Chapter 7 Dynamics. 7.1 Newton-Euler Formulation of Equations of Motion

Transcription

1 Itroduto to Robots,. arry Asada Chapter 7 Dyams I ths hapter, we aalyze the dyam behavor of robot mehasms. he dyam behavor s desrbed terms of the tme rate of hage of the robot ofgurato relato to the ot torques exerted by the atuators. hs relatoshp a be expressed by a set of dfferetal equatos, alled equatos of moto, that gover the dyam respose of the robot lkage to put ot torques. I the ext hapter, we wll desg a otrol system o the bass of these equatos of moto. wo methods a be used order to obta the equatos of moto: the Newto-Euler formulato, ad the Lagraga formulato. he Newto-Euler formulato s derved by the dret terpretato of Newto's Seod Law of Moto, whh desrbes dyam systems terms of fore ad mometum. he equatos orporate all the fores ad momets atg o the dvdual robot lks, ludg the ouplg fores ad momets betwee the lks. he equatos obtaed from the Newto-Euler method lude the ostrat fores atg betwee adaet lks. hus, addtoal arthmet operatos are requred to elmate these terms ad obta explt relatos betwee the ot torques ad the resultat moto terms of ot dsplaemets. I the Lagraga formulato, o the other had, the system's dyam behavor s desrbed terms of work ad eergy usg geeralzed oordates. hs approah s the exteso of the dret method dsussed the prevous hapter to dyams. herefore, all the workless fores ad ostrat fores are automatally elmated ths method. he resultat equatos are geerally ompat ad provde a losed-form expresso terms of ot torques ad ot dsplaemets. Furthermore, the dervato s smpler ad more systemat tha the Newto-Euler method. he robot s equatos of moto are basally a desrpto of the relatoshp betwee the put ot torques ad the output moto,.e. the moto of the robot lkage. As kemats ad stats, we eed to solve the verse problem of fdg the eessary put torques to obta a desred output moto. hs verse dyams problem s dsussed the last seto of ths hapter. Effet algorthms have bee developed that allow the dyam omputatos to be arred out o-le real tme. 7. Newto-Euler Formulato of Equatos of Moto 7... Bas Dyam Equatos I ths seto we derve the equatos of moto for a dvdual lk based o the dret method,.e. Newto-Euler Formulato. he moto of a rgd a be deomposed to the traslatoal moto wth respet to a arbtrary pot fxed to the rgd, ad the rotatoal moto of the rgd about that pot. he dyam equatos of a rgd a also be represeted by two equatos: oe desrbes the traslatoal moto of the etrod (or eter of mass), whle the other desrbes the rotatoal moto about the etrod. he former s Newto's equato of moto for a mass partle, ad the latter s alled Euler's equato of moto. We beg by osderg the free dagram of a dvdual lk. Fgure 7.. shows all the fores ad momets atg o lk. he fgure s the same as Fgure 6.., whh desrbes the stat balae of fores, exept for the ertal fore ad momet that arse from the dyam moto of the lk. Let be the lear veloty of the etrod of lk wth referee v Departmet of Mehaal Egeerg Massahusetts Isttute of ehology

2 Itroduto to Robots,. arry Asada to the base oordate frame O-xyz, whh s a ertal referee frame. he ertal fore s the gve by m v, where m s the mass of the lk ad v s the tme dervatve of v. Based o D Alembert s prple, the equato of moto s the obtaed by addg the ertal fore to the stat balae of fores eq.(6..) so that f f + m g m v,, (7..),, + 0, where, as Chapter 6, f, ad f, + are the ouplg fores appled to lk by lks - ad +, respetvely, ad g s the aelerato of gravty. f, v ω f, + N, + N, Lk r, C r, C O z O - m g x O y Jot Jot + Fgure 7.. Free dagram of lk moto Rotatoal motos are desrbed by Euler's equatos. I the same way as for traslatoal motos, addg ertal torques to the stat balae of momets yelds the dyam equatos. We beg by desrbg the mass propertes of a sgle rgd wth respet to rotatos about the etrod. he mass propertes are represeted by a erta tesor, or a erta matrx, whh s a 3 x 3 symmetr matrx defed by I {( y y ) + ( z z ) } ρ dv ( x x )( y y ) ρ dv ( z z )( x x ) ρ dv ( x x )( y y ) ρ dv {( z z ) + ( x x ) } ρ dv ( y y )( z z ) ρ dv ( z z)( x x ) ρ dv ( y y)( z z) ρ dv {( x x + ) ( y y) } ρ dv (7..) where ρ s the mass desty, x, y, z are the oordates of the etrod of the rgd, ad eah tegral s take over the etre volume V of the rgd. Note that the erta matrx vares wth the oretato of the rgd. Although the heret mass property of the rgd Departmet of Mehaal Egeerg Massahusetts Isttute of ehology

