Singularity-Free Dynamic Modeling Including Wheel Dynamics for an Omni-Directional Mobile Robot with Three Caster Wheels

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1 86 Interntonl Je Heon Jornl Chng, of Byng-J Control, Atomton, Y, Whee Kk nd Km, Systems, nd Seog-Yong vol. 6, no., Hn , Febrry 008 Snglrty-Free Dynmc Modelng Incldng Wheel Dynmcs for n Omn-Drectonl Moble Robot wth hree Cster Wheels Je Heon Chng, Byng-J Y, Whee Kk Km, nd Seog-Yong Hn Abstrct: Most of the revosly emloyed dynmc modelng roches, ncldng Ntrl Orthogonl Comlement Algorthm, hve lmttons on ther lcton to the moble robot, secfclly t snglr confgrtons. Also, n ther dynmc modelng of moble robots, wheel dynmcs s slly gnored ssmng tht ts dynmc effect s neglgbly smll. As remedy for ths, snglrty-free oertonl sce dynmc modelng roch bsed on Lgrnge s form of the D Alembert rncle s roosed, nd the snglrty-free chrcterstc of the roosed dynmc modelng s dscssed n the rocess of nlytcl dervton of the roosed dynmc model. hen n ccrte dynmc model tkng nto ccont the wheel dynmcs of the omndrectonl moble robot s derved, nd throgh smlton t s mnfested tht the effect of the wheel dynmcs on the whole dynmc model of the moble robot my not be neglgble, bt rther n some cses t s sgnfcntly lrge, ossbly ffectng the oertonl erformnces of dynmc model-bsed control lgorthms. Lstly, the mortnce of ts ccrte dynmc model s frther llstrted throgh mlse nlyss nd ts smlton for the moble robot. Keywords: Dynmcs, mct, knemtcs, omn-drectonl moble robot.. INRODUCION Most moble robots hve closed knemtc chns lke rllel robots. One rt of ech wheel of the moble robot s nterfced wth the grond, nd the other rt s connected to the body of the moble robot. hs, ech wheel cn be treted s serl sb-chn n rllel mechnsm [,3]. Prtclrly, for moble robot to hve omn-drectonl chrcterstcs on the lnr srfce, ech wheel ttched to the moble robot mst hve three degrees of freedom. Ether the cster wheel or the Swedsh wheel cn be modeled knemtclly s three-degrees-of-freedom serl chn. However, t s known tht both the Swedsh wheel nd most other tyes of omndrectonl wheels re very senstve to rod Mnscrt receved November 3, 006; revsed My 8, 007; cceted October, 007. Recommended by Edtorl Bord member Sooyong Lee nder the drecton of Edtor Je Bok Song. hs work ws sorted by GRRC roject of Gyeong-g rovnce goverment, Reblc of Kore ( ). Je Heon Chng nd Byng-J Y re wth the School of Electrcl Engneerng nd Comter Scence, Hnyng Unversty, 7 S -dong, Ansn, Gyeongg 46-79, Kore (e-mls: mdng@ems.com, bj@hnyng.c.kr). Whee Kk Km s wth the Dertment of Control nd Instrmentton Engneerng, Kore Unversty, Kore (e-ml: wheekk@kore.c.kr). Seog-Yong Hn s wth the Dertment of Mechncl Engneerng, Hnyng Unversty, Kore (e-ml: syhn@ hnyng.c.kr). Corresondng thor. condtons nd ths ther oertonl erformnces re more or less lmted, comred to conventonl wheels. In contrst, the ctve cster wheel s reltvely esy to bld nd nsenstve to rod condtons, even beng ble to overcome smll bms encontered n neven floors. Cmon, et l. [] ddressed the fct tht the omndrectonl moble robot wth three cster wheels mst se more thn for motors to vod snglrty, nd tht s n dmssble confgrton, two motors on two of the three wheels shold be sed. However, ther work does not rovde ny closed-form dynmcs for omn-drectonl moble robots. Sh nd Angeles [4] roosed n orthogonl comlementbsed, closed-form dynmc model for the DOF dfferentlly drven moble robot. Y, et l. [5] extended ths methodology to 3 DOF omndrectonl moble robot wth three cster wheels. However, ths roch sffers from lgorthmc snglrty, deendng on the choce of mnmm coordntes for whch the system dynmcs s referenced or exressed. o coe wth ths roblem, the set of mnmm coordntes shold be chnged from one to nother. However, ths s lso nconvenent nd ths snglrty-free dynmc formlton s demnded. In generl robotc felds, dynmc model-bsed control s demnded to ensre more enhnced system erformnce of systems. U to now, the effect of the wheel dynmcs on the whole dynmcs of the moble system hs not been exmned ntensvely. Accrte

