Google Scholar
Euler, L., “Theoria motus corporum solidorum seu rigidorum, [Treatise on the motion of solids or rigid bodies],” Opera omnia II, vol. 9, Rostock, 1765, pp. 84–98; also in “Formulae generales pro translatione quacumque corporum rigidorum, [General formulae for the motion of rigid bodies],” Novi Comentarii Academiae Scientiarum Petropolitanae, [New memoirs of the imperial academy of sciences in St. Petersburg], vol. 20, 1776, pp. 189–207.
Feuerbach, K. W., Grundriss zu analytischen Untersuchungen der dreieckigen Pyramide [Foundations of the analytic theory of the triangular pyramid], Nüremberg, 1827.
Friberg, O., “Computation of Euler Parameters from Multipoint Data,” Journal of Mechanisms, Transmissions, and Automation in Design, ASME Transactions, vol. 110, June 1988, pp. 116–121.
Gergonne, J. D., Annales de mathématique pures et appliquées [Annals of pure and applied mathematics] better known as Annales de Gergonne, Neimes, 1824–27.
Goldstein, H., Classical Mechanics, Addison-Wesley Publishing Co., Inc., Reading, MA, 2nd ed., 1980.Google Scholar
Hamilton, W. R., Lectures on quaternions containing a systematic statement of a new mathematical method: of which the principles were communicated in 1843 to the Royal Irish Academy and which has since formed the subject of successive courses of lectures delivered in 1848 and subsequent years, in the halls of Trinity College, Dublin: with numerous illustrative diagrams, and with some geometrical and physical applications, Hodges and Smith, Dublin, 1853.Google Scholar
Maxwell, E. A., General Homogeneous Coordinates in Space of Three Dimensions, Cambridge University Press, London, 1951. An excellent general reference, reissued in 2008.Google Scholar
Möbius, A. F., “Der barycentrische Calcul, [Barycentric calculus]”, Crelle's Journal für die reine und angewandte Mathematik [Crelle's Journal for Pure and Applied Mathematics], Leipzig, 1827.
Mozzi, G., “Discorso matematico sopra il rotamento momentaneo dei corpi [Mathematical treatise on temporally revolving bodies],” Stamperia di Donato Campo, Napoli, 1763.
Plücker, J., Neue Geometrie des Raumes gegründet auf die Betrachtung der geraden Linie als Raumelement [A new geometry of space based on consideration of the straight line as the spatial element], B. G. Teübner, Leipzig, 1868–69, pp. 1–374.
Rodrigues, O.. “Des lois géométriques qui régissent les déplacements d'un système solide dans l'espace, et de la variation des coordonnées provenant de ces déplacements considérés indépendamment des causes qui peuvent les produire [Geometric laws that govern the movement of a solid system in space, and the change of coordinates from these displacements considered independently of the causes that produce them],” Journal de Mathématiques Pures et Appliquées [Journal of pure and applied mathematics], Bachellier, Paris, 5, 1840, pp. 380–440, better known as Annales de Gergonne, described in detail in H. Cheng and K. C. Gupta, “An Historical Note on Finite Rotations,” Journal of Applied Mechanics, ASME Transactions, vol. 56, no. 1, 1989, pp. 139–45.
Stuelpnagel, J., “On the Parametrization of the Three-Dimensional Rotation Group,” SIAM Review, vol. 6, no. 4. (Oct., 1964), pp. 422–30.CrossRefGoogle Scholar

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