Teaching Spatial Kinematics for Engineers

Transcription

1 845 Teachng Spatal Knematcs for Engneers Author: Hellmuth Stachel, Insttute of Dscrete Mathematcs and Geometry, Venna Unversty of Technology, Abstract Spatal knematcs s a challengng feld because of ts real world applcatons at seral and parallel manpulators However, t s not easy to teach as on the one hand the students spatal abltes need to be well developed And on the other hand famlarty wth calculus and vector algebra s substantal After several years of experence I learned that one can expect from students to work wth dual vectors Only at frst sght they look too abstract But students soon estmate the advantages of ths method: () Though lookng lke common vectors, they allow to handle orented lnes Wth dot product and cross product complcated D-geometrc problems can be solved n a very bref and elegant way () Besde the correspondance between dual unt vectors and orented lnes, the general dual unt vectors symbolze screws, e, nstantaneous motons As each dual vector s a dual multple of a dual unt vector, any nstantaneous moton s a helcal moton wth well defned axs, angular velocty and translatory velocty () The spatal Three-Pole-Theorem states, that the screw of the relatve moton s the dfference of the screws of the absolute moton and the gudng moton Therefore the nstantaneous moton of the endeffector equals the sum of the screws of the nstanteous rotatons between consecutve lnks Ths leads drectly to the Jacob-matrx of seral robots wth the dual vectors of the revolute axes as columns And t allows to solve, eg, calbraton problems at parallel robots, thus revealng that the purely abstract lookng dual vector algebra s a powerful tool for a plenty of real world applcatons Index Terms dual vector, forward knematcs, lne coordnates, parallel manpulator, spatal knematcs, seral manpulator, screw, Jacob matrx DUAL UNIT VECTORS REPRESENTING ORIENTED LINES IN THE -SPACE There s a tght connecton between spatal knematcs and the geometry of lnes n the Eucldean -space we start wth recallng the use of approprate lne coordnates (cf [4], p 55 ff)) Any orented lne (spear) g wth drecton vector g and passng through pont a, e, g = a + IR represented by the par of vectors IE Therefore g can be unquely ( g, g ), the drecton vector g and the momentum vector g, accordng to the defntons g g = and g := a g whch mply g g = Conversely, any par ( g, g ) of vectors obeyng g g = and g g = defnes a unque spear g because a := g g s a pont of ths lne, the pedal pont of g wth respect to the orgn It makes sense to replace the par ( g, g ) by the dual vector g : = g + ε g, () where the dual unt ε obeys the rule ε = We extend the usual dot product of vectors to dual vectors and notce g g = ( g + ε g ) ( g + ε g ) = g g + ε g g = + ε = () Hence we call g a dual unt vector In ths sense the set of orented lnes n unt sphere The representaton of orented lnes g n IE can be seen as the dual extenson of the IE by dual unt vectors g brngs about several advantages, and from now on we do not dstngush between orented lnes g and ther representng dual vector g as well as for ponts and ther coordnate vector x THEOREM : For two gven orented lnes g, h n IE let n denote the common normal endowed wth an arbtrary orentaton If the helcal moton along n whch transforms g nto h (see Fgure ) has the angle ϕ of rotaton and the length ϕ of translaton and we combne them n the dual angle ϕ = ϕ + ε ϕ, then the followng equatons hold true: g h = cosϕ = cosϕ εϕ snϕ and g h = snϕ n = sn ϕ n + ε(snϕ n + ϕ cos ϕ n ) () Internatonal Conference on Engneerng Educaton July 5 9, 5, Glwce, Poland