2 Snglrty-Free Dynmc Modelng Incldng Wheel Dynmcs for n Omn-Drectonl Moble Robot wth hree 87 dynmc model tkng nto ccont the wheel dynmcs of the omn-drectonl moble robot wold be benefcl. In lght of these fcts, ths er ntrodces snglrty-free, ccrte dynmc model ncldng the wheel dynmcs for the omn-drectonl moble robot hvng three ctve cster wheels. he snglrty-free chrcterstc of the roosed dynmc modelng methodology wll be shown n the rocess of nlytcl dervton of the roosed dynmc model. hrogh smlton, the dscrency of the ncomlete dynmc model s shown by comrson wth the exct dynmc model.. KINEMAIC MODELING Moblty s known s the nmber of mnmm nt rmeters reqred to secfy ll the loctons of the system reltve to nother. Grübler s forml descrbng moblty s gven by [3] J M = NL ( ) ( N F), () = where N, L, J, nd F denote the dmensons of the llowble moton sce snned by ll jonts, the nmber of lnks, the nmber of jonts, nd the moton degree of freedom of the th jont, resectvely. Consder moble robot wth three cster wheels shown n Fg. (). Assme tht every wheel mntns ont contct wth the grond nd tht the wheel does not sl n the horzontl drecton whle beng llowed to rotte bot the vertcl xs. hen, the moton of ech wheel mechnsm cn be modeled s two revolte jonts nd one rsmtc jont s shown n Fg. (b). he frst revolte jont descrbes the grond-wheel nterfce. It denotes the rottonl moton of the wheel nd offset steerng lnk bot the vertcl xs ssng throgh the grond contct ont. he rsmtc jont descrbes the trnsltonl moton of the center of the wheel. he second revolte jont reresents the rotton of the moble ltform reltve to the offset steerng lnk. Here, the x-xs of the moble ltform s gven s the reference lne. Moblty for ths mechnsm s esly obtned s 3 from (). Note tht holonomc constrnts hve the sme dfferentl form s non-holonomc constrnts. hs, moblty cn be obtned by exmnng the nstntneos moton of the moble robot (.e., velocty level) t the crrent confgrton nless t s n snglr confgrton. It s remrked tht the oston of the nterfce wth the grond (.e., the oston of bse of the moble robot) s contnosly movng s shown n Fg. (b), whle the ostons of sl rllel mnltors hvng holonomc constrnts re sttonry. Fg. dects the knemtc descrton of n omndrectonl moble robot. hs system conssts of three wheels, three offset steerng lnks, nd moble ltform. Frst, ssme tht the ott moton of the moble robot occrs n the lnr domn. XYZ reresents the globl reference frme, nd xyz denotes locl coordnte frme ttched to the moble ltform;, j, nd k re the nt vectors of the xyz coordnte frme. C denotes the orgn of the locl coordnte. We defne θ s the rottng ngle of the Wheel #3 l A 3 () Z X Y O 3 θ A ϕ A d θ j C ω ϕ 3 v c k ˆwheel b Wheel # ϕ O O Wheel # η (b) Fg.. Instntneos knemtc model of wheels. Fg.. Knemtc descrton of the omn-drectonl moble robot.

3 88 Je Heon Chng, Byng-J Y, Whee Kk Km, nd Seog-Yong Hn wheel nd ϕ s the steerng ngle between the steerng lnk nd the locl x-xs. η denotes the nglr dslcement of the wheel reltve to the X- xs of the reference frme. r nd d denote the rds of the wheel nd the length of the offset steerng lnk, resectvely. Defne the ott velocty of the moble robot s = v ω, c () where vc = vcx vcy nd ω reresents the trnsltonl velocty vector of the ltform center C nd the nglr velocty of the body frme bot the vertcl xs, resectvely. In the followng nlyss, we ssme tht the wheel contcts the grond t ont nd tht the rottonl moton of the wheel s llowed bot the xs ssng throgh the center of the wheel nd the contct ont. he lner veloctes t the center of ech of the three wheels cn be exressed s vo = θ snϕ+ cos ϕj rk, (3) v = θ snϕ + cos ϕ j rk, (4) nd ( ) ( ) o ( ) v = θ snϕ + cos ϕ j rk. (5) o he lner velocty t C of the moble robot cn be descrbed, for ech of the three wheels, resectvely, s vc = vo+ ηk OA + ωk AC = θ( snϕ+ cosϕj) rk + ηk ( dcosϕ+ dsnϕj+ hk) (6) l + ωk + j, vc = vo + ηk OA + ωk AC = θ( snϕ+ cosϕj) rk + ηk ( dcosϕ+ dsnϕj+ hk) (7) l + ωk + j, nd vc = vo3 + η3k OA ωk AC 3 = θ3 ϕ3 + ϕ3 + η ϕ + ϕ + + ω ( sn cos j) rk 3k ( dcos 3 dsn 3j hk) k (-b j), (8) where ω reresentng the nglr velocty of the moble ltform cn lso be descrbed, for ech of the three wheels, resectvely, s nd ω η ϕ (9) ω η ϕ (0),, ω = η + ϕ () 3 3. vc = vcx vcy n (6) throgh (8) nd ω n (9) throgh () cn be exressed n one mtrx form gven by v dsnϕ rcosϕ η cos sn, cx vcy = d ϕ+ l r ϕ l θ ω 0 ϕ () vcx dsnϕ rcosϕ η vcy = dcosϕ l rsnϕ l θ, ω 0 ϕ (3) nd v dsnϕ + b rcosϕ b η cos sn 0. cx vcy = d ϕ3 r ϕ 3 θ 3 ω 0 ϕ3 (4) Eqtons ()-(4) reresent the frst-order knemtcs of the moble robot, nd they re nstntneosly eqvlent to tht of tycl rllel robot tht s connected to fxed grond. he vrble ( η ) s defned s n bsolte nglr dslcement of the wheel bot the vertcl xs wth resect to the globl frme. In fct, ths vrble lys the role of nterfcng the wheel to the grond. Prctclly, ()- (4) reresent the forwrd velocty eqtons for the three wheels of the moble robot nd ech mtrx n the rght hnd sde s exressed s G φ for =,,3. Now, the ntermedte coordnte trnsfer method [6], whch s systemtc nd effectve n nlyss nd modelng of rllel robots, wll be emloyed to derve the forwrd knemtc relton. kng the nverse of ()-(4), we hve η θ = [ G φ ] (5) ϕ l rsnϕ rcosϕ rcosϕ rsnϕ l = dcosϕ dsnϕ dsnϕ+ dcos ϕ, dr l rsnϕ rcosϕ dr + rsnϕ rcosϕ

