Combined state and parameter estimation of vehicle handling dynamics

Transcription

1 Loughborough Unverty Inttutonl Repotory Combned tte nd prmeter etmton of vehcle hndlng dynmc h tem w ubmtted to Loughborough Unverty' Inttutonl Repotory by the/n uthor. Ctton: BES, M.C. nd GORDON,.J.,. Combned tte nd prmeter etmton of vehcle hndlng dynmc. IN: Proceedng of of the 5th Interntonl Sympoum on Advnced Vehcle Control (AVEC), Ann Arbor, USA, Augut, pp Addtonl Informton: h conference pper. Metdt Record: Veron: Accepted for publcton Publher: c Socety of Automotve Engneer of Jpn (JSAE) Plee cte the publhed veron.

2 h tem w ubmtted to Loughborough Inttutonl Repotory ( by the uthor nd mde vlble under the followng Cretve Common Lcence condton. or the full text of th lcence, plee go to:

3 44 COMBINED SAE AND PARAMEER ESIMAION O VEHICLE HANDLING DYNAMICS Mtthew C Bet nd mothy J Gordon Loughborough Unverty, UK Abtrct h pper conder n extended form of the wellnown Klmn flter oberver, to recontruct dynmc tte from mll enor et, but lo to rpdly dpt elected prmeter n the nonlner dynmc model whch le t the hert of the oberver. A generc procedure decrbed for contructng the extended Klmn flter n uch wy tht ny combnton of model prmeter cn be dentfed. he tudy crred out n multon, ung two dfferent vehcle dynmc model, one to ct the tet vehcle, the other formng the nucleu of the oberver. he umpton tht whle n-vehcle tetng mot derble for provng mny controller lgorthm, here we need true reference tte nformton, to exmne Klmn flter ccurcy. A number of experment re crred out to prove the ytem dentfcton properte nd lo to compre t performnce wth more conventonl Klmn flter, bed on lner hndlng model. he reult demontrte hgh level of performnce nd gnfcnt robutne to degn prmeter uch prmeter dptton peed nd ntcpted enor noe. Mot gnfcntly, the oberver lo operte well nd cpble of prmeter dptton when model nd enor covrnce nformton not vlble uully retrctng fctor n prctcl Klmn flter etmtor degn. he only gnfcnt cvet tht we re buyng excellent dynmc trcng from mll enor et, t ome computtonl expene.. Introducton Recent reerch h gven re to number of modelbed vehcle control trtege, whch rely on utble low order dynmc model nd/or tte oberver. Method ext for model dentfcton (eg Bet nd Gordon, 999) nd lo oberver degn, (eg Kence nd D, 997) but reerch by the uthor ugget tht the mot effcent route to optml dynmc chrcterton by combnng the two procee. Rel-tme etmton precrbe need for low order model, nd Klmn flter method re ttrctve n explotng thee. However, the performnce nd optmlty of the Klmn flter fundmentlly retrcted by model ccurcy. hu the uggeted oluton for the vehcle hndlng ytem tht () complete model poble hould be ncorported n the flter, nd (b) ey model prmeter hould be dentfed low-vryng tte. he frt requrement led to the ue of nonlner model, nd hence n extended Klmn flter (EK). h cn be degned n the form : xˆ & = f ( x, ˆ η) + K( h( x, ˆ η)) () for tte x, prmeter η nd enor, nd where f nd h re generl nonlner model for the tte dervtve nd enor repectvely. h tructure lend telf to meet the econd requrement (b) bove, by extenon of the tte vector to nclude ubet of prmeter η. Although f unnown for thee prmeter tte, the expectton tht they wll be drven by the econd term n equton (), to mprove the nnovton error (enor error, h( x, ˆ η) ), provded the optml feedbc mtrx K utbly formulted. he pper thu precrbe generc oberver/dentfer degn proce whch ccommodte ny choce of η. In ccordnce wth requrement () bove, the model h three degree of freedom nd four ndependent combned-lp Pcej tyre model; pror experence ugget tht the model mut nclude the combned effect of lterl nd longtudnl to vod tedy-tte prmeter etmton error (ee Bet nd Gordon, 998). However, thee tyre model me the ytem too complex for rel-tme etmton, o the emph here to explore the lmt of ccurcy nd flexblty of the new lgorthm, rther thn provde n mmedtely relble oberver oluton. he clcl Klmn flter combne enor nd model nformton, ung both of thee ource to contruct tte vector. In th reved tructure, vrble model prmeter cue ome hft wy from the model n ndependently relble ource, but provde very flexble wy of formng combned oberver/dentfer whereby both tte nd prmeter re vred to bet expln the enor.. Modellng

