FOUR-WHEEL VEHICLE SUSPENSION MODELING FOR CONTROL SYSTEM DEVELOPMENT

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Basil Mosley
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1 Poeedings of COEM 5 Copyigh 5 by CM 8h Inenaional Congess of Mehanial Engineeing Novembe 6-, 5, Ouo Peo, MG FOUR-WHEEL VEHICLE SUSPENSION MODELING FOR CONROL SYSEM DEVELOPMEN Cláudio Civellao DN Suual Soluions Inenaional azil Division (Dana Indúsias Lda) v. Pesidene Médii, 939, Zip ode 668-, Osaso, SP, azil laudio.ivellao@dana.om Edilson Hioshi amai Sae Univesiy of São Paulo (USP) Esola Poliénia DEM Zip ode 558-9, São Paulo, SP, azil edhamai@usp.b Déio Cisol Donha Sae Univesiy of São Paulo (USP) Esola Poliénia DEM Zip ode 558-9, São Paulo, SP, azil dedonha@usp.b bsa. In een yeas seveal woks have been published egading aive and semi-aive suspension sysem fo vehiles in geneal; howeve few ailes have deal wih an adequae fou-wheel suspension model fo onol developmen. When a fou-wheel model is used, seveal onol poblems may appea, suh as he pesene of nonminimum phase ansmissions zeos, he lak of obsevabiliy and onollabiliy, and ill-ondiioned sysems. In his wok a seven-degee of feedom fou-wheel vehile model is pesened, and a balaned sae-spae ealizaion and model eduion is poposed o ge a fully onollable and obsevable model fo vehile suspension onolle design. In addiion a obus onol (LQG/LR) is designed and simulaions esuls ompaing passive and aive sysems ae pesened. Keywods: Vehile suspension, fou-wheel model, balaned ealizaion, spae-sae, obus onol.. Inoduion he vehile mahemaial modeling is he base of seveal onol saegies applied o vehile suspension sysems. In een yeas seveal woks have been published egading aive and semi-aive suspension sysem fo numeial simulaions. Some woks dealing wih diffeen kind of onol appoah used jus a quae-a model wih wo degee of feedom (Rao and Pahlad, 995; Suz and Rohinha, 5). Ohe woks aimed a ohe vehiles movemens suh as oll o pih ahe hen wih he onol appoah have pesened half-a models wih fou degee of feedom (Ha e al., 996; sao and Chen, ; Simon and hmadian, ). Ohe woks suh as Cuz e al., 3, and Yamamua and Masada, 979, have used fou-wheel modeling fo MGLEV vehile onolle developmen, bu did no use a MIMO (muliple inpu muliple oupu) onol appoah, alhough Cuz e al., 3, had o deal wih he ase of an ove auaed sysem, solved wih an opimal disibuion of foes among he auaos designed o minimize he maximum foe. he goal of his wok is o popose an alenaive saegy of vehile modeling, epesenaive of a full fou-wheeled vehile, whih is suiable fo MIMO onol developmen. he MIMO onol appoah an onside he global pefomane of vehile movemens o deemine he bes onol aion, whih is, a pioi, a bee hoie han he use of seveal SISO loal onolles woking independenly. In addiion, a sae-spae heoy is an elegan way o appoah a onol poblem, mainly egading MIMO sysems whih ae naually deal. his heoy has given impoan oneps suh as obsevabiliy and onollabiliy, and leaded o seveal onol design mehods, suh as linea quadai egulao (LQR), pole plaemen, opimal H, and obus onol design mehods as LQG-LR (Doyle and Sein, 98; Cuz, 996) and H (Glove and Doyle, 989). he hoie of MIMO appoah fo suspension sysem onol developmen bings up seveal onol issues as fo example he pesene of non-minimum phase ansmission zeos. igh half plane zeo gives an uppe bound o he ahievable bandwidh. he bandwidh deeases wih deeasing fequeny of he zeo. I is hus moe diffiul o onol sysems wih slow zeos. On he ohe hand, when he vehile body is modeled as a igid body, he fou independen suspension sysem (one fo eah wheel) an only onol hee (heave, pih and oll) fom he six degee of feedom. In his ase, i is vey ommon o use fou auaos one fo eah wheel and jus hee linea-independen signals ould be measued o deemine uniquely he posiion of vehile body (heave, oll and pih movemen o is aeleaion sae (heave, oll and pih aeleaion. In his way, a fou dimensional onol signal anno influene some movemens of vehile body, whih anno be epesened in a hee dimensional spae, fo example, when he onol signals exie a wis mode of a body suue. his siuaion is exended o he sysem oupu when, fo example, he veial aeleaion

