Variable-poled Tracking Control of a Two-wheeled Mobile Robot Using Differential Flatness

Transcription

1 Jornal of Roboics Neworing Arificial Life Vol No (Jne 04) -6 Variable-pole racing Conrol of a wo-wheele Mobile Robo Using Differenial Flaness Liming Chen Yingmin Jia he Sevenh Research Division Beihang Universi (BUAA) Beijing 009 China s: clmes@6com mjia@baaecn Absrac his paper invesigaes he racing conrol of a wo-wheele mobile robo in boh inemaic namic moels Differenial flaness PD-specral heor are se for conroller esign Base on ifferenial flaness he original ssem is ransforme via a sae prolongaion a sae ransformaion ino a normal form o appl feebac lineariaion hen sing PD-specral heor variable poles of racing error namics are assigne o realie he sabili of rajecor racing Simlaion resls are presene o emonsrae he effeciveness of he propose meho Kewors: mobile robo nonlinear conrol rajecor racing ifferenial flaness Inrocion Wheele mobile robos have been proven o be one of he mos acive areas of research since he are mch sefl in varieies of applicaions ranging from insrial seings o miliar ssems o home roboics ec One of he main lines of research is he rajecor racing problem which is concerne wih riving a mobile robo as close as possible o a esire eplici rajecor An he racing conrol approaches incle bacsepping sliing moe conrol lineariaion 3 neral newor-base conrol 4 f conrol 5 ifferenial flaness-base conrol 6-8 which is also se in his paper Differenial flaness has been inroce b Fliess e al 9 I is a ver sefl ool for nonlinear conroller esign Roghl speaing a ssem is iffereniall fla if here eiss variables of he same imension as inps calle fla op sch ha saes inps can be algebraicall epresse in erms of fla op is erivaives Moreover his mapping is inverible he ssem is eqivalen o a linear one If he esire rajecor of fla op is given hen b performing a feebac lineariaion esigning a ime invarian conroller for he linearie ssem aron he esire rajecor sable racing error namics are achieve Kinemaic moel namic moel of a wo-wheele mobile robo have been proven o be iffereniall fla b choosing he cener posiion of he wheel ale of he robo as he fla op In Refs 7 8 conrollers are esigne in his scheme However in hese conrollers he parameers of he racing error namics are consan which means ha he convergence spee of racing error is fie his brings limiaion for his meho In his paper he se of PD-specral heor of linear ime-varing (LV) ssems is propose for racing conroller esign he PD-specral heor has been evelope b Zh 0 which can be seen as a naral eension of he convenional eigenvale-eigenvecor heor for linear ime-invarian (LI) ssems Afer he sae ransformaion base on ifferenial flaness he Pblishe b Alanis Press Coprigh: he ahors

2 L Chen Y Jia applicaion of PD-specral heor becomes mch easier B assigning ime-varing poles more generalie racing error namics can be obaine An he poles can be change a an ime as we wan he res of his paper is organie as follows: In secion he inemaic moel namic moel of a wo-wheele mobile robo are erive anale wih ifferenial flaness Secion 3 presens he esign of conrol law in boh moels Simlaion resls are shown in secion 4 Finall concling remars are given in secion 5 Moels of wo-wheele Mobile Robo wih Differenial Flaness Kinemaic moel Fig he configraion of a wo-wheele mobile robo wih no slip Fig shows ha he robo's configraion in Caresian coorinaes is given b q [ ] where ( ) is he coorinaes of he cener of he wheel ale is he heaing angle of he robo Wih he assmpion of noslip coniion a he wheel conac poins he veloci of he wheel ceners are parallel o he heaing orienaion hen he inemaic moel can be wrien as cos 0 v q S( q) v sin 0 () 0 where v is he heaing spee rning spee o oline how he inemaic moel of he wowheele mobile robo is iffereniall fla we nee o selec siable fla ops epress all sae variables inps in erms of he fla ops heir erivaives he imensions of fla ops shol be eqal o ha of he inps Here we can choose he cener posiion of he wheel ale ( ) as he fla ops ( ) Wih he chosen fla ops he saes can be epresse as arcan he wo inps can be wrien as v I can be noice ha he epression of sae conains he firs orer erivaives of boh he fla ops Accoring o ifferenial flaness heor he ssem () nee o be eene o 4 imensions Appl one prolongaion of v b consiering i as an aiional sae hen he eene ssem is given b vcos v () vsin where are new inps of he eene ssem he new inps can be calclae as (3) he eene ssem () can be ransforme via a sae ransformaion [ ] v [ ] ino a normal form cos(arcan ) sin(arcan ) (4) sin(arcan ) cos(arcan ) Since he epressions of new inps in (3) are erive from he ssem eqaions () he sae ransformaion ensres ha (3) saisfies he eqaions (4) B replacing in (3) wih respecivel: (5) where are parameers o be esigne sbsiing (5) ino he ssem eqaions (4) one ges (6) Acall if onl he saes of he original ssem () [ v ] can be esimae sing he relaionship of sae ransformaion he inps (5) can be rewrien as cos sin ( cos sin ) / v (7) Pblishe b Alanis Press Coprigh: he ahors 3