3 Itroduto to Robots,. arry Asada 3 does ot hage whe vewed from a frame fxed to the, ts matrx represetato whe vewed from a fxed frame,.e. ertal referee frame, hages as the rotates. he ertal torque atg o lk s gve by the tme rate of hage of the agular mometum of the lk at that stat. Let ω be the agular veloty vetor ad I be the etrodal erta tesor of lk, the the agular mometum s gve by I ω. Se the erta tesor vares as the oretato of the lk hages, the tme dervatve of the agular mometum ludes ot oly the agular aelerato term Iω, but also a term resultg from hages the erta tesor vewed from a fxed frame. hs latter term s kow as the gyrosop torque ad s gve by ω ( I ω ). Addg these terms to the orgal balae of momets (4-) yelds N ( C ) ( C ) ( ), N, + r, + r, f, + r, f, + Iω ω ( I ω ) 0,,, (7..3) usg the otatos of Fgure 7... Equatos () ad (3) gover the dyam behavor of a dvdual lk. he omplete set of equatos for the whole robot s obtaed by evaluatg both equatos for all the lks,,, Closed-Form Dyam Equatos he Newto-Euler equatos we have derved are ot a approprate form for use dyam aalyss ad otrol desg. hey do ot expltly desrbe the put-output relatoshp, ulke the relatoshps we obtaed the kemat ad stat aalyses. I ths seto, we modfy the Newto-Euler equatos so that explt put-output relatos a be obtaed. he Newto-Euler equatos volve ouplg fores ad momets f, ad N,. As show eqs.(6..) ad (6..), the ot torque, whh s the put to the robot lkage, s luded the ouplg fore or momet. owever, s ot expltly volved the Newto-Euler equatos. Furthermore, the ouplg fore ad momet also lude workless ostrat fores, whh at terally so that dvdual lk motos oform to the geometr ostrats mposed by the mehaal struture. o derve explt put-output dyam relatos, we eed to separate the put ot torques from the ostrat fores ad momets. he Newto-Euler equatos are desrbed terms of etrod velotes ad aeleratos of dvdual arm lks. Idvdual lk motos, however, are ot depedet, but are oupled through the lkage. hey must satsfy erta kemat relatoshps to oform to the geometr ostrats. hus, dvdual etrod posto varables are ot approprate for output varables se they are ot depedet. he approprate form of the dyam equatos therefore ossts of equatos desrbed terms of all depedet posto varables ad put fores,.e., ot torques, that are expltly volved the dyam equatos. Dyam equatos suh a explt put- output form are referred to as losed-form dyam equatos. As dsussed the prevous hapter, ot dsplaemets q are a omplete ad depedet set of geeralzed oordates that loate the whole robot mehasm, ad ot torques are a set of depedet puts that are separated from ostrat fores ad momets. ee, dyam equatos terms of ot dsplaemets q ad ot torques are losed-form dyam equatos. Example 7. Fgure 7.. shows the two dof plaar mapulator that we dsussed the prevous hapter. Let us obta the Newto-Euler equatos of moto for the two dvdual lks, ad the derve losed-form dyam equatos terms of ot dsplaemets θ adθ, ad ot torques ad. Se the lk mehasm s plaar, we represet the veloty of the etrod of Departmet of Mehaal Egeerg Massahusetts Isttute of ehology

