Tutoral (T9) Geometry and Screw Theory for Robotcs Stefano Stramgol and Herman Bruynnckx March 15, 2001
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Contents 1 Moton of a Rgd Body 5 1.1 Introducton... 5 1.2 The Eucldean Space... 6 1.2.1 Ponts versus vectors.... 6 1.2.2 The scalar product...... 7 1.2.3 Measurng the dstance between ponts... 7 1.2.4 Orthogonalty... 7 1.2.5 Vector Product... 8 1.2.6 Coordnate Systems..... 8 1.3 Confguratons of a rgd body... 10 1.3.1 A Le Group approach: SE(3)... 11 1.3.2 The fnte twst descrpton... 18 1.4 Veloctes for a rgd body: Twsts... 23 1.4.1 Twsts as elements of se(3)... 23 1.4.2 Twsts as applcatons on screws... 25 1.5 Forces appled to rgd bodes: Wrenches... 26 1.5.1 Wrenches as elements of se(3)... 26 1.5.2 Wrenches as applcatons on screws... 27 1.6 Dynamcs of a rgd body...... 29 1.6.1 Knetc co-energy and energy... 29 1.6.2 Coordnate changes for nertas... 30 1.6.3 Euler equaton of a rgd body... 32 1.7 Exponental of Le algebras..... 33 1.7.1 Exponental of elements of so(3)... 33 1.7.2 Exponental of elements of se(3)... 33 2 Seral knematc chans 35 2.1 Confguraton knematcs...... 35 2.1.1 Knematc pars... 35 2.1.2 Forward poston knematcs of seral chans and Brockett s Product of Exponentals... 38 2.1.3 Inverse poston knematcs... 39 2.2 Dfferental knematcs of seral chans... 39 2.2.1 The geometrc Jacoban... 39 2.2.2 Sngulartes... 40 2.2.3 Redundancy... 40 2.3 Dynamcs of seral chans..... 41 2.3.1 The dynamc equatons of a seral manpulator... 41 2.3.2 Redundancy resoluton... 43 2.3.3 Ideal constrants... 43 3
4 CONTENTS 3 Interacton and Control 45 3.1 Ports and Interconnecton..... 45 3.1.1 Power Ports... 45 3.1.2 Generalzed Port-Controlled Hamltonan Systems... 46 3.1.3 Interconnecton of GPCHSs... 50 3.2 The proposed control archtecture... 51 3.2.1 The IPC... 51 3.2.2 The Supervsor... 52 3.3 IPC Graspng... 53 3.3.1 The Sprngs - Spatal Complance... 55 3.3.2 Masses... 57 3.3.3 Dampers - Energy Dsspaton... 58 3.3.4 The control Scheme..... 58 3.4 Summary... 60 A Projectve geometry and knematcs 63 B Introducton to Le groups 67 B.1 Matrx Le groups... 68 B.1.1 Matrx Group Actons.... 70 B.1.2 Adjont representaton... 70
1 Moton of a Rgd Body Ths Chapter shows the fundamental dfferences between the moton propertes of a pont mass and of a rgd body. The rgd body does not lve n a Eucldean space. Hence, ths Chapter explans whch relevant concepts from non-eucldean (.e., dfferental) geometry are needed to descrbe the knematcs and dynamcs of a rgd body. 1.1 Introducton The moton of a partcle mass s easy to descrbe because ts confguraton can be assocated to a pont of the three-dmensonal Eucldean space. After havng chosen coordnates, each pont can be assocated to a trple of real numbers n R 3 ; but the most mportant thng s that the algebrac and topologcal propertes of R 3 correspond to real physcal propertes of the moton of the partcle: forces can be added; velocty vectors too; magntudes of vectors correspond to magntudes of forces and veloctes; orthogonalty of a force and velocty vector gves zero power; the velocty and acceleraton vectors are the tme dervatves of the pont s poston vector; Newton s Laws lnk a three-dmensonal force vector to a three-dmensonal acceleraton vector, through the scalar quantty mass, for pont masses, as well as sphercally-symmetrc rgd bodes such as planets and canon balls. In contrast to the smplcty of the pont mass moton propertes, the moton and the dynamcs of a rgd body are much more complex. A rgd body s composed of an nfnte number of pont masses, whch are constraned not to move wth respect to each other. It turns out that the dmenson of the space necessary to descrbe the confguraton of a rgd body s sx: three dmensons for orentaton, and three for translaton. And the force acceleraton relaton s now a full sx-by-sx matrx, and not a scalar anymore. Moreover, the acceleraton nvolved n ths dynamc relaton s not just the second-order tme dervatve of the poston/orentaton vector of the rgd body. Even the short overvew above should make t clear that t s wrong to treat the sx poston/orentaton coordnates of a rgd body n the same way as one treats the three poston coordnates of a pont: the geometrcal propertes of rgd bodes are fundamentally dfferent from the geometrcal propertes of pont masses. For example, f one contnuously ncreases one of these sx numbers (.e., one that corresponds to orentaton representaton), the rgd body arrves at the same confguraton after every rotaton over 360 degrees; ths curvature property does not occur when one ndefntely ncreases any of the three coordnates of a pont confguraton. Locally (.e., n the neghborhood of a specfc confguraton) t s possble to descrbe a confguraton usng sx real numbers, but ths descrpton s not an ntrnsc property of 5
6 1. MOTION OF A RIGID BODY the moton. (An ntutve defnton of an ntrnsc property s: any property that does not change f one changes the coordnate representaton; the total mass of a pont or body s such an ntrnsc property, at least n non-relatvstc dynamcs.) A lot of powerful tools are avalable whch allow to descrbe the moton of rgd bodes n a geometrcal and global way. These methods are related to the geometry of lnes and screws and to the dfferental geometrc concept of a Le group. Ths text makes ample use of these (and other) mathematcal concepts, because ther mathematcal propertes correspond perfectly to the (deal) physcal propertes of movng rgd bodes. The qualty of engneerng results depends crtcally on the qualty and fathfulness of the mathematcal models on whch the engneerng s based: especally n robotcs, engneers have made (and are stll makng) fundamental errors n ther reasonng, for the sole reason that that reasonng was based on mathematcal propertes of the coordnates n ther model for whch no correspondng physcal propertes exst. In a nutshell: t s not because one can use n numbers as coordnates on a gven space, that the objects n that space have exactly the same propertes as the n-tuples n R n! 1.2 The Eucldean Space The space of rgd body moton (poston and orentaton; translatonal and angular velocty; and translatonal and angular acceleraton) has dfferent propertes than the well-known three-dmensonal Eucldean space we lve n. A Eucldean space s a contnuous set of ponts together wth an extra structure whch s able to descrbe orthogonalty and to measure length. Ths extra structure s the scalar product. We ndcate an Eucldean space of dmenson n wth E(n). The Eucldean space E(n) s therefore the set of ponts of the space. We ndcate the three-dmensonal space we lve n wth E(3) or smply E. Sometmes we omt the nformaton about the dmenson and just ndcate the Eucldean space as E. It s mportant to realze that snce a pont can be fxed or movng wth respect to an observer, for a gven Eucldean space we need to specfy an observer whch does not move wthn t. We suppose that there exsts an nertal observer we consder as reference n whch the ponts belongng to the Eucldean space are not movng. It s possble to see n (Stramgol 2001, Stramgol & Bruynnckx 2001) that the hypothess we use consderng a reference Eucldean space hdes mportant consequences. 