Jian dai a historical review of the theoretical development of rigid body displacements from rodrigues..., 12p
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Mechanism and Mechanism and Machine Theory 41 (2006) 41–52 Machine Theory 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 WC2R 2LS, UK Received 5 November 2004; received in revised form 30 March 2005; accepted 28 April 2005 Available online 1 July 2005Abstract The development of the ﬁnite twist or the ﬁnite screw displacement has attracted much attention in theﬁeld 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 toRodriguesÕ 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 thetheoretical 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 quaternionsand Rodrigues parameterization and relates the work to Cliﬀord biquaternions and to StudyÕs dual angleproposed in the late 19th century. The review of the work from these mathematicians concentrates on thedescription and the representation of the displacement and transformation of a rigid body, and on themathematical formulation and its progress. The paper further relates this historic development to the contemporary development of the ﬁnite screwdisplacement 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: jian.dai@kcl.ac.uk 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
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42 J.S. Dai / Mechanism and Machine Theory 41 (2006) 41–52Keywords: Theoretical kinematics; Rotations; Transformation group; Screw; Finite twist; Finite screw displacement;Rigid body displacement; Mathematics; History; Review1. Introduction Position (translation) and orientation (rotation), together known as location, hold interest in thestudy of mechanisms and machines and of their motion capabilities. Orientation may be measuredin 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 beestablished 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 moredemanding problem of describing the ﬁnite rigid body displacement, or the ﬁnite twist consistingof 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 rigidbody. 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. Theﬁrst period was in the early and the main part of the 19th century when mathematicians startedfocusing on general applications to the physical world which was also a source of mathematicalprogress. The second period was in the late 19th century and the early 20th century when Studyused 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. Thethird period was in the second part of the 20th century when kinematicians revisited the theoriesdeveloped 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 importantperiod 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 solvingkinematics problems and of obtaining solutions for mechanisms. While before this period, mostscientists made very few statements regarding the physical application of their theory and steeredclear 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 ﬁnitetwist in the 1990s.2. Chasles motion and Rodrigues parameters In the early 19th century in Europe, new professional status of mathematics was fostered [13] bythe creation of new universities or equivalent institutions and the reinvigoration of certain oldones. 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 tothe physical world.
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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 motionsof 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 displacementor the ﬁnite twist. The rotation axis can be taken the same direction as the translation. The threerotational degrees of freedom correspond to the two angles needed to deﬁne the direction of therotation axis and to the amount of rotation about that axis. Any such a resultant ﬁnite twist maybe deﬁned by means of the angle of rotation, the direction and position of the axis, and the pitchor the translation along that axis. This rotation angle is unique provided that it is conﬁned tovalues 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 Ecole Normale, worked on transformation groupsto study the composition of successive ﬁnite rotations by an entirely geometric method. In1840, Rodrigues published a paper on the transformation groups. Rodrigues parameters [18] thatintegrate the direction cosines of a rotation axis with the tangent of half the rotation angle werepresented with three quantities. The angles of the rotations appear as half-angles which occurredfor 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 representationof 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 resultantaxis and the geometrical value of the resultant angle of rotation from the given angles and axisorientations of the two successive rotations. This led to the Rodrigues formula [20] for a generalscrew displacement producing not only the rotation matrix but also the translation distance. Theformula can be written in vector form as in [6–8,21]. Rodrigues work is the ﬁrst treatment ofmotion 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 rotationmatrix [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 CarlFriedrich 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 wasunderstood 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 meansof a complex number in three dimensions. This let him produce a three-number expression of a
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44 J.S. Dai / Mechanism and Machine Theory 41 (2006) 41–52complex 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 rotationsand 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 byintegrating 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 allthe 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 ofthe 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 forthe multiplication of quaternions, leading to the revelation of the group properties of the set of allorthogonal 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 formulaedemonstrated the enormous importance of quaternion in the rotation group as brought to lightby 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 UniversityCollege London professor of mathematics and mechanics, and scientiﬁc philosopher WilliamKingdon Cliﬀord (1845–1879) invented in 1873 dual numbers for concise manipulation of theanalysis, and applied the dual numbers to kinematics. The operator e was acquired by Cliﬀordto transform rotation about an axis into translation parallel to the axis. He derived the theoryof biquaternions [38–41] (now favorably called dual quaternions) and associated them speciﬁcallywith linear algebra to represent a general displacement of a rigid body and to model the group ofrigid body displacements. The primary part of the dual quaternion is Euler–Rodrigues parame-ters; the dual part of it is the quaternion product of the vector quaternion of a translation vectorand 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, devel-oped the important notion of a dual angle [42], which was composed of the projected angle be-tween two lines as its primary part, and the perpendicular distance between the two lines as itsdual part. The dual angle has a remarkable property that trigonometrical identities for ordinaryangles are all valid.
