Screw and Lie group theory in multibody kinematics

Transcription

1 Multbody Syst Dyn DOI /s Screw and Le group theory n multbody knematcs Moton representaton and recursve knematcs of tree-topology systems Andreas Müller 1 Receved: 10 Aprl 2016 / Accepted: 16 June 2017 The Author(s Ths artcle s publshed wth open access at Sprngerlnk.com Abstract After three decades of computatonal multbody system (MBS dynamcs, current research s centered at the development of compact and user-frendly yet computatonally effcent formulatons for the analyss of complex MBS. The key to ths s a holstc geometrc approach to the knematcs modelng observng that the general moton of rgd bodes and the relatve moton due to techncal jonts are screw motons. Moreover, screw theory provdes the geometrc settng and Le group theory the analytc foundaton for an ntutve and compact MBS modelng. The nherent frame nvarance of ths modelng approach gves rse to very effcent recursve O(n algorthms, for whch the so-called spatal operator algebra s one example, and allows for use of readly avalable geometrc data. In ths paper, three varants for descrbng the confguraton of tree-topology MBS n terms of relatve coordnates, that s, jont varables, are presented: the standard formulaton usng body-fxed jont frames, a formulaton wthout jont frames, and a formulaton wthout ether jont or body-fxed reference frames. Ths allows for descrbng the MBS knematcs wthout ntroducng jont reference frames and therewth renderng the use of restrctve modelng conventon, such as Denavt Hartenberg parameters, redundant. Four dfferent defntons of twsts are recalled, and the correspondng recursve expressons are derved. The correspondng Jacobans and ther factorzaton are derved. The am of ths paper s to motvate the use of Le group modelng and to provde a revew of dfferent formulatons for the knematcs of tree-topology MBS n terms of relatve (jont coordnates from the unfyng perspectve of screw and Le group theory. Keywords Rgd bodes Multbody systems Knematcs Relatve coordnates Recursve algorthms Screws Le groups Frame nvarance Dedcated to Prof. Peter Maßer B A. Müller a.mueller@jku.at 1 Insttute of Robotcs, Johannes Kepler Unversty, Altenberger Str. 69, 4040 Lnz, Austra

2 A. Müller 1 Introducton Computatonal multbody system (MBS dynamcs ams at mathematcal formulatons and effcent computatonal algorthms for the knetc analyss of complex mechancal systems. At the same tme the modelng process s supposed to be ntutve and user frendly. Moreover, the effcency of MBS algorthms and the complexty of the actual modelng process s largely determned by the way the knematcs s descrbed. Ths concerns the core ssues of representng rgd body motons and descrbng the knematcs of techncal jonts. Both ssues can be addressed wth concepts of screw and Le group theory. Spatal MBS perform complcated motons, and n general rgd bodes perform screw motons that form a Le group. Although the theory of screw motons s well understood, screw theory has almost completely been gnored for MBS modelng wth only a few exceptons. The latter can be grouped nto two classes. The frst class uses of the fact that the velocty of a rgd body s a screw, referred to as the twst. The propagaton of twsts wthn an MBS s thus descrbed as a frame transformaton of screw coordnates. Ths gave rse to the so-called spatal vector formulaton ntroduced n [23, 24] and to the so-called spatal operator algebra, whch was formalzed n [64] and used for O(n forward dynamcs algorthms, for example, n [25, 32, 33, 38, 63, 65]. Screw notatons are also used n the formulatons presented n [5, 36, 37, 74]. Further MBS formulatons were reported that use screw notatons uncommon for the MBS communty [26]. All these approaches only explot the algebrac propertes of screws as far as relevant for a compact handlng of veloctes, acceleratons, wrenches, and nerta. The second class goes one step further by recognzng that fnte motons form the Le group SE(3 wth the screw algebra as ts Le algebra se(3. Moreover, screw theory provdes the geometrc settng and Le group theory the analytc foundaton for an ntutve and effcent modelng of rgd body mechansms. Some of the frst publcatons reportng Le group formulatons of the knematcs of an open knematc chan are [15, 28, 29] and[19, 20]. In ths context the term product of exponentals (POE s beng used snce Brockett used t n [15]. Unfortunately, these publcatons have not reached the MBS communty, presumably because of the used mathematcal concepts that dffer from classcal MBS formalsms. The basc concept s the exponental mappng that determnes the fnte relatve moton of two adjacent bodes connected by a lower par jont n terms of a screw assocated wth the jont. The product of the exponental mappngs for all consecutve jonts determnes the overall moton of the chan. Wthn ths formulaton, twsts are naturally represented as screws, and jont motons are descrbed n terms of screw coordnates. Motvated by [15], Le group formulatons for MBS dynamcs were reported n a few publcatons, for example, [44, 46, 56 60]. It should be mentoned that the basc elements of a screw formulaton for MBS dynamcs were already presented n [39] but dd not receve due attenton. A crucal feature of these geometrc approaches s ther frame nvarance, whch allows for arbtrary representatons of screws and for freely assgnng reference frames, whch drastcally smplfy the knematcs modelng and also provde a drect lnk to CAD models. Moreover, the POE, and thus the knematcs, can even be formulated wthout the use of any jont frame, whch bascally resembles the zero reference formulaton reported for a robotc arm n [27]. On the other hand, classcal approaches to the descrpton of jont knematcs are the Denavt Hartenberg (DH [3, 22, 35] (n ts dfferent forms and the Sheth Ucker two-frame conventon [69]. Such two-frame conventons are used n most of the current MBS dynamcs smulatons packages that use relatve coordnates. The Le group descrpton, on the other hand, not only allows for arbtrary placement of jont frames but makes them dspensable altogether.

