Control and Path Planning of a Walk-Assist Robot using Differential Flatness

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1 he 00 EEE/RSJ nenaona Coneene on negen Robos and Sses Oobe 8-, 00, ape, awan Cono and Pah Pannng o a Wak-Asss Robo usng Deena Faness Chun-Hsu Ko and Sun K. Agawa Absa Wh he gowh o ede popuaon n ou soe, ehnoog w pa an poan oe n povdng unona ob o huans. n hs pape, we popose a obo wakng hepe wh boh passve and ave ono odes o gudane. Fo he pespeve o huan sae, he passve ode adops he bakng ono aw on he whees o deena see he vehe. he ave ode an gude he use een when he passve ono wh use-apped oe s no adequae o gudane. he heo o deena aness s used o pan he aeo o ono gans whn he poposed shee o he onoe. Sne he use npu oe s no known a-po, he heo o ode pedve ono s used o peoda opue he aeo o hese ono gans. he suaon esus show ha he wakng asss obo, aong wh he suue o hs poposed ono shee, an gude he use o a goa eeve.. NRODUCON As he ede popuaon gows apd, wakng asss obos w onnue o be an poan eseah op n ou soe [-]. hs eseah as o povde asssane o he ede dung wakng. o enhane he sae and onvenene, negen wakng ads ae needed. Robos onans sevea useu ehnoog oponens, e.g., sensng, auaon, oon ono, and opue negene. s e o desgn obo wakng hepes o he ede b negang hese ehnooges. Man obo wakng hepes have been poposed o asss huan wakng [-]. Sses ha suppo useu unons suh as gudane [,5,7-0], obsae avodane [,8], and heah onong [,7] have been deveoped. o ono a obo wakng hepe, Spenko e a. [7] used a vaabe dapng ode o nease he wakng sab. Haa e a. [8] poposed an adapve oon ono agoh o obsae/sep avodane and gav opensaon. Chu e a. [] used he passve behavo o enhane he neaon beween he use and he ave suppo sse. Agawa e a. [0] poposed a wo-phase passve ono agoh o gudng a use o aan desed poson and oenaon, whe aowng o sa eos. n genea, he obo wakng hepes an be assed no hs wok was suppoed n pa b he Naona Sene Coun o awan, ROC. unde he Gan NSC and NSC E MY. he seond auho aso gaeu aknowedges paa suppo o Wod Cass Unves poga. C. H. Ko s wh he Depaen o Eea Engneeng, -Shou Unves, Kaohsung, awan ROC (e-a: hko@su.edu.w. Sun K. Agawa s wh he Depaen o Mehana Engneeng, Unves o Deawae, Newak, USA (oespondng auho; phone: ; ax: ; e-a: agawa@ude.edu. hese wo pes: ( ave and ( passve. he ave obo wakng hepes [-5] use sevo oos o povde gudane o he use, whe ave addng eneg o he sse. he passve wakng hepes [8-] ove on b use supped oes. Conoed bakes ae used o see he wake whe onsan exang eneg o he sse. Wh hs pope o dsspaon o eneg, he ae nheen sae and end o avod bud up o eneg. Howeve, a use woud ke o nease he wakng speed, see he hepe, o wak uph wh he wake, age use-apped oes ae needed. Fuheoe, passve ode a no be enough o aheve eeve aeo pannng and ono. Hene, n ode o eeve hep a use, s poan o onsde boh ave and passve ono saeges wh use apped oes o see o he goa. n hs pape, we popose usng passve and ave ono odes n eeve gudane o obo wakng hepe. Passve ode, wh bakng ono aw on he whees o deena see he vehe, s s uzed. When he passve ono wh use-apped oes s no suen o auae gudane, ave ode s hen used. Boh wh passve and ave odes, he