4 Snglrty-Free Dynmc Modelng Incldng Wheel Dynmcs for n Omn-Drectonl Moble Robot wth hree 89 η θ = [ G φ ] (6) ϕ l rsnϕ rcosϕ rcosϕ rsnϕ l = dcosϕ dsnϕ dsnϕ + dcosϕ, dr l r snϕ r cosϕ dr + r snϕ + r cosϕ nd η3 θ 3 = [ 3G φ ] (7) ϕ3 rsnϕ3 rcosϕ3 brsnϕ3 = dcosϕ3 dsnϕ3 bdcos ϕ 3. dr rsnϕ3 rcosϕ3 dr brsnϕ3 As shown n (5)-(7), there re nne jont vrbles. Among these nne jont vrbles only three jont vrbles cn be selected s ndeendent jont vrbles snce the moble robot hs moblty three. he remnng sx jont vrbles cn be exressed n terms of the ndeendent jont vrbles de to the knemtc constrnts of the moble robot. Note tht the desred ctve nt vector cn be selected ot of the remnng sx jont vrbles excldng the three ntrnsclly ssve vrbles ( η ), whch cnnot be ctted n jont sce. For bbrevton, the forwrd nd nverse mtrces re gven n ()-(4), nd (5)-(7) re exressed s G φ nd, G φ resectvely. he ccelerton reltonsh between the ott vector nd the nt jont vector of ech wheel cn be exressed s = G φ φ + φ [ φφ] φ, H (8) where H φφ s three dmensonl rry nd denotes the vrton of the Jcobn (.e., Hessn) wth resect to the jont ngle nd s defned s (Aendx A.) ( φ ). H φφ = G φ he Hessn H φφ descrbes tht t ffects the velocty of the oertonl sce set () on the ccelerton of the th chn jont vrble, nd t hs M N N dmensons. M nd N denote the nmber of the ott nd the nmber of the nt jonts, resectvely. he nverse reltonsh of (8) s gven by φ φ φ = G + H, (9) where ( φφ ) φ φ φ φ = H G G H G nd the oertor ' ', clled Generlzed Sclr Dot Prodct (Aendx A.), s emloyed to smlfy the fnl form of ths second-order nverse knemtc model. For detled descrton of the knemtc modelng method for rllel mechnsms sed n ths secton, refer to Freemn nd esr [6] nd Y nd Freemn [7]. As frther nottons, φ G nd φ H j;; denotng the jth row of the Jcobn nd the j th lne of the Hessn mtrx t the th chn, resectvely, wll be doted n the followng sectons. 3. DYNAMIC MODELING In generl, three mn methodologes, ctegorzed s the Recrsve Newton-Eler formlton [4,8], the Lgrnge-Eler method [9], nd Lgrnge s form of the Generlzed Prncle of D Alembert (oen-tree strctre method) [3,6,0,] hve been extensvely nvestgted for the dynmc modelng of robots. All these efforts hve contrbted to the rogress n dynmc modelng for robots. In ths secton, two dynmc modelng roches, the Ntrl Orthogonl Comlement Algorthm [4] nd the dynmc modelng roch emloyng Lgrnge s form of the D Alembert rncle [6], re dscssed to derve the dynmc model of the omndrectonl moble robot hvng three cster wheels. 3.. he ntrl orthogonl comlement lgorthm Sh nd Angeles [4] emloyed the concet of orthogonl comlement modelng lgorthm sng the mtrx of non-holonomc constrnts to develo closed-form dynmc model for dfferentl-drven DOF moble robot. Lter, Y, et l. [5] extended ths concet to derve the dynmc model of 3 DOF omn-drectonl moble robot wth three cster wheels. As mentoned before, ths moble robot ossesses three knemtc chns nd ech chn conssts of two bodes, wheel, nd steerng lnk s shown n Fg. 3. he ntrl orthogonl comlement lgorthm converts the dynmc model derved n terms of Lgrngn coordntes nto tht n terms of mnmm coordntes by embeddng non-holonomc constrnts. Drng the rocess of the dynmc modelng, t s j;