4 Itroduto to Robots,. arry Asada 4 eah lk by a -dmesoal vetor v ad the agular veloty by a salar veloty ω. We assume that the etrod of lk s loated o the eter le passg through adaet ots at a dstae from ot, as show the fgure. he axs of rotato does ot vary for the plaar lkage. he erta tesor ths ase s redued to a salar momet of erta deoted by I. From eqs. () ad (3), the Newto-Euler equatos for lk are gve by f m 0, f, + g mv 0, N N + r f r f I ω 0 (7..4) 0,,,, 0, 0, Note that all vetors are x, so that momet N -, ad the other vetor produts are salar quattes. Smlarly, for lk, f m, + g mv 0, N r f I ω 0 (7..5),,, y V I, m θ, I, m f, Lk 0 O θ, x Fgure 7.. Mass propertes of two dof plaar robot o obta losed-form dyam equatos, we frst elmate the ostrat fores ad separate them from the ot torques, so as to expltly volve the ot torques the dyam equatos. For the plaar mapulator, the ot torques ad are equal to the ouplg momets: N,,, (7..6) Substtutg eq.(6) to eq.(5) ad elmatg f, we obta Departmet of Mehaal Egeerg Massahusetts Isttute of ehology

5 Itroduto to Robots,. arry Asada 5 Smlarly, elmatg f 0, yelds, r m v + r m g I ω 0 (7..7),, r m v r m v + r m g + r m g I ω 0 (7..8) 0, 0, 0, 0, Next, we rewrte v, ω, ad r, + usg ot dsplaemets θ ad θ, whh are depedet varables. Note that ω s the agular veloty relatve to the base oordate frame, whle θ s measured relatve to lk. he, we have θ, ω θ θ ω + (7..9) he lear velotes a be wrtte as θ sθ v θ osθ { sθ + s( θ + θ)} θ s( θ + θ)} θ v (7..0) { osθ + os( θ + θ)} θ + os( θ + θ)} θ Substtutg eqs. (9) ad (0) alog wth ther tme dervatves to eqs. (7) ad (8), we obta the losed-form dyam equatos terms of θ adθ : where (7..-a) θ + θ hθ hθθ + G θ + θ + h θ + G (7..-b) ( I (7..-a) + (7..-b) m + I + m + + osθ) + m I m( + osθ) + I m sθ m g osθ + m g{ os( θ + θ) + osθ os( ) m g θ + θ (7..-) h (7..-d) G } (7..-e) G (7..-f) he salar g represets the aelerato of gravty alog the egatve y-axs. More geerally, the losed-form dyam equatos of a -degree-of-freedom robot a be gve the form + q + h q q + G,,, (7..3) k k k Departmet of Mehaal Egeerg Massahusetts Isttute of ehology

6 Itroduto to Robots,. arry Asada 6 where oeffets, h k, ad G are futos of ot dsplaemets q, q,, exteral fores at o the robot system, the left-had sde must be modfed aordgly Physal Iterpretato of the Dyam Equatos q. Whe I ths seto, we terpret the physal meag of eah term volved the losedform dyam equatos for the two-dof plaar robot. he last term eah of eqs. (-a, b), G, aouts for the effet of gravty. Ideed, the terms G ad G, gve by (-e, f), represet the momets reated by the masses m ad m about ther dvdual ot axes. he momets are depedet upo the arm ofgurato. Whe the arm s fully exteded alog the x-axs, the gravty momets beome maxmums. Next, we vestgate the frst terms the dyam equatos. Whe the seod ot s mmoblzed,.e. θ 0 ad θ 0, the frst dyam equato redues to θ, where the gravty term s egleted. From ths expresso t follows that the oeffet aouts for the momet of erta see by the frst ot whe the seod ot s mmoblzed. he oeffet gve by eq. (-a) s terpreted as the total momet of erta of both lks refleted to the frst ot axs. he frst two terms, m +, eq. (-a), represet the momet of erta of lk I wth respet to ot, whle the other terms are the otrbuto from lk. he erta of the seod lk depeds upo the dstae L betwee the etrod of lk ad the frst ot axs, as llustrated Fgure he dstae L s a futo of the ot agle θ ad s gve by L + + osθ (7..4) Usg the parallel axes theorem of momet of erta (Goldste, 98), the erta of lk wth respet to ot s m L +I, whh s osstet wth the last two terms eq. (-a). Note that the erta vares wth the arm ofgurato. he erta s maxmum whe the arm s fully exteded ( θ 0 ), ad mmum whe the arm s ompletely otrated ( θ π ). O L θ L θ θ ' Fgure 7..3 Varyg erta depedg o the arm ofgurato Departmet of Mehaal Egeerg Massahusetts Isttute of ehology