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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 ofscrews, a new system of mathematics was created by Sir Robert Stawell Ball (1840–1913), theLowndean Chair of Astronomy and Geometry at Cambridge University and a former professorof applied mathematics and mechanism of the Royal College of Science in Dublin. In 1900, hepublished [12] a treatise on the theory of screws and established [46] a broad mathematical foun-dation that integrates both rotational and linear quantities into a single geometrical element, thescrew. A screw is an elegant geometric entity and the system of screws covers all kinematics andmechanics of a rigid body. This system can be used to incorporate the previous development fromChasles motion to Rodrigues formulae and to HamiltonÕs quaternions in the study of rigid bodydisplacements. For instance, the ﬁrst number of HamiltonÕs quaternion is what we would term thepitch 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 acomposition of two displacements. KleinÕs Ôsimultaneous invariantÕ [50] of two linear complexescan 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 purelygeometrical researches. From the discovery of dual angles and in parallel to BallÕs creation of the new system ofmathematics, 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 bodydisplacement, the line coordinates of the displacement, i.e. the screw axis, can be extracted asderived by Hunt [52]. In StudyÕs work, the half-angles were again used and the rigid body dis-placement was investigated in the projective seven-space. The hyperquadric in this projectiveseven-space are remarkable analogous to KleinÕs hyperquadric [17,53] for lines in the projectiveﬁve-space developed by German mathematician Felix Christian Klein (1849–1925) of Munichwho obtained in 1868 his doctorate from German mathematician Julius Plucker (1801–1868) ¨[17,54,55]. A point on StudyÕs hyperquadric presents all information of the rigid body location includingboth position and orientation. A point which is not on StudyÕs hyperquadric, resulting from thediagonal of the Hamilton operator [56] being replaced by some non-zero quantities [51], pre-sents a Ôsimilarity transformationÕ involving a change of scale which was discussed by Davidsonand Hunt [34] with a scale factor other than +1. Any chosen similarity transformation with itsparticular scale factor has its corresponding point in the projective seven-space, in which Studyused ÔSomaÕ to describe a displaced body in the similarity transformation. The ÔSomaÕ started, asdiscussed by Bottema and Roth [20], from a six-parameter representation. The ﬁrst three ofthese parameters are components of a rotation triplet; the second three are components of atranslation vector. In StudyÕs work of eight homogeneous numbers, this second three are rep-resented with four numbers of which the ﬁrst number represents the scalar product of a trans-lation vector with the rotation triplet. The remaining three numbers give the vector product ofthis triplet with the translation vector after deducting from the translational vector weightedwith 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 asits components can be used to relate the primary part to the dual part.
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46 J.S. Dai / Mechanism and Machine Theory 41 (2006) 41–525. Finite screw displacements After a long dormant period, an increasing amount of the study of the screw displacement of arigid body began to thrive from contemporary kinematicians. Dimentberg [57] in 1965 presentedthe ﬁnite screw displacement with a screw axis [12] attached with the tangent of the dual angle ofhalf the rotation. A general screw displacement that is a ﬁnite displacement was given and wasanalogous to RodriguesÕ original formula but in a dual form. The screw displacement of a rigidbody through a dual angle of half the rotation about an axis whose resultant screw is equivalent totwo successive half-revolutions executed about two screws which intersect the axis of the resultantscrew 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 displacementby premultiplying a dual line vector with the dual quaternion acting as a screw operator. Thescrew displacement was completed about the screw axis of the operator that has the common per-pendicular with the line vector. Consider the screw operator as a function of time, continuous spa-tial motion can also be obtained as that in BlaschkeÕs work [59] in 1958. The correspondingtransformation between coordinate frames in the ﬁnite screw displacement was described by Yuanand Freudenstein [60] in 1971. Further to this, Bottema [61] investigated in 1973 the displacementsof a row of points and of a line. The axes of the screw displacements which complete the formerform a regulus as a cylindroid or hyperbolic paraboloid, and the axes of screw displacementswhich 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 1882 by French mathematician George Henri Halphen (1844–1889), wassubstantiated and complemented by Roth [63] in 1967. With this work, a resultant screw displace-ment can be formed from two given constituent ﬁnite displacements by using the screw trianglewhose name was originated by Roth [63] and which is constructed by three axes of screws asits vertices and three common perpendiculars of the axes as its sides. The method is equivalentto the decomposition of a screw displacement into two line reﬂections [20]. Further, the screw axisgeometry of ﬁnitely separated positions based on ﬁve geometric elements was investigated by Tsaiand 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 displace-ment. In 1985, Pennock and Yang [65] investigated the use of dual number matrices for transfor-mation of coordinates of lines to solve the inverse kinematics problem of robot manipulators. Inthe 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 formof the Jacobian of a manipulator. He further applied quaternions to the study of spherical chainsand 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 manipula-tors. They converted the dual joint angles to real numbers through a suitable mapping that arobot can approximate the conﬁguration required to produce the desirable location within thelimits of its workspace. It demonstrated that for certain manipulators, the real-part mappingproduces 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 studyindicated that this action is restricted to the required action on the Klein hyperquadric and splits