3 Screw theory n MBS knematcs The benefts of geometrc modelng have been recognzed already n robotcs. Recently, at least n robotcs, the text books [40, 52, 67] have reached a wder audence. Modern approaches to robotcs make extensve use of screw and Le group theoretcal concepts. Ths s, also supported by the Unversal Robot Descrpton Format (URDF that s used, for nstance, n the Robot Operatng System (ROS, rather than DH parameters. In MBS dynamcs the benefts of geometrc mechancs are slowly beng recognzed. Interestngly, ths manly apples to the modelng of MBS wth flexble bodes undergong large deformatons [8, 72]. Ths s not surprsng snce geometrcally exact formulatons requre correct modelng of the fnte knematcs of a contnua. The dsplacement feld of a Cosserat beam, for nstance, s a proper moton n E 3 and thus modeled as moton n SE(3. Ths s an extenson of the orgnal work on geometrcally exact beams and shells by Smo [70, 71], where the dsplacement feld s modeled on SO(3 R 3. Another topc where Le group theory s recently appled n MBS dynamcs s the tme ntegraton. To ths end, Le group ntegraton schemes were modfed and appled to MBS models n absolute coordnate formulaton [17], where the motons of ndvdual bodes are descrbed as a general screw moton that are constraned accordng to the nterconnectng jonts. It shall be remarked that, despte the current trend to emphasze the use of Le group (basc concepts, the bascs formulatons for nonlnear flexble MBS were already reported by Borr et al. [10 12]. The am of ths paper s to provde a comprehensve summary of the basc concepts for modelng MBS n terms of relatve coordnates usng jont screws and to relate them to exstng formulatons that are scattered throughout the lterature. Wthout loss of generalty the concepts are ntroduced for a knematc chan wthn an MBS wth arbtrary topology [34, 45]. It s also the am to show that MBS can be modeled n a user-frendly way wthout havng to follow restrctve modelng conventons and that ths gves rse to O(n formulatons. The latter are not the topc of ths paper. The paper s organzed as follows. In Sect. 2, the MBS confguraton s descrbed n terms of jont varables, used as generalzed coordnates, wth the jont geometry parameterzed by jont screw coordnates. Ths classcal approach of usng body-fxed jont frames to descrbe relatve confguratons s extended to a formulaton that does not nvolve jont frames. The correspondng relatons for the MBS velocty are derved n Sect. 3. A formulaton s ntroduced for each of the four dfferent defntons of rgd body twsts found n the lterature. The latter are called the body-fxed, spatal, hybrd, and mxed twsts. They dffer by the reference pont used to measure the velocty and by the frame n whch the angular and translatonal veloctes are resolved. The dfferent twst representatons are ntroduced n Sect. A.2. Recursve relatons for the respectve Jacobans are derved, and the computatonal aspects are dscussed wth emphasze on ther decomposton. The presented formulaton allows for an effcent modelng of the MBS knematcs n terms of readly avalable geometrc data. Throughout the paper, only a few basc concepts from Le group theory are requred, whch are summarzed n Appendx A. The used nomenclature s summarzed n Appendx B. As for all Le group formulatons, the bggest hurdle for a reader (who may be already be famlar wth MBS formulatons s the notaton. The reader not famlar wth screws and Le group modelng may want to consult Sect. A.1 before readng Sect. 2 and Sect. A.2 before readng Sect. 3. Ths paper s amed to provde a reference and cannot replace an ntroductory textbook lke [40, 52, 67]. A begnner s recommended to consult [40]. Yet there s no text book that treats the topc from an MBS perspectve. Readers not nterested n the dervatons could smply use the man relatons that are dsplayed wth a black border.

4 A. Müller Table 1 n-dmensonal moton subgroups of SE(3 n Subgroup Moton 1 R 1-dm. translaton along some axs 1 SO(2 1-dm. rotaton about arbtrary fxed axs 1 H p screw moton about arbtrary axs wth fnte ptch 2 R 2 2-dm. planar translaton 2 SO(2 R translaton along arbtrary axs & rotaton along ths axs 3 R 3 spatal translatons 3 SO(3 spatal rotatons about arbtrary fxed pont 3 H p R 2 translaton n a plane + screw moton to ths plane (ptch h 3 SO(2 R 2 = SE(2 planar motons 4 SO(2 R 3 = SE(2 R spatal translatons + rotaton about axs wth fxed orentaton (Schönfles moton 6 SE(3 spatal moton 2 Confguraton of a knematc chan In ths secton the knematcs modelng usng jont screw coordnates s presented. For smplcty, a sngle open knematc chan s consdered comprsng n movng bodes nterconnected by n 1-DOF lower par jonts. To smplfy the formulaton, but wthout loosng generalty, hgher-dof jonts are modeled as combnaton of 1-DOF lower par jonts. Bodes and jonts are labeled wth the same ndces = 1,...,n, whereas the ground s ndexed wth 0. Wth the sequental numberng of bodes and jonts of the knematc chan, jont connects body to ts predecessor body 1. A body-fxed reference frame (BFR F s attached to body of the MBS. The body s then knematcally represented by ths BFR. 2.1 Jont knematcs It has been the standard approach n MBS modelng to represent hgher-dof jonts by combnaton of 1-DOF lower par jonts, that s, usng ether revolute, prsmatc, or screw jonts. Ths wll be adopted n the followng although ths s not the way n whch MBS models are mplemented n practce, but t smplfes the ntroducton of the presented approach wthout compromsng ts generalty. The justfcaton of ths approach s that most techncal jonts are so-called lower knematc pars (also called Reuleaux pars characterzed by surface contact [61, 62], that s, they are the mechancal generators of moton subgroups of SE(3 [67]. However, not all moton subgroups are generated by lower pars. The 10 subgroups are lsted n Table 1. So-called macro jonts are frequently used n MBS modelng to generate moton subgroups by combnaton of lower pars. Table 2 shows the correspondence of moton subgroups wth lower pars and macro jonts. Mssng n ths lst are jonts relevant for MBS modelng such as unversal/hook and constant velocty jonts snce they are not lower knematc pars. They can be modeled by combnaton of lower par jonts. The classcal approach to descrbe jont knematcs s to ntroduce an addtonal par of body-fxed jont frames (JFR for each jont (Fg. 1[74]. Denote by J 1, the JFR for jont on body 1 and by J, the JFR on body. The relatve moton of adjacent bodes s represented by the frame transformaton between the respectve JFRs that can be descrbed n terms of screw coordnates (Sect. A.1.

5 Screw theory n MBS knematcs Table 2 Mechancal generators of the n-dmensonal subgroups of SE(3. A moton subgroup can be generated by a lower par or by a macro jont,.e., a combnaton of jonts wth smaller DOF n Subgroup Lower par Macro jont 1 R Prsmatc jont 1 SO(2 Revolute jont 1 H p Screw jont 2 R 2 combnaton of two nonparallel prsmatc jonts 2 SO(2 R Cylndrcal jont 3 R 3 combnaton of three nonparallel prsmatc jonts 3 SO(3 Sphercal jont 3 H p R 2 planar jont + screw jont wth axs normal to plane 3 SO(2 R 2 = SE(2 Planar jont 4 SO(2 R 3 = SE(2 R planar jont + prsmatc jont wth axs normal to plane 6 SE(3 free jont Fg. 1 Descrpton of the knematcs of jont connectng body wth ts predecessor body 1. A body-fxed JFR J, s ntroduced on body, and J 1,, on body 1, respectvely. A revolute jont s shown as an example Lower par 1-DOF jonts restrct the nterconnected bodes so to perform screw motons wth a certan ptch h. Revolute jonts have ptch h = 0, and prsmatc jonts have ptch h =, whereas proper screw jonts have a nonzero fnte ptch. Denote by ( 1 Z = 1 e 1 z 1 e + 1 e h (1 the unt screw coordnate vector of jont expressed n the JFR J 1, on body 1, where 1 z s the poston vector of a pont on the jont axs measured n the JFR J 1,,and 1 e s the unt vector along the jont axs resolved n JFR J 1,. Assumpton 1 It s assumed throughout the paper that the two JFRs concde n the reference confguraton q = 0. Ths assumpton can be easly relaxed f requred. Denote wth q the jont varable (angle, translaton. Wth Assumpton 1, the confguraton of the JFR J, on body relatve to the JFR J 1, on body 1sgvenbythe exponental n (69asD (q := exp( 1 Z q.