7 Itroduto to Robots,. arry Asada 7 Let us ow exame the seod terms o the rght had sde of eq. (). Cosder the stat whe θ θ 0 ad θ 0, the the frst equato redues to θ, where the gravty term s aga egleted. From ths expresso t follows that the seod term aouts for the effet of the seod lk moto upo the frst ot. Whe the seod lk s aelerated, the reato fore ad torque dued by the seod lk at upo the frst lk. hs s lear the orgal Newto-Euler equatos (4), where the ouplg fore -f l, ad momet -N, from lk are volved the dyam equato for lk. he ouplg fore ad momet ause a torque about the frst ot axs gve by t t N, r 0, I ω r 0, f, m v { I + m ( + osθ )} θ (7..5) where N, ad f l, are evaluated usg eq. (5) for θ θ 0 ad θ 0. hs agrees wth the seod term eq. (-a). hus, the seod term aouts for the terato betwee the two ots. he thrd terms eq. () are θ proportoal θ θ to the square of the ot velotes. We osder the stat whe 0 ad 0, as show Fgure 7..4-(a). I ths ase, a etrfugal fore ats upo the seod lk. Let f et be the etrfugal fore. Its magtude s gve by f et m Lθ (7..6) where L s the dstae betwee the etrod C ad the frst ot O. he etrfugal fore ats the dreto of posto vetor r O,C. hs etrfugal fore auses a momet et about the seod ot. Usg eq. (6), the momet et s omputed as et r f (7..7), et m θ hs agrees wth the thrd term h θ eq. (-b). hus we olude that the thrd term s aused by the etrfugal effet o the seod ot due to the moto of the frst ot. Smlarly, rotatg the seod ot at a ostat veloty auses a torque of hθ due to the etrfugal effet upo the frst ot. f et f Cor O r 0, θ C (a) r, O y O θ θ x y b O b (b) θ θ + x θ b Fgure 7..4 Cetrfugal (a) ad Corols (b) effets Departmet of Mehaal Egeerg Massahusetts Isttute of ehology

8 Itroduto to Robots,. arry Asada 8 Fally we dsuss the fourth term of eq. (-a), whh s proportoal to the produt of the ot velotes. Cosder the stat whe the two ots rotate at velotes θ ad θ at the same tme. Let O b -x b y b be the oordate frame attahed to the tp of lk, as show Fgure 7..4-(b). Note that the frame O b -x b y b s parallel to the base oordate frame at the stat show. owever, the frame rotates at the agular veloty θ together wth lk. he mass etrod of lk moves at a veloty of θ relatve to lk,.e. the movg oordate frame O b -x b y b. Whe a mass partle m moves at a veloty of v b relatve to a movg oordate frame rotatg at a agular veloty ω, the mass partle has the so-alled Corols fore gve by m( ω v ). Let f Cor be the fore atg o lk due to the Corols effet. he Corols fore b s gve by m θ θ os( θ + θ) f Cor (7..8) m θ θ s( θ + θ) hs Corols fore auses a momet C or about the frst ot, whh s gve by Cor r f θ (7..9) 0, Cor m θθ s he rght-had sde of the above equato agrees wth the fourth term eq. (-a). Se the Corols fore gve by eq. (8) ats parallel wth lk, the fore does ot reate a momet about the seod ot ths partular ase. hus, the dyam equatos of a robot arm are haraterzed by a ofguratodepedet erta, gravty torques, ad terato torques aused by the aeleratos of the other ots ad the exstee of etrfugal ad Corols effets. 7.. Lagraga Formulato of Robot Dyams 7... Lagraga Dyams I the Newto-Euler formulato, the equatos of moto are derved from Newto's Seod Law, whh relates fore ad mometum, as well as torque ad agular mometum. he resultg equatos volve ostrat fores, whh must be elmated order to obta losedform dyam equatos. I the Newto-Euler formulato, the equatos are ot expressed terms of depedet varables, ad do ot lude put ot torques expltly. Arthmet operatos are eeded to derve the losed-form dyam equatos. hs represets a omplex proedure that requres physal tuto, as dsussed the prevous seto. A alteratve to the Newto-Euler formulato of mapulator dyams s the Lagraga formulato, whh desrbes the behavor of a dyam system terms of work ad eergy stored the system rather tha of fores ad momets of the dvdual members volved. he ostrat fores volved the system are automatally elmated the formulato of Lagraga dyam equatos. he losed-form dyam equatos a be derved systematally ay oordate system. Let q,,q be geeralzed oordates that ompletely loate a dyam system. Let ad U be the total ket eergy ad potetal eergy stored the dyam system. We defe the Lagraga L by Departmet of Mehaal Egeerg Massahusetts Isttute of ehology