1.2.1 Ponts versus vectors The ponts of E are not vectors but ponts: they do not have a versus and a drecton and as such t makes no sense to, for example, sum them! We can assocate a vector to them only once we have chosen a reference pont: f we choose such a reference pont o E, we can assocate to every pont p E a vector gong from o to p whch we ndcate wth (p o). It s not here the place to go nto detals, but ths assocaton s possble snce the space s Eucldean: for curved spaces ths s not possble. For example, what would be the meanng of the sum of two ponts on the surface of the earth? However, once we have vectors, we can sum them; for example, t makes sense to sum the veloctes or acceleratons that a movng pont mass gets under the nfluence of separate forces actng on the pont. And for ponts, the velocty and acceleraton are ndeed vectors, because they represent the tmed dstance between consecutve
1.2. THE EUCLIDEAN SPACE 7 p m (p q) = (m l) q l Fgure 1.1: The dstance between ponts are represented by vectors. postons of the same pont. These vectors are called free vectors, because ther physcal meanng does not change f one moves them parallel to themselves to another place n the space. Mathematcally speakng, we use as a vector space the quotent space of all dfferent vectors, usng parallelsm and equal length as equvalence relatons. For example, as shown n Fgure 1.1, we consder the vector (p q) equvalent to the vector (m l) where p, q, l, m E. The set of these free vectors are denoted wth E (n) or E f t s not necessary to show the dmenson n. 1.2.2 The scalar product We can now ntroduce the scalar product whch characterzes a Eucldean space. It s a map whch gves a real number once we gve two vectors belongng to E as nput:, : E E R ; (v, w) v, w. The scalar product satsfes the followng propertes: 1. v, w = w, v (symmetry). 2. v, v 0 (postveness). 3. αv 1 + βv 2,w = α v 1,w + β v 2,w (lnearty). 1.2.3 Measurng the dstance between ponts We can defne the dstance d(p, q) between the ponts p and q as the length of (p q) E. Ths dstance defnes a norm: : E R ; v v, v. 1.2.4 Orthogonalty Orthogonalty of vectors v, w 0s also defned. We say that two vectors wth non-zero length are orthogonal f ther scalar product s zero: v w v, w =0. The cosne of an angle s also a consequence of the scalar product. It can be defned usng only the scalar product: cos v w := v, w v w.
8 1. MOTION OF A RIGID BODY 1.2.5 Vector Product It s also possble to defne a vector product (after a choce of orentaton (Stramgol 2000)!): : E (3) E (3) E (3); (v, w) v w. The vector product s skew-symmetrc,.e., v w = (w v). The followng tlde-relaton s a one-to-one relaton between a pont and a mappng of ponts: x = x 1 x 2 x = 0 x 3 x 2 x 3 0 x 1. (1.1) x 3 x 2 x 1 0 Usng ths notaton, If we choose a coordnate system Ψ o and express v, w E (3) n Ψ o as v o,w o R 3, we can wrte: v o w o =ṽ o w o, where ṽ o s now the matrx correspondng to the above-mentoned mappng. We can therefore express the vector product of two vectors as the product of a skew-symmetrc matrx and a vector. The followng equaltes hold: v o w o =ṽ o w o = w o v o = w o v o = w T v o. Issues about the vector product wll be further explaned n Remark. 4, p.14. 1.2.6 Coordnate Systems If we want to work wth a Eucldean space, we need to defne coordnate systems n order to be able to represent ponts and vectors by means of numbers to work wth. Formally, we defne a coordnate system for an n-dmensonal Eucldean space E(n) as an (n +1)-tuple composed of a pont o E(n) and n lnear ndependent vectors belongng to E (n): Ψ o := (o, e 1,e 2,...,e n ) E(n) E (n) E (n). }{{} n tmes The symbol captal Ps s used to ndcate a coordnate system. A specal knd of coordnate systems s called orthonormal. systems are defned as those coordnates systems for whch Orthonormal coordnate e =1 (unt vectors), and e,e j =0 j (orthogonalty). An orthonormal coordnate system s a system for whch ts defnng vectors are orthogonal wth respect to each other, and have all unty length. As an example consder Fgure 1.2 for the case of E(2). Part a) represents the set Ψ o representng a coordnate system and composed of o, e 1,e 2 ; b) shows t n the way we are used to use t shftng the vectors to o; c) shows the parallelogram rule to see how to calculate the coordnates
1.2. THE EUCLIDEAN SPACE 9 v w v o o w a) b) p p v v o w o w c) d) Fgure 1.2: Example of the use of coordnates for a pont p; and d) shows an orthonormal frame f we suppose that the length of e 1 and e 2 s equal to unty. We say that an orthonormal coordnate system (o, ˆx, ŷ, ẑ) for E(3) s rght handed f, and only f, ˆx ŷ =ẑ. We mplctly use only rght-handed coordnate systems. Gven a coordnate system, we can assocate to each pont or vector a set of real numbers representng ts coordnates n the gven frame. The coordnates x of a pont p E are: The coordnates of a vector v E are: x = (p o),e R, x = v, e R In 3D we work wth E(3) and we ndcate sometmes for an orthonormal coordnate system ˆx := e 1, ŷ := e 2 and ẑ := e 3. From now on we wll mplctly always use orthogonal, postvely-orented coordnate systems... Coordnate mappng For any Cartesan coordnate system Ψ =(o, ˆx, ŷ, ẑ ), we can defne a coordnate mappng ψ whch assocates to each pont a set of numbers n the followng way: ψ : E(3) R 3 ; p (p o ), ˆx (p o ), ŷ. (p o ), ẑ We ndcate wth ψ the coordnate map assocated to the coordnate frame Ψ =(o, ˆx, ŷ, ẑ ).