6 A. Müller Fg. 2 Defnton of body-fxed RFR F and JFR J, and J 1, for jont Remark 1 It s a common practce to locate the JFRs wth ther orgns at the jont axs (as n Fg. 2, so that z = 0. Then the jont screw coordnates for the three types of 1-DOF jonts are Z revolute = ( ( e e, Z screw =, Z prsmatc = 0 eh 2.2 Recursve knematcs usng body-fxed jont frames ( 0. (2 e The absolute confguraton of body, that s, the confguraton of ts BFR F relatve to the nertal frame (IFR F 0 s represented by C SE(3. Therelatve confguraton of body relatve to body 1sC 1, := C 1 1 C. The confguraton of a rgd body n the knematc chan can be determned recursvely by successve combnaton of the relatve confguratons of adjacent bodes as C = C 0,1 C 1,2 C 1,. For jont, denotebys 1, the constant transformaton from JFR J 1, to the RFR F 1 on body 1, and by S, the constant transformaton from JFR J, to the RFR F on body (Fg. 2. Then the relatve confguraton s C 1, = S 1, D (q S 1,. Denote by q V n the vector of jont varables that serve as generalzed coordnates of the MBS. The jont space manfold s V n = R n P T n R for an MBS model comprsng n P prsmatc and n R revolute/screw jonts (n P + n R = n. The absolute confguraton (.e. relatve to the IFR of body n the chan s C (q = S 0,1 D 1 (q 1 S 1 1,1 S 1,2D 2 (q 2 S 1 2,2... S 1,D (q S 1,. (3 Ths formulaton requres the followng modelng steps: Introducton of body-fxed JFR J, at body wth relatve confguraton S, ; Introducton of body-fxed JFR J 1, at body 1 wth relatve confguratons S 1, ; The screw coordnate vector 1 Z of jont represented n JFR J 1, at body 1. Expresson (3 s the standard MBS formulaton for the knematcs of an open chan n terms of relatve coordnates, that s, jont angles or translatons. For 1-DOF jonts, the JFR s usually orented so that ts 3-axs ponts along the jont axs (as n Fg. 2. Then the screw

7 Screw theory n MBS knematcs coordnates are 1 Z = (0, 0, 1 s, 0, 0,s + h (1 s T,wheres = 1 for prsmatc jont, and s = 0 for a screw jont wth fnte ptch h (for revolute jonts, h = 0. Remark 2 The matrx C s used to represent the confguraton of body ; hence the symbol. Frequently, the symbol T s used [40, 74], whch refers to the fact that these matrces descrbe the transformaton of pont coordnates (Sect. A.1. Remark 3 It s mportant to emphasze that the Le group formulaton (3 s merely another approach to the standard matrx formulaton of MBS knematcs amng at compact expressons that smplfy the mplementaton wthout compromsng the effcency. It also ncludes varous conventons used to descrbe the jont knematcs. An excellent overvew of classcal matrx methods (also wth emphass on how they can be employed for synthess can be found n [74]. For nstance, S 1, and S, can be parameterzed n terms of the constant part of the Denavt Hartenberg (DH parameters [74]. Formulaton (3 n partcular resembles the Sheth Ucker conventon (whch was ntroduced to elmnate the ambguty of the DH parameters [69, 74]. In that notaton the matrces S 1, and S, are called the shape matrces of jont. However, the Sheth Ucker conventon stll presumes certan algnment of jont axes. For example, a revolute axs s supposed to be parallel to the 3-axs of the JFRs. A recent dscusson of these notatons can be found n [9]. An expresson smlar to (3 was also presented n [54], where no restrcton on the jont axs s mposed. A recursve formulaton of the MBS moton equatons usng homogeneous transformaton matrces was also presented n [36, 37]. Remark 4 (Mult-DOF jonts The descrpton for 1-DOF jonts n terms of a screw coordnate vector Z can be generalzed to jonts wth more than one DOF. For a jont wth DOF ν, the relatve confguraton of the JFRs can alternatvely be descrbed n terms of ν jont varables q 1,...,q ν by D (q 1,...,q ν := exp( 1 Z 1 q Z ν q ν or D (q 1,...,q ν := exp( 1 Z 1 q 1... exp( 1 Z ν q ν. For a sphercal jont, for nstance, the varables n the frst form are the components of the rotaton axs tmes angle n (64, and n the second form, these are three angles correspondng to the order of 1-DOF rotatons (e.g. Euler-angles. For lower par jonts, n the frst case, q 1,...,q ν are canoncal coordnates of the frst knd on the jont moton subgroup, and n the second case, they are canoncal coordnates of the second knd [52]. The Z 1,...,Z ν form a bass on the subalgebra of the moton subgroup generated by the jont. 2.3 Recursve knematcs wthout body-fxed jont frames The ntroducton of jont frames s a tedous step wthn the MBS knematcs modelng. Moreover, t s desrable to mnmze the data requred to formulate the knematc relatons. In ths regard the frame nvarance of screws s benefcal. The two constant transformatons from the JFR to the BFR on the respectve body can be summarzed usng (76 as C 1, (q = S 1, D (q S 1, = S 1,1 S 1, S,D (q S 1, = B exp ( X q, (4 so that the relatve confguraton splts nto only one constant and a varable part. The constant part B := S 1, S 1, = C 1, (0 s the reference confguraton of body wth respect to body 1whenq = 0. The varable part s now gven n terms of the constant screw

8 A. Müller Fg. 3 Descrpton of the knematcs of jont wthout body-fxed JFRs n the zero-reference relatve confguraton wth q = 0. The vector x s used when the jont screw coordnates are represented n the BFR F on body, and x 1, s used when the jont screw coordnates are represented n the BFR F 1 on body 1 coordnate vector of jont represented n BFR F : ( X = Ad 1 S, Z = e x, e + h. (5 e The matrx Ad S, defned n (73 transforms screw coordnates represented n J, 1 to those represented n F accordng to ther relatve confguraton descrbed by S,. As ndcated n Fg. 3, here e s the unt vector along the axs of jont resolved n the BFR F,and x, s the poston vector of a pont on the axs of jont, measured and resolved n F. Ths s ndeed the same screw as n (1 but expressed n the BFR on body. The jont screw can alternatvely be represented n F 1.Then C 1, (q = S, 1 D (q S 1, = S, 1 D (q S 1, 1 S, 1S 1, = exp ( 1 X q B (6 wth the jont screw coordnate vector ( 1 X = Ad 1 S 1, Z = Ad B X = 1 e 1 x 1, 1 e + 1, (7 e h now expressed n the BFR F 1 at body 1, where 1 x 1, s the poston vector of a pont on the axs of jont measured n F 1. Successve combnaton of the relatve confguratons yelds C (q = B 1 exp (1 X 1 q 1 B2 exp (2 X 2 q 2... B exp ( X q = exp ( 0 X 1 q 1 B1 exp ( 1 X 2 q 2 B2... exp ( 1 X q B. (8 Thefrstformof(8 was reported [60], and both forms n [56, 57]. It wll be called the body-fxed Product-of-Exponentals (POE formula n body-fxed descrpton snce the jont knematcs s expressed by exponentals of jont screws. It seems to be more convenent to work wth the screw coordnates X.Alson[5], two varants of the knematc descrpton of a seral chan were presented usng a BFR on body 1or, respectvely.