4 . Source Model he tudy crred out ung reference ource model to provde true tte trjectore nd enor meurement. h model multe full order moton of rgd vehcle body, wth ndependent upenon freedom, though wheel-hop mode re uppreed through the umpton of nert-le wheel. A Pcej tyre model mplemented n both longtudnl nd lterl xe, ncorportng frcton lmtng ellpe, nd the tyre force hve nonlner lod nd cmber dependent frcton. Longtudnl dynmc mode nclude mple engne torque model, nd rod lod. he prece equton for the ource model re omtted here, for brevty, but lo we wll ee n Secton 4. becue the detl nd even to ome extent the ccurcy of the ource dt of econdry mportnce; the tudy hould revel mlr reult for ny utbly formulted hgh order model, or ndeed for n ctul tet vehcle.. Oberver Model he oberver model formulted from the populr three degree of freedom yw/delp/roll model, decrbed ung vehcle centred SAE xe wth roll moton umed to be contrned bout n nclned roll x (gure ). A fourth, longtudnl freedom decrbe forwrd peed. he Newton nd Euler equton of moton re therefore for delp, Mv& + Mhp& = for forwrd peed, for yw, for roll, I =, 4 y Mur Mu& = x + Mrv + Mhrp zz r& I xz =, 4 p& = b c y =, = 3,4 I r& xz + Mhv& + I p& xx = Mhur ( B f + Br ) p + ( Mgh K f K r ) φ + ( h h f ) y + ( h hr ) & θ = p =, y = 3,4 y () (3) (4) Here the force x, y ctng on the body, re gven from tyre force : x, y, = = tx, ty, coδ coδ ty, tx, nδ, nδ, x3,4 y3,4 = = tx3,4 ty3,4 (5) (6) nd the tyre force re derved ung the Pcej mgc formul Ω (ee for exmple Mllen nd Mllen, 995) : Cα tnα K x ( w u coα ) = Ω tx, tx,, Sp (7) μz μz u coα where v br α = α = + δ u rc v α 3 = α 4 = u ( ) Vertcl lod trnfer due to roll then ncluded by z, z3,4 cmg = ( b + c) bmg = ( b + c) ± ± ( h K θ B p) y, ( h K θ B p) y3,4 he oberver model therefore completely decrbed n term of fve tte vrble, 3 contnt prmeter, nd two nput, ummred n ble. u p roll x v h f h rf t f r rf rr f r h h b r f h r h rr gure : Klmn flter model x ytem nd roll x geometry he Klmn flter requred to etmte the fve tte, nd dpt number of the prmeter to mot ccurtely decrbe the vrton of et of enor. hee mut therefore fully repreent the vehcle dynmc, nd lthough rte enor re ncluded for ome experment, the bc enor et mde up of four ccelerometer. hee re orented n two pr, one pr bove the front xle t heght h f =.3m bove the roll x, the other bove the rer xle t h r =.5m. Wthn ech pr, the frt enor lgned longtudnlly nd the econd lterlly. hey re modelled 4 ( v + h p) = u& r f + ε = v& + h p& + br& f + ru + ε 3 = u& r( v + hr p) + ε3 = v& + h p& cr& + ru + ε r 4 c (8) (9) ()