obo wake s onoed o anan a onsan wakng speed. he ehod o deena aness s used o see e vang aeoes o he ono gans so ha he vehe aheves a desed na poson and oenaon [4, ]. n hs appoah, he nonnea suue o he vehe dna equaons unde he ono aw s expoed o seek he pope o deena aness. he saes and ono npus ae epesened as unons o a oupus and he devaves. he a oupus ae hen paaeezed b a se o ode unons ha he bounda ondons. Wh deena aness appoah, a easbe aeo wh passve o ave ono ode an be eas ound b usng he nonnea pogang. Sne he use apped oes ae no known apo, ode pedve ono (MPC s used o nd he souon, epeve. he eande o hs pape s oganzed as oows: Seon desbes he dna equaons o he obo wakng hepe. n Seon, deena aness appoah s addessed. aeo pannng and ode pedve ono ae desbed n Seons V and V. Seon V pesens he suaon esus Fna, onudng eaks ae gven n Seon V /0/$ EEE 606

2 Fg.. A obo wakng hepe. Fg.. he onguaon o a obo wakng hepe.. HE ROBO WALKNG HELPER AND S MODEL he obo wakng hepe [] s shown n Fg.. onans he suppo ae, wo whees dven b oos wh enodes, wo passve ases, an uason senso aa, a oe senso, and a onoe. he oos povde he pope oques o onong he oon o he obo wakng hepe. he enodes ae used o easue he whee speed, whe he oe senso dees he use apped oe. o desgn he onoe, a sped dna ode s desbed nex. Fg. shows he obo on a sope n wod oodnaes xz, [0,] gven b [ x,, z, θ ] q =, ( whee x,, z ae he oodnaes o he ene o ass and θ s he headng ange o he obo. he sope ange s assued o be a unon o, α = α (. When he obo oves on he sope, he bod oodnaes x z oae θ abou z axs and α abou he x axs. he oaon ax R s auaed as osθ sn θ 0 R = osα snθ osα osθ sn α ( sn α sn θ sn α osθ osα Wh he assupon o no-sp ondon a he whee ona pons, he veo o he whee enes ae paae o he headng deon. Hene, q an be expessed as x z osθ = vxˆ ' = v sn θ osα sn θ sn α ( and θ = ω (4 whee v s he headng speed and ω he unng speed. he angua veo an be expessed as ω = α xˆ + ω zˆ' = [ α ω snα ω osα ] (5 whee α = α '(. he equaons o oon wh he use-apped oe F, oo apped oques, he gav oe on a sope and he no-sp onsan oe/oen ae wen as τ + τ [ x z ] = ( F + xˆ' gzˆ + λˆ' + pzˆ ' (6 and τ τ ω + ω ω = zˆ' + M x xˆ' + M ˆ' (7 Hee, s he obo ass, he oen o nea o he obo, s he whee adus, b ha dsane beween he wo whees, τ and τ he oo oques on he gh and e whees, λ he onsan oe, p he noa eaon oe and M x, M he onsan oens. he oen o nea an be auaed as = R = bod R, bod (8 Wh he assupon o se abou he pane x z, he vaues o and ae zeos. B deenang Eq. (, [ x,, z ] vxˆ' v xˆ = + '. On subsung hs [ x ],, z no Eq. (6 and pe-upng b x' ˆ, we ge τ + τ v = F + g snθ snα (9 Fuheoe, b deenang Eq. (5, ω = α ω sn α ω osαα ω osα ω sn αα (0 [ ] On subsung Eq. (0 no Eq. (7 and pe-upng b ˆ, we ge τ τ = θω ( ω α osθ snθ ( α sn Fo Eqs. (, (4, (9, and (, he sae equaons o he obo on a sope ae expessed as x = v osθ = v snθ osα z = v snθ sn α θ = ω τ + τ F v = + g snθ sn α τ τ ω = α osθ sn θ α snθω ( 607