2 84 Here we use the dual extenson of dfferentable functons whch s defned by f ( x) = f ( x + εϕ ) = f ( x) + ε x f ( x) Ths s the begnnng of a Taylor seres where due to ε = all hgher powers are vanshng Ths guarantees that denttes lke cos x + sn x = are preserved under the dual extenson as they are vald for the power seres, too The notaton ε orgnates from the fact that the dual unt can be seen as such a small number that ts square s neclectable Note that only dual numbers x wth non-vanshng real part have an nverse x FIGURE : DUAL ANGLE ϕ = ϕ + εϕ BETWEEN g AND h FIGURE : k : = cosϕ g + snϕ h On the other hand we use n Theorem the dual extenson of the vector product accordng to g h = ( g + εg ) ( h + εh ) = g h + ε ( g h + g h ) (4) Proof of Theorem : Suppose g := a g and h := b h Then the shortest dstance between g and h reads g h ϕ = ( b a) = [det( b, g, h) det( a, g, h)] = [ ( b h) g ( a g) h)] = ( g h + g h ) snϕ snϕ snϕ snϕ If a and b are supposed to be the ntersecton ponts of g and h wth the common normal n, then snϕ n = a sn ϕ n = a ( g h) = ( a h) g ( a g) h + [( a b) g] h = ( a h) g ( b g) h The expresson n brackets vanshes and could therefore be added wthout changng the value On the other hand due to standard formulas from vector algebra we see ( g h) + ( g h ) = [( a g) h] + [ g ( b h)] = ( a h) g ( g h) a + ( g h) b ( g b) h = = ( a h) g ( b g) h + ( g h)( b a) = snϕ n + ϕ cosϕ n By dual extenson t s possble to convert formulas from sphercal geometry nto formulas on spears The followng example should llustrate ths so called prncple of transference (German: Uebertragungsprnzp) whch dates back to E Study A frst example s gven n THEOREM : Let g and h be two orthogonally ntersectng spears wth the common perpendcular n Then k : = cosϕ g + snϕ h (5) s the mage of g under the helcal moton along n through the dual angle ϕ (see Fgure ) Proof: From the perpendcular ntersecton of g and h we conclude g h = and n := g h These equatons mply k k=cos ϕ ( g g) + sn ϕ ( h h ) =, and from Theorem we conclude g k : = sn ϕ ( g h) = snϕ n as stated Another standard example s shown n [5]: The statement that n any sphercal trangle wth atmost one rght angle the alttudes are concurrent leads by dual extenson to a statement of J Hjelmslev and F Morley (898): At any skew hexagon wth only rght angles the common perpendculars of opposte sdes ntersect a common lne perpendcularly, provded atmost one par of opposte sdes s parallel Fnally, n [] the dfferental geometry of ruled surfaces s based on Frenet equatons whch are exactly the dual extensons of the Frenet equatons for sphercal curves Here the curvature center of a sphercal curve s replaced by the curvature axs (or Dstel axs) of the ruled surface In the sequel we need Internatonal Conference on Engneerng Educaton July 5 9, 5, Glwce, Poland

3 847 THEOREM : Any dual vector v = v + ε v s a dual multple of a dual unt vector, e, v = λ g wth g g = In the case v o the dual unt vector g s unquely determned up to ts sgn Proof: We have to fulfll the equaton v + ε v = ( λ + ε λ)( g + ε g ) = λg + ε( λg + λg ) Frst we note that v v mples v v + ε ( v v ) = λ + ε λλ Ths gves v = λg, v = λg + λg and v v = λλ For λ = ± v we get λ g g = λλ g = v and g = v v () λ λ λ as the soluton In the case λ =, e, v = ε v, we set λ = v, λ g = v and choose an arbtrary ĝ under λ, otherwse the unt vector g can be chosen arbtrarly, too DUAL VECTORS REPRESENTING SCREWS In ths secton we demonstrate the use of dual vectors for descrbng nstantaneous motons (for an ntroducton see also [,,]): Let a rgd body representng the movng system Σ perform a one-parameter moton aganst the frame, the fxed system Σ We assume that cartesan coordnate frames are attached to each system Σ, =,, and we use the subscrpt to ndcate related coordnate vectors Then the movement of Σ aganst Σ can analytcally be descrbed by the coordnate transformaton whch at each nstant t gves the Σ -coordnate vector x of any pont whch wth respect to the movng system Σ has the coordnate vector x Ths coordnate transformaton reads Σ / Σ : x = u ( t) + A( t) x wth A( t) A( t) T = I and det A ( t ) = (7) Here I denotes the unt matrx, u ( t) s the Σ -coordnate vector of the orgn of Σ, and A ( t) s an orthogonal matrx, e, ts transposed A ( t) T s at the same tme ts nverse A ( ) t In order to fgure out the dstrbuton of velocty vectors v of ponts attached to the movng system Σ we dfferentate and replace x by x due to (7) We thus obtan after droppng the parameter t T T v = x = u + A x = ( u A A u ) + A A x (8) because of x = o The matrx T A A s skew as dfferentaton of T A A = I gves A A T + A A T = A A T + ( A A T ) T = O = zero matrx There s a dual vector q = q + εq such that A A T x = q x for all x and T q : = u A A u (9) We call ths dual vector the nstantanous screw as accordng to (8) t rules the dstrbuton of those velocty vectors whch the gven moton nstantaneously assgns to each pont Σ We have v = q + ( q x ) () FIGURE : INSTANTANEOUS MOTION WITH SCREW q = ω p FIGURE 4: PORTION OF PLÜCKER S CYLINDROID Due to Theorem the dual vector q s a dual multple of a dual unt vector, e, q = ω p or explctely q + εq = ( ω + εω )( p + εp ) = ω p + ε ( ω p + ω p ) () Internatonal Conference on Engneerng Educaton July 5 9, 5, Glwce, Poland