5 90 Je Heon Chng, Byng-J Y, Whee Kk Km, nd Seog-Yong Hn #3 #7 exressed s E C M t = WMt + w + w. (5) necessry to obtn the nternl knemtc reltonsh between the ndeendent (or mnmm) coordntes ( φ ) nd the deendent coordntes ( φ ). It s gven by J φ = J φ, (0) where J s lwys sqre mtrx. hen, the mng relton between φ nd φ s exressed s φ = J J φ, () # where the nvertblty of J s ssocted wth snglrty. he detled dervtons of (0)-() re shown n the Aendx A.3. he velocty (or twst t ) of ech ndvdl body cn be wrtten s lner trnsformton (, ) G G of the jont rtes of ech knemtc chn, nd the congregton of the twsts of ll rgd-bodes cn be denoted s generlzed twst, gven by t = G φ + G φ, () where t = t,, t 7. Sbstttng the relton gven n () nto (), the generlzed twst () t nd the chnge of twst cn be wrtten n terms of the ndeendent jont rtes ( φ ) s t = Gφ, (3) t = G φ + G φ, (4) #5 where G = G + GJ J. Now, the non-holonomc constrnt gven by G wll be embedded nto the system dynmcs exressed n terms of the Lgrngn coordntes. he nconstrned Newton-Eler eqton cn be # Fg. 3. Dsssembled moble robot. #4 #6 Sbstttng (3)-(4) nto the nconstrned Newton- Eler eqton, the nconstrned Newton-Eler eqton cn be exressed s E C ( ) φ MG φ = MG + WMG + w + w (6) n terms of the ndeendent coordntes, where M, W, E C w nd w denote mtrx of generlzed mss, mtrx of generlzed nglr velocty, generlzed externl wrench nd nonworkng constrnt wrench, resectvely. By lyng the constrned relton, the constrned dynmc eqton s exressed s [4] ( ), = + + (7) G MGφ G MG WMG φ τ E where G w = τ nd G w = 0 denote the jont torqe vector nd non-workng term, resectvely. Rerrngng (7) yelds the Eler-Lgrnge dynmc eqton n terms of the ndeendent coordntes: ( ) C( ) τ = I φ φ + φ, φ φ, (8) ( ) G MG, I φ = (9) C, G MG WMG, (30) ( φφ ) = ( + ) where φ = φ, φ denotes the Lgrngn coordntes of the system. ( ) C I φ nd (, ) C φφ denotes the nert mtrx term nd centrfgl nd Corols terms, resectvely. he shortcomng of ths lgorthm comes from the nverson of J n (). Y nd Km [] ddressed tht n lgorthmc snglrty wold hen n the omn-drectonl moble robot hvng three cster wheels, when the recrocl screws of the three ndeendent jonts meet t one common oston or t nfnty. It s observed from (7) tht ths knemtc snglrty rogtes to dynmcs snce the dynmc model reqres the nverson of J. When emloyng the ntrl orthogonl comlement lgorthm n the dynmc modelng of the moble robot, the set of ndeendent coordntes needs to be dted freqently by the other rorte sets tht re not n snglrty confgrton to ensre tht the dynmc model s vld. However, t doesn t seem ntrl or convenent. 3.. Lgrnge s Form of the D Alembert Prncle A dynmc modelng roch emloyng Lgrnge s form of the D Alembert rncle [6] s emloyed to resolve the snglrty roblem. he omn-drectonl moble robot s closed-chn

6 Snglrty-Free Dynmc Modelng Incldng Wheel Dynmcs for n Omn-Drectonl Moble Robot wth hree 9 system tht conssts of three knemtc chns. As frst ste to derve the dynmc model of the closedchn system, the system s converted nto severl oen-tree strctre by cttng rorte jonts of the closed chns. hen, by sng the Lgrnge dynmc formlton, the dynmc model for ech of the serl chns s evlted. Lstly, by sng the vrtl work rncle, the oen chn dynmcs cn be drectly ncororted nto closed chn dynmcs (for nstnce, nto jont-sce dynmcs or nto oertonl-sce dynmcs). Fg. 4 shows the oen-tree strctre of the moble robot, formed by cttng jonts of closed chns. Intlly, the Lgrnge formlton to obtn the dynmc model for ech of the serl sb-chns s descrbed. Lgrngn s defned s ( θθ) = ( θθ) ( θ) L, k,. (3) In (3), the otentl energy ( θ ) cn be gnored snce t s ssmed tht the moble robot moves only n the lnr domn. Reflectng ths fct, the Lgrnge dynmc eqton cn be exressed s d k k = τ, dt () θ θ where τ s the nⅹ jont torqe vector nd k reresents the totl knetc energy of ech oen chn, whch s the sm of the knetc energes of ll rgd bodes n the corresondng oen chn: n = k = k. (33) he knetc energy of the th rgd body, k, cn be exressed s C k = mv C v, C + ω I ω (34) where the frst term nd the second term reresent the # #5 #3 Fg. 4. An oen-chn model of the moble robot. #7 #4 # #6 trnsltonl nd rottonl knetc energy of the rgd body, resectvely. v C nd ω reresent bsolte velocty of the mss center of the th lnk nd the nglr velocty of the th lnk wth resect to locl frme fxed t the mss center, resectvely. m nd C I re the mss nd locl nert mtrx of the th lnk, resectvely. Note tht ech serl sb-chn reresentng one wheel of the moble robot conssts of two bodes: wheel nd n offset lnk s shown n Fg. 4. he lner velocty t the center of ech wheel (,, nd 3) nd the nglr velocty of the corresondng wheel cn be descrbed s ( cos - sn j) vc r ϕ ϕ θ = (35) nd ω = η k + θ ϕ + ϕ j (36) ( sn cos ) c resectvely. he lner velocty t the center of ech offset lnk (5, 6, nd 7) nd the nglr velocty of the corresondng lnk cn be descrbed s nd ( ) c O v = v + η k cosϕ + snϕ j (37) ω c = η k, (38) resectvely. he rmeter denotes the dstnce from the center of ech wheel to the center of mss of ech offset lnk. he knetc energes of the wheel nd the offset lnk of the th chn re exressed, resectvely, s k = m vc v c + ω c I ωc (39) nd k = m vc v c + ω c I ωc. (40) he totl knetc energy of the th chn s k k k. = + (4) Now, sbstttng (4) nto (3) yelds n oen chn dynmcs of the th chn s [6] φ = φφ φ + φ φφφ φ ext where I P, (4) m + Iz+ I 0 0 z 3 I φφ = 0 mr + mr 0, (43)