3 o ake he obo passve and dsspave, he ono aw s hosen as τ = K θ, τ = K θ ( whee θ and θ ae he angua speeds o he gh and he e whees, espeve. K and K ae non-negave paaees. Wh he no-sp ondon, he angua speeds θ, θ an be auaed as θ = ( v + bω /, θ = ( v bω / (4 Subsung Eqs. ( and (4 no Eq. (, we an oban he dna equaons o he obo gven b q = SV (5 V = AK + B whee osθ 0 v snθ osα 0 V =, S =, ω snθ snα 0 0 v + bω v bω = K (6 A v + bω v bω, K =, K F / g snθ snα B = α osθ snθ α snθω he enes o K an now be egaded as he new ono npus whh need o be panned o see he vehe o s uen sae o he desed goa sae. non-negave vaues o he K ae hosen, he obo wake s onoed wh a dsspave/passve ode. Ohewse, an ave ode s used n onong he obo wake. o desgn he eedbak onoe, we uhe anso he dna equaons o be sa and/o dna eedbak neazabe b usng he deena aness appoah, desbed n he nex seon.. DFFERENAL FLANESS n he deena aness appoah [,4], we s see suabe a oupus and hen expess a sae vaabes and npu n es o he a oupus and he devaves. Hee, we nd ha he obo sse s deena a wh he a oupus (, =(x,. Wh a gven sope hegh z(, he sope ange and s devaves an be expessed as α = an ( z" α = + z + z" α = + z" ( + Dene s as he engh o he sope, hen (7 and s = / osα + α s = osα + α + α + α + α s = osα s θ = aan( v = + s s s ω = θ = + s + ss v = + s ω = ( snθ + osθ s v ω / v (8 (9 Wh Eqs. (6-9, he ono npus K an be expessed wh he a oupus and he devaves, gven b v K = K( = A,,,,,, B (0 ω he above equaon an be used n pannng a desed aeo wh he ave o passve odes. V. RAJECORY PLANNNG o peo he aeo pannng [,], he bounda ondons o ( (, ( ae s auaed o he na and na pons. hen, a se o unons ae hosen o ng he aeo ha an pass hough hese wo end pons. Fna, we nd a aeo b sovng he nonnea onsaned equaons. he na ondons 0, (0, (0, (0, (0, (, (, (, (0 and he na ondons,, ( o he aeo an be obaned wh he gven sope hegh z(, he na saes x ( 0, (0, θ (0, v(0, ω(0, and he na saes x,, θ, v(, ω( b usng he oowng expessons = x, =, = v osθ, = v snθ osα, = v osθ v snθω = v snθ osα + v osθω osα v snθ snαα' ( Noe ha he above expessons ae used o auang he vaues o ( 0, (0, (, bu he vaues o he paaees v ( 0, v ae s needed. Sne v ( 0, v ae no speed n he pannng, hese vaues an be aba seeed. he aeo s hen ed wh he oowng o 608

4 onsans o he non-negave vaues n K, expessed as n J a, b, v (0, v, sube o K( a, b, v (0, v, 0 (7 τ τ ( a, b, v (0, v, τ One he desed aeo s ound, he oques an be obaned o onong he obo o aheve he age. Fg.. Mode pedve ono o he obo. = Φ + k = = Φ + k = a φ b φ ( whee Φ, Φ,φ ae he ode unons and k he ode nube. Hee, Φ, Φ s a aeo ha passes hough he na and na pons. Moeove, φ ( ae he unons wh φ ( 0, φ (0, φ (0, φ, φ, φ zeo. n hs pape, Φ, =, ae hosen o be he oowng ponoa unons o e [5] Φ = ( + 5, =, he oeens k (=,,k=,,5 ae soved usng he gven bounda ondons 0, (0, (0,,,. Wh he hoe o ( Φ (, k ae eas obaned as = (0, = ( = ( = (0.5 = (0, 0 = 0.5 (0 / / 4 / Fuheoe, he ode unons φ ( ae seeed as [5] (4 φ = + (5 hese ode unons possess he pope o havng zeo devaves up o he ode a 0 and. One he a oupus ae paaeezed wh Eq. (, he saes and npus ae aso epesened as unons o a, b, v (0, v,. aeo geneaon hen an be aheved b sovng a nonnea onsaned opzaon pobe. Wh he ave ode, he opzaon pobe s desbed as n, J a, b, v (0, v sube o τ τ ( a, b, v (0, v, τ (6 whee J s use-dened obeon unon and τ he axu oque o he bake