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 demonstrated 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 rotation 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. 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.

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 development 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 mechanisms 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 development, 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, follows 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, quaternions, 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 translation 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 parameterization 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 numbers 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 dimensions 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 orientation 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. The parameters are sometimes referred to as Euler parameters but Rodrigues should take all the credit [33,34]. This is the reason that the four parameters are sometimes called Euler–Rodri- gues parameters [8]. The four Euler–Rodrigues parameters led to Rodrigues parameterization of the quaternion [32,33] and were equivalent to HamiltonÕs system of quaternions as noted by Klein [35]. Based on these four parameters, Rodrigues further derived other composition formulae [18,8] for finite rotations along with a full physical meaning for combining rotations and for construct- ing a rotation matrix. The two vector-form composition formulae [8] constitute the theorem for the multiplication of quaternions, leading to the revelation of the group properties of the set of all orthogonal rotations, the full orthogonal group SO(3) as it is now called. Although Hamilton [31] made the same formulae as the foundation of his calculus of quaternions, Rodrigues formulae demonstrated the enormous importance of quaternion in the rotation group as brought to light by Cayley [36,37] in the composition of rotations. These parallel developments from both Ham- ilton and Rodrigues were recorded by Klein [35] in 1884. After the discovery of quaternions, a former KingÕs College London student, young University College London professor of mathematics and mechanics, and scientific philosopher William Kingdon Clifford (1845–1879) invented in 1873 dual numbers for concise manipulation of the analysis, and applied the dual numbers to kinematics. The operator e was acquired by Clifford to transform rotation about an axis into translation parallel to the axis. He derived the theory of biquaternions [38–41] (now favorably called dual quaternions) and associated them specifically with linear algebra to represent a general displacement of a rigid body and to model the group of rigid body displacements. The primary part of the dual quaternion is Euler–Rodrigues parameters; the dual part of it is the quaternion product of the vector quaternion of a translation vector and that of the primary part of the quaternion.

4. StudyÕs dual angle, BallÕs treatise on screws and KleinÕs hyperquadric

The study on rigid body displacements moved on. In the late 19th century, Eduard Study (1862–1930), a teacherÕs son who obtained his doctorate from the University of Munich, developed the important notion of a dual angle [42], which was composed of the projected angle between two lines as its primary part, and the perpendicular distance between the two lines as its dual part. The dual angle has a remarkable property that trigonometrical identities for ordinary angles are all valid. J.S. Dai / Mechanism and Machine Theory 41 (2006) 41–52 45

In the same time, following decades of continuing effort [43–45] in developing the theory of screws, a new system of mathematics was created by Sir Robert Stawell Ball (1840–1913), the Lowndean Chair of Astronomy and Geometry at Cambridge University and a former professor of applied mathematics and mechanism of the Royal College of Science in Dublin. In 1900, he published [12] a treatise on the theory of screws and established [46] a broad mathematical foundation that integrates both rotational and linear quantities into a single geometrical element, the screw. A screw is an elegant geometric entity and the system of screws covers all kinematics and mechanics of a rigid body. This system can be used to incorporate the previous development from Chasles motion to Rodrigues formulae and to HamiltonÕs quaternions in the study of rigid body displacements. For instance, the first number of HamiltonÕs quaternion is what we would term the pitch of the screw. Hamiltonian system [47] of rays can be developed into a conoidal cubic sur- face—cylindroid [44,48,49] which plays a fundamental part in the theory of screws and gives a composition of two displacements. KleinÕs Ôsimultaneous invariantÕ [50] of two linear complexes can be explained based on the virtual coefficient of the two screws reciprocal to the complexes. Highly significantly, it is the theory of screws that attaches a physical significance to those purely geometrical researches. From the discovery of dual angles and in parallel to BallÕs creation of the new system of mathematics, Study presented the rigid body displacement in eight homogeneous coordinates [51], which are actually identical with a dual quaternion. In Study coordinates of a rigid body displacement, the line coordinates of the displacement, i.e. the screw axis, can be extracted as derived by Hunt [52]. In StudyÕs work, the half-angles were again used and the rigid body displacement was investigated in the projective seven-space. The hyperquadric in this projective seven-space are remarkable analogous to KleinÕs hyperquadric [17,53] for lines in the projective five-space developed by German mathematician Felix Christian Klein (1849–1925) of Munich who obtained in 1868 his doctorate from German mathematician Julius Plu¨cker (1801–1868) [17,54,55]. A point on StudyÕs hyperquadric presents all information of the rigid body location including both position and orientation. A point which is not on StudyÕs hyperquadric, resulting from the diagonal of the Hamilton operator [56] being replaced by some non-zero quantities [51], presents a Ôsimilarity transformationÕ involving a change of scale which was discussed by Davidson and Hunt [34] with a scale factor other than +1. Any chosen similarity transformation with its particular scale factor has its corresponding point in the projective seven-space, in which Study used ÔSomaÕ to describe a displaced body in the similarity transformation. The ÔSomaÕ started, as discussed by Bottema and Roth [20], from a six-parameter representation. The first three of these parameters are components of a rotation triplet; the second three are components of a translation vector. In StudyÕs work of eight homogeneous numbers, this second three are represented with four numbers of which the first number represents the scalar product of a translation vector with the rotation triplet. The remaining three numbers give the vector product of this triplet with the translation vector after deducting from the translational vector weighted with a factor of the triplet. The similarity transformation is at the extent of Hamilton quaternions. When the scale factor is +1, Hamilton operator [30,56] which is a 4 · 4 skew symmetric matrix with a translation vector as its components can be used to relate the primary part to the dual part. 46 J.S. Dai / Mechanism and Machine Theory 41 (2006) 41–52