7 9 Je Heon Chng, Byng-J Y, Whee Kk Km, nd Seog-Yong Hn = F (44) ( OA rk) ext ext In (4), I φφ nd φφφ P denote the nert mtrx nd the nert ower rry referenced to the Lgrngn coordnte set, resectvely. In (43), I z nd I z denote the Z-drectonl mss moment of nert mtrx of I nd I, resectvely. Note tht every comonent of I φφ s constnt vle. hs, P φφφ reflectng the chnge of I φφ wth resect to the jont ngles becomes zero rry. F ext denotes the vector sm of the externl forces exerted on the wheel t the contct ont between the wheel of the th chn nd the grond. he nmberng of the sbsecton shold tke the bove form Oertonl-sce dynmcs he oertonl sce dynmc model cn be drectly obtned from the dynmcs n Lgrnge coordnte ( φ ) set by sng the rncle of vrtl work gven by 3 δ = φ δ φ, (45) where nd φ denote the oertonl force vector nd the effectve jont torqe vector of the th chn, resectvely. (45) cn be rrnged s 3 φ = φ + l = G (46) by emloyng knemtc relton φ G between the Lgrnge coordnte set ( φ ) nd the oertonl sce coordnte set () of the gven moble robot. l denotes the oertonl force vector led to the ltform of the moble robot. Now, by sbstttng the oen chn dynmcs, gven by (4), of the three chns nto (46), the oertonl sce dynmc model of the whole moble robot cn be derved s [6] = I + P F ext, (47) where the nert mtrx I nd the nert ower rry P wth resect to the oertonl sce coordnte set () cn be obtned resectvely s 3 φ φ = φφ + l = I G I G I, (48) P G G P G where φ φ φ = φφφ = φ φ G I Hφφ G + l P, (49) m 0 0 l l I = 0 ml 0, (50) 0 0 mlb φ φ = φφ I G I G, (5) nd 3 φ ext = ext = F G. (5) Ech of the frst terms n (48) nd (49) come from ech oen chn dynmcs, nd the second terms, li nd, l P denote the nert mtrx nd the nert ower rry of the moble ltform, resectvely. In (50), m l nd b reresent the mss nd the rds of the moble ltform, resectvely. As shown n (48), the oertonl-sce dynmc formlton lso ses the mtrx nverson φ G tht corresonds to the nverse Jcobn of ech chn gven n (5)-(7). Note tht these nverse reltons re not snglr nless the offset dstnce or the rds of the wheel s zero []. hs, the dynmc modelng roch bsed on Lgrnge s form of the D Alembert rncle s snglrty-free, whch s benefcl s comred to the orthogonl comlement bsed lgorthm [5]. Also note tht the derved dynmc model of the omn-drectonl moble robot ncorortes the wheel dynmcs of the moble robot nto the whole dynmcs of the system, whch hs been gnored freqently n revos works. 4. IMPULSE MODELING he oertonl-sce dynmc model ncldng the wheel dynmcs cn be emloyed to derve the mlse model when moble robot colldes wth n externl envronment. Most generlly, the mct s rtlly elstc n the rnge of 0< e <. When the coeffcent of resttton e s known, the velocty of colldng bodes cn be obtned mmedtely fter the mct. he comonent of the ncrement of reltve velocty long vector n tht s norml to the contct

8 Snglrty-Free Dynmc Modelng Incldng Wheel Dynmcs for n Omn-Drectonl Moble Robot wth hree 93 srfce s gven by [] ( v v ) = ( + e)( v v ) n n, (53) where v nd v re the bsolte veloctes of the colldng bodes mmedtely before mct nd v nd v re the velocty ncrements mmedtely fter mct. he externl mct modelng methodology for the serl tye system s ntrodced by Wlker [3]. When robot system ntercts wth n envronment, the dynmc model of generl robot systems s gven s = I + P F ext, (54) where F ext s the mlsve externl force t the contct ont. Integrton of the dynmc model gven n (54) over contctng tme ntervl gves t0 + t t0 t t0 t = + I t + + P 0 t0 t0 t0 + t extdt. t0 dt dt dt F (55) Snce the oston nd veloctes re ssmed fnte ll the tme drng mct, the ntegrl term nvolvng P becomes zero s t goes to zero, s does the term nvolvng ctton nt. hs, we obtn the followng smle exresson ( ( t t) ( t) ) ˆ + = I F ext, (56) where t0 + t t0 Fˆ ext = F extdt s defned s the externl mlse t the contct ont. hs, the velocty ncrement t the contct ont s = I ˆ F ext. (57) Assmng tht the robot mcts on fxed sold srfce, sbsttton of (57) nto (53) gves ext e ˆ I F n= ( + ) n, (58) where t shold be noted tht the bsolte velocty ( v ) wth resect to the ltform coordnte s gven s nd tht the velocty ncrement of the fxed v = v = Imlse srfce s lwys zero ( ) 0. lwys cts t the contct ont n the drecton of the srfce norml vector (n) nder the ssmton tht no frcton exts on the contctng srfce. hs, we hve Fˆext = ˆ ext. F n (59) Sbstttng (59) nto (58), we derve the mgntde of the externl mlse s follows: ( e) ˆ + n Fext =. - n I n 5. SIMULAION (60) In order to verfy the beneft of the dynmc model of the moble robot ncldng wheel dynmcs, severl smltons were crred ot. he rmeters emloyed n smltons re gven n bles nd. he moble robot trvels long the crclr th wth the rds of R, gven n Fg. 5(). It rottes n the conterclockwse drecton. he ntl nd fnl ostons re the sme, nd the ntl nd fnl veloctes re gven s zero. β denotng the ngle between the globl X-xs nd the locl nt vector s gven s ffth-order olynoml wth resect to tme sch s () t 0 t t 3t 4t 5 t. β = (6) For the gven ntl condtons, we hve = 0, = 0, = 0, = 0 π / 60, = π 4 5 = π 5 30 / 60, / 60. Fgs. 6() nd 6(b) denote the th of ech jont. hey re obtned by nmercl ntegrton of (5)- (7). Fgs. 6(c) nd 6(d) re obtned by the nverse knemtcs. In order to vldte the dynmcs of the whole moble robot dynmcs, we nclded the smlton reslt of the knetc energy of the moble robot s shown n Fg. 7. he knetc energy of the model wth ble. Knemtc rmeters. Lnk l r D [m] Lnk A b H [m] ble. Dynmc rmeters. Mss Wheel Lnk Pltform Kg Inert tensor kg m ( =,, 3) 4 Wheel dg [ 0.783,.565, 0.783] 0 6 Offset lnk dg [ 8.438,.04,.604] 0 3