2 in fou poins of he vehile body (eah poin ove is espeive wheel) is measued insead of heave, oll and pih aeleaion measuemens. hus a hee dimensional spae is epesened in a fou dimensional spae, meaning ha he ue oupu epesens only a hype-plane in a fou-dimensional oupu. In his siuaion, a sysem model podues an ill-ondiioned ansfe funion maix, whih is no adequae o onol design. In addiion, i would be vey ineesing fo he onolle designe if he onolle aion ould be foused in a pa of he sysem whee he enegy applied is bee used o ge a desied sysem pefomane. In ohe wods, i is desiable a speial ealizaion whee he sysem model is sepaaed ino wo pas: one whee he saes ae easily onolled by he inpus and easily esimaed fom he sysem oupus, and ohe whee he saes ae no onolled o pooly onolled by he inpus, and also, hese saes ae no esimaed o badly esimaed fom he oupus. Fo minimal ealizaions, i is possible o find a so-alled balaning similaiy ansfomaion suh ha he onollabiliy and obsevabiliy gamians ae boh equal Σ diag(σ, σ,..., σ n ), σ σ... σ n (fo minimal sysems σ n > ). σ i ae he Hankel singula values, and gives a peise measue of how obsevable and onollable he saes ae, in view of he enegy inepeaions of gamians. his enegy inepeaion show ha eah Hankel singula value is a measue of he amoun of enegy injeed ino he i-h sae x i, and he amoun of enegy of x i in he oupu enegy. Fuhemoe, he fa ha Σ is diagonal indiaes ha, in his ealizaion, he diffeen sae omponens ae unoelaed. I is easonable, heefoe, o disad he saes oesponding o he smalles σ i, when is impoan o edue he ode of he model, while peseving as muh as possible he inpu-oupu behavio (Mooe, 98; Shelfhou, 996). Depending on he hoie of saes o be epesened by he mahemaial vehile model and wha signals ae measued and he posiion of onol foes, he sysem ould no be ompleely obsevable and/o onollable, and in his ase, i is onsideed a non-minimum sysem in he MMillan sense (Mooe, 98). he minimizaion of suh sysems, sepaaing he onollable and obsevable pa fom he non-onollable and non-obsevable one, is he neessay ondiion fo applying a unaed alaned Realizaion (R) poedue, as desibed above. Illondiioned onollabiliy and/o obsevabiliy maies an esul in a nealy non-minimum sysem, whih mus be minimized even having a full ank, in ode o avoid numeial poblems in R algoihm. One a balaned edued ealizaion of he sysem is available, i is he ime o design he onolle. In his wok a LQG/LR design mehod is used. he onolle designed is hen applied o he oiginal full-ode sysem and is pefomane is measued by he RMS values of pih, oll and heave aeleaions, while he sysem is exied by disubanes geneaed by oad unevenness. hese esuls ae afewads ompaed wih a passive suspension, whee dampes wih fixed damping oeffiiens eplae he aive auaos.. Mahemaial modeling.. Fou-wheel vehile model Figue. Vehile physial model

3 Poeedings of COEM 5 Copyigh 5 by CM 8h Inenaional Congess of Mehanial Engineeing Novembe 6-, 5, Ouo Peo, MG he omplee physial vehile model is shown in Fig.. he model has a igid body epesening he vehile spung body and ohe fou masses epesening he fou wheels. he vehile spung body has hee degees of feedom: heave, oll and pih. Eah wheel has only he veial movemen, esuling in a seven degee of feedom model in a whole. Vaiables z ( ) and z ( ) define veos of he fou elaive displaemens beween he gound and eah wheel and beween eah wheel and a poin in he vehile body whee eah suspension is onneed. In he same way z ( ) and z ae he elaive veloiy. z ( ) is he veo of veial veloiy, applied by gound in eah wheel and z ( ) is he veo of he veial veloiy of he eah wheel. he vaiables desibing he vehile body posiion ae z CG (), ϕ () and θ (), fo heave, pih and oll movemens, espeively. hus a veo [ ] ( ) zcg ϕ( ) θ is defined. he foes ha he suspension sysem applies o he vehile spung body is as follows: F + C Kz z () In Eq.(), K diag( k, k, k, k ) epesens he fou sping siffness, and C diag(,,, ) epesens he f f f f fou damping oeffiiens, whee index f means fon and means ea. he esulan foes in wheels ae epesened as: C z + C F F + Kz z () In Eq.(), K diag( k, k, k, k ) gives he fou ie siffness, and C diag(,,, ) gives he fou ie damping oeffiien. hus he dynamial vehile model an be wien as follows: L F M (3) z F M (4) In Eq. (3) and Eq. (4), M diag ( m, J ϕ, J ) epesens he mass and momens of ineia of he spung vehile θ diag( m, m, m, m body, M ) epesens he fou wheel masses, L is a ansfomaion maix ha elaes he fou veial displaemens of he spung vehile body a he suspension onneion poins o heave, pih and oll displaemens given by: L l l l l (5) d d d d Defining he disubane inpu as w ( ) z ( ) and he sae vaiables as follows: x z( ) z( ) z ( ) ( ) (6) equaions () and () ae ewien as follows: F R x (7) C w( ) F Rx + (8) [ K C C L ] R (9) [ K K ( C C ) C L ] R ()