3 Wheele Robo s racing Conrol Dnamic moel B ignoring he mass of he wheels he eqaions of moion can be erive sing Eler-Lagrange meho as M( qq ) C( qqq ) E( q) C ( q) (8) where m 0 m sin r M 0 m m cos l m sin m cos m I (9) 0 0 m cos cos cos C 0 0 m sin E sin sin r b b Here m is he robo mass I he momen of ineria of he robo abo is cener of mass he isance beween he cener of mass he cener of he wheel ale r he wheel rais b half isance beween he wo wheels l r he moor orqes on he wheels he consrain force B iffereniaing () one ges q S vsv hen b sbsiing q ino (8) pre-mlipling b S sing he proper S C 0 one can have v ( S MS) S ( MS CS) v( S MS) S E (0) his can finall be calclae as v A( v) Bτ where mr mr A( v ) m v B b b m I r( m I) r( m I) Afer inrocing an inp ransformaion [ ] A( v) Bτ he namic moel can be wrien as q S( q) v v () Here we can also choose he cener posiion of he wheel ale ( ) as he fla ops ( ) hen all sae variables inps can be epresse in erms of he fla ops heir erivaives he epression of sae conains he secon orer erivaives of boh he fla ops which means ha he ssem () nee o be eene o 6 imensions so ha i can be ransforme ino a normal form via a sae ransformaion On appling one prolongaion of he eene ssem is given b vcos v () vsin where are new inps of he eene ssem he sae ransformaion can be wrien as [ ] v [ ] Now we calclae he hir orer erivaives of he fla ops irecl sing () as C D (3) where cos sin sin v cos v C D sin cos cos v sin v (4) B replacing wih his iels 3 Design of Conrol Law ( C) D 3 Inrocion of PD-specral heor 0 respecivel (5) Consier SISO LV ssems represene b he n horer scalar LV namical ssems of he form: ( n) ( n) () () () 0 (6) n I can be convenienl represene as D {} 0sing he scalar polnomial ifferenial operaor (SPDO) n n D n () () () (7) where / is he erivaive operaor he facoriaion of SPDO can be represene as D ( n ( )) ( ( ))( ( )) (8) where a collecion { ( )} n is calle a series D- specrm(sd-specrm) for D an n-parameer famil { ( ) ( )} n is calle a parallel D- specrm(pd-specrm) for D where () are n pariclar solions for () saisfing some nonlinear inepenen consrains Acall { ( ) ep( ( ) )} n consies a fnamenal se of solions o D {} 0 he solion o D {} 0 is niforml asmpoicall sable if (i) all PD-eigenvales are of polnomial orer or slower ha is an ineger m 0 eiss sch ha () lim 0 ; m Pblishe b Alanis Press Coprigh: he ahors 4