9 Screw theory n MBS knematcs Fg. 4 Descrpton of the knematcs of jont wth respect to the spatal IFR n the zero-reference confguraton wth q = 0 In summary, ths body-fxed POE formulaton does not requre ntroducton of JFRs. It only requres the followng readly avalable nformaton: The relatve reference confguraton B of the adjacent bodes connected by jont for q = 0; The screw coordnates X of jont represented n the BFR F at body, or alternatvely the screw coordnates 1 X represented n the BFR F 1 at body 1. The form (8 smplfes the expresson for the jont knematcs. Its man advantage s that t only nvolves the reference confguraton B of BFRs. 2.4 Recursve knematcs wthout body-fxed jont frames and screw coordnates Thanks to the frame nvarance, the jont screw coordnates can even be descrbed n the spatal IFR, that s, wthout reference to any body-fxed frames. To ths end, (8 s wrtten as C (q = B 1 exp (1 X 1 q 1 B 1 1 B 1 B 2 exp (2 X 2 q 2 B 1 2 B B 1 B exp ( X q B 1 B 1 1 B 1 B. (9 Relaton (76 yelds B exp(q XB 1 = exp(qb XB 1 = exp(qad B X,sothat C (q = exp(y 1 q 1 exp(y 2 q 2... exp(y q A. (10 Here A := B 1 B = C (0 = ( R (0 r (0 0 1 (11 s the absolute reference confguraton (.e. relatve to IFR of body, and ( Y j = Ad j e Aj X j = j y j e j + h j e j (12

10 A. Müller Fg. 5 Model of the RCM mechansm dsclosed n [77]. The model was create wth the MBS tool Alaska s the screw coordnate vector of jont j represented n the IFR F 0 n the reference confguraton wth q = 0 (Fg. 4. The drecton unt vector e j and the poston vector y j of a pont on the jont axs are expressed n the IFR F 0 (the leadng superscrpt 0 omtted. The transformaton (12 relates the body-fxed to the spatal representaton of jont screw n the reference confguraton. The product of the exp mappngs n (10 descrbes the moton of an RFR on body, whch at q = 0 concdes wth the IFR relatve to the IFR. The relaton to the actual BFR s acheved by the subsequent transformaton A. Such a zero reference formulaton has been frst reported by Gupta [27] n terms of frame transformaton matrces and was latter ntroduced by Brockett [15] as the POE formula for robotc manpulators. Formulaton (10 was then used n [16] for MBS modelng. It should be remarked that n the classcal lterature on screws, the spatal representaton of a screw s denoted by the symbol $ [31, 66]. All data requred for ths spatal POE formulaton s represented n the spatal IFR: Absolute reference confguratons A = C (0, that s, the reference confguraton of body wth respect to the IFR F 0 for q = 0. Jont screw coordnates Y 0 Y 0 n spatal representatons, that s, measured and resolved n the IFR F 0 for q = 0. The result (10 s remarkable snce t allows for formulatng the MBS knematcs wthout body-fxed jont frames. From a modelng perspectve ths has proven very useful snce no jont transformatons S,, S 1, or B are needed. Only the absolute reference confguratons A wth respect to the IFR and the reference screw coordnates (12, that s, e and p, resolved n the IFR, are requred. Ths s n partcular advantageous when processng CAD data. Moreover, f n the reference (constructon confguraton the RFR of the bodes concde wth the IFR (global CAD reference system, that s, all parts are desgned wth respect to the same RFR, then A = I and Y j = j X j. 2.5 Example Fgure 5 shows a surgcal devce that conssts of a robot arm and a remote center of moton (RCM mechansm. Ths was dsclosed n the patent [77]. The robot arm consstng of bodes 1, 2, 3 s used to poston the RCM mechansm consstng of bodes 4 and 5. The surgcal nstrument s mounted n the socket at the remote end of body 5. The axes of jonts 4 and 5 and of the nstrument ntersect at one pont. Ths allows the nstrument to freely pvot around an ncson pont. The reference confguraton s shown n Fg. 5. The IFR s located at the base of the mechansm. The jont screw coordnates n spatal representaton are determned by the geometrc

11 Screw theory n MBS knematcs Fg. 6 Descrpton of the geometry of the RCM mechansm parameters shown n Fg. 6. The poston vectors y and unt vectors e n (12are y 1 = (0, 0, 0 T, y 2 = ( d 2, 0, 0 T, y 3 = (d 3, 0, 0 T, y 4 = (d 4, 0,h 4 T, y 5 = (d 5, 0, 0 T, e 1 = e 2 = e 3 = e 5 = (0, 0, 1 T, e 4 = ( (1/ 2, 0, 1/ 2 T. Snce any pont on the jont axes can be used, the 3-components n y, = 1, 2, 3, 5, are set to zero. An arbtrary pont on the axs of jont 4 s chosen as ndcated. Thus the jont screw coordnates (12are Y 1 = (0, 0, 1, 0, 0, 0 T, Y 2 = ( {0, 0, 1, 0,d 2, 0 T, Y3 = (0, 0, 1, 0, d 3, 0 T, Y 4 = ( 1 2, 0, 1/ 2, 0, d 4 / 2 h 4 / 2, 0 T, Y 5 = (0, 0, 1, 0, d 5, 0 T. The reference confguratons (11 of the bodes are determned by 1/ 2 0 1/ 2 R 1 (0 = R 2 (0 = R 3 (0 = R 5 (0 = I, R 4 (0 = / 2 0 1/, 2 r 1 (0 = ( x 1, 0,z 1 T, r 2 (0 = ( x 2, 0, z 2 T, r 3 (0 = (x 3, 0,z 3 T, r 4 (0 = (x 4, 0,z 4 T, r 5 (0 = (x 5, 0,z 5 T. Therewth the confguraton of all bodes are determned by the POE (10. For nstance, c 123 s d 2 c 1 + (d 2 + d 3 c 12 + (x 3 d 3 c 123 C 3 (q = s 123 c d 2 s 1 + (d 2 + d 3 s 12 + (x 3 d 3 s z