4 .. uao model In his wok i is assumed ha he auao is a devie able o geneae axial foes and ha has a esidual damping oeffiien ( a ). he auao ime esponse is onsideed 5ms, so ha he auao dynamis is fas enough o be disegad in he model. he auao foes (u i ) ae applied in he same ime in boh spung and unspung masses, as aion and eaion foes; i.e., if he foe in he spung mass is posiive han he foe in he unspung mass will have he same ampliude bu will be negaive. Figue. uao model s shown in Fig., o eplae a passive suspension sysem wih an aive one, an aive auao eplaes he shok absobe (dampe). Maix C diag(,,, ) is han eplaed by C diag(,,, ) fo aive onolled f f a a a a sysem. In geneal, a is onsideed a leas abou fou imes smalle han f and..3. Spae Sae Dynami Sysem Repesenaion he spae-sae epesenaion of he sysem is given by Eq. (), Eq. () and Eq. (3). x + w ) u( ) x + () ( C x + D w ) D u( ) y + () ( C x + D w ) D u( ) y + (3) ( 4x 4 4 x4 I M M I R L R 4x3 4x3 (4) I M C 3x4 (5) M M L [ I ] C (7) x3 (6)

5 Poeedings of COEM 5 Copyigh 5 by CM 8h Inenaional Congess of Mehanial Engineeing Novembe 6-, 5, Ouo Peo, MG C [ L M L ] R (8) D D D (9) D [ L M L] In hese equaions, he inpu u() is a veo of he fou auao foes u i (), i,,3 and 4, one fo eah wheel. he oupu y ()z () is he elaive ie displaemens beween he gound and he ene of wheel. he oupu y () is a veo of aeleaions a fou poins in vehile body ove eah wheel. s a measue of y () is vey diffiul, i was no onsideed in he onol feedbak, i was jus onsideed hee o show he ie behavio of he sysem unde onol ompaed o he passive one. Wheeas y () is easily measued by a simple aeleao devie used o measue he veial aeleaion only..4. Fequeny Domain ppoah Sine he onol design is based on a fequeny domain appoah, he dynamial sysem desibed above mus be epesened as a ansfe Funion Maix (FM). hus he vehiula sysem is epesened by he FM G( (Eq. ()) and he onolle will be epesened by he FM K( as follows: G( G ( G ( () G ( G ( () y y ( w( G( ( u( () u ( s ) K ( s ) y ( s ) (3) 3. Model Reduion he main goal of model eduion is o ge a edued sysem able o geneae a oupu vey simila o y () fom a espeively inpu u(), and ha have good popeies fo mulivaiable onol developmen. he poedue begins wih he oiginal fom of G (, whih is analised by is Fequey Response Funion (FRF). I emeged fom his analysis ha G ( is ill-ondiioned, sine one of is singula values is almos zeo along all fequeny speum of inees. he main ause of his ill ondiion is beause he sysem has hee vaiables (heave, oll and pih) o be onolled by fou auaos and measued by fou aeleomees. So he fis sep is o eliminae his sysem misondiion. 3.. Eliminaion of wis Mode If he vehile body was flexible, he fou auao would be able o onol he wis mode of vehile body vibaion. u sine he vehile body is igid, i have leaded o he ill-ondiioned fom of G (. o avoid his undesiable siuaion he apaiy of onol o a ove he wis mode mus be eliminaed. o do his, a new oodinae sysem is used o epesen he FM. S is a base of a new subspae, whee eah olumn epesen a movemen of he vehile body (oll, heave, pih and wis espeively), and is a ohonomal maix given by: S (4) he ansfomaion S mus be applied as follows: S ; S ; C S C ; D S D S ; D D (5)