5. Finite screw displacements

After a long dormant period, an increasing amount of the study of the screw displacement of a rigid body began to thrive from contemporary kinematicians. Dimentberg [57] in 1965 presented the finite screw displacement with a screw axis [12] attached with the tangent of the dual angle of half the rotation. A general screw displacement that is a finite displacement was given and was analogous to RodriguesÕ original formula but in a dual form. The screw displacement of a rigid body through a dual angle of half the rotation about an axis whose resultant screw is equivalent to two successive half-revolutions executed about two screws which intersect the axis of the resultant screw at right angles and form a dual angle of half the rotation with one another. Yang and Freudenstein [58] in 1964 applied dual quaternions to obtain the screw displacement by premultiplying a dual line vector with the dual quaternion acting as a screw operator. The screw displacement was completed about the screw axis of the operator that has the common perpendicular with the line vector. Consider the screw operator as a function of time, continuous spatial motion can also be obtained as that in BlaschkeÕs work [59] in 1958. The corresponding transformation between coordinate frames in the finite screw displacement was described by Yuan and Freudenstein [60] in 1971. Further to this, Bottema [61] investigated in 1973 the displacements of a row of points and of a line. The axes of the screw displacements which complete the former form a regulus as a cylindroid or hyperbolic paraboloid, and the axes of screw displacements which complete the latter form a line congruence of order 3. The geometric relationship concerning the combination of two finitely separated displacements, first suggested [62] in 1882by French mathematician George Henri Halphen (1844–1889), was substantiated and complemented by Roth [63] in 1967. With this work, a resultant screw displacement can be formed from two given constituent finite displacements by using the screw triangle whose name was originated by Roth [63] and which is constructed by three axes of screws as its vertices and three common perpendiculars of the axes as its sides. The method is equivalent to the decomposition of a screw displacement into two line reflections [20]. Further, the screw axis geometry of finitely separated positions based on five geometric elements was investigated by Tsai and Roth [64] in 1973 and the property of the finite screw cylindroid was presented. In addition to line geometry, dual number matrices were used to investigate the screw displacement. In 1985, Pennock and Yang [65] investigated the use of dual number matrices for transformation of coordinates of lines to solve the inverse kinematics problem of robot manipulators. In the following year, the property of the dual orthogonal matrix was revealed by McCarthy [66], leading to the development of a dual form of the Denavit–Hartenberg matrix [67] and a dual form of the Jacobian of a manipulator. He further applied quaternions to the study of spherical chains and the dual angles and dual quaternions to that of spatial open and close chains [68]. In 1990, Pohl and Lipkin [69] investigated the way of implementing the dual angles for robotic manipulators. They converted the dual joint angles to real numbers through a suitable mapping that a robot can approximate the configuration required to produce the desirable location within the limits of its workspace. It demonstrated that for certain manipulators, the real-part mapping produces a minimization of the end-effector location error. A dual orthogonal matrix can be represented as a six-dimensional representation of group E(3), an action on a line of the projective five-space as described by Selig and Rooney [70]. Their study indicated that this action is restricted to the required action on the Klein hyperquadric and splits J.S. Dai / Mechanism and Machine Theory 41 (2006) 41–52 47 the Klein hyperquadric into two orbits: the lines at infinity and the finite lines. The isotropy group of a finite line consists of the rotations with that line as axis, together with the translations along the line, leading to the Lie group version of the finite screw displacement. In 1991, Samuel, McA- ree and Hunt [71] unified the dual transformation matrices and screw geometry through the use of invariant properties of orthogonal matrices and demonstrated the equivalence of screw geometry with the matrix representations of the Euclidean group, providing a complete expression for the finite screw motion in terms of the entire dual number transformation matrix. The finite screw demonstrates to be suitable for trajectory planning and a concise expression is developed to give the transformation matrix describing the displacement at each point along the path of the finite screw motion. To represent the finite screw, Hunt [72] specified in 1987 the axis and pitch of the finite screw displacement by considering the geometric form in the point-line-plane system and by describing a body in two generally disposed locations. While maintaining the two sets of ratios of directed planes and directed lines of a body in two locations, five necessary conditions were proposed to construct six equations to determine the homogenous Plu¨cker line coordinates of the axis of the finite screw. In the application of finite screws, Young and Duffy [73] applied in 1986 the finite displacement to identify the extreme positions of manipulators. Angeles [74] developed in 1986 an algorithm based on the concept of the principal values and directions of the second-moment tensor of three non-collinear points of a rigid body in finitely separated positions.