10 1. MOTION OF A RIGID BODY Change of coordnates If we take two coordnate systems, we can consder the mappng representng the change of coordnates from one system to the other. A change of coordnates from p 1 (coordnates wth respect to Ψ 1 )top 2 (coordnates wth respect to Ψ 2 ) can be expressed as: Or, n mappng form: p 2 =(ψ 2 oψ 1 1 )(p1 ). R 3 ψ 1 1 E (3) ψ2 R 3 ; p 1 p p 2. Projectve coordnates A lot of concepts of mportance n knematcs and dynamcs are better explaned n projectve terms. It s not here the place to treat projectve geometry n detals, but we wll just say a few thngs n order to use t as a tool. For more mathematcal detals the reader can consult Appendx A. Frst of all, the set PR n denotes an n-dmensonal projectve real space. Each of the elements of PR n s an (n +1)-tuple of real numbers. Two of these (n +1)vectors of real numbers a and b n PR n are related by the projectve equvalence relaton f a s a multple of b: a b λ 0:a = λb. It s possble to study three-dmensonal Eucldean space usng projectve coordnates PR 3. A representatve of a fnte pont p E can then be: x P 1 := y z. 1 Any pont λp 1 wth λ 0s also an equvalent projectve representaton of the same pont. For reasons whch are explaned n Appendx A, a pont whose numercal representaton has the last component equal to zero, represents a pont at nfnty. Usng projectve geometry, t s therefore possble to consder ponts at nfnty n a consstent way. 1.3 Confguratons of a rgd body A lot of geometrcal and analytcal methods have been developed over the last three centures to descrbe the (changes of) confguraton of rgd bodes. The frst dfferental geometrcal nterpretaton seems to be due to Study (Selg 1996). In ths course we wll treat the fnte twsts representaton and the Le group representaton of the socalled Specal Eucldean group SE(3). Another possble analytcal representaton are b-quaternons. Detals about ths topcs can be found n (Selg 1996, Stramgol 2001, Karger & Novak 1978, Abraham & Marsden 1994). The prevously mentoned methods are all global, geometrcal, well-defned methods. Ths cannot be sad of descrptons lke Euler angles whch are only vald locally, and are not as powerful as the methods descrbed n the followng sectons.
1.3. CONFIGURATIONS OF A RIGID BODY 11 1.3.1 A Le Group approach: SE(3) Gven a certan body n space, we can descrbe the moton of ths body from a certan poston to a new one. Ths set of motons are characterzed by the fact that they are what are called n mathematcs orentaton preservng sometres. Isometres means that the moton does not change the dstance between ponts and orentaton preservng mples that the moton does not map a rght-handed coordnate frame onto a lefthanded frame. Coordnates changes versus motons It s a general fact that f we consder a global map of ponts for an n-dmensonal Eucldean space E whch s bjectve (one-to-one and onto), we can always assocate to t an equvalent change of coordnates n the followng sense. Suppose to have a bjectve mappng h: h : E E s.t. h(p 1 )=h(p 2 ) p 1 = p 2. (1.2) Once we have chosen a global 1 coordnate functon ψ : E R n, (1.3) we can consder the numercal representaton of h n the coordnates ψ by defnng: If we then defne a new coordnate map h n : R n R n ; x (ψ ohoψ 1 )(x). (1.4) ψ : E R n ; p (ψ oh)(p), (1.5) the numercal representaton h n of h s clearly equvalent to a change of coordnate from Ψ to Ψ. If, for hypothess h s a postve sometry, the converse s clearly only true f the coordnate functons are sometres from E to R n. Coordnate functons satsfyng ths hypothess are the set of rght-handed frames or the set of left-handed frames, but not a combnaton of them. For ths reason, we wll frst study coordnate changes and then defne motons based on the prevous arguments. Rotatons We start by consderng changes of coordnates n 2D. Consder two coordnate systems, wth ther orgn n common, but rotated over an angle θ wth respect to one another, as shown n Fgure 1.3. What s the relaton between the coordnates of any pont p E(2) n Ψ 1 and Ψ 2? The defntons above yeld the followng for the coordnates of the pont p n the frst frame: ( ) ( ) p 1 x1 (p o), ˆx1 = = ψ y 1 (p) =, 1 (p o), ŷ 1 or, equvalently, 1 A global coordnate functon exsts snce the space E s Eucldean. (p o) =x 1ˆx 1 + y 1 ŷ 1, (1.6)
12 1. MOTION OF A RIGID BODY ŷ 1 ŷ 2 ˆx 2 p θ o ˆx 1 Fgure 1.3: Change of coordnates under rotaton. wth x 1,y 1 R (scalar coordnates) and ˆx 1, ŷ 1 E (2) (unt poston vectors n 2D). Smlarly, the coordnates p 2 R 2 of the pont p wth respect to the second frame are: ( ) ( ) p 2 x2 (p o), ˆx2 := =, y 2 (p o), ŷ 2 The expresson for (p o) n Ψ 1, Eq. (1.6), gves: ( ) ( ) p 2 x1 ˆx = 1 + y 1 ŷ 1, ˆx 2 x1 ˆx = 1, ˆx 2 + y 1 ŷ 1, ˆx 2. x 1ˆx 1 + y 1 ŷ 1, ŷ 2 x 1 ˆx 1, ŷ 2 + y 1 ŷ 1, ŷ 2 In matrx form, ths gves: ( ) ( )( ) x2 ˆx1, ˆx = 2 ŷ 1, ˆx 2 x1, y 2 ˆx 1, ŷ 2 ŷ 1, ŷ 2 y 1 or p 2 = R1 2 p 1, where R1 2 s the rotaton matrx, defned as: ( ) ( ) R1 2 := ˆx1, ˆx 2 ŷ 1, ˆx 2 cos θ sn θ =. ˆx 1, ŷ 2 ŷ 1, ŷ 2 sn θ cos θ Remark 1 (Orthonormal matrces) The matrx R1 2 representng the change of coordnates between two coordnate systems rotated wth respect to one another, has the followng propertes: det(r 2 1)=1; R 1 2 =(R 2 1) 1 =(R 2 1) T ; the columns and rows vectors of R 2 1 have length 1 and are orthogonal. These mathematcal propertes correspond to physcal propertes of rotatons: a rotaton doesn t change the volume of a set of three ndependent vectors; the nverse of a rotaton conssts of reversng the projecton axes; and, the projectons of a set of three orthogonal unt vectors gve rse to an orthogonal matrx. In 3D, f the coordnates we use are all rght handed, we have a smlar result as n 2D for a general rotaton around the orgn: x 2 y 2 = ˆx 1, ˆx 2 ŷ 1, ˆx 2 ẑ 1, ˆx 2 ˆx 1, ŷ 2 ŷ 1, ŷ 2 ẑ 1, ŷ 2 x 1 y 1. (1.7) z 2 ˆx 1, ẑ 2 } ŷ 1, ẑ 2 {{ ẑ 1, ẑ 2 } z 1 R1 2
1.3. CONFIGURATIONS OF A RIGID BODY 13 R 2 1 s therefore almost dentcal to the two-dmensonal case, but now we are dealng wth a 3 3 matrx. Remark 2 It s mportant to note that: The columns of R 2 1 are the unt vectors along the axes of Ψ 1, expressed n the coordnate system Ψ 2. The rows of R 2 1 are the the unt vectors along the axes of Ψ 2, expressed n the coordnate system Ψ 1. A drect consequence of ths s that the nverse of R1 2,.e., R1 2, s equal to the transpose of R1. 2 Ths property holds for any rotaton matrx R: R 1 = R T The followng denttes are useful propertes when rotatng any vectors v, w R 3 : and where the tlde operator s defned n Eq. (1.1). R(v w) =(Rv) (Rw), (1.8) R wr T = (Rw), (1.9) As an example of a rotaton n E(3), consder a change of coordnates due to a rotaton of θ around ŷ 1. The matrx representng ths change of coordnates looks lke: cos(θ) 0 sn(θ) R1 2 = 0 1 0. sn(θ) 0 cos(θ) It s possble to see that the set of matrces satsfyng R 1 = R T s a three-dmensonal matrx Le group, whch s called the orthonormal group and ndcated wth O(3). Frst, t fulflls all propertes of a group: Assocatvty: R 1,R 2,R 3 O(3) (R 1 R 2 )R 3 = R 1 (R 2 R 3 ). Identty: I O(3). Inverse: R O(3) R 1 O(3),RR 1 = I. Moreover, t s a Le group, because the composton operaton s contnuous. Snce R T R = I, R O(3), and snce the determnant of a product s the product of the determnants, the determnant of any matrx n O(3) must be ±1. Ths shows that O(3) s composed of two dsjont components, one whose matrces have determnant equal to 1, and one consstng of the matrces wth determnant equal to +1. The former s not a group by tself, because the product of two such matrces has determnant +1. The latter s agan a Le group, called the specal orthonormal group, and denoted by SO(3): SO(3) = {R R 3 3 ; R 1 = R T, det R =1}. Remark 3 Note that the prevous rotatons are all around a specfc pont n space, namely the common orgn of the coordnate systems. Ths mples that SO(3) can only be used to descrbe rotatons around a specfc pont and not any rotaton! Ths s not explctly mentoned n most of the robotcs lterature.