9 Itroduto to Robots,. arry Asada 9 L q, q ) ( q, q ) U ( q ) (7..) ( Note that the potetal eergy s a futo of geeralzed oordates q ad that the ket eergy s that of geeralzed velotes q as well as geeralzed oordates q. Usg the Lagraga, equatos of moto of the dyam system are gve by d dt L q L q Q,,, (7..) where Q s the geeralzed fore orrespodg to the geeralzed oordate q. Cosderg the vrtual work doe by o-oservatve fores a detfy the geeralzed fores atg o the system. 7.. Plaar Robot Dyams Before dsussg geeral robot dyams three-dmesoal spae, we osder the dof plaar robot, for whh we have derved the equatos of moto based o Newto-Euler Formulato. Fgure 7.. shows the same robot mehasm wth a few ew varables eeded for the Lagraga Formulato. y V V ω ω I θ,, m O θ, I, m x Fgure 7.. wo dof robot he total ket eergy stored the two lks movg at lear veloty veloty ω at the etrods, as show the fgure, s gve by v ad agular ( m v + I ω ) (7..3) where v represets the magtude of the veloty vetor. Note that the lear velotes ad the agular velotes are ot depedet varables, but are futos of ot agles ad ot Departmet of Mehaal Egeerg Massahusetts Isttute of ehology

10 Itroduto to Robots,. arry Asada 0 agular velotes,.e. the geeralzed oordates ad the geeralzed velotes that loate the dyam state of the system uquely. We eed to rewrte the above ket eergy so that t s wth respet to θ ad θ. he agular velotes are gve by ω + (7..4) θ, ω θ θ he lear veloty of the frst lk s smply θ v (7..5) owever, the etrodal lear veloty of the seod lk v eeds more omputato. reatg the etrod C as a edpot ad applyg the formula for omputg the edpot veloty yeld the etrodal veloty. Let be the x Jaoba matrx relatg the etrodal veloty vetor to ot velotes. he, v J J q q J J q (7..6) where q ( θ θ ). Substtutg eqs.(4~6) to eq.(3) yelds θ + θ θ + θ ( θ ) θ θ θ (7..7) where oeffets are the same as the oes eq.(7..). ( θ m I m( + osθ) + I ( θ) m + I + m + + osθ ) + I ( ) (7..-a) + (7..-b) (7..-) Note that oeffets ad are futos of θ. he potetal eergy stored the two lks s gve by U mg sθ + mg{ sθ + s( θ + θ)} (7..8) Now we are ready to obta Lagrage s equatos of moto by dfferetatg the above ket eergy ad potetal eergy. For the frst ot, L U [ m g osθ + m g{ os( θ + θ) + osθ}] G q q (7..9) L θ + θ q d L θ θ θ + + θ + θ dt q θ θ (7..0) Departmet of Mehaal Egeerg Massahusetts Isttute of ehology

11 Itroduto to Robots,. arry Asada Substtutg the above two equatos to eq.() yelds the same result as eq.(7..-a). he equato of moto for the seod ot a be obtaed the same maer, whh s detal to eq.(7..-b). hus, the same equatos of moto have bee obtaed based o Lagraga Formulato. Note that the Lagraga Formulato s smpler ad more systemat tha the Newto-Euler Formulato. o formulate ket eergy, velotes must be obtaed, but aeleratos are ot eeded. Remember that the aelerato omputato was omplex the Newto-Euler Formulato, as dsussed the prevous seto. hs aelerato omputato s automatally dealt wth the omputato of Lagrage s equatos of moto. he dfferee betwee the two methods s more sgfat whe the degrees of freedom rease, se may workless ostrat fores ad momets are preset ad the aelerato omputato beomes more omplex Newto-Euler Formulato Ierta Matrx I ths seto we wll exted Lagrage s equatos of moto obtaed for the two d.o.f. plaar robot to the oes for a geeral d.o.f. robot. Cetral to Lagraga formulato s the dervato of the total ket eergy stored all of the rgd bodes volved a robot system. Examg ket eergy wll provde useful physal sghts of robot dyam. Suh physal sghts based o Lagraga formulato wll supplemet the oes we have obtaed based o Newto-Euler formulato. As see eq.(3) for the plaar robot, the ket eergy stored a dvdual arm lk ossts of two terms; oe s ket eergy attrbuted to the traslatoal moto of mass m ad the other s due to rotato about the etrod. For a geeral three-dmesoal rgd, ths a be wrtte as where ω mv v + ω Iω,,, (7..) ad I are, respetvely, the 3x agular veloty vetor ad the 3x3 erta matrx of the -th lk vewed from the base oordate frame,.e. ertal referee. he total ket eergy stored the whole robot lkage s the gve by (7..) se eergy s addtve. he expresso for the ket eergy s wrtte terms of the veloty ad agular veloty of eah lk member, whh are ot depedet varables, as metoed the prevous seto. Let us ow rewrte the above equatos terms of a depedet ad omplete set of geeralzed oordates, amely ot oordates q [q,..,q ]. For the plaar robot example, we used the Jaoba matrx relatg the etrod veloty to ot velotes for rewrtg the expresso. We a use the same method for rewrtg the etrodal veloty ad agular veloty for three-dmesoal mult- systems. v ω L J q A J q (7..3) Departmet of Mehaal Egeerg Massahusetts Isttute of ehology