6 fewads, he las olumn of maixes, he las line of maix C, and boh las olumn and las line of maix D mus be eliminaed. he las olumn of maix S mus also be eliminaed in ode o mainain he maix mulipliaion mah. 3.. Sysem Ode Minimizaion he ank of onolabiliy maix [ n- ] is alulaed and if he sysem uns o be no ompleely onolable, a minimizaion poedue mus be applied. his poesses onsiss o ge a similaiy ansfomaion suh ha: ; ; ; C C (6) nd he ansfomed sysem has he fom pesened in he sequel: n x C [ Cn C ] ; ; ; (7) he algoihm used o ge he maix was he Saiase lgoihm of Rosenbok (968). he las opeaion onsiss in sepaae he onollable pa o be used in onolle design. he esuling sysem is: G ( C ( si ) (8) simila, bu dual, poess mus be applied in ase of lak of obsevabiliy unaed alaned Realizaion he R poedue as fis pesened in Mooe (98) is eneed aound infomaion obained fom he onolabiliy Gammian W and he obsevabiliy Gammian W o, whih an be obained fom solving he wo Lyapunov equaions below: W + W ; W + W C C o o (9) One solved he Gammians, he algoihm onsiss in six moe seps:. ompue Cholesky faos W L L, W o L o L o ;. ompue he singula value deomposiion (SVD) of Cholesky podu UΣV L o L, whee Σ is diagonal posiive and U, V have ohonomal olumns; 3. ompue de balaning ansfomaion as pesened nex: L V ; U L o (3) 4. fom he balaned ealizaion as in following equaions: ; L ; ; C C b b b (3) 5. so Σ in desending ode of Hankel singula values and oganize b, b and C b in he same ode; 6. unae b, b and C b o fom he edued ealizaion by eliminaion of ow and olumns elaed o negleed Hankel singula values. In his wok, he oiginal 5 h ode sysem was edued o a ompleely onollable and obsevable balaned h ode sysem. he maix L in Eq. (3) was used o design he Kalman File in he LQG/LR poedue, whih is suinly desibed in he nex seion. 4. Conol Design he aim of onolle design is o find a FM K(, whih impoves he omfo of he vehile, i. e., edues he aeleaion ampliude in vehile body movemens. hus, as an be seen in Fig. 3, he onolle design onsiss o

7 Poeedings of COEM 5 Copyigh 5 by CM 8h Inenaional Congess of Mehanial Engineeing Novembe 6-, 5, Ouo Peo, MG alulae he maixes H and G. K( an han be wien as follows: u ( si + G + H C H D G) H S K( S S G (3) b n b n Figue 3. lok diagam of onolled suspension sysem. o alulae he maies H and G, i is applied a LQG/LR mehodology as desibed in Civellao and amai (3). lso in ha wok, a FRF nomalizaion is poposed by he use of he maix S u ((D. D) ½ ) -, whih was used o ge maix n b. S u and D n D. S u. he esul of his onolle design is shown in Fig. 4 hough he FRF of G (jω) and G (jω) fo passive suspension sysem and aive suspension sysem in losed loop. Figue 4. FRF of ( G ( w( and ( G ( w( fo passive sysem and aive losed loop sysem. y y s showed in Fig. 4, he aeleaion ampliude in vehile body movemens is signifianly deeased along all fequeny ange analyzed, whih means a good impovemen in vehile omfo. On he ohe hand, he aive losed loop sysem pefomane esul in an impovemen of wheel hop aound Hz, wha an ause los of adheene beween wheel and gound, impaing negaively in vehile handling pefomane. his issue mus be he fous fo suspension onolle design impovemens.