oo. Fuheoe, he opzaon pobe wh he passve ode nudes he V. MODEL PREDCVE CONROL o peen hs suue whn he appaon o obo wakng hepe, ode pedve ono appoah s uzed [6-8], as shown n Fg.. n aeo pannng based on MPC, gven he age pon, he uen easued saes, use-apped oes and sope anges, a easbe aeo and he ono npus ae peded o a peod. Sne MPC s based on a onsan vaue o uen easued use-apped oe, whe n ea use-apped oe ae e vang, he MPC on uses he pedve npus o ono he obo ove a peod Δ ( Δ <. Ae e Δ, MPC needs o ead he saes and use-apped oes, and ake a new aeo o he nex peod agan. he poedue w onnue un he sop eons ae sased. Noe ha a ed sope unon z( s used o MPC n eah peod. Gven he uen poson (, z, he age poson (, z, he easued uen sope ange α, and he age sope ange α, a pedve sope unon z( s hen ed wh he oowng o z( = d 0 + d + d ( ( + d ( (8 Wh he bounda ondons z( =z, z ( =an(α, z( =z, and z ( =an(α, he oeens d (=0,, ae soved as d = z, d = an α d d 0 = ( = z z ( an α + an α ( ( z z ( (an α + an α ( + (9 o asss he use n wakng sab, he passve ode s s seeed o pan he aeo o he pesen sae o he goa pon b sovng Eq. (7. no easbe souon an be ound, he headng ange a he na pon s se as a desgn vaabe n Eq. (7. Howeve, no easbe souon an be ound agan, he ave ode s used o pan he aeo wh Eq. (6. Wh above poedues, he passve ode an be used as uh as possbe n he gudane. V. SMULAON RESULS n ode o deonsae he eeveness o he poposed appoah, he MPC was apped o a gudane pobe o he wake. he paaee vaues o he obo n suaons ae =5 kg, = kg, =.5 kg, = kg, =0. kg, =0.075, and b=0.4. he sa and end pons (x,, θ 609

5 Fg. 4. he aeo o he obo wh =0 N. Fg. 7. he aeoes o he obo wh =6, 0, and 6 N. Fg. 8. he aeo o he obo on a sope wh =6 N and h=0.. Fg. 5. he sae vaabes x,, θ, v, and ω wh =0 N. Fg. 6. he vaues o K, K, τ, and τ wh =0 N. ae (0, 0, π/ ad and (6, 6, π/ ad, espeve. he ondon o sabe wakng s s onsdeed,.e., he speed and angua speed (v, ω o he sa and end pons ae Fg. 9. he sae vaabes x,, θ, v, and ω o he obo wh =6 N and h=0.. se o he sae vaues ( /s, 0 ad/s. he use-apped oe s assued as a unon o e (+0.sn(π/. Fg. 4 shows he aeo wh =0 N. We obseve ha he 600

6 o he obo wakng hepe sne use apped oes ae no known po. he suaon esus show ha he obo wakng hepe wh ode pedve ono an auae gude he use o a goa, deonsang he een o he poposed ono shee. Fg. 0. he vaues o F, F g, K, K, τ, and τ wh =6 N and h=0.. passve ode s hus os used dung he aneuve and he ave ode s used o appoahng he age. he aeoes o x,, θ, v, ω and he vaues o K, K ae shown n Fgs. 5 and 6, espeve. he obo s aos onoed wh a onsan speed o /s. When he obo appoahes he goa poson, he ave ode wh a age oque s used o auae seeng he obo o he headng ange o goa pon. Fg. 7 shows he aeoes wh =6, 0, 6 N, espeve. he esus show ha he obo eahes he na pon auae wh a age oe ( =6 N. Howeve, wh a sa oe ( =6, 0 N, he obo eques he passve/ave ode o eahng he goa. he gudane o he obo on a sope s uhe onsdeed. he sope hegh z s assued o be a unon o, ( z ( = 0. /( + e. Fgs. 8-0 show he aeoes and he vaues o he paaees K, K on a sope hegh h=0. wh a use-apped oe =6 N. We ound ha he ave ode s used aound = due o a age sope ange and a