4 848 wth p p = In order to fgure out the meanng of the dual scalar ω and the orented lne p we use a pont s of ths lne and set p = s p Then accordng to () and () we get v = q + ( q x ) = ( ω p + ω p ) + [ ω p x ] = ω p + ω[ p ( x s )] () Ths reveals that v s the velocty vector of x under a helcal moton about the nstantaneous axs (ISA) p wth angular velocty ω and translatory velocty ω (see Fgure ) We call ω the dual angular velocty of the nstantaneous moton In ths sense the dual unt vectors are at the same tme the screws for nstantaneous rotatons wth angular velocty Now t s clear why due to Theorem the axs p s unquely determned only under ω : Otherwse the nstantaneous moton s a translaton and here only the drecton of the axs s determned, but not the axs tself THEOREM 4: When the nstantaneous screw q s expressed as a multple q = ω p of a dual unt vector, then p s the nstantaneous axs and ω the dual angular velocty of the nstantaneous helcal moton One has to be careful wth dual vectors when coordnate transformatons are appled as the two components of a dual vector behave dfferent: THEOREM 5: If n Σ the coordnates x of pont are replaced by x accordng to x = b + B x wth orthogonal B, then the components of any screw q = q + εq (nclusve spears) have to be replaced by q = B q and q = B q + ( b B q ) () Proof: Because of the geometrc meanng of the screw q = ω p the coordnate transformaton does affect the underlyng spear, but not the dual angular velocty, e, q = ω p And p + εp s transformed nto p + εp by p = B p and p = s p = ( b + B s ) B p = ( b B p ) + B ( s p ) = ( b B p ) + B p The formulas for q n (9) refer to coordnates n the fxed system Σ However, for the sake of smplcty we avod the more precse notaton q Eqs () could be used to represent the screw also wth respect to Σ, and then the coordnate vector should be denoted by q THEOREM : For any nstantaneous moton wth the screw q the path-normals, e, the lnes n perpendcular to v and passng through, consttute a lnear lne complex as n = n + ε n obeys the lnear homogeneous equaton q n + q n = ( q n IR) (4) Ths equaton s ndependent from Hence any lne ± n, whch s a path-normal at one of ts ponts, s a path-normal at each pont Proof: Due to () and n = x n we have = v n = [ q + ( q x )] n = q n + det( q, x, n) = q n + q ( x n) = q n + q n By () and () eq (4) s equvalent to ω p n = ω cosα IR, e, ( ω + εω )(cosα εα sn α) IR or ω cosα ω α snα = Hence the path normals n of the nstantaneous moton are characterzed by ω α tanα ω = (5) wth α denotng the dual angle between the ISA p and any orentaton of n The quotent on the left hand sde s the ptch of the helcal moton Fnally, we need the spatal Three-Pole-Theorem THEOREM 7: If for three gven systems Σ, Σ, Σ the dual vectors q and q are the nstantaneous screws of Σ / Σ and Σ / Σ, respectvely, then q = q q, e, ωp = ω p ω p () s the nstantaneous screw of the relatve moton Σ / Σ The three correspondng lnear lne complexes are ncluded n a pencl of lne complexes Proof: Accordng to () we have Internatonal Conference on Engneerng Educaton July 5 9, 5, Glwce, Poland