8 5. Oiginal Model Conolle Simulaion Simulaions wee aomplished using Runge-Kua 4h and 5h ode mehods. whie noise poessed by a suiable shaping file was used o simulae he disubane signal geneaed by he oad. he values of aeleaion of he oupu of he plan wee evaluaed as is RMS values. In he simulaions, he vehile speed was 6 km/h, duing.8 seonds, whih is equivalen o 8 mees un. I is impoan o highligh ha he onolle was applied in an oiginal plan, insead of he balaned model used as he base of onolle design. able shows simulaion esuls ompaing passive sysem and aive losed loop sysem pefomane. able. Simulaion esuls RMS values. Heave a. (m/s ) Pih a. (m/s ) Roll a. (m/s ) Passive sysem ive sysem I is han lea ha he poposed onol mehodology ahieved is goals. 6. Conlusions he modeling saegy pesened in his wok is a good appoah fo suspension sysems onolle developmen, whih has bough he following advanages: a simple model wih only fou sensos (aeleomee and fou auaos; a edued ode model, and onsequenly a edued onolle ode; he R appoah povides a bee use of he sysem enegy, sine i is based on Gammian appoah. 7. Refeenes Civellao, C.; amai, E. H.; 3, Sudy of Ride and Handling Impovemen of Pikup Ligh uk Vehiles Using Robus Semi-ive Suspension Conol ased on Magneo-Rheologial Dampes, Poeedings of COEM 3, he 7h Inenaional Congess of Mehanial Engineeing, S. Paulo, azil, COEM Cuz, J. J. da; ia,.; Cosa, E.. da; Sales, R. M.; 3, Conol and Opimizaion of he Eleomagnei Suspension Opeaion of a Maglev Vehile, Poeedings of COEM 3, he 7h Inenaional Congess of Mehanial Engineeing, S. Paulo, azil, COEM3-. Cuz, J. J. da; 996, Conole Robuso Mulivaiável, Edusp, São Paulo, azil. Doyle, J.C.; Sein, G.; 98, Mulivaiable Feedbak Design: Coneps fo a Classial/Moden Synhesis, IEEE ansaions on uomai Conol, vol. C-6, No., Febuay, pp Glove, K.; Doyle, J.C.; 989, Sae-spae ppoah on H Opimal Conol., Volume 35 of Leue Noes in Conol and Infomaion Sienes. Spinge-Velag. Ha,.; Youn, I.; Chen, H. H.; 996, Conol of Suspensions fo Vehiles wih Flexible odies Pa I: ive Suspensions, ansaions of he SME, Vol. 8, Sepembe, pp Mooe,.; 98, Pinipal Componen nalysis in Linea Sysems: Conollabiliy, Obsevabiliy, and Model Reduion, IEEE ansaions on uomai Conol, vol. 6-, Febuay, pp Rao, M. V. C.; Pahlad, V.; 997, unable Fuzzy Logi Conolle fo Vehile-aive Suspension Sysems, Fuzzy Ses and Sysems 85, 65-4/97, Elsevie Siene.V.,pp. -. Shelfhou, G.; 996, Model Reduion fo Conol Design, PHD hesis, Caholi Univesiy of Leuven - Faulei oegepas Weenshappen, enbegkaseel, -3 Hevelee (elgium). Simon, D. E.; hmadian, M.;, n lenaive Semi-aive Conol Mehod fo Spo Uiliy Vehiles, Poeedings Ins. Mehanial Enginees, Vol. 6 Pa D: Jounal uomobile Eng., ImehE, pp Sus, L..; Rohinha, F..; 5, Compaison of Conol Saegies fo Magneoheologial Vehile Suspension Sysems, Poeedings of XI DINME, 8 h Febuay-4 h Mah, Ouo Peo, MG, azil. amai, E. H.; 995, Esudo do Poblema de Conole da Suspensão de ens de la Veloidade Leviados Magneiamene, PHD hesis, Sae of São Paulo Univesiy USP DEM, S. Paulo, azil. sao, Y. J.; Chen, R.;, he design of an ive Suspension Foe Conolle Using Genei lgoihms wih Maximum soke onsains, Poeedings Ins. Mehanial Enginees, Vol. 5 Pa D: Jounal uomobile Engineeing, ImehE, pp Yamamua, S.; Ohnishi, K; Masada, E; 979, heoy of Conol Sysem of Eleomagneially Leviaed ogie uk, Eleial Engineeing in Japan, Vol. 99, No. 6, pp Responsibiliy noie he auhos ae he only esponsible fo he pined maeial inluded in his pape.

Velocity and Acceleration Simulation of a Vehicle with a Continuously Variable Power Split Transmission

Velocity and Acceleration Simulation of a Vehicle with a Continuously Variable Power Split Transmission Wold Aademy of Siene, Engineeing and Tehnology 55 009 eloiy and Aeleaion Simulaion of a ehile wih a Coninuously aiable Powe Spli Tansmission A. Babaei, N. Choupani Absa A oninuously vaiable ansmission