On the application of Quaternions to the Theory of Rotation

Abstract

In a paper published in the Philosophical Magazine, February 1845, [20], I showed how some formulae of M. Olinde Rodrigues relating to the rotation of a solid body might be expressed in a very simple form by means of Sir W. Hamilton's theory of quaternions. The property in question may be thus stated. Suppose a solid body which revolves through an angle θ round an axis passing through the origin and inclined to the axes of coordinates at angles a, b, c. Let and write let X, y, z be the coordinates of a point in the body previous to the rotation, iPj, yι. •2' 1 those of the same point after the rotation, and suppose then the coordinates after the rotation may be determined by the formula viz., developing the second side of this equation.

An Historical Note on Finite Rotations

“…Therefore, the finite rotation formula (1) or (4) and (5) is nothing but a vector or matrix form of the formulae (8) obtained by Euler more than two hundred years ago. Our conclusion also seems to be supported by Cayley (1846), Benedikt (1944), andHill (1945). If one uses Euler angles, 4>, 0, and i/s the following rotation matrix can be obtained (Euler, 17?…”

“…The corresponding rotational elements derived by Rodrigues are shown in Appendix All. Rodrigues parameters (10) were used by the Irish mathematician Arthur Cayley (1821-1895) to study the motion of a rigid body in Cayley (1843), 1846, and (1848). One disadvantage of using Rodrigues parameters 10is that the rotation matrix (116) is singular when <j> = w. For the general computer implementation of multibody dynamic systems, the following four parameters are extensively used:…”

“…I, Note 20, p. 586), the quaternion formulae (21) itself was first derived by Hamilton. However, physical meaning of the quaternion form (21) in the context of finite rotation, utilizing Rodrigues parameters (10), was first discovered by Cayley [Cayley (1843)]. When expanded, the quaternion form (21) leads to formulae about which Cayley asserted in Cayley (1845) that "In fact the formulae are precisely those given for such a transformation by M. Olinde Rodrigues" in Rodrigues (1840) shown in Appendix AIL…”

Desarrollo de una Nueva Solución Compacta a la Cinemática de Manipuladores Robóticos basada en Cuaterniones Duales

“…Los trabajos de Hamilton (1853), Cayley (1889), y posteriormente Tait et al (1890) como base de los cuaterniones permiten desarrollar en este trabajo, una estrategia para obtener una solución práctica como el producto de matrices pero sin las singularidades y con un ahorro de tiempo en el cálculo en operaciones como se demuestra en Salamin (1995).…”

Euler–Rodrigues formula variations, quaternion conjugation and intrinsic connections

“…In 2012, Kovács [41] gave a new derivation of the Euler-Rodrigues formula based on a matrix transformation of three continuous rotations. In the same year, Pujol [15,42] investigated the relation between the composition of rotations and the product of quaternions [43] and related the work to Cayley's early contribution [11,44] through Euler-Rodrigues parameters. Following various studies, the use of the Euler-Rodrigues formula and of the Euler-Rodrigues-parameters formulated unit-quaternion has been extended to a broad range of research topics including vector parameterization of rotations [33,40,45], rational motions [46][47][48][49], motion generation [50][51][52] and planning [53], kinematic mapping [54,55], orientation [56] and attitude estimation [57][58][59], mechanics [60], constraint analysis [61,62], reconfiguration [63,64], mechanism analysis [65][66][67] and synthesis [68,69], sensing [70] and computer graphics [71][72][73] and vision [74].…”