6. The ﬁnite twist representation and ﬁnite screw systems

The study raised much interest in the 1990s particularly in the representation of the finite twist. In 1990, while studying the invariant property of a rigid body undergoing a finite twist displacement, Parkin [75] proposed a finite twist representation. Deriving from difference screws of both initial and final positions of a rigid body, the direction component, translation component, sign and angle of rotation of the finite twist were presented. The particular form in terms of the finite twist was proposed based on the coordinate transformation with line triplets of the initial and final locations of the body, on the condition that the axis of the finite twist remains invariant in space while the twist motion takes place and that the axis is perpendicular to, and reciprocal to, each of the difference screws from the two line triplets. Parkin [76] further identified in 1991 the finite displacement screws of a compound body and presented the 2-system of finite displacement screws of the point-lines. The quasi-pitch (q-pitch) was then proposed as the ratio of half the translation distance over tangent of half the rotation angle. The q-pitch contains the essential feature of the rigid body displacement and presents as an intrinsic part of a finite twist. With this proposed q-pitch, Parkin [77] demonstrated in 1992that the axes of finite twist displacements of a point-line object have a similar conformation with linear combinations of screws and presented the finite twist cylindroid as linear combinations of two basis finite screws. At the same time, Hunt [52] demonstrated in 1992that the q-pitch of the finite twist can be extracted from the Study coordinates. Huang [78] in 1994 and 1995 investigated the finite screw system of the third order [79] associated with kinematic chains and identified [80] in 1997 the cylindroid associated with the Bennett mechanism. In 1995, Dai, Holland and Kerr [81] further 48 J.S. Dai / Mechanism and Machine Theory 41 (2006) 41–52 investigated the finite twist representation and transformation and its ordered combination for serial manipulators. They revealed that the motion imparted by joints to the end-effector link can be represented by an ordered set of finite twist transformations and this ordered combination based on the Lie group operation gives a resultant finite twist of an end-effector relative to a datum point. Naturally, the study of the finite screw displacement and the development of the q-pitch of the finite twist progressed into the study of finite screw systems in conjunction with the study of theory of screw systems by Gibson and Hunt [82,83], Rico and Duffy [84–86], and Dai and Rees Jones [87–89]. In 1994, Huang and Roth [90] applied an analytical approach to investigate the finite screw systems. In addition to the case in which two points are specified, which is the same as the case in which a line with an associated point on it is specified [77], they demonstrated that the screw systems resulting from the other four incompletely specified displacement problems defined by Tsai and Roth in 1973 [64] can also be represented by linear systems or their nonlinear subsets. The first case was demonstrated by substituting the q-pitch into the screw triangle and rearranging the resultant displacement screw to form a 2-system by extracting two new basis screws whose elements were known parameters. The technique was then extended to four-systems to pres- ent the analytical representations of the finite twist systems using linear spaces. Almost at the same time, Hunt and Parkin [91] identified two particular linear two-systems for the three possible combinations of two geometrical elements and demonstrated the axes of the finite screws with the q- pitch are the generators of a cylindroid of a point-line object. The finite twist system of point-line displacements was further identified to be a general two-systems specified in [83] and the finite twist system of plane-line displacements was identified to be a fourth special two-system specified in [83]. The line-displacement three-system was identified as quadric surfaces which are hyperboloids in the most general case and become hyperbolic paraboloids in some special three-systems. In closely parallel to the study of the finite screw displacement, Borri, Trainelli and Battasso [92] revisited in 2000 the representations and parameterizations of motion from Cayley formula and Rodrigues parameters in a great length. They examined the projection of a point of a unit circle from the pole onto a y-axis that produces a stereographic projection defined as the tangent of half the rotation angle. They related the projection to the structure of CayleyÕs rotation vector and rigid displacement vector and then to Rodrigues parameters, leading to the revelation of the inherent structure of rigid body motion. The concept of the finite twist is thus known, as are the transformation matrices used to accom- plish the relevant elemental rigid body transformations, namely rotations and translations. The representation and correspondence between a finite twist and its transformation are, on one hand, the algebra of translational and rotational transformations as applied to screw quantities; on the other hand, the new systematic representation of these motions.

7. Conclusions

The work of Rodrigues, Cliﬀord and Study has all provided analytical means of describing the ﬁnite displacement. It is in many respects desirable to use a form of representation that has the same number of parameters as there are degrees of freedom, that is six in spatial cases, three in planar cases. J.S. Dai / Mechanism and Machine Theory 41 (2006) 41–52 49

In reviewing the historical progress in the study of mathematical means of describing finite displacements, this paper presented intrinsic relationships between different developments in the progress and explores the coherent theme in the two-century-long development of the theories which constitute a broad and solid mathematical science foundation for our todayÕs study of the finite twist motion of a rigid body.

Acknowledgements

The author thanks Professor Gene Fichter of Oregon for the suggestion when they met in Milan in 1995 on writing a review of the development of the theory of rigid body displacements and the ﬁnite twist and the encouragement. Thanks are also given to the staﬀ in Maughan library of KingÕs College London, University of London, for providing a substantial number of interli- brary loans from across the world.

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