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J.S. Dai / Mechanism and Machine Theory 41 (2006) 41–52 47the Klein hyperquadric into two orbits: the lines at inﬁnity and the ﬁnite lines. The isotropy groupof a ﬁnite line consists of the rotations with that line as axis, together with the translations alongthe 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 ofinvariant properties of orthogonal matrices and demonstrated the equivalence of screw geometrywith the matrix representations of the Euclidean group, providing a complete expression for theﬁnite screw motion in terms of the entire dual number transformation matrix. The ﬁnite screwdemonstrates to be suitable for trajectory planning and a concise expression is developed to givethe transformation matrix describing the displacement at each point along the path of the ﬁnitescrew motion. To represent the ﬁnite screw, Hunt [72] speciﬁed in 1987 the axis and pitch of the ﬁnite screwdisplacement by considering the geometric form in the point-line-plane system and by describing abody in two generally disposed locations. While maintaining the two sets of ratios of directedplanes and directed lines of a body in two locations, ﬁve necessary conditions were proposedto construct six equations to determine the homogenous Plucker line coordinates of the axis of ¨the ﬁnite screw. In the application of ﬁnite screws, Young and Duﬀy [73] applied in 1986 the ﬁnite displacementto identify the extreme positions of manipulators. Angeles [74] developed in 1986 an algorithmbased on the concept of the principal values and directions of the second-moment tensor of threenon-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 displace-ment, Parkin [75] proposed a ﬁnite twist representation. Deriving from diﬀerence screws of bothinitial and ﬁnal positions of a rigid body, the direction component, translation component, signand angle of rotation of the ﬁnite twist were presented. The particular form in terms of the ﬁnitetwist was proposed based on the coordinate transformation with line triplets of the initial andﬁnal locations of the body, on the condition that the axis of the ﬁnite twist remains invariantin space while the twist motion takes place and that the axis is perpendicular to, and reciprocalto, each of the diﬀerence screws from the two line triplets. Parkin [76] further identiﬁed in 1991the ﬁnite displacement screws of a compound body and presented the 2-system of ﬁnite displace-ment screws of the point-lines. The quasi-pitch (q-pitch) was then proposed as the ratio of half thetranslation distance over tangent of half the rotation angle. The q-pitch contains the essential fea-ture of the rigid body displacement and presents as an intrinsic part of a ﬁnite twist. With thisproposed q-pitch, Parkin [77] demonstrated in 1992 that the axes of ﬁnite twist displacementsof a point-line object have a similar conformation with linear combinations of screws and pre-sented the ﬁnite twist cylindroid as linear combinations of two basis ﬁnite screws. At the same time, Hunt [52] demonstrated in 1992 that the q-pitch of the ﬁnite twist can be ex-tracted from the Study coordinates. Huang [78] in 1994 and 1995 investigated the ﬁnite screw sys-tem of the third order [79] associated with kinematic chains and identiﬁed [80] in 1997 thecylindroid associated with the Bennett mechanism. In 1995, Dai, Holland and Kerr [81] further
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48 J.S. Dai / Mechanism and Machine Theory 41 (2006) 41–52investigated the ﬁnite twist representation and transformation and its ordered combination forserial manipulators. They revealed that the motion imparted by joints to the end-eﬀector linkcan be represented by an ordered set of ﬁnite twist transformations and this ordered combinationbased on the Lie group operation gives a resultant ﬁnite twist of an end-eﬀector relative to adatum point. Naturally, the study of the ﬁnite screw displacement and the development of the q-pitch of theﬁnite twist progressed into the study of ﬁnite screw systems in conjunction with the study of the-ory of screw systems by Gibson and Hunt [82,83], Rico and Duﬀy [84–86], and Dai and ReesJones [87–89]. In 1994, Huang and Roth [90] applied an analytical approach to investigate theﬁnite screw systems. In addition to the case in which two points are speciﬁed, which is the sameas the case in which a line with an associated point on it is speciﬁed [77], they demonstrated thatthe screw systems resulting from the other four incompletely speciﬁed displacement problems de-ﬁned by Tsai and Roth in 1973 [64] can also be represented by linear systems or their nonlinearsubsets. The ﬁrst case was demonstrated by substituting the q-pitch into the screw triangle andrearranging the resultant displacement screw to form a 2-system by extracting two new basis screwswhose 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 sametime, Hunt and Parkin [91] identiﬁed two particular linear two-systems for the three possible com-binations 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-linedisplacements was further identiﬁed to be a general two-systems speciﬁed in [83] and the ﬁnite twistsystem 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 themost 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 formulaand Rodrigues parameters in a great length. They examined the projection of a point of a unitcircle from the pole onto a y-axis that produces a stereographic projection deﬁned as the tangentof half the rotation angle. They related the projection to the structure of CayleyÕs rotation vectorand rigid displacement vector and then to Rodrigues parameters, leading to the revelation of theinherent 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. Therepresentation 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 theother hand, the new systematic representation of these motions.7. Conclusions The work of Rodrigues, Cliﬀord and Study has all provided analytical means of describing theﬁnite displacement. It is in many respects desirable to use a form of representation that has thesame number of parameters as there are degrees of freedom, that is six in spatial cases, three inplanar cases.