14 1. MOTION OF A RIGID BODY The Le group SO(3) s the approprate mathematcal model of the orentatons of a rgd body. Also the angular veloctes of rgd bodes must get a good mathematcal model. Ths s gven by the so-called Le algebra of SO(3), whch s denoted wth so(3). A Le algebra descrbes the local ( frst order ) propertes of ts parent Le group (rrespectve of what the Le group represent), whle mantanng the geometrc propertes. From the theory on left and rght nvarant vector felds, Eq. (B.5) and Eq. (B.6), we can see that elements of the algebra should look lke T L = R 1 Ṙ, or lke T R = ṘR 1, for any curve R(t) SO(3). Because R 1 = R T, dfferentatng the equaton R T R = I yelds: Ṙ T R + R T Ṙ =0 R T Ṙ = (R T Ṙ) T, whch means that the matrx T L s skew-symmetrc. Smlarly, T R s also skew-symmetrc, so that so(3) s the vector space of skew-symmetrc matrces. Note that due to the structure of the matrx commutator of the algebra, the commutaton of skew-symmetrc matrces s stll skew-symmetrc. There s a bjectve relaton between 3 3 skew-symmetrc matrces and 3 vectors as t was shown n Eq. (1.1). It s straghtforward to see that the algebra matrx commutator of so(3) corresponds to the vector product of the correspondng three vectors: [ ω x, ω y ]= ω x ω y ω x,ω y R 3. The elements of so(3) correspond to the angular velocty of the frames whose relatve change of coordnates s represented by the matrx R1 2 SO(3). We wll analyse n more detal the general case,.e., angular and translatonal veloctes. A lot of coordnate representatons of rotatons exsts, besdes the above-mentoned skew-symmetrc matrx representaton of SO(3); for example, the unt quaternons, whch are somorphc to another Le group, namely SU(2) (the Specal Untary group. They are both double covers of SO(3) (.e., each rotaton n SO(3) corresponds to two elements n the double cover group). In contrast to SO(3), they are both topologcally speakng smply connected. SO(3) s nstead not contractble and t s a typcal example of a non smply connected manfold wthout holes. Ths means that f we defne a contnuous closed curve n SO(3) (a rng), there s not always a way to contnuously make ths rng smaller and smaller n order to let t become a pont. Remark 4 (Vector product) It s a msconcepton to nterpret the vector product n three space as an extra structure on E (the scalar product beng the other structure). Ths s NOT the case, because the vector product s a consequence of the fact that for E the Le group of rotatons SO(3) can be ntrnscally defned. As we have just seen, the Le algebra so(3) has a commutator defned on t and therefore, after a choce of orentaton of space, the ntrnsc bjecton between so(3) and E carres the commutaton operaton defned n so(3) over to the vector product n E. It s possble to fnd two bjectons n the followng way. A vector of E s characterzed by a drecton d, an orentaton v and a module m. An element of so(3) represents an angular velocty. It s possble to see that any angular moton leaves a lne d nvarant. Ths s the frst step n ths bjecton: the vector of E whch we assocate to a vector n so(3) wll have as drecton the lne left nvarant by ths rotaton (d = d ). We stll need a module and an orentaton. It s possble to show that there exsts a postve defnte metrc on so(3), called the Kllng form (Selg 1996), whch defnes the magntude of a rotaton n a unque, coordnate-ndependent way. We can therefore choose as m the module of ths angular velocty vector calculated usng the metrc on so(3). The only choce we are left
1.3. CONFIGURATIONS OF A RIGID BODY 15 ŷ 1 p o 1 ˆx 1 ŷ 2 ˆx 2 o 2 Fgure 1.4: A general change of coordnates. wth s that of the orentaton. We clearly have two possble choces to orent the lne. If we look at the rotaton moton around ts axs as a clockwse moton, we can orent the lne as gong away from us or as comng toward us. In the frst case we say that we choose a rght-handed orentaton and n the second case a left-handed orentaton. Ths s clearly related to the orentaton chosen for the Eucldean space because f we consder a pont x movng due to the descrbed rotaton, we can consder a vector x(t) o and a vector x(t + dt) o where o s the orthogonal projecton 2 of x on the axs of rotaton. The drecton of the vector (x(t) o) (x(t + dt) o) s the assocated drecton of rotaton. Notce that f we look at ths moton through a mrror, the orentaton s changed because, under the same rotatonal moton, a lne orented toward the mrror wll be seen through the mrror as a lne orented n the opposte drecton. Ths s called a reflecton. Translatons Translatons are much smpler than rotatons. If we have two coordnate systems dsplaced wth respect to each other, the change of coordnates s smple: p =(Ψ o Ψ 1 j )(p j )=p j + p j, (1.10) where p j represents the poston of the orgn of Ψ j wth respect to the orgn of Ψ, and expressed as a numercal vector n one of the two frames (ether one, because they are not rotated wth respect to each other). Obvously ths set of translatons s also a Le group, due to the well-known addtvty of translaton vectors. General changes of coordnates It s now possble to consder a general transformaton ncludng any rotaton and/or translaton. Fgure 1.4 shows the change of coordnates from a 2D frame Ψ 1 to another frame Ψ 2. p 2 R 3 contans the coordnates of the vector (p o 2 ) E expressed n the frame Ψ 2. Takng the coordnates of the followng vector equalty yelds (p o 2 )=(p o 1 )+(o 1 o 2 ), p 2 = R 2 1 p1 + p 2 1, 2 We can consder an orthogonal projecton snce we are n a Eucldean space
