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Technical Brief

# The Rodrigues Equations for the Composition of Finite Rotations: A Simple Ab Initio Derivation and Some Consequences

[+] Author and Article Information
Jose Pujol

Department of Earth Sciences,
The University of Memphis,
Memphis, TN 38152

For Euler, translations included rotations.

Gauss introduced half-angles in 1819 as part of several unpublished notes that show he had derived the basic features of quaternions. These notes, and others, were published posthumously much later [11]. For further details see Ref. [12].

Manuscript received May 21, 2013; final manuscript received September 3, 2013; published online October 8, 2013. Editor: Harry Dankowicz.

Appl. Mech. Rev 65(5), 054501 (Oct 08, 2013) (6 pages) Paper No: AMR-13-1037; doi: 10.1115/1.4025356 History: Received May 21, 2013; Revised September 03, 2013

## Abstract

The well-known Rodrigues' equations for the composition of two finite rotations were introduced in 1840 without a detailed derivation. Although later derivations are available, the one presented here is very simple, is based on rotation results due to Euler, is essentially self-contained, and makes use of analysis techniques familiar to engineering students. As the Rodrigues equations have a strong connection to quaternions, this matter is considered in some detail. Then the equations are used to derive some general results concerning the angle and axis of the composite rotation. If α, β, θ and a, b, d indicate the rotation angles and axes for the two individual rotations and for the composite rotation, respectively, some of the results are as follows. (a) The value of θ does not depend on the order of the two rotations. (b) The vector d is equal to the sum of two vectors, one in the plane Π spanned by a and b, and one perpendicular to Π. Moreover, the vector in Π is always between a and b. (c) The rotation axes for the two possible composite rotations are symmetric with respect to Π. (d) The angle θ decreases from α + β to α − β as the angle between a and b increases from 0 to π. Finally, the Rodrigues equations are used to derive the equations corresponding to the composition of infinitesimal rotations, which are also derived using alternative approaches.

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## References

Rodrigues, O., 1840, “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,” J. Math. Pures Appl., 5, pp. 380–440.
Altmann, S., 1989, “Hamilton, Rodrigues, and the Quaternion Scandal,” Math. Mag., 62, pp. 291–308.
Ames, J., and Murnaghan, F., 1958, Theoretical Mechanics, Dover, New York, Chap. 2, p. 462.
Altmann, S., 1986, Rotations, Quaternions, and Double Groups, Oxford University, London, Chap. 9, p. 317.
Paul, B., 1963, “On the Composition of Finite Rotations,” Am. Math. Monthly, 70(8), pp. 859–862.
Euler, L., 1776a, “Formulae Generales pro Translatione Quacunque Corporum Rigidorum,” Novi Comment. Acad. Sci. Petrop., 20, pp. 189–207. Available at http://eulerarchive.maa.org (Enestrom number E478).
Euler, L., 1776b, “Nova Methodus Motum Corporum Rigidorum Determinandi,” Novi Comment. Acad. Sci. Petrop., 20, pp. 208–238. Available at http://eulerarchive.maa.org (Enestrom number E479).
Pujol, J., 2009, “Tutorial on Rotations in the Theories of Finite Deformation and Micropolar (Cosserat) Elasticity,” Bull. Seismol. Soc. Am., 99(2B), pp. 1011–1027. Available at https://umdrive.memphis.edu/jpujol/public/Quaternion/Pujol-2009.pdf
Jansen, L., and Boon, M., 1967, Theory of Finite Groups. Applications in Physics, North-Holland, Amsterdam, p. 367.
Palais, B., and Palais, R., 2007, “Euler's Fixed Point Theorem: The Axis of a Rotation,” J. Fixed Point Theory Appl., 2, pp. 215–220.
Gauss, C., 1900, “Mutationen des Raumes,” Carl Friedrich Gauss Werke, Vol. 8, Königlichen Gesellschaft der Wissenschaften, Göttingen, Germany, pp. 357–362. Available at https://umdrive.memphis.edu/jpujol/public/Quaternion/Gauss-1900.pdf
Pujol, 2012, “Hamilton, Rodrigues, Gauss, Quaternions, and Rotations: A Historical Reassessment,” Commun. Math. Anal., 13(2), pp. 1–14. Available at https://umdrive.memphis.edu/jpujol/public/Quaternion/Hamilton-Rodrigues-Gauss-paper.pdf
Cheng, H., and Gupta, K., 1989, “An Historical Note on Finite Rotations,” J. Appl. Mech., 56, pp. 139–145.
Hamilton, W., 1847, “Theory of Quaternions,” Proc. R. Ir. Acad., 3, pp. 1–16. Available at https://umdrive.memphis.edu/jpujol/public/Quaternion/Hamilton-1847.pdf
Shabana, A., 1994, Computational Dynamics, Wiley, New York, Chap. 5, p. 464.
Tenenbaum, R., 2004, Fundamentals of Applied Dynamics, Springer-Verlag, New York, Chap. 3, p. 713.
Routh, E., 1905, The Elementary Part of a Treatise on the Dynamics of a System of Rigid Bodies, 7th ed., MacMillan and Co., London, Chap. 5, p. 443. Available at http://archive.org/details/elementaryrigid05routrich
Hamilton, W., 1853, Lectures on Quaternions, Hodges and Smith, Dublin, p. 736. Available at http://books.google.com/books/about/Lectures_on_Quaternions.html?id=TCwPAAAAIAAJ
Schaub, H., Tsiotras, P., and Junkins, J., 1995, “Principal Rotation Representations of Proper N × N Orthogonal Matrices,” Int. J. Eng. Sci., 33( 15), pp. 2277–2295.

## Figures

Fig. 1

Counterclockwise rotation of angle ψ of a vector r about an axis identified by a unit vector v. The result is the vector r′. The arc joining r and r′ is in a plane perpendicular to v.

Fig. 2

Rotation of a vector r with a particular realization of the two composite rotations in Eq. (42). In both plots the shaded area is a portion of the plane Π spanned by the vectors a and b. The rotation angles and axes were determined using Eqs. (21) and (28), and the corresponding rotations were performed using Eq. (34) which resulted in the vectors labeled r″. The rotation axes have been labeled d1 and d2. The rotation angles are equal (see text). The same vantage point was used for the two plots.

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