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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 dis-placements, this paper presented intrinsic relationships between diﬀerent developments in the pro-gress and explores the coherent theme in the two-century-long development of the theories whichconstitute a broad and solid mathematical science foundation for our todayÕs study of the ﬁnitetwist motion of a rigid body.Acknowledgements The author thanks Professor Gene Fichter of Oregon for the suggestion when they met inMilan in 1995 on writing a review of the development of the theory of rigid body displacementsand the ﬁnite twist and the encouragement. Thanks are also given to the staﬀ in Maughan libraryof KingÕs College London, University of London, for providing a substantial number of interli-brary loans from across the world.References [1] L. Euler, Problema algebraicum ob aﬀectiones prorsis singulares memorabili, 1770, Opera Omnia, I 6 (1770) 287– 315. [2] L. Euler, Formulae generales pro traslatione quacunque corporum rigidorum, Novi Commentari Academiae Imperialis Petropolitanae 20 (1775) 189–207, Leonhardi Euleri Opera Omnia, Series Secunda, Opera Mechanica Et Astronomica, Basileae MCMLXVIII, 9, 84–98. [3] H. Goldstein, Classical Mechanics, Addison-Wesley Pub. Co., 1950. [4] L. Euler, Nova Methodus Motum Corporum Rigidorum Determinandi, Novi Commentari Academiae Imperialis Petropolitanae 20 (1775) 208–238, Leonhardi Euleri Opera Omnia, Series Secunda, Opera Mechanica Et Astronomica, Basileae MCMLXVIII, 9, 99–125. [5] L. Euler, De Motu Corporum Circa Puncum Fixum Mobilium, Commentatio 825 indicis ENESTROEMIANI, Opera postuma 2 (1862) 43–62, Leonhardi Euleri Opera Omnia, Series Secunda, Opera Mechanica Et Astronmica, Basileae MCMLXVIII, 9, 413–441. [6] J.W. Gibbs, in: E.D. Wilson (Ed.), Vector Analysis, Scribner, New York, 1901, and Yale University Press, New Haven, 1931. [7] K.E. Bisshopp, RodriguesÕ formula and the screw matrix, Transactions of ASME, Journal of Engineering for Industry 91 (1969) 179–185. [8] H. Cheng, K.C. Gupta, An historical note on ﬁnite rotations, Transactions of ASME, Journal of Applied Mechanics 56 (1989) 139–145. [9] M.F. Beatty, Vector analysis of ﬁnite rigid rotations, Transactions of ASME, Journal of Applied Mechanics 44 (1977) 501–502.[10] J.J. Craig, Introduction to Robotics, second ed., Addison-Wesley, Reading, MA, 1989.[11] O.A. Bauchau, L. Trainelli, The vectorial parameterization of rotation, Nonlinear Dynamics 32 (2003) 71–92.[12] R.S. Ball, A Treatise on the Theory of Screws, Cambridge University Press, Cambridge, 1900.[13] I. Grattan-Guinness, The Fontana History of the Mathematical Sciences, Fontana Press, An Imprint of HarperCollins Publishers, 1997.[14] G. Mozzi, Discorso Matematico Sopra Il Rotamento Momentaneo Dei Corpi, Stamperia di Donato Campo, Napoli, 1763.[15] M. Ceccarelli, Screw axis deﬁned by Giulio Mozzi in 1763 and early studies on helicoidal motion, Mechanism and Machine Theory 35 (2000) 761–770. ´ ´ ´ ´ ´[16] M. Chasles, Note sur le proprietes generales du systeme de deux corps semblables entrÕeux et places dÕune maniere ´ ´ quelconque dans lÕespace; et sur le deplacement ﬁni ou inﬁniment petis dÕun corps solide libre, Bulletin Des Sciences Mathematiques 14 (1830) 321–326.
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