9 94 Je Heon Chng, Byng-J Y, Whee Kk Km, nd Seog-Yong Hn Y j C β R = m X () rjectory. () Drvng ngle. (b) Vrton of β wth resect to tme. (b) Steerng ngle. Fg. 5. rjectory of the moble robot. wheel dynmcs s lrger thn tht of the model wthot wheel dynmcs. As mentoned before, most revos stdes on dynmcs of the moble robot n the ltertre gnore the wheel dynmcs. However, n some cses, the effects of wheel dynmcs on the whole dynmcs of the moble robot my not be gnorble, nd resltntly, cttor szng or control lgorthms bsed on the ncomlete lnt model my reslt n degrded erformnce of the system. hs, we wold lke to show the dscrency between the ncomlete dynmc model nd the snglrty-free dynmc model ncldng wheel dynmcs tht s derved n ths er. Intlly, the chrcterstc of the nert mtrx I obtned from the oertonl sce roch wll be comred for the two cses, one ncldng the wheel dynmcs nd the other not ncldng the wheel dynmcs. Secfclly, drng the crclr moton of the moble robot, the moble robot kees the orgnl confgrton wth resect to the body-fxed coordnte frme. hs, the dynmc model mntns the sme vle wth resect to the body-fxed coordnte frme. When the content of the nert mtrx of the dynmc model gnorng the wheel dynmcs gven by (c) Drvng velocty. (d) Pltform nglr velocty. Fg. 6. Moton hstory of moble robot.

10 Snglrty-Free Dynmc Modelng Incldng Wheel Dynmcs for n Omn-Drectonl Moble Robot wth hree 95 Fg. 7. Comrson of knetc energy. () Wth wheel dynmcs I = s comred to tht of the dynmc model ncldng the wheel dynmcs gven by I = t confgrton, t cn be observed tht there exst dfferences between those two models. Prtclrly, t cn be noted tht there exsts sbstntl dscrency eseclly n the y-drecton t ths secfc confgrton. Another observton cn be mde by comrson of the oertonl forces gven n (47) for the two cses. As shown n Fgs. 8 nd 9, there re sgnfcnt dfferences t F x, F y, nd M Z between the two models. hs fct tells s tht the ncomlete dynmc model neglectng wheel dynmcs my deterorte the reqred control erformnce of some dynmc model-bsed control lgorthms. In order to rovde gdelne to the oertor or desgner of moble robots whether the effect of the wheel dynmcs of consderton wll be of sgnfcnce or not, more detled nlyss shold be erformed. Fgs. 0- show the oertonl forces nd moment n x, y, nd θ z drectons when the moble robot follows crclr trjectory wth rds of 5m for erod of 60sec. It s ssmed tht the ltform mss s 5kg nd the whole mss of three wheels vres from 0kg to 3kg, reresentng mxmlly 60% of the ltform mss n smlton. From these lots t cn be observed tht the oertonl force nd moment n y nd θz drectons re ffected lrgely by the wheel dynmcs, bt the x- drectonl oertonl force s not ffected tht mch. (b) Wthot wheel dynmcs. Fg. 8. X nd Y drectonl nertl forces n the oertonl sce. () Wth wheel dynmcs. (b) Wthot wheel dynmcs. Fg. 9. Z-drectonl nertl moment n the oertonl sce.

11 96 Je Heon Chng, Byng-J Y, Whee Kk Km, nd Seog-Yong Hn Fg. 0. X-drectonl nertl force n the oertonl sce t the crclr trjectory. Fg. 3. X-drectonl nertl force n the oertonl sce t the elltc trjectory. Fg.. Y-drectonl nertl force n the oertonl sce t the crclr trjectory. Fg. 4. Y-drectonl nertl force n the oertonl sce t the elltc trjectory. Fg.. Z-drectonl nertl moment n the oertonl sce t the crclr trjectory. Fg. 5. Z-drectonl nertl moment n the oertonl sce t the elltc trjectory.

12 Snglrty-Free Dynmc Modelng Incldng Wheel Dynmcs for n Omn-Drectonl Moble Robot wth hree 97 () Ellse. () Ellse. (b) Moble confgrton. Fg. 6. Steerng ngle (0,0,0 ). (b) Moble confgrton. Fg. 8. Steerng ngle (40,0,0 ). () Ellse. () Ellse. (b) Moble confgrton. Fg. 7. Steerng ngle (45,45,45 ). (b) Moble confgrton. Fg. 9.Steerng ngle (60,35,35 ).