5 849 v = v v = [ q + ( q x )] [ q + ( q x )] = ( q q ) + [( q q ) x ] for each pont / Let a lne n ntersect the ISAs p of Σ Σ and p of Σ Σ orthogonally, e, p n = p n = Then we obtan ωp n = ( ω p n ω p n ) = whch means that the lne n does ntersect the axs p of Σ / Σ orthogonally, too, provded ω Let two skew axes p and p be gven, e, p p R When for the correspondng dual veloctes we replace ω by cω wth a constant c varyng n R, then the axes p of the relatve motons Σ / Σ consttute a cylndrod or Plücker conod (see [4], p 8) Fgure 4 gves an mpresson of the cylndrod by showng some generators 'between' p and p Now the prncple of transference can be used to convert theorems from sphercal knematcs nto those of spatal knematcs One example s the dual extenson of the Euler-Savary-formula (see []) It dates back to M Dstel (94) and deals wth the curvature axes of ruled surfaces whch are traced by lnes under a spatal moton Other examples concernng overconstraned spatal mechansms derved from sphercal ones are presented n [5] APPLICATIONS EAMPLE : Infntesmal forward and nverse knematcs of R robots: A seral robot s an open knematc chan of lnks Σ, Σ,, Σ Any two consecutve lnks are connected by a revolute jont wth the axs p, p,, p, respectvely Gven: Any posture of a R robot wth nstantaneous angular veloctes ω, ω,, ω R of the relatve motons about the axes All axes are gven n the same coordnate system Requred: What s the nstantaneous moton of the endeffector Σ? Ths nstantaneous moton s defned by the screw q From the Three-Pole-Theorem (Theorem 7) we conclude that ω ω ω q = p + p + + p Ths can also be wrtten n matrx form: We combne the coordnates,, the axes as columns n a -matrx J Then the resultng screw reads q = p = J Ω ω = / p p of wth Ω as the column vector of gven angular veloctes J s called Jacob matrx In the regular case ( det J ) we can also solve the nverse problem: For gven q we get the correspondng angular veloctes ω,, ω by solvng a lnear system of equatons If there s a rank defcency of J, then the nstantaneous degree of freedom of Σ Σ s less than Just n ths case the columns of J are lnearly dependent Ths s equvalent to the statement that the sx axes are ncluded n a lnear lne complex EAMPLE : Calbraton of Stewart-Gough-Platforms: Gven: Any posture of a Stewart-Gough-Platform, e, a parallel manpulator where the platform Σ s connected wth the frame Σ by sx telescopc legs The anchor ponts n Σ are denoted by a,, a, those n the platform Σ by b,, b We assume that for all these ponts the nstantaneous coordnates are gven n the same coordnate system Requred: Suppose that by precse measurements a mslocaton of the platform aganst the frame has been detected How to fgure out whch leg s manly responsble for ths devaton There s a (small) helcal moton whch transports the actual posture nto the target posture After solvng ths regstraton problem we get the axs p, the angle ϕ of rotaton and the length ϕ of translaton Let us assume that ths movement s performed wthn say second Ths defnes a screw q, and the nstantaneous helcal moton of Σ assgns to each of ts anchor ponts b,, b a velocty vector accordng to () The component of ths vector n drecton of the leg ab gves the correspondng varaton n length whch has to be carred out wthn second So, the leg wth the largest velocty component should be most responsble for the mslocaton Of course, the relablty on ths result needs to be fostered by terated measurements n dfferent postures How to compute these varatons? Let d denote the dstance b a Then the carrer lne of the leg orented n the drecton ab has the coordnate components l = b a and l = b l d / Internatonal Conference on Engneerng Educaton July 5 9, 5, Glwce, Poland

6 85 From d = ( b a ) we obtan be dfferentaton dd = ( b a ) b, hence d = l b = l [ q + ( q b )] = l q + det( l, q, b ) = l q + ( b l ) q = l q + l q We form a -matrx J wth rows consstng of the coordnates of ( l, l ) wrtten n ths order Then we get D = J q wth D denotng the column of d So we get the varaton of leg lengths by multplyng ths Jacob matrx J wth the screw In sngular postures, whch are characterzed by det J =, there are nfntesmal self motons of the platform whle the lengths of all telescopc legs reman fxed Just n sngular postures the rows n J are lnearly dependent and therefore the sx lnes ab ncluded n a lnear lne complex REFERENCES [] Blaschke, W, "Knematk und Quaternonen", VEB Deutscher Verlag der Wssenschaften, Berln 9 [] Husty, M, Karger, A, Sachs, H, Stenhlper, W, "Knematk und Robotk", Sprnger-Verlag, Berln Hedelberg 997 [] Müller, HR, "Knematk", Sammlung Göschen, Walter de Gruyter & Co, Berln 9 [4] Pottmann, H, Wallner, J, "Computatonal Lne Geometry",, Computatonal Lne Geometry, Sprnger Verlag, Berln, Hedelberg [5] Stachel, H, "Eucldean lne geometry and knematcs n the -space", NK Artémads, NK Stephands (eds): Proceedngs of the 4th Internatonal Congress of Geometry, Thessalonk 997, 8-9 [] Stachel, H, "Instantaneous spatal knematcs and the nvarants of the axodes", Proceedngs Ball Symposum, Cambrdge, no Internatonal Conference on Engneerng Educaton July 5 9, 5, Glwce, Poland