12 A. Müller Fg. 7 Descrpton of the nstantaneous knematcs of a knematc chan wth c 123 := cos(q 1 + q 2 + q 3 etc. The expressons for C 4 (q and C 5 (q are rather complcated and are omtted here. Instead of deducng them from the geometry n Fg. 6, the body-fxed representaton of the jont screw coordnates can be determned by relaton (12. Ths yelds 1 X 1 = (0, 0, 1, 0, x 1, 0 T, 2 X 2 = (0, 0, 1, 0,d 2 x 2, 0 T, 3 X 3 = (0, 0, 1, 0, d 3 + x 3, 0 T, 4 X 4 = (0, 0, 1, 0, d 4 / 2 h 4 / 2 + x 4 / 2 + z 4 / 2, 0 T, 5 X 5 = (0, 0, 1, 0, d 5 + x 5, 0 T. Ths example shows the smplcty of the approach. 3 Velocty of a knematc chan In ths secton, recursve relatons are derved for the four forms of twsts ntroduced n Sect. A.2, namely the body-fxed, spatal, hybrd, and mxed twsts [18]. 3.1 Body-fxed representaton of rgd body twsts Body-fxed twsts The body-fxed twst of body, denoted V b = (ω b, vb T, s the aggregate of the angular velocty ω b := ω of ts BFR and the translatonal velocty v b := v of ts orgn relatve to the IFR (Sect. A.2. The twst of body n a knematc chan s the sum of twsts of the jonts connectng t to the ground. Represented (measured and resolved n ts BFR F (Fg. 7, ths s ( ( ( V b = q e 1 1 b,1 e q e 2 e 1 h 2 1 b,2 e q e e 2 h 2 b, e +. e h (13

13 Screw theory n MBS knematcs Here b,j s the nstantaneous poston vector of a pont on the axs of jont j measured n the BFR F,and e j s the unt vector along the axs resolved n the BFR. The nstantaneous jont screw coordnates n (13 are confguraton dependent and related to the jont screws (5 and(12 (deduced from reference confguraton by a frame transformaton Body-fxed Jacoban and recursve relatons The body-fxed twst s determned by (80 n terms of the confguraton C(t.Usng(4 and (77, from C = C 1 C 1, there follows the recursve relaton (notce that C 1 1, = C, 1 = C 1 C 1 V b = Ad C, 1 V b 1 + X q = Ad 1 B exp( X q Vb 1 + X q. (14 The frame transformatons due to the relatve motons C, 1 of adjacent bodes propagate the twsts wthn the knematc chan. The frst term on the rght-hand sde of (14 s the twst of body 1 represented n the BFR F on body, and the second term s the addtonal contrbuton from jont. The confguraton C of body depends on the jont varables q j, j. The body-fxed twst (80 can thus be expressed as V b = j Ĵb,j q j where Ĵ b,j := C 1 C q j.thepoe(8, together wth (77 and(76, yelds C q j = exp ( X q B 1 exp ( j+1 X j+1 q j+1 B 1 j+1 j X j exp ( ( j+1 X j+1 q j+1 exp X q C 1 ( = C 1 C j j X j C 1 j C = Ad C,j j X j, j. (15 Usng (12, ths yelds the followng relatons: J b,j = Ad C,j j X j = Ad C,j A 1 Y j j = Ad C,j S j 1 j,j Z j, j. (16 The J b,j are the screw coordnate vectors n (13 obtaned va a frame transformaton (73of j X j n (5, or Y j n (12, to the current confguraton. The body-fxed twst s hence V b = J b,j q j = J b q. (17 j The 6 n matrx J b = ( J b,1,...,jb,, 0...,0 (18 s called the geometrc body-fxed Jacoban of body [52]. It s the central object n all formulatons that use body-fxed twsts and Le group formulatons [46, 56, 57, 60]. The geometrc Jacoban appears n the lterature under dfferent names. For nstance, n [41, 42, 46], t s called the knematc basc functon (KBF snce t s the pvotal object for (recursve evaluaton of MBS knematcs.

14 A. Müller Expresson (16 gves rse to the recursve relaton { J b,j = AdC, 1 J b 1,j, j <, X, j =. (19 Ths s essentally another form of recurson (14 usng(17. Remark 5 (Dependence on jont varables Wth (8, respectvely (10, t s clear that the Jacoban J b of body can only depend on q 1,...,q. Moreover, notng n (16 that C,j = C 1 C j s ndependent from q 1, t follows that J b depends on q 2,...,q, that s, t s ndependent from the frst jont n the chan. Ths s obvous from a knematc perspectve snce V b s the sum of twsts of the precedng bodes n the chan expressed n the BFR on body. Ths only depends on the confguraton of the bodes relatve to body but not on the absolute confguraton of the overall chan, whch s determned by q 1. Remark 6 (Requred data The second form n (16 n conjuncton wth (10 allows for computaton of the body-fxed Jacoban wthout ntroducng body-fxed JFRs. The only nformaton needed s the jont screw coordnates Y j represented n the IFR and the reference confguratons A j. Remark 7 (Change of reference frame When another BFR on body s used, whch s related to the orgnal BFR by S SE(3, ts confguraton s gven by C = C S. Thecorrespondng body-fxed twst follows from (80 as V b = C 1 Ċ = S 1 C 1 Ċ S = Ad 1 S ( V b and, n vector form, V b = Ad 1 S Vb. (20 On the other hand, the body-fxed twst s nvarant under a change of IFR, whch s gven by C = SC. Body-fxed twsts are therefore called left-nvarant vector felds on SE(3 snce left multplcaton of C wth any S SE(3 does not affect V b. Remark 8 (Applcaton of body-fxed representaton The recursve relatons for body-fxed twst and Jacoban are the bass for the MBS dynamcs algorthms n [1, 2, 7, 25, 30, 38, 39, 56, 57, 60, 75]. In [1, 2] the adjont transformaton matrx n (14 was called the shft matrx, and X was called the moton map matrx. However, the geometrc background was rarely exploted as n [56, 57, 60] and[39]. Remarkably, Lu [39] already presented all relevant formulatons n terms of screws Body-fxed system Jacoban and ts decomposton The body-fxed twsts are summarzed n the overall twst vector V b = (V b 1,...,Vb n T.Recurson (14 can then be wrtten n the matrx form V b = D b V b + X b q (21 wth Ad C2,1 0 0 D b := 0 Ad C3.2 0, X b := dag (1 X 1,..., n X n. ( Ad Cn,n 1 0