13 98 Je Heon Chng, Byng-J Y, Whee Kk Km, nd Seog-Yong Hn Fgs. 3-5 show the oertonl forces nd moments n x, y, nd θz drectons when the moble robot follows n elltc trjectory wth long rds of 3m nd short rds of m. he ltform mss s ssmed to be 0kg nd the whole mss of three wheels vres from 0kg to 5kg, reresentng mxmlly 5% of the ltform mss. he reslts show tht s the wheel mss ncreses, the oertonl forces n x, y, nd θ z drectons ncrese drstclly comred to the crclr trjectory. he seed of the moton lso ffects the dynmc forces. he elltc trjectory wth erod of 30sec. yelds lrger oertonl forces nd moment, comred to the cse of crclr trjectory wth erod of 60sec. It cn be smmrzed from these smltons tht the effect of wheel dynmcs cold ffect the dynmc behvor of the moble robot s the nert of the wheels ncreses, mlyng tht the wheel dynmcs shold not be crelessly neglected n dynmc nlyss of the moble robot. Frthermore, the mortnce of ccrte dynmc model of the moble robot cn be vslzed throgh the qntttve nlyss of the mct geometry. When the moble robot colldes wth n envronment, the mlse chrcterstc of the moble robot cn be stded by nlyzng the ellse geometry. he ellse denotes the mont of normlzed mlse tht my be exerenced by the moble robot colldng wth some object from the crrent confgrton to ny rbtrry drecton wth nt velocty. o nvestgte mlse chrcterstcs n the oertonl sce of the moble robot, the externl mlse mesre gven n (60) wll be emloyed n smlton. In the followng nvestgton, t s ssmed tht the coeffcent of resttton e s 0.8 nd the velocty of the orgn of the locl coordnte s gven s m / s. he ellses of Fgs. 6-9 show the confgrtons of steerng ngles ( ϕ, ϕ, ϕ 3) for ech moble robot nd ther corresondng mlse geometres. Note, rtclrly, tht the confgrton gven n Fg. 8 shows nform ellsod snce the wheel dynmcs contrbtes to the mlse geometry symmetrclly, jst lke the cse of gnorng the wheel dynmcs tht lwys genertes crclr she. However, t s observed from Fgs. 6 nd 7 tht the mont of mlse s greter n the movng drecton s comred to those n the other drectons, confrmng tht the wheel dynmcs ndeed nflences sgnfcntly n the nlyss of the mct geometry. 5. CONCLUSIONS he contrbton of ths er cn be smmrzed s follows. Frstly, n lgorthmc snglrty-free dynmc modelng roch s roosed. Secondly, comlete dynmc model ncldng the wheel dynmcs s sggested s closed form. he vldty of the roosed method hs been shown throgh severl smltons. Conclsvely, t s remrked tht the roosed snglrty-free, ccrte dynmc model ncldng wheel dynmcs ensres snglrty-free oerton of the moble system nd fclttes the model-bsed control nd mct nlyss for the moble robot nvolvng collsons wth externl envronments. A. Hessn mtrx It s defned s ( ) ( φ) ( φ) = H φφ APPENDIX descrbes tht t ffects the velocty of the oertonl sce set () on the ccelerton of the jont vrbles nd t hs M N N dmensons. M nd N denote the nmber of the ott nd the nmber of the nt jonts, resectvely. he hyscl Hessn mtrx reresents the centretl nd Corols ccelerton of the lnk. he second order KIC mtrx (Hessn Mtrx) s oerted n lne by lne fshon corresondng to the th ott. Poston Hessn mtrx s symmetrc nd rottonl Hessn mtrx s lwys nll n the lne. However, rottonl Hessn mtrx s n er trnglr mtrx n the sce. A. Generlzed sclr dot rodct ( ) [ ] [ ] = [ ] A B C, (A) where [ A ] s (PxQ) mtrx, [ B ] nd [ ] C s (QxMxN) rry nd (PxMxN) rry, resectvely. In tensor form, (A) cn be exressed s c = b, (A) kl j jkl j where sbscrts of mtrx c,, j, nd k reresent the corresondng lne, row, nd colmn, resectvely. he oerton ws frst emloyed n robot dynmc modelng formlton by Freemn nd esr [7]. It lys rmry role for the systemtc develoment of somorhc trnsfer technqes. A.3 Hgher-order loo method he hgher-order constrnt eqtons cn be obtned drectly t the velocty level by sng common ntermedte coordnte set. G φ φ = G φ φ, (A3) G φ φ = 3G φ φ 3. (A4)