5 Unmodelled meurement noe multed ung n ndependent Gun whte noe gnl wth equl mgntude on ech enor : (,[ 8 ] ) ε N ρ () where ρ =.. hu the RMS noe cn be nterpreted % of reltvely hgh lterl ccelerton of 8m/. Stte, x u forwrd velocty (m/) v delp velocty (m/) p roll ngulr velocty (rd/) r yw ngulr velocty (rd/) θ roll ngle (rd) prmeter, η (defult vlue) I xx roll moment of nert (6 gm ) I zz yw moment of nert (3 gm ) I xz roll/yw cro moment of nert (8 gm ) m m (83 g) b longtudnl Dtnce of C of G to front xle (. m) c longtudnl Dtnce of C of G to rer xle (.4 m) h C of G heght bove roll x (.347 m) h ground plne to roll x dtnce below CofG (.3 m) h f roll x to x x vert. dtnce t front xle (.45 m) h r roll x to x x vert. dtnce t rer xle (.5 m) t f front trc (.5 m) t r rer trc (.5 m) K f front roll tffne (34 Nm/rd) K r rer roll tffne (.5 Nm/rd) BBf front roll dmpng (7 Nm/rd) BBr rer roll dmpng (7 Nm/rd) Sp Pcej tyre model hpe coeffcent (.74,.4,., -.) C α zero lterl lp cornerng tffne (35 N) K x zero longtudnl tyre lp rte (35 N) μ tyre frcton coeffcent (.) Input, u δ front wheel teer ngle (rd) w front wheel peed referenced to forwrd peed (wheel ngulr velocty multpled by rollng rdu, m/) ble : Model nomenclture 3. ormulton of Klmn lter 3. Lner Klmn lter A b for evlutng the extended Klmn flter we frt derve the equvlent lner tme-nvrnt form. he lner model formed from equton (),(4) nd (5) for the reduced tte vector z = [ r,v,p, φ], wth u umed contnt nd replcng the nonlner tyre model wth lner (cornerng tffne) multpler of α from equton (8). Exponentl dcretton of thee lner equton then gve model A, B, C, D whch cn be relted to the true ource tte z by z y + = Az = Cz + Bδ + ω + Dδ + υ () Here proce nd meurement error re denoted ω nd υ repectvely, nd n optml flter cn ely be derved, provded the followng expectton E( ) pply: E E ( ω ) =, E( υ ) =, ( ω ω ) =, E( υ υ ) =, ( j) j he mplet form of the Klmn flter lo ume E( ω υ ) =, but th cn not be expected here; for ccelerometer the enor model C, D correlte wth A, B, nd th enure tht ω nd υ hre common modellng error component. he Klmn flter thu pecfed n the more generl form (eg from Gelb, 974), ( A KC) zˆ + ( B KD) Ky + = δ z ˆ + (3) wth the (tme-nvrnt) gn mtrx K found through clculton of the ntcpted tte error covrnce mtrx P, whch the oluton to the lgebrc Rcct equton : P = GPG GPC [ CPC + R] CPG + Q (4) K = APC + S CPC + R where ( )( ) G = A SR Q = Q SR S C j ( ) (5) ω Q S nd e =, E e e = (6) υ S R 3. Extended Klmn lter he extended flter employ the fully nonlner model of equton () (), whch cn be wrtten for contnuou tte propgton, nd mpled enor dt n the generl form, x& ( t) = f ( x ( t), η, u( t) ) + ω() t (7) y = h x, η, u t + υ ( ()) Gelb (974) propoe etmton of x ˆ by the ue of truncted ylor ere for f ( x() t ) n the contnuou model nd tte error covrnce mtrx etmte. he oberver then derved n contnuou-dcrete form, ung contnuou model propgton combned wth mpled meurement whch provde dcrete tte

6 correcton. he fnl form then fully dcreted to prctcble form. Aumng tme-nvrnt noe covrnce mtrce, nd gn ncorportng the generl form for correlted model nd enor error, (whch expounded n Bet et l, ) the reultng lgorthm provde the followng model for tte nd tte error covrnce mtrx dervtve : x& ˆ() t = f ( xˆ () t, η, u) (8) P& t = G t P t + P t G t + where () () () () () Q G () t SR H = (t) (t) Q = Q SR S nd the enor provde flter nd tte correcton t ech dcrete tme tep, to mprove the vrble etmte from the (-) to (+) form : xˆ P where K P ( + ) = xˆ ( ) + K [ y h( xˆ ( ), η, u) ] ( + ) = [ I K H( xˆ ( ) )] P ( ) = ( ) H ( xˆ ( ) )[ H( xˆ ( ) ) P ( ) H ( xˆ ( ) ) + R] (9) In the bove, mtrce nd H re Jcobn reultng from the ylor ere expnon, evluted t the tte etmte : f ( x(t), η, u(t) ) (t) = x(t) x(t) = xˆ (t) () h( x(t), η, u(t) ) H(t) = x(t) x(t) = xˆ (t) hee equton conttute the complete extended Klmn flter n contnuou form, wth dcrete enor correcton. Provded utbly low mplng ntervl choen, the dcrete tte updte cn be pproxmted from equton (8) by Euler ntegrton xˆ xˆ + x& (t), P = P + P& (t) () = ˆ + + It hould be noted t th pont tht (t) nd H(t) would normlly be decrbed nlytclly, nd ndeed thee dervtve cn theoretclly be derved for the model ued here, but the tyre nonlnerte me the reultnt formule prtculrly complex. In prctce, equton () