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An Historical Review of the Theoretical Development of Rigid Body Displacements from Rodrigues Parameters to the finite Twist

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An Historical Review of the Theoretical Development of Rigid Body Displacements from Rodrigues Parameters to the finite Twist Mechanism and Machine Theory Mechanism and Machine Theory 41 (2006) 41–52 www.elsevier.com/locate/mechmt An historical review of the theoretical development of rigid body displacements from Rodrigues parameters to the finite twist Jian S. Dai * Department of Mechanical Engineering, School of Physical Sciences and Engineering, King’s College London, University of London, Strand, London WC2R2LS, UK Received 5 November 2004; received in revised form 30 March 2005; accepted 28 April 2005 Available online 1 July 2005 Abstract The development of the finite twist or the finite screw displacement has attracted much attention in the field of theoretical kinematics and the proposed q-pitch with the tangent of half the rotation angle has dem- onstrated an elegant use in the study of rigid body displacements. This development can be dated back to RodriguesÕ formulae derived in 1840 with Rodrigues parameters resulting from the tangent of half the rota- tion angle being integrated with the components of the rotation axis. This paper traces the work back to the time when Rodrigues parameters were discovered and follows the theoretical development of rigid body displacements from the early 19th century to the late 20th century. The paper reviews the work from Chasles motion to CayleyÕs formula and then to HamiltonÕs quaternions and Rodrigues parameterization and relates the work to Clifford biquaternions and to StudyÕs dual angle proposed in the late 19th century. The review of the work from these mathematicians concentrates on the description and the representation of the displacement and transformation of a rigid body, and on the mathematical formulation and its progress. The paper further relates this historic development to the contemporary development of the finite screw displacement and the finite twist representation in the late 20th century. Ó 2005 Elsevier Ltd. All rights reserved. * Tel.: +44 (0) 2078482321; fax: +44 (0) 2078482932. E-mail address: [email protected] URL: http://www.eee.kcl.ac.uk/mecheng/jsd. 0094-114X/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2005.04.004 42 J.S. Dai / Mechanism and Machine Theory 41 (2006) 41–52 Keywords: Theoretical kinematics; Rotations; Transformation group; Screw; Finite twist; Finite screw displacement; Rigid body displacement; Mathematics; History; Review 1. Introduction Position (translation) and orientation (rotation), together known as location, hold interest in the study of mechanisms and machines and of their motion capabilities. Orientation may be measured in a number of ways including the use of Euler angles [1–3] proposed in 1775 by German and Rus- sian mathematician Leonhard Euler (1707–1783, a Swiss native), and the rotation matrix may be established using Euler finite rotation formula [4,5] whose matrix form can be seen in [6–11]. While rotations can be characterized by means of Euler or Bryant angles, or Euler parameters, none of these representations of rotations lends themselves directly or by extension to the more demanding problem of describing the finite rigid body displacement, or the finite twist consisting of an arbitrary rotation about an axis passing through a point and a translation along the axis. This resulted in a need of the concept of the generalized finite twist displacement of a rigid body. The study with the emphasis on its pure analytical content and the mathematical develop- ment has been progressing for the past two centuries and can be summarized in three periods. The first period was in the early and the main part of the 19th century when mathematicians started focusing on general applications to the physical world which was also a source of mathematical progress. The second period was in the late 19th century and the early 20th century when Study used ÔSomaÕ to describe the body displacement. With the emergence of BallÕs treatise [12] in 1900, an elegant system of mathematics on theory of screws was formed for rigid body mechanics. The third period was in the second part of the 20th century when kinematicians revisited the theories developed by mathematicians and astronomers, applied the theories to kinematics and mecha- nisms and continued the effort to develop and complement the theories. This was the important period of the theoretical kinematics. The importance of the last period is the practical use and continuing effort of theory develop- ment, of amalgamating theories and approaches into new theories and approaches, of solving kinematics problems and of obtaining solutions for mechanisms. While before this period, most scientists made very few statements regarding the physical application of their theory and steered clear of the philosophical aspects of their work. This paper reviews the progress of the study of rigid body displacements in these periods, fol- lows the development of the theories, and associates this development with the study of the finite twist in the 1990s. 2. Chasles motion and Rodrigues parameters In the early 19th century in Europe, new professional status of mathematics was fostered [13] by the creation of new universities or equivalent institutions and the reinvigoration of certain old ones. A massive growth was there in publishing mathematics in books and journals. In that time, algebra became algebras and the theory of equations was joined by differential operators, quater- nions, determinants and algebraic logic. That was the time when mathematicians began moving to the physical world. J.S. Dai / Mechanism and Machine Theory 41 (2006) 41–52 43 In 1830, after Italian Mathematician Giulio MozziÕs (1730–1813) revelation of the instanta- neous motion axis [14,15], French mathematician and historian of mathematics Michel Chasles (1793–1880), suggested that in terms of end-point locations, all finite and infinitesimal motions of a rigid body could be duplicated by means of a rotation about an axis, together with a trans- lation along that axis. A motion [16,17] is known favorably now as the finite screw displacement or the finite twist. The rotation axis can be taken the same direction as the translation. The three rotational degrees of freedom correspond to the two angles needed to define the direction of the rotation axis and to the amount of rotation about that axis. Any such a resultant finite twist may be defined by means of the angle of rotation, the direction and position of the axis, and the pitch or the translation along that axis. This rotation angle is unique provided that it is confined to values in the range of Àp and p. A short while after ChaslesÕ work, French mathematician Olinde Rodrigues (1794–1851, Por- tuguese origin, also a banker and a social reformer), the son of a Jewish banker and who was awarded a doctorate in mathematics from E´ cole Normale, worked on transformation groups to study the composition of successive finite rotations by an entirely geometric method. In 1840, Rodrigues published a paper on the transformation groups. Rodrigues parameters [18] that integrate the direction cosines of a rotation axis with the tangent of half the rotation angle were presented with three quantities. The angles of the rotations appear as half-angles which occurred for the first time in the study of rotations. The half-angles are an essential feature of the param- eterization of rotations and are the measure of pure rotation for the most elegant representation of rotations in kinematics. Based on these three parameters, Rodrigues composition formulae [10,18–20] were proposed for two successive rotations to construct the orientation of the resultant axis and the geometrical value of the resultant angle of rotation from the given angles and axis orientations of the two successive rotations. This led to the Rodrigues formula [20] for a general screw displacement producing not only the rotation matrix but also the translation distance. The formula can be written in vector form as in [6–8,21]. Rodrigues work is the first treatment of motion in complete isolation from the forces that cause it. The Rodrigues parameters were further taken by English mathematician Arthur Cayley (1821– 1895, a graduate and later Sadleirian professor of pure mathematics at Cambridge University) to comprise a skew symmetric matrix which then formed CayleyÕs formula [22] for a rotation matrix [23]. 3. HamiltonÕs quaternions, Rodrigues parameterization and Clifford biquaternions In this period, huge interest was in algebras and eventually led to the invention of quaternions. This stemmed from the study of complex numbers. With GaussÕ (German mathematician Carl Friedrich Gauss, 1777–1855) suggestion in 1831, the complex plane [24,25] with the complex num- bers started to gain favor. In this study, Irish mathematician and astronomer William Rowan Ham- ilton (1805–1865) suggested a new algebraic version [26] in 1833, in which the complex number was understood as an ordered pair of real numbers satisfying the required algebraic properties. From this development, more important and famous extension to algebra was on the way. HamiltonÕs own work on algebraically describing mechanics led him seek an algebraic means of a complex number in three dimensions. This let him produce a three-number expression of a 44 J.S. Dai / Mechanism and Machine Theory 41 (2006) 41–52 complex number and nurtured a new finding. On a walk into Dublin on 16 October 1843, Ham- ilton discovered a four-number expression. This unexpected venture into four algebraic dimen- sions gave Hamilton the breakthrough and established the theory of Quaternions [27–31]. Hamilton came to this discovery algebraically [32]. The quaternion was used to represent the ori- entation of a rigid body with four quantities identical to Euler–Rodrigues parameters of rotations and was further applied to representing spherical displacements. A few years early than HamiltonÕs discovery, in the same paper published in 1840 where Rodri- gues developed his three parameters, Rodrigues explicitly defined other four parameters by pre- senting a scalar with the cosine of half the rotation angle and further three numbers by integrating the direction cosines of the rotation axis with the sine of half the rotation angel.
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