14 Snglrty-Free Dynmc Modelng Incldng Wheel Dynmcs for n Omn-Drectonl Moble Robot wth hree 99 Reresentng the frst-order KIC mtrces by corresondng colmn vectors, g j whch denotes the jth colmn t the Jcobn mtrx of the th chn, (A3) nd (A4) cn be rewrtten s [ g g g 3] φ = [ g g g 3] [ g g g ] φ = [ g g g ], φ (A5) φ (A6) Eqtons (A5) nd (A6) cn be gmented nto sngle mtrx eqton tht cn be exressed s g g3 g g3 0 0 g g 0 0 g g g g 0 = φ. g 0 3g Now, (A7) cn be exressed smly s where φ (A7) J φ = J φ, (A8) φ nd φ denote the deendent nd ndeendent jont velocty, resectvely, gven by = 3 3 = 3. φ η ϕ η ϕ η θ φ θ θ ϕ, (A9) Drect nverson of the sqre mtrx J, whch s ssmed to be nonsnglr, gves φ φ φ. (A0) = J J = G REFERENCES [] C. Cmon, G. Btn, nd B. D Andre-Novel, Strctrl roertes nd clssfcton of knemtcs nd dynmcs models of wheeled moble robot, IEEE rns. on Robot nd Atomton, vol. 4, no., , 987. [] B.-J. Y nd W. K. Km, he knemtcs for redndntly ctted omn-drectonl moble robots, Jornl Robotc Systems, vol. 9, no. 6, , 00. [3] W. K. Km, B.-J. Y, nd D. J. Lm, Knemtc modelng of moble robots by trnsfer method of gmented generlzed coordntes, Jornl of Robotc Systems, vol., no. 6,. 30-3, 004. [4] S. K. Sh nd J. Angels, Knemtcs nd dynmcs of three wheeled -DOF AVG, Proc. of IEEE Int. Conf. Robotcs nd Atomton, , 989. [5] B.-J. Y, W. K. Km, nd S. Prk, Knemtc/ dynmc modelng nd nlyss of omndrectonl moble robots wth redndnt ctton, torl (S) for vehcle mechnsms hvng ctton redndncy nd dtble knemtc strctre, Proc. of IEEE/RSJ Int. Conf. Intellgent Robots nd Systems,. -5, 999. [6] R. A. Freemn nd D. esr, Dynmc modelng of serl nd rllel mechnsms/ robotc systems, Prt I-Methodology, Prt II- Alctons, Proc. of the 0th Conf. ASME Bennl Mechnsm,. 7-7, 988. [7] B.-J. Y nd R. A. Freemn, Geometrc nlyss of ntgonstc stffness n redndntly ctted rllel mechnsms, Jornl of Robotc Systems, vol. 0, no. 5, , 993. [8] M. W. Wlker nd D. E. Orn, Effcent dynmc comter smlton of robotc mechnsms, ASME Jornl of Dynmc Systems, Mesrement, nd Control, vol. 04, no. 3,. 05-, 98. [9] C.-J. L, A new Lgrngn formlton of dynmcs for robot mnltors, ASME Jornl of Dynmc Systems, Mesrement, nd Control, vol., , 989. [0] W. Cho nd D. esr, he dynmcs nd stffness modelng of generl robotc mnltor systems wth ntgonstc ctton, Proc. of IEEE Int. Conf. Robotcs nd Atomton, , 989. [] F. C. Prk, J. C. Cho, nd S. R. Ploen, Symbolc formlton of closed chn dynmcs n ndeendent coordntes, Mechnsm nd Mchne heory, vol. 34, no. 5, , 999. [] J. Wttenbrg, Dynmcs of Systems of Rgd Bodes, Stttgrt, B.G. ebner, 977. [3] I. D. Wlker, he se of knemtc redndncy n redcng mct nd contct effects n mnlton, Proc. of IEEE Int. Conf. Robotcs nd Atomton, , 990. Je Heon Chng receved the B.S. degree n Control nd Instrmentton Engneerng from Chosn Unversty n 998 nd the M.S. degree n Electroncs, Electrcl, Control nd Instrmentton Engneerng from Hnyng Unversty n 003. Crrently, he s workng towrd Ph.D. degree n the Dertment of Electroncs, Electrcl, Control nd Instrmentton Engneerng, Hnyng Unversty. Hs reserch nterests nclde knemtc nd dynmc modelng of hybrd mechnsm, rllel mnltor, moble robot, desgn nd control of srgcl robot, nd mltle mct modelng.

15 00 Je Heon Chng, Byng-J Y, Whee Kk Km, nd Seog-Yong Hn Byng-J Y receved the B.S. degree from the Dertment of Mechncl Engneerng, Hnyng Unversty, Seol, Kore n 984, nd the M.S. nd Ph.D. degrees from the Dertment of Mechncl Engneerng, Unversty of exs t Astn, n 986 nd 99, resectvely. From Jnry 99 to Agst 99, he ws Post- Doctorl Fellow wth the Robotcs Gro, Unversty of exs t Astn. From Setember 99 to Febrry 995, he ws n Assstnt Professor n the Dertment of Mechncl nd Control Engneerng, Kore Insttte of echnology nd Edcton (KIE), Chonn, Chngnm, Kore. In Mrch 995, he joned Hnyng Unversty, Ansn, Gyeongg-do, Kore s n Assstnt Professor n the Dertment of Control nd Instrmentton Engneerng. Crrently, he s Professor wth the School of Electrcl Engneerng nd Comter Scence, Hnyng Unversty. He styed t Johns Hokns Unversty s Vstng Professor from Jnry 004 to Jnry 005. Hs reserch nterests nclde desgn, control, nd lcton of srgcl robot, rllel mnltor, mcromnltor, htc devce, nd nthroomorhc mnltor systems. Whee Kk Km receved the B.S. degree from the Dertment of Mechncl Engneerng, Kore Unversty, Seol, Kore n 980, nd the M.S. nd Ph.D. degrees from the Dertment of Mechncl Engneerng, Unversty of exs t Astn, n 985 nd 990, resectvely. From Jnry n 990 to Jnry n 99, he ws Post-Doctorl Fellow wth the Robotcs Gro, Unversty of exs t Astn. Snce 99 he hs been n the Dertment of Control nd Instrmentton Engneerng, Kore Unversty t Chochwon, Chngnm, Kore. Crrently, he s Professor n the sme dertment. Hs reserch nterests nclde desgn of rllel robots, knemtc/dynmc modelng nd nlyss of rllel/ moble/wlkng robots. Seog-Yong Hn receved the B.S. degree from the Dertment of Mechncl Engneerng, Hnyng Unversty, Seol, Kore n 98, nd the M.S. nd Ph.D. degrees from the Dertment of Mechncl Engneerng, Oregon stte nversty t Oregon, n 984 nd 989, resectvely. Snce 995 he hs been n the Dertment of Mechncl Engneerng, Hnyng Unversty t Seol, Kore. Crrently, he s Professor n the sme dertment. Hs reserch nterests nclde sold mechncs, frctre mechncs, otmm desgn.