12 Itroduto to Robots,. arry Asada where J L ad J A are, respetvely, the 3 x Jaoba matres relatg the etrod lear veloty ad the agular veloty of the -th lk to ot velotes. Note that the lear ad agular velotes of the -th lk are depedet oly o the frst ot velotes, ad hee the last - olums of these Jaoba matres are zero vetors. Substtutg eq.(3) to eqs.() ad () yelds L q J L A J q q J A ( m IJ q q q + ) where s a x matrx gve by L L ( m J J + J A I J A ) (7..4) (7..5) he matrx orporates all the mass propertes of the whole robot mehasm, as refleted to the ot axes, ad s referred to as the Mult-Body Ierta Matrx. Note the dfferee betwee the mult- erta matrx ad the 3 x 3 erta matres of the dvdual lks. he former s a aggregate erta matrx ludg the latter as ompoets. he mult- erta matrx, however, has propertes smlar to those of dvdual erta matres. As show eq. (5), the mult- erta matrx s a symmetr matrx, as s the dvdual erta matrx defed by eq. (7..). he quadrat form assoated wth the mult- erta matrx represets ket eergy, so does the dvdual erta matrx. Ket eergy s always strtly postve uless the system s at rest. he mult- erta matrx of eq. (5) s postve defte, as are the dvdual erta matres. Note, however, that the mult- erta matrx volves Jaoba matres, whh vary wth lkage ofgurato. herefore the mult- erta matrx s ofgurato-depedet ad represets the stataeous omposte mass propertes of the whole lkage at the urret lkage ofgurato. o mafest the ofgurato-depedet ature of the mult- erta matrx, we wrte t as (q), a futo of ot oordates q. Usg the ompoets of the mult- erta matrx { }, we a wrte the total ket eergy salar quadrat form: q q (7..6) Most of the terms volved Lagrage s equatos of moto a be obtaed dretly by dfferetatg the above ket eergy. From the frst term eq.(), d dt q d dt d ( q ) q + q dt (7..7) he frst term of the last expresso, q, omprses the dagoal term as well as off- q dagoal terms q, represetg the dyam teratos amog the multple ots due to aeleratos, as dsussed the prevous seto. It s mportat to ote that a par of ots, ad, have the same oeffet of the dyam terato,, se the mult- erta matrx s symmetr. I vetor-matrx form these terms a be wrtte olletvely as Departmet of Mehaal Egeerg Massahusetts Isttute of ehology

13 Itroduto to Robots,. arry Asada 3 q > q q (7..8) > q q It s lear that the teratve ertal torque aused by the -th ot aelerato upo the - th ot has the same oeffet as that of alled Maxwell s Reproty Relato. q q aused by ot upo ot. hs property s he seod term of eq.(7) s o-zero geeral, se the mult- erta matrx s ofgurato-depedet, beg a futo of ot oordates. Applyg the ha rule, d dt dq k k qk dt k qk q k (7..9) he seod term eq.(), Lagrage s equato of moto, also yelds the partal dervatves of. From eq.(6), q q k q q q q k (7..0) q k k k k Substtutg eq.(9) to the seod term of eq.(7) ad ombg the resultat term wth eq.(0), let us wrte these olear terms as h k C k q q (7..) k where oeffets C k s gve by C k q k q k (7..) hs oeffet C k s alled Chrstoffel s hree-idex Symbol. Note that eq.() s olear, havg produts of ot velotes. Eq.() a be dvded to the terms proportoal to square ot velotes,.e. k, ad the oes for k : the former represets etrfugal torques ad the latter Corols torques. h Cq + Ckq q k (Cetrfugal) + (Corols) (7..3) k Departmet of Mehaal Egeerg Massahusetts Isttute of ehology