16 1. MOTION OF A RIGID BODY where p 2 1 are the coordnates of the vector (o 1 o 2 ) expressed n the frame Ψ 2. In matrx form, ths change of coordnates becomes: ( ) ( )( ) p 2 R 2 = 1 p 2 1 p 1 1 0 T. 3 1 1 The transformaton matrx n ths equaton s often called the homogeneous (transformaton) matrx H1 2 R4 4 : ( ) H1 2 R := 2 1 p 2 1 0 T, 3 1 because t represents a coordnate transformaton between two frames by a smple matrx multplcaton. Ths matrx can also be nterpreted as a projectve coordnate change snce we can consder the orgnal three-dmensonal vector as an element belongng to PR 3. We can recognze a couple of sub-cases, such as a pure translaton: ( ) H1 2 I p 2 := 1 0 T, 3 1 and a pure rotaton around the orgn: H 2 1 := ( R 2 1 0 3 0 T 3 1 The homogeneous transformaton matrx can represent rotatons around a pont dfferent from the orgn: subsequent coordnate changes compose accordng to the so-called chan rule ). H 4 1 = H 2 1 H 3 2 H 4 3, (1.11) for any ntermedate frames Ψ 2 and Ψ 3. So, a rotaton around any gven pont p can now be expressed as a translaton followed by a rotaton and an nverse translaton: ( )( )( ) ( ) I p R 0 I p R p Rp =. (1.12) 0 1 0 1 0 1 0 1 The rghtmost transformaton on the left-hand ste transports the coordnates of the rgd body that rotates about p from ther expresson wth respect to the orgn to ther expresson wth respect to the pont p; the mddle transformaton does the rotaton about p; and the leftmost transforms all coordnates back to the orgn agan. It s clear that only the pont p s mapped on tself for all the possble R, hence the composte transformaton s a rotaton about p. The nverse of an homogeneous transformaton matrx s smple: ) H 1 2 =(H2 1 ) 1 = ( (R 2 1 ) T (R 2 1) T p 2 1 0 T 3 1 = ( R 1 2 R 1 2p 2 1 0 T 3 1 ). The set of matrces SE(3) := {( ) } R p s.t. R SO(3),p R 3 0 1 s called the Specal Eucldean matrx group, whch s of great mportance n robotcs. SE(3) s a Le group, as proven by the same arguments as for SO(3) above. Hence,
1.3. CONFIGURATIONS OF A RIGID BODY 17 h ŷ 1 ŷ 2 ˆx 1 Fgure 1.5: A representaton of a moton. ˆx 2 veloctes Ḣj map from the tangent space T H j SE(3) to se(3) ether wth left or rght translatons (see Appendx B). The elements of se(3) are of the followng form: se(3) = {( ) Ω v 0 0 ; Ω so(3),v R 3 }. (1.13) Ths s proven from the defnton by consderng H(t) SE(3) as a parameterzed curve, and seeng (Appendx B) that elements of the form ḢH 1 and H 1 Ḣ belong to se(3), and have the form gven n Eq. (1.13). The Le algebra se(3) has ts commutator: Furthermore, the algebra commutator of se(3) s such that snce ) ( ) ( ω v T := se(3) [ 0 0 T 1, T [ ω1, ω 2 ]= 2 ] ω 1 v 2 ω 2 v 1 se(3). 0 0 Motons versus changes of coordnates The relatonshp between a change of coordnates and a moton follows from Fgure 1.5: the moved ponts n the moved coordnate system Ψ 2 have the same coordnates as the orgnal ponts n the orgnal coordnate system Ψ 1. In other words: ψ 2 ohoψ 1 1 = I, where I s the 4 4 dentty matrx f ψ are projectve coordnates. From ψ 2 ohoψ1 1 = ψ 2 oψ1 1 }{{} H1 2 oψ 1 ohoψ 1 1 = I, t follows that ψ 1 ohoψ 1 1 = H 1 2, whch mples that:
18 1. MOTION OF A RIGID BODY The representaton of the moton from Ψ 1 to Ψ 2 n the coordnate Ψ 1 (ψ 1 ohoψ 1 s the nverse of the coordnate change from Ψ 1 to Ψ 2 (ψ 2 oψ1 1 ). 1 ) The same moton expressed n Ψ 2 s: ψ 2 ohoψ2 1 = ψ 2 oψ1 1 oψ 1 }{{} I To summarze: oho ψ1 1 oψ 1 oψ2 1 = H1 2 }{{} H1 2 H1 2 = H1 2. I A general change of Cartesan coordnates n E(3) from Ψ to Ψ j can be expressed wth a matrx of the form: ( H j R j = p j ) O3 T SE(3), 1 where R j SO(3) and pj R3. For a rgd moton h : E E; p q, whch maps Ψ to Ψ j, we have that q = Hj p, and q j = Hj pj, 1.3.2 The fnte twst descrpton Ths descrpton s the geometrcal one used n Screw Theory (Ball 1900), and t s based on Chasles Theorem. Chasles Theorem (whch wll be proved later on usng Le groups) says that any moton of a rgd body can be acheved as a rotaton around a geometrcal lne l together wth a pure translaton along l. The lne l s called the (screw) axs of the moton. For a pure translaton, ths axs s at nfnty. It can furthermore be proven that a pure non-trval rotaton and a pure non-trval translaton are commutatve f and only f they are around and along the same geometrcal lne. Therefore, t s not mportant f we consder frst the rotaton and then the translaton, or vce versa. The screw axs The axs of the moton s a lne. The set of fnte lnes n space can be parameterzed by a mnmum of four scalars; for example by the so-called Denavt-Hartenberg lne coordnates. Many other non-mnmum coordnate sets are avalable for lnes, the most common of whch are the so-called Plücker coordnates. These also allow drectly to descrbe lnes at nfnty, whch descrbe pure translatons. We wll use these coordnates n what follows. We can defne a lne as the lnear locus passng through two ponts p 1 and p 2. Once we have chosen a coordnate frame Ψ, we can consder ther projectve coordnates: x 1 x 2 P1 = y 1 and P 2 = y 2. z 1 s 1 z 2 s 2
1.3. CONFIGURATIONS OF A RIGID BODY 19 We can consder the followng sx-dmensonal real vector whch contans the Plücker ray coordnates for the lne jonng p 1 and p 2 wth respect to frame Ψ : ( l x := 1 x 2 s 1 s 2, y 1 y 2 s 1 s 2, z 1 z 2 s 1 s 2, y 1 y 2 z 1 z 2, z 1 z 2 x 1 x 2, x 1 x 2 T y 1 y 2 ), (1.14) defned usng 2-by-2 determnants of the coordnates of the ponts. The queston arses mmedately wether l s a meanngful numercal representaton of the lne jonng p 1 and p 2 n the coordnate Ψ? The answer s yes f we consder t as an element belongng to PR 5 nstead of as an element belongng to R 6 (as we wll explan shortly), and the advantage of ths representaton s that t automatcally allows to represent lnes at nfnty wthout any extra defntons! The prevous coordnates are well posed and useful for the followng reasons: If nstead of takng P1 and P 2 we take two other equvalent projectve coordnates αp1 and βp2 we obtan the same projectve coordnates for the lne, namely αβl l. If nstead of takng P1 we take another pont on the lne, namely P 1 +α(p 1 P 2 ), the propertes of determnants assure that the lne representaton does not change. If both ponts are at nfnty (.e., s 1 = s 2 = 0), l s stll properly defned and characterzed by ts frst three components beng equal to 0. It s easly possble to see that Eq. (1.14) can be wrtten as: ( ) ( ) l s2 p = 1 s 1 p 2 s2 p = 1 s 1 p 2. (1.15) p 1 p 2 p 1 (p 2 p 1 ) From ths last expresson, t s possble to consder two cases: the frst one when the defnng ponts are fnte (s 1,s 2 0) and the second when the ponts are at nfnty (s 1,s 2 =0) and therefore descrbe a lne at nfnty. In the frst case, we have as coordnates of a fnte lne a vector of the form: ( ) ω, (1.16) r ω where ω s a vector pontng from p 1 to p 2 and therefore spannng the lne, and r can be any pont on the lne snce r = p 1 + α(p 2 p 1 ) r ω = p 1 ω. Furthermore, snce the representaton of the lne s projectve, for any α 0, ( ) ω α r ω represents the same lne. Ths fnte lne can be assocated to a rotaton axs. If furthermore, besdes the rotaton axs we defne a drecton of the lne to consder the postve rotaton around the axs and a magntude representng the rotaton angle, we can completely determne a fnte rotaton. We can gve a compact descrpton of a rotaton at once f we leave the projectve nature of the lne descrpton and we consder ω as a vector wth a magntude and a drecton. We can then talk about a unt lne by consderng an ˆω such that ˆω =1and then call a unt lne an element of the form: ( ) ˆω. (1.17) r ˆω