4 L Chen Y Jia (ii) he eene means of real pars of PD-eigenvales 0 ha is em(re ( )) lim Re ( ) are all 0 negaive 3 Conrol law of inemaic moel If he esire rajecor of fla ops ( ) are given b ( ( ) ( )) racing error can be inroce as e [ e e e e] [ ] Using (6) parameers can be esigne as () e () e (9) () e () e which iels e () e () e 0 (0) e () e () e 0 PD-specral heor is se o esign hese ime-varing conrol gains o ensre he error namics o be niforml asmpoicall sable Firs appropriae ime-varing PD-eigenvales which saisf (i) (ii) are esigne as hen corresponing SD-eigenvales can be calclae as Using (8) we obain () () () () can be esigne in he same wa 33 Conrol law of namic moel Similarl if he esire rajecor of fla ops are given b ( ( ) ( )) racing error e [ e e e e e e] [ ] hen parameers can be esigne as () e () e3() e () () e () e3() e B esigning appropriae PD-eigenvales 3 SD-eigenvales can be calclae as V V 3 V 3 3 where V V V V 3 V3 e hen conrol gains are obaine b () 3 3 () () () () () 3 can be esigne in he same wa 4 Simlaion Resl he esire rajecor is given b () (m) () (m) over [0 0](s) In he inemaic moel iniial saes are se as [ (0) (0) v(0) (0)] [ ] I shol be poine o ha he aiional sae v is in he conroller PD-eigenvales are selece as ( 05 ) ( 05 ) i ( 05 ) ( 05 ) i Fig (a) shows ha he mobile robo graall converges o Fig Desire rajecor racing rajecor: (a)kinemaic moel (b)dnamic moel Fig3 Kinemaic moel: (a)racing error e (b)racing error e (c)heaing spee v ()rning spee Pblishe b Alanis Press Coprigh: he ahors 5

5 Acnowlegemens Wheele Robo s racing Conrol his wor was sppore b he Naional Basic Research Program of China (973 Program: 0CB800 0CB80) he NSFC ( ) Fig4 Dnamic moel: (a)racing error e (b)racing error e (c)ransforme inp ()ransforme inp he esire rajecor finall moves along i Fig 3 emonsraes he racing errors e e which converge o ero he conrol inps v In he namic moel [ (0) (0) v(0) (0) (0) (0)] [ ] Conroller is esigne wih PD-eigenvales (05 ) ( 05 i ) (05 ) ( 05 ) i 3 05 Fig (b) shows he racing rajecor Fig 4 epics he racing errors he ransforme inps 5 Conclsion In his paper we have presene a novel meho for rajecor racing conrol of a wo-wheele mobile robo in is inemaic moel namic moel Base on ifferenial flaness he ssems can be ransforme ino normal forms o se feebac lineariaion he applicaion of PD-specral heor ensres he sabili of he racing error namics esablishes ajsable poles o change as we wan References R Fierro F V Lewis Conrol of a non-holonomic mobile robo: bacsepping inemaics ino namics J Robo Ss 4(3) (997) J M Yang J H Kim Sliing moe conrol for rajecor racing of nonholonomic wheele mobile robos IEEE rans Robo Aom 5(3) (999) D H Kim J H Oh racing conrol of a wowheele mobile robo sing inp-op lineariaion Conr Eng Prac 7(3) (999) V Boqee R Garcia R Barea M Mao Neral conrol of he movemens of a wheel-chair J Inell Robo Ss 5(3) (999) Das I N Kar Design implemenaion of an aapive f logic-base conroller for wheele mobile robos IEEE rans Conr Ss echnol 4(3) (006) Chn-Hs Ko Snil K Agrawal Wal-assis robo: a novel approach o gain selecion of a braing conroller sing ifferenial flaness American Conrol Conference (Marrio Waerfron Balimore 00) pp Chin Pei ang Differenial flaness-base inemaic namic conrol of a iffereniall riven wheele mobile robo in Proc IEEE In Conf Roboics Biomimeics (Gilin China 009) pp Ji-Chl R Snil K Agrawal Differenial flanessbase robs conrol of a wo-wheele mobile robo in he presence of slip in Proc DSCC ASME Dnamic Ssems Conrol Conference (Ann Arbor Michigan USA 008) pp -7 9 M Fliess J Lévine Ph Marin P Rochon Flaness efec of non-linear ssems: inrocion heor eamples Inernaional Jornal of Conrol 6(6) (995) J Zh A nifie specral heor for linear ime-varing ssems--progress challenges in Proc IEEE Conf Decision Conrol (New Orleans LA 995) pp Pblishe b Alanis Press Coprigh: he ahors 6