15 Screw theory n MBS knematcs On the other hand, the recursve expresson for the Jacoban (19 reads n the matrx form V b = J b q = A b X b q, (23 where the 6n n matrx J b = A b X b s the system Jacoban n body-fxed representaton,and I Ad C2,1 I 0 0 A b := Ad C3,1 Ad C3,2 I Ad Cn,1 Ad Cn,2 Ad Cn,n 1 I (24 s the screw transformaton matrx. Comparng (21and(23showsthatA b = (I D b 1.In fact, D b s nlpotent so that the von-neumann seres A b = ( I D b 1 = I + D b + ( D b ( D b n (25 termnates wth (D b n+1 = 0. That s, A b s the 1-resolvent of D b, whch s the fundamental pont of departure for many O(n algorthms. Moreover, (25 s another form of the recursve coordnate transformatons. Hence the nverse (A b 1 = (I D b. Remark 9 (Overall nverse knematcs soluton The above result allows for a smple transformaton from body-fxed veloctes to the correspondng jont rates. When the twsts of all bodes are gven, (23 s an overdetermned lnear system n q. It has a unque soluton as long as the twsts are consstent wth the knematcs. Premultplcaton of (23 wth (I D b followed by (X b T and ((X b T X b 1 yelds q = (( X b T X b 1( X b T ( I D b V b. (26 The n n dagonal matrx (X bt X b 1 = dag(1/ 1 X 1 2,...,1/ n X n 2 has full rank. Due to the block dagonal structure, ths yelds the solutons q = X T (Vb Ad C, 1 V b 1 / X 2 for the ndvdual jonts. Ths s ndeed the projecton of the relatve twst of body wth respect to body 1 onto the axs of jont. It s an exact soluton of the nverse knematcs for the overall MBS, presumng that the twsts are compatble, that s, satsfy (14. If ths s not the case, (26 s the unque pseudonverse soluton of system (23 of6n equatons for the n unknowns q mnmzng the resdual error. Ths can be consdered as the generalzed nverse knematcs problem: gven desred twsts of all ndvdual lnks, fnd the jont rates that best reproduce these twsts. Ths can be appled, for nstance, to the nverse knematcs of human body models processng moton capture data (estmated poston and orentaton of body segments and when nosy data s processed. Although soluton (26 seems straghtforward, t should be remarked that there s no frame nvarant nner product on se(3, that s, no norm of screws that s nvarant under a change of reference frame can be defned [67]. The correctness of (26 follows by regardng the transposed jont screw coordnates as coscrews, and X T X s the parng of screw and coscrew coordnates rather than an nner product.

16 A. Müller 3.2 Spatal representaton of rgd body twsts Spatal twsts A representaton of the body twst, whch s less common n MBS modelng but frequently used n mechansm theory, s the so-called spatal twst, denoted V s = (ωs, vs T. Ths s the twst of body represented n the IFR. It conssts of the angular velocty of the BFR of body measured and resolved n the IFR and of the translatonal velocty v s := ṙ ω s r of the (possbly magnary pont on the body that s momentarly travelng through the orgn of the IFR measured and resolved n the IFR (Sect. A.2. Wth the notaton n Fg. 7, the spatal twst of body s geometrcally readly constructed as ( ( ( V s = q e 1 e 1 + q 2 s 1 e 1 + h 1 e q 1 s 2 e 2 + h 2 e 2 e s e + h e, (27 where s j s the poston vector of a pont on the jont axs j expressed n the IFR. The screw coordnates n (27 are confguraton dependent. They are equal to Y j n the reference confguraton q = 0,wheres = y Spatal Jacoban and recursve relatons To derve an analytc expresson, usng the POE, defnton (80 of the spatal twst s appled. As apparent from (27, the nonvanshng nstantaneous jont screws are dentcal for all bodes. Ths s clear snce the IFR s the only reference frame nvolved. The spatal twst can thus be expressed as V s = j Js j q j wth Ĵ s j := C q j C 1.UsngthePOE,astraghtforward dervaton analogous to (15 yelds J s j = Ad C j j X j = Ad Cj A 1 Y j j = Ad Cj S j 1 j,j Z j, j. (28 The J s j s the nstantaneous screw coordnate vector of jont j n (27 n spatal representaton, that s, represented n the IFR. The matrx J s = ( J s 1,...,Js, 0,...,0 (29 s called the spatal Jacoban of body. The relatons (27 and(28 yeld the followng recursve expresson for the spatal twsts of bodes n a knematc chan: V s = Vs 1 + Js q. (30 Remark 10 The spatal representaton has remarkable advantages. The velocty recurson (30 s the smplest possble snce the twsts of ndvdual bodes can smply be added wthout any coordnate transformaton. An mportant observaton s that J s j s ntrnsc to jont j. The nonzero screw vectors n the Jacoban (29 are thus the same for all bodes. Thssa consequence of usng a sngle spatal reference frame.

17 Screw theory n MBS knematcs Spatal system Jacoban and ts decomposton The overall spatal twst V s = (V s 1,...,Vs n T of the knematc chan s determned as V s = J s q, (31 where the spatal system Jacoban possesses the factorzatons J s = A s Y s = A sb X b = A sh X h. (32 Theren t s Y s = dag(y 1,...,Y n, X h n (44, and Ad C Ad C1 Ad C2 0 0 A sb := Ad C1 Ad C2 0 0, Ad C1 Ad C2 Ad Cn 1 Ad Cn Ad r Ad r1 Ad r2 0 0 A sh := Ad r1 Ad r Ad r1 Ad r2 Ad rn 1 Ad rn A s := A sb dag ( Ad 1 A 1,...,Ad 1 A n. (33 All nonzero entres n a column of these matrces are dentcal. Hence the constructon of these matrces only requres determnaton of the n entres n the last row that are coped nto the upper trangular block matrx. The factorzaton (32 gves rse to an expresson for ts nverse. Notng that A sb = dag(ad C1,...,Ad Cn A b, the relaton for the nverse of A b n terms of matrx D b n (22 yelds ( A sb 1 ( = I D b dag(ad C1,...,Ad Cn Ad 1 C Ad C2 Ad 1 C = 0 Ad C3 Ad 1 C Ad Cn Ad 1 C n (34 and (A s 1 accordngly. Remark 11 (Dependence on jont varables Smlarly to the body-fxed twst, snce J s = Ad C X = Ad C 1 Ad B exp X q X = Ad C 1 X s ndependent from q, t follows that the spatal Jacoban of body only depends on q 1,...,q 1. Indeed, the moton of jont does not change ts screw axs about whch body s movng. Remark 12 (Change of reference frame The spatal twst s called a rght-nvarant vector feld on SE(3 because t does not change when C s postmultpled by any S SE(3,