14 Itroduto to Robots,. arry Asada 4 hese etrfugal ad Corols terms are preset oly whe the mult- erta matrx s ofgurato depedet. I other words, the etrfugal ad Corols torques are terpreted as olear effets due to the ofgurato-depedet ature of the mult- erta matrx Lagraga formulato Geeralzed Fores Fores atg o a system of rgd bodes a be represeted as oservatve fores ad o-oservatve fores. he former s gve by partal dervatves of potetal eergy U Lagrage s equatos of moto. If gravty s the oly oservatve fore, the total potetal eergy stored lks s gve by U m g (7..4) r 0, where r 0, s the posto vetor of the etrod C that s depedet o ot oordates. Substtutg ths potetal eergy to Lagrage s equatos of moto yelds the followg gravty torque see by the -th ot: G U q 0, m q r L g m g J (7..5), where L J, s the -th olum vetor of the 3 x Jaoba matrx relatg the lear etrod veloty of the -th lk to ot velotes. No-oservatve fores atg o the robot mehasm are represeted by geeralzed fores Q Lagraga formulato. Let δ Work be vrtual work doe by all the o-oservatve fores atg o the system. Geeralzed fores Q assoated wth geeralzed oordates q, e.g. ot oordates, are defed by δ Work Q δq (7..6) If the vrtual work s gve by the er produt of ot torques ad vrtual ot dsplaemets, δq ++ δq, the ot torque tself s the geeralzed fore orrespodg to the ot oordate. owever, geeralzed fores are ofte dfferet from ot torques. Care must be take for fdg orret geeralzed fores. Let us work out the followg example. Example 7. Cosder the same d.o.f. plaar robot as Example 7.. Istead of usg ot agles θ ad θ as geeralzed oordates, let us use the absolute agles, φ ad φ, measured from the postve x-axs. See the fgure below. Chagg geeralzed oordates etals hages to geeralzed fores. Let us fd the geeralzed fores for the ew oordates. Departmet of Mehaal Egeerg Massahusetts Isttute of ehology

15 Itroduto to Robots,. arry Asada 5 y θ, φ δφ O φ θ, x δφ Fgure 7.. Absolute ot agles φ ad φ ad dsoted lks As show the fgure, ot torque ats o the seod lk, whose vrtual dsplaemet s δφ, whle ot torque ad the reato torque at o the frst lk for vrtual dsplaemetδφ. herefore the vrtual work s δ Work ( δφ + δφ (7..7) ) Comparg ths equato wth eq.(6) where geeralzed oordates are φ q, φ q, we a olude that the geeralzed fores are: Q (7..8), Q he two sets of geeralzed oordates θ ad θ vs. φ ad φ are related as φ + (7..9) θ, φ θ θ Substtutg eq.(9) to eq.(7) yelds δ Work ( δθ + δθ (7..30) ) δθ + δ ( θ + θ) hs ofrms that the geeralzed fores assoated wth the orgal geeralzed oordates,.e. ot oordates, are ad. No-oservatve fores atg o a robot mehasm lude ot oly these ot torques but also ay other exteral fore F ext. If a exteral fore ats at the edpot, the geeralzed fores Q(Q,, Q ) assoated wth geeralzed oordates q are, vetor form, gve by ext δwork δq + F δp ( + J F ) δq Q δq ext Q + J F (7..3) ext Departmet of Mehaal Egeerg Massahusetts Isttute of ehology

16 Itroduto to Robots,. arry Asada 6 Whe the exteral fore ats at posto r, the above Jaoba must be replaed by dr J r (7..3) dq Note that, se geeralzed oordates q a uquely loate the system, the posto vetor r must be wrtte as a futo of q aloe. Departmet of Mehaal Egeerg Massahusetts Isttute of ehology