age gav oe Fg. he posve oques ae apped b he obo wake. he poposed passve/ave ono ode s eeve o hepng he use wakng on he sope wh a onsan speed. V. CONCLUSON n hs pape, we have pesened a nove ehod o aeo pannng o he e-vang ono gans o a obo wakng hepe. he dna ode o he vehe on a sope wh passve/ave ono aw was s esabshed. he aeoes o he gans wee deveoped b usng appoahes o he heo o deena aness. Aodng, easbe aeoes wh he passve/ave ehod wee ound b sovng a nonnea poga. Fuheoe, ode pedve ono was aso peened REFERENCES [] H. Yu, M. Spenko, and S. Dubowsk, An adapve shaed ono sse o an negen ob ad o he ede, Auon. Robos, vo. 5, no., pp. 5 66, 00. [] O. Chu, Y. Haa, Z. Wang, and K. Kosuge, A ono appoah based on passve behavo o enhane use neaon, EEE ans. Robos, vo., no. 5 pp , O [] A. J. Renshe, R. A. Coope, B. Bash, and M. L. Bonnge, negen wakes o he ede: Peoane and sae esng o VA-PAMAD obo wake, J. Rehab. Res. Dev., vo. 40, no. 5, pp. 4 4, 00. [4] G. Wasson, P. Sheh, M. Awan, K. Ganaa, A. Ledoux, and C. Huang, Use nen n a shaed ono aewok o pedesan ob ads, n Po. EEE/RSJ n. Con. ne. Robos Ss., pp , 00. [5] B. Ga, An adapve gudane sse o obo wakng ads, Jouna o Copung and noaon ehnoog, vo. 7, no., pp. 09 0, 009. [6] A. M. Saban, V. Genovese, and E. Paheo, A ob ad o he suppo o wakng and obe anspoaon o peope wh oo paens, n Po. EEE/RSJ n. Con. ne. Robos Ss., pp , 00. [7] M. Spenko, H. Yu, and S. Dubowsk, Robo pesona ads o ob and onong o he ede, EEE ansaons on Neua Sses and Rehabaon Engneeng, vo. 4, no., pp. 44-5, Sep [8] Y. Haa, A. Haa, and. K. Kosuge, Moon ono o passve negen wake usng sevo bakes, EEE ans. Robos, vo., no. 5, pp ,O [9] N. Neabakhsh and K. Kosuge, Opa Gudane b Ondeona Passve Mob Ad Sse, EEE/RSJ nenaona Coneene on negen Robos and Sses, pp , 006. [0] J. C. Ru, K. Pahak, and S. K. Agawa, Cono o a passve ob asss obo, Jouna o Meda Deves, vo., pp. 000 (7 pages, Mah 008. [] S. H. Yu, C. H. Ko, and K. Y. Young, On he desgn o a obo wakng hepe wh huan nenson and envoonena sensng, n Po. CACS n. Auo. Con. Con., 008. [] K. Pahak and S. K. Agawa, An negaed pah-pannng and ono appoah o nonhoono unes usng swh oa poenas, EEE an. on Robos, vo., no. 6, pp. 0-08, Deebe 005. [] J. C. Ru, S. K. Agawa, and J. Fanh, Moon pannng and ono o a ao wh a seeabe ae usng deena aness, Jouna o Copuaona and Nonnea Dnas, vo., pp. 000 (8 pages, Ju 008. [4] H. Sa-Raez and S. K. Agawa, Deena Fa Sses. s ed., New Yok: Mae Dekke, 004. [5] S. K. Agawa and. Veeakaew, A hghe-ode ehod o dna opzaon o a ass o nea sses, Jouna o Dna Sses, Measueen, and Cono, vo. 8, pp , Deebe 996. [6] M. Benosan and K. Y. Lu, Onne eeenes eshapng and ono eaoaon o nonnea au oean ono, EEE ans. Con. Ss. eh., vo. 7, no., pp , Ma [7] D. Gu and H. Hu, Reedng hozon akng ono o wheeed obe obos, EEE ans. Con. Ss. eh., vo. 4, no. 4, pp , Ju 006. [8] L. Magn and R. Saon, Mode pedve ono o onnuous-e nonnea sses wh peewse onsan ono, EEE ans. Auo. Con., vo. 49, no. 6, pp , June

5-1. We apply Newton s second law (specifically, Eq. 5-2). F = ma = ma sin 20.0 = 1.0 kg 2.00 m/s sin 20.0 = 0.684N. ( ) ( )

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