18 A. Müller representng a change of body-fxed RFR. Under a change of IFR accordng to C = SC, the spatal twsts transform as V s = Ad S V s. (35 Remark 13 (Applcaton of spatal representaton The spatal twst s used almost exclusvely n mechansm knematcs (often wthout mentonng t but s becomng accepted for MBS modelng snce t was ntroduced n [23, 24]. For knematc analyss of mechansms, t s common practce to (nstantaneously locate the global reference frame so that t concdes wth the frame where knetostatc propertes (twsts, wrenches are observed, usually at the end-effector. For a seral robotc manpulator, the end-effector frame s located at the termnal lnk of the chan, so that A n = I,andV s n s then the spatal end-effector twst. From ther defnton t follows that the spatal and hybrd twst (see next secton of body are numercally dentcal when the BFR F overlaps wth the IFR F 0. The most promnent use of the spatal representaton n dynamcs s the O(n forward dynamcs method by Featherstone [23, 24]. Ths has not yet been wdely appled n MBS dynamcs. Ths may be due to use of an uncommon choce of reference pont (the IFR orgn at whch the spatal enttes are measured, so that results and nteracton wrenches must be transformed to body-fxed reference frames. The spatal representaton of twsts must not be confused wth the spatal vector notaton proposed n [23, 24]. The latter s a general expresson of twsts as 6-vectors (lke body-fxed and spatal but wthout reference to a partcular frame n whch the components are resolved. Ths allows for abstract dervaton of knematc relatons, but these relatons must eventually be resolved n a partcular frame, and ths eventually determnes the computatonal effort. A notable applcaton of the spatal twst s the modelng and numercal ntegraton of nonlnear elastc MBS, where t s called the base pole velocty [10] orfxedpolevelocty [13], and the ntrnsc couplng of translatonal and angular velocty (accordng to the screw moton was dscussed. The correspondng momentum balance and conservaton propertes are dscussed n [11, 12] (seealso[50]. Remark 14 As n Remark 9, relaton (34 gves rse to an overall nverse knematcs soluton. For gven spatal twsts V s, ths reads n components as q = X T Ad 1 C (V s Vs 1 / X Hybrd form of rgd body twsts Hybrd twsts In varous applcatons, t s benefcal to measure the twst of a body n the body-fxed BFR but resolve t n the IFR. Ths s commonly referred to as the hybrd twst [18, 52], denoted V h = (ωs, ṙ T. The geometrc constructon (Fg. 7 yelds ( ( ( V h = q e 1 e 1 + q 2 b,1 e 1 + h 1 e q 1 b,2 e 2 + h 2 e 2 e b, e + h e. (36 As n (13, b,j s the poston vector of a pont on the axs of jont j measured from the BFR F of body,ande j s the unt vector along the axs, but now expressed n the IFR F 0.Ths was orgnally ntroduced n [76]and[78] and s used n varous O(n dynamcs algorthms (Remark 15.

19 Screw theory n MBS knematcs Hybrd Jacoban and recursve relatons The hybrd twst s merely the body-fxed twst resolved n the IFR. Usng (75, ths transformaton s V h = Ad R V b where R s the rotaton matrx of body. Then(17 leads to V h = j J h,j q j = J h q (37 wth the columns J h,j := Ad R J b,j of the hybrd Jacoban Jh = (Jh,1,...,Jh,, 0...,0. The recursve expressons (19 and(14 reman vald when all screw coordnate vectors are resolved n the IFR. The jont screw coordnates are then confguraton dependent. The screw coordnate vector of jont j measured n the BFR F j on body j and resolved n the IFR F 0 s related to (5 va ( 0 X j j = Ad R j j X j = e j x j,j e j + h j e j. (38 As n (5, the poston vector x j,j of a pont on the axs of jont j s measured from the BFR F j but now resolved n the IFR. The relatons Ad C = Ad r Ad R and r,j = r j r lead to Ad R J b,j = Ad R Ad C,j j X j = Ad R Ad C,j Ad 1 R j 0 X j j = Ad R Ad 1 C Ad Cj Ad 1 R j 0 X j j = Ad 1 r Ad rj 0 X j j = Ad r Ad rj 0 X j j = Ad r j r 0 X j j = Ad r,j 0 X j j. (39 Therewth the columns of the hybrd Jacoban of body are J h,j = Ad r 0,j X j j, j. (40 The J h,j s the nstantaneous screw coordnate of jont j n (36 measured at BFR on body and resolved n the IFR. In the hybrd form, all vectors are resolved n the IFR. That s, the screw coordnates 0 X j j depend on the current confguraton q even though the jont axs s constant wthn body j. The hybrd twst s resolved n the IFR. Snce the screw coordnates 0 X j j are already resolved n the IFR, the transformaton to the current confguraton, to determne the nstantaneous jont screws J h,j (q, only requres translatons of orgns. Ths s obtaned by shftng the reference pont accordng to r,j, whch s why the matrx Ad r,j s also called the shft dyad [21]. Ths s not a frame transformaton. Relaton (40 gves rse to the recursve relaton for the hybrd Jacoban and, analogously to (14, for the hybrd twsts, { J h,j = Adr, 1 J h 1,j, j <, 0 X, j =. (41 V h = Ad r, 1 V h X q. (42 The advantage of the hybrd form over the body-fxed s that (41 only nvolves the relatve dsplacement r, 1 n contrast to the complete relatve confguraton C, 1 n (19. It must be recalled, however, that the vectors e j and r j,j must be transformed to the IFR. Furthermore, when formulatng equatons of moton, the nerta propertes of the body must be resolved n the IFR so that they become confguraton dependent [50].

20 A. Müller Remark 15 (Applcaton of hybrd representaton The hybrd form was used n [76] for forward knematcs calculaton of seral manpulators and n [4, 5] to compute the moton equatons and the nverse dynamcs soluton. It s used n many recursve O(n forward dynamcs algorthms such as [6, 38, 53, 54, 64], where relatons (42and(40 play a central role. In the so-called spatal operator algebra [64], hybrd screw enttes are called spatal vectors. The hybrd form s deemed computatonally effcent snce the transformatons only nvolve translatons. The actual confguraton of the chan s not dscussed n these publcatons, but t enters va the vectors e (q and r (q, respectvely d,j (q. In[38] the nverse transformaton Ad 1 r,j was denoted by j X (not to be confused wth (5, and the screw coordnate vector 0 X j j n (38byφ j.in[64], Ad 1 r,j was denoted by φ,j T,and0 X j j wth. The transposed matrces appear snce they arse from the transformaton of wrenches. H T j Hybrd system Jacoban and ts decomposton The hybrd system Jacoban, whch determnes the overall hybrd twst vector V h = (V h 1,...,Vh n T accordng to V h = J h q = A h X h q, (43 s decomposed n terms of I Ad r2,1 I 0 0 A h := Ad r3,1 Ad r3,2 I 0, X h := dag (0 X ,...,0 Xn n. (44.. Ad rn,1 Ad rn,2 Ad rn,n 1 I In analogy to (25, A h can be resolved as power seres usng the relaton A h = (I T h 1 wth the 6n 6n matrx Ad r2,1 0 0 T h := 0 Ad r3,2 0. ( Ad rn,n 1 0 Ths leads to the nverse (A h 1 = (I T h and a soluton q of (43 of the form ( Mxed form of rgd body twsts Mxed twsts When formulatng the Newton Euler equatons of rgd bodes, t can be benefcal to use the body-fxed angular velocty and the translatonal velocty measured at the body-fxed BFR but resolved n the IFR. Ths s called the mxed twst denoted by V m = (ω b, ṙ T.Itsusedn MBS dynamcs modelng [68], bascally because when usng the mxed twst, the Newton Euler equatons wth respect to the COM are decoupled and because the body-fxed nerta tensor s constant (see also the companon paper [50]. The mxed twst s readly found as ( ( ( V m = q e q e 2 b,1 e 1 + e 1 h q e 1 b,2 e 2 + e 2 h. (46 2 b, e + e h