20 1. MOTION OF A RIGID BODY It s then possble to descrbe what s called a rotor as a trple (l, ˆω, ω ) wth an orented axs and a magntude. A pure rotaton of θ around an axs orented 3 and drected by ˆω and passng through r can be expressed wth the followng rotor: ( ) ( ) ( ) ˆω θˆω ω θ = =, (1.18) r ˆω r θˆω r ω whch should be nterpreted as a trple (l, d, θ) where l s a lne, d s a chosen postve drecton and θ s an assocated magntude. Ths can also be nterpreted as a lne vector: a vector ω that can move along ts spannng lne. In the second case, for a lne at nfnty, we obtan a representaton of the form ( 0, (1.19) v) where v := p 1 (p 2 p 1 ). Ths can also be ncely nterpreted as a rotaton at nfnty whch turns out to be representatve of a pure translaton as explaned hereafter. Suppose we want to consder a rotaton around an axs whch we want to move toward nfnty: αr wth α. In order to fnd a sensble representaton, we need to consder a lne set parameterzed by: ( 1 α ω ) (αr) ( 1 α ω) = ( 1 α ω ) r ω (1.20) For any value of α, the moment r ω must be nterpreted as the lnear moton of a pont passng through the orgn of the coordnate system and rgdly attached to the space whch s rotatng around the axs passng through αr. The larger α, the closer wll be the lnear veloctes of the ponts close to the orgn. In the extreme, for α, all the fnte ponts wll have a lnear moton equal to v := r ω. Ths mples that a rotaton at nfnty corresponds to a pure translaton and t s represented by the followng lne at nfnty: ( 0. v) The ptch It s possble to show that there s an ntrnsc measure for rotatons (the Kllng form of SO(3)) and therefore t s possble to quantfy a rotaton around the axs wth a number θ (wth physcal, dmensonless unts radans). In the same way, beng n a Eucldean space, we can measure a translaton dsplacement t of a moton ntrnscally wthout usng any coordnates, but only the metrc of the space; ts dmensons are, for example, n meters. The rato between the rotaton and the translaton s called the ptch λ of the fnte moton: λ := t θ. It s a coordnate-ndependent property of the moton. 3 Clearly, mplctly, we have assocated a postve drecton wth a postve rotaton; ths can be done n two ways by consderng a rght or left conventon and corresponds to the orentaton of space appearng n the term r ω.
1.3. CONFIGURATIONS OF A RIGID BODY 21 The Screw Geometrcally, a screw S s nothng else than a lne l together wth a scalar ptch λ: S := (l, λ). Snce the dmenson of the space of lnes s four, the dmenson of the space of screws s fve. Note that a screw, as a lne, does not have any module nformaton a pror. If ths module nformaton s gven as we dd wth the lne, we get a sx-dmensonal space of measurable screws. Chasles theorem and fnte twsts Chasles theorem states that any fnte rgd body moton can be expressed as a rotaton around an axs plus a translaton along the same axs. We wll now prove ths theorem by frst relatng the fnte motons to Le groups. One of the propertes of Le groups s the possblty to relate vectors belongng to the Le algebra of a Le group to elements of the Le group (see Appendx B). For our goal, t s suffcent to realze that for any matrx H SE(3), there exsts always at least one 4 matrx T se(3) such that: H = e T, where the exponental s the usual exponental of matrces, defned as: e A = I + A + 1 2! A2 + 1 3! A3 +... Ths mples that for any moton whch can be represented by a matrx H SE(3) (see Sect. 1.3.1), we can consder a new matrx representaton that expresses an element of se(3) n the form: ) ( ω v T =. (1.21) 0 1 We can defne an equvalent vector representaton of T : ( ω T :=. v) Now suppose for the moment that ω 0. In ths case, t s easy to see that we can always fnd a vector r and a scalar λ such that: ( ( ) ( ω ω 0 = + λ, (1.22) v) r ω ω) where λω s the v component along ω and r ω s the one orthogonal to ω. It s also easy to fnd an expresson for r and h n the followng way. The second part of the Chasles decomposton can be wrtten as: and therefore, takng the vector product wth ω we obtan: 4 In certan cases there could be two solutons. v = r ω + λω, (1.23) ω v = ω (r ω). (1.24)
22 1. MOTION OF A RIGID BODY Wth some straghtforward geometrc consderatons, t s furthermore possble to see that ω (r ω) =r ω, and therefore we fnd an expresson for r: r = ω v ω 2. (1.25) The calculaton of the ptch λ s even smpler. By takng the scalar product of ω wth Eq. (1.23) we obtan: ω T v = ω T (r ω)+λω T ω, (1.26) and snce the frst term on the rght sde s zero (from the property of the mxed product), we obtan: λ = ωt v ω 2. (1.27) Equvalently to Eq. (1.22), usng the matrx form, t s always possble to decompose T as: T = T r + λ T t, where ) ( ω ωr T r := 0 0 and T t := ( ) 0 ω. 0 0 The elements T r and T t are both matrces belongng to se(3). It s therefore possble to check whether these two matrces are commutatve and therefore whether: T r Tt = T t Tr, whch s true f and only f the Le algebra commutator (see Appendx B) s zero,.e.: [ T r, T t ]=0. Ths can be easly seen to be satsfed for the gven decomposton and therefore, thanks to the Campbell-Baker-Hausdorff formula (Selg 1996), t s possble to see that we can wrte: e T = e T r e λ T t = e λ T t e T r. Wth some calculatons, t s furthermore possble to see that: ( ) H r := e T r R (Rr r) = and H 0 1 t := e λ T t = where R = e ω SO(3) s a rotaton matrx. We fnally obtan: H = H r H t = H t H r, ( ) I λω. 0 1 whch shows that the two motons are commutatve as expected. Usng wha has been explaned n Sect. 1.3.1, t s possble to see that H r represents a pure rotaton of θ := ω around an axs passng through r and along ω, and H r represents a pure translaton of θλ along the drecton ω. We have therefore proven Chasles theorem, snce we have shown that any moton can be descrbed as a rotaton around an axs after a translaton along the same axs or vce versa.
1.4. VELOCITIES FOR A RIGID BODY: TWISTS 23 In the case n whch ω =0, followng the same arguments as above, we can see that ths represents a pure translaton along v. Furthermore, we have also seen that the sx-dmensonal vector T can be nterpreted exactly as the sum of the Plücker ray coordnate vector of a fnte lne and an nfnte lne: T = T r + λt t, where T t s clearly related to T r by Eq. (1.22). The nfnte lne T t s called the polar of the fnte lne T r. Ths mples that we can express any moton by specfyng a fnte lne, and a scalar λ whch s exactly the nterpretaton of a screw wth an assocated module! For pure translatons we only have a lne at nfnty whch can be easly nterpreted as a screw wth nfnte ptch usng an argument smlar to the one used n Eq. (1.20). For a pure rotaton we only have a fnte lne whch can be nterpreted as a screw wth ptch equal to zero. We have also shown that the lnear combnaton of lnes seen as elements of PR 5 s n general not a lne, but a screw! We can therefore express any lnear combnaton of lnes as a screw and therefore as a fnte lne plus the ptch tmes ts polar. Note that, as wth the lne, screws are projectve enttes whch mples that they do not n general have a drecton and a module. Once we assocate a drecton and a module, we can use them drectly to express what s called a fnte twst n order to express any rgd body moton. In ths case, a fve-dmensonal screw becomes sx-dmensonal, due to the extra nformaton about the magntude. 1.4 Veloctes for a rgd body: Twsts Once agan, also for veloctes, we can gve ether a Le group or a screw nterpretaton. We wll analyse them both and fnd relatons as n the case of fnte motons. 1.4.1 Twsts as elements of se(3) Elements of se(3) (see Eq. (1.13)) are called twsts n mechancs and they represent the velocty of a rgd body moton geometrcally (.e., coordnate-free). In order to understand ths, we must look at the acton of elements of SE(3) on ponts of R 3. Consder any pont p not movng wth respect to the reference frame Ψ. If we ndcate wth p ts numercal representaton n Ψ, ths means that ṗ =0. Take now a second reference Ψ j possbly movng wth respect to Ψ. By lookng at the change of coordnates and dfferentatng, we obtan: P j = Ḣj P, where we denoted P k := ( p k 1 ), k =, j, and H j SE(3). We can now transport Ḣj to the dentty ether by left or rght translaton. If we do so, we obtan the two possbltes reported n Table 1.1. In the case we
24 1. MOTION OF A RIGID BODY Transport T Ḣ j P j Notaton Left Rght T := H j Ḣ j T := Ḣ j H j Ḣ j = Hj T Ḣ j = TH j P j = H j ( TP ) P j = T (H j P ),j T j,j T Table 1.1: The used notaton for twsts. consder the left translaton, workng out the terms, we obtan: (ṗj ) ( P j = H j }{{ ( T P R j )= = p j )( ) )( ω v p ṗ } 0 0 1 0 0 1 j = R j (ω p )+R j v Ḣ j and usng the rght translaton nstead we obtan: (ṗj ) )( P ( ω j = T (H j v P R j )= = p j )( ) p ṗ }{{} 0 0 0 0 1 1 j = ω (R j p + p j )+v Ḣ j From the prevous two expressons and Table 1.1, t s possble to see that Ta b,c represents the moton of Ψ a wth respect to Ψ c expressed n frame Ψ b. Note that v s not the relatve velocty of the orgns of the coordnate systems! (Ths would not gve rse to a geometrcal entty.) v s the velocty of an magnary pont connected to the movng body and nstantaneously passng through the orgn at the nstant under consderaton. Changes of coordnates for twsts Usng the left and rght map, we have seen that: It can be easly seen that T,j = H jḣj and T j,j = Ḣj H j. (1.28) T j,j = H j T,j H j, (1.29) and ths gves an expresson for changes of coordnates of twsts. Ths clearly corresponds to the adjont group representaton ntroduced n Appendx B on pag. 70. It s easer to work wth the sx-dmensonal vector form of twsts and t s possble to see that we can fnd a matrx expresson for the adjont representaton: T j,j It s possble to proof that ths matrx representaton s: ( R j ) Ad H j := 0. = Ad H j T,j. (1.30) p j Rj By dfferentaton of the prevous matrx as a functon of H j we can also fnd an expresson for ts tme dervatve and the adjont representaton of the algebra ad (see Appendx B). It can be shown that: ( ) Ad H j = Ad H j ad T,j wth T,j := H jḣj, (1.31) R j
1.4. VELOCITIES FOR A RIGID BODY: TWISTS 25 ω r ω r λ Ψ Fgure 1.6: Intuton of a Twsts where ad T k,j := k,j ( ω 0 ṽ k,j ω k,j ) s the adjont representaton we were lookng for. Ths can be easly checked by testng the relaton proved n Eq. (B.9) for a general matrx Le group. It s furthermore easly possble to prove that: ad T k,j T k, l whch can be useful to analyse mechansms. =[T k,j,t k, l ], 1.4.2 Twsts as applcatons on screws Wth the same lne of reasonng as we dd for fnte motons, we can gve an nterpretaton to elements of se(3) as a twst appled on a geometrcal screw. In a smlar way, we could state Chasles theorem, but now appled to nstantaneous veloctes nstead of to fnte twsts. Smlarly as before, the theorem says that any element of se(3) can be descrbed as a pure nstantaneous rotaton around an axs plus a pure nstantaneous translaton along the same axs: ( ( ) ( ω ˆω 0ˆω) = ω, +α v) r ˆω }{{} rotaton }{{} translaton where ω = ω ˆω. Analyzng the prevous formula and wth reference to Fgure 1.6, t s possble to see that v s the velocty of an magnary pont passng through the orgn of the coordnate system n whch the twst s expressed and movng together wth the object. The sx-vector representng the rotaton s what we called prevously a rotor and can be assocated to a geometrcal lne, namely the lne passng through r and spanned by ω whch s left nvarant by the rotaton.
26 1. MOTION OF A RIGID BODY Ψ 1 Ψ 3 Ψ 4 Ψ 2 Fgure 1.7: Interpretaton of the ndces for the notaton used n twsts. Once agan, the theorem of Chasles s one of the two theorems on whch screw theory s based because t gves to elements of se(3) a real tensoral geometrcal nterpretaton. Ths nterpretaton s the one of a motor or screw whch are enttes characterzed by a geometrcal lne and a scalar called the ptch. Ths ptch relates the rato of translaton and rotaton along and around the lne. If we furthermore consder the change of coordnates expressed n Eq. (1.30), ths can be nterpreted as a change of Plücker ray coordnates of the geometrcal lne assocated to the twst usng Chasles theorem. Ths also mples that n the notaton T k,j we use, the frames Ψ j can be any frame fxed to the consdered body of reference and ths holds also for Ψ. Wth reference to Fgure 1.7, we can prove what has just been sad by notcng that snce H 2 1 and H 4 3 are constant, the followng s true: and that T 1,1 4 = Ḣ1 4 H1 4 = Ḣ1 3 H4 3 H3 4 H1 3 = }{{} I T 1,1 3, T 2,1 3 = H1 2 1,1 T 3 H2 1 = H1 2 Ḣ1 3 H1 3 H2 1 = (H 2 1 H1 3 )(H3 1 H2 1 )= T 2,2 3. (1.32) We can therefore talk about T j as a geometrc entty modeled as a screw wthout the necessty to express t numercally n a certan frame Ψ k. 1.5 Forces appled to rgd bodes: Wrenches The concept of dualty s fundamental when workng n mechancs. Wthn the theory of Le groups, we know that se(3) s a vector space, and as such, we can consder the dual se (3) whch s the vector space of lnear operators on se(3). Ths corresponds to the space of wrenches whch are a sx-dmensonal generalzaton of three-dmensonal forces to rgd bodes. 1.5.1 Wrenches as elements of se(3) Twsts are the generalzaton of veloctes and are elements of se(3). The dual vector space of se(3) s called the dual Le algebra and denoted wth se (3). It s the vector space of lnear operators from se(3) to R. Ths space represents the space of forces for
1.5. FORCES APPLIED TO RIGID BODIES: WRENCHES 27 rgd bodes whch are called wrenches. The applcaton of a wrench on a twsts gves a scalar representng the power suppled by the wrench. A wrench n vector form wll be a 6 dmensonal row vector snce t s a co-vector (lnear operator on vectors) nstead than a vector. W = ( m f ) where m represents a torque and f a lnear force. For what just sad we have: Power = WT where T s a twst of the object on whch the wrench s appled. Clearly, to calculate the power, the wrench and the twst have to be numercal vectors expressed n the same coordnates and result Power = WT = mω + fv. Another representaton of a wrench n matrx form s: ( ) f m W =. 0 0 How do wrenches transform changng coordnate systems? twsts: We have seen that for T j, = Ad H j T, where Ad H j ( R j := 0 p j Rj R j ). Suppose to supply power to one body attached to Ψ j represented n Ψ j s W j. by means of a wrench whch Changng coordnates from Ψ j to Ψ the expresson for the suppled power should stay constant and ths mples that: W j T j, j = W j Ad H j T, j =(Ad T (W j ) T ) T T, H j j = W T, j whch mples that the transformaton of wrenches expressed n vector form s: (W ) T = Ad T (W j ) T. H j Note that f the mappng Ad H j was mappng twsts from Ψ to Ψ j, the transposed maps wrenches n the opposte drecton: from Ψ j to Ψ! Ths s a drect consequence of the fact that wrenches are duals to twsts. 1.5.2 Wrenches as applcatons on screws The concept of dualty s nstead not drectly used n the theory of screws. Ths has often brought to useless dscussons whch could be avoded just by realzng the dualty structure between twsts and wrenches. For reasons whch wll not be explaned here (Stramgol, Maschke & Bdard 2000) and dependng on the exstence of what s called the Klen form, the decomposton result for twsts gven by Chasles theorem,
28 1. MOTION OF A RIGID BODY F r F r λ Ψ Fgure 1.8: Intuton of a Wrench can be gven for wrenches by ts dual analogous theorem called Ponsot s theorem. Ths theorem states that any element of se (3) can be splt as a sum of two terms: ( ) m = f ( ) ( r f f + λ. (1.33) f 0) It can be drectly seen that we have somehow nverted the the role of the frst three and last three components wth respect to Chasles theorem as shown n Eq. (1.22). Ths fact as profound explanatons whch can be found n (Stramgol, Maschke & Bdard 2000). The components of Eq. (1.33) can stll be nterpreted as lne components whch are now called Plücker axs coordnates. For smplcty, the axs coordnates can be just consdered as ray coordnates n whch we nverted the frst and the last three components. There are actually much deeper projectve geometrcal reasons for the relaton between ray and axs coordnates whch have to do wth the self dualty of lnes n projectve space (Lpkn 1985). Also wth reference to Fgure 1.8, the frst element of Eq. (1.33) s representng a pure force appled along the lne passng through r. The second s nstead a pure momenta whch does not need to be assocated to a fnte lne but to an nfnte lne whch s agan the polar of the lne representng the pure force. As we dd wth twsts, we can descrbe a wrench as appled on a screw. Ths s true because once we have defned a lne, a ptch and a magntude, any wrench can be expressed as a lne wth a magntude plus the ptch tmes ts polar as shown n Eq. (1.33). Note that the role of angular veloctes for twsts s taken by lnear forces n wrenches snce both elements are assocable to a pure lne: the frst as a rotaton axs and the second as an applcaton lne of a force. In a smlar way, the role of pure translatons s taken by pure moments. Ths also mples that a wrench appled on a screw wth zero ptch corresponds to a pure force and a wrench appled on a screw wth nfnte ptch corresponds to a pure moment.