21 Screw theory n MBS knematcs As n (36, e j s the unt vector along the axs of jont j measured and resolved n the IFR F 0, and b,j s the poston vector of a pont on the axs measured n the BFR F of body and resolved n the IFR. The mxed twst s related to the body-fxed, spatal, and hybrd form va ( ( I 0 R V m = V b 0 R = T 0 0 I Mxed Jacoban and recursve relatons Expresson (46 s wrtten as ( R V h = T 0 V s r I. (47 V m = j J m,j q j = J m q, (48 where the mxed Jacoban of body s ntroduced as J m = ( J m,1,...,jm,, 0...,0. (49 The elements n the nstantaneous jont screw coordnate vectors J m,j n (46 are not consstently resolved n one frame. Rather e j s resolved n BFR F,ande j n the IFR. The mxed Jacoban can thus not be derved va frame transformatons. Startng from the body-fxed Jacoban yelds ( R J m,j = T 0 r,j I 0 X j j, j, (50 where 0 X j j are the screw coordnates of jont j measured n frame F j andresolvednthe IFR,gvenn(38. The dfference to (40 s that the angular and translatonal part are resolved n dfferent frames. Expresson (50 can be wrtten n the recursve form ( R, 1 0 J J m,j = r, 1 R I h 1,j, j <, ( R T ( I 0 X, j =. Ths drectly translates to a recursve relaton for the mxed twsts wthn a knematc chan ( V m R, 1 0 = V m r, 1 R I 1 + Jm, q. ( Mxed system Jacoban and ts decomposton The overall mxed twst vector V m = (V m 1,...,Vm n T can be expressed n terms of the system Jacoban J m as V m = J m q = A m X m q (53 wth X m := X h and the matrx A m as n (44 but wth the Ad r,j replaced by the matrx n (50. Ths allows for a closed-form nverson of A m analogous to that of A h.

22 A. Müller Table 3 Transformaton of the dfferent representatons of twsts and jont screw coordnates V s V b V h V m V s I Ad C (54 Ad r (58 (47 V b Ad 1 C I Ad R T (47 V h Ad r Ad R (55 I (47 V m (47 (47 (47 I Y 0 X X Y I Ad r (57 Ad A (12 0 X Ad r I Ad R (56 X Ad 1 A Ad R T I 3.5 Relaton of dfferent forms The ntroduced twsts are related by certan (not necessarly frame transformatons, and t s occasonally desrable to swtch between them. From ther defntons n (80 t s clear that body-fxed and spatal twsts, and thus the correspondng Jacobans, are related by V s = Ad C V b, Js = Ad C J b. (54 Evaluatng ths n the reference confguraton q = 0 leads to the relaton of jont screw coordnates (12. The body-fxed twst s related to ts hybrd verson by a coordnate transformaton determned by the rotaton R matrx, algnng the body frame wth the IFR, ( V h = R 0 V b 0 R = Ad R V b. (55 Transformaton (55 apples to a general hybrd screw and n partcular to the jont screws (5 and(38 and Jacobans (16 and(40: 0 X j j = Ad R j j X j, J h = Ad R J b. (56 Combnng (56 and(12 yelds the relaton of hybrd and spatal versons of jont screws Y j = Ad r 0 X j j (57 wth the current poston vector r of body n C.From(54 and(55 t follows that and thus V s = Ad C Ad 1 R V h = Ad r V h (58 J s j = Ad r J h,j, j. (59 Ths descrbes the change of reference pont from the BFR of body to the IFR. The transformatons between the dfferent forms of twsts and jont screws are summarzed n Table 3. It should be fnally mentoned that the screw coordnates X and 0 X are just dfferent coordnates for the same geometrc object, namely of the nstantaneous jont screw of jont measured n the BFR at body but resolved ether n ths BFR or n the IFR. The vector Y on the other hand s a snapshot of the jont screw coordnates of jont n spatal representaton at the reference q = 0. Remark 16 (Computatonal effcency It s clear from (14, (30, (42, and (52 that the number of numercal operatons dffer between the four dfferent representatons of twsts.

23 Screw theory n MBS knematcs Ths allows for selectng the most effcent one when a knematc analyss s envsaged. In [55] the problem of determnng the twsts of the termnal body n a knematc chan (robot end-effector was analyzed for body-fxed, spatal, and hybrd forms. Ths study suggests that the spatal representaton s computatonally most effcent. A conclusve analyss of all four forms has not yet been reported. Moreover, the general stuaton ncludes the dynamc analyss. Ths was partly addressed n [73, 79]. 3.6 Example (contnued The Jacoban n body-fxed and spatal representaton s determned for the example n Sect The nstantaneous screw coordnates n body-fxed representaton are readly found wth (16. For nstance, the nstantaneous screw coordnate vector of jont 1 expressed n the body-fxed frame on body 3 s J b 3,1 = Ad C 3,1 A1 1 Y 1 = Ad 1 C3,1 X 1 = ( 0, 0, 1,(d 2 + d 3 s 2 + (x 1 + x 3 d 3 s 23,d 2 x 1 (d 2 + d 3 c 2 + (d 3 x 1 x 3 c 23, 0 T. Proceedng analogously for the other jonts yelds the body-fxed Jacoban of body 3 as J b 3 (q = (d 2 + d 3 s 2 + (x 1 + x 3 d 3 s 23 s 3 (x 2 + x 3 d 2 d d 2 x 1 (d 2 + d 3 c 2 + (d 3 x 1 x 3 c 23 c 3 (d 2 + d 3 x 2 x 3 d 3 x 2 x 3 d The body-fxed twst of body 3 s therewth V b 3 = Jb 3 (q q. Agan detals for body 4 and 5 are omtted because of space lmtaton. The spatal representaton of the screw coordnates of jont 1,...,4, for nstance, s found wth (28: J s 1 (q = (0, 0, 1, 0, 0, 0T, J s 2 (q = (0, 0, 1, d 2s 1,d 2 c 1, 0 T, J s 3 (q = ( 0, 0, 1,(d 2 + d 3 s 23 d 2 s 1,d 2 c 1 (d 2 + d 3 c 12, 0 T, J s 4 (q = 1 ( c123, s 123, 1,(d 2 + d 3 s 12 d 2 s 1 + (d 4 + h 4 d 3 s 123, 2 d 2 c 1 (d 2 + d 3 c 12 + (d 3 d 4 h 4 c 123,d 2 s 23 (d 2 + d 3 s 3 T. That for jont 5 s omtted agan. These J s consttute the spatal Jacobans Js n (29. 4 Conclusons and outlook Screw and Le group theory gves rse to compact formulatons of the equatons governng the MBS knematcs n terms of relatve (jont coordnates. Ths s benefcal for the actual modelng process and for the mplementaton of MBS algorthms and ther computatonal propertes. The frame nvarance of these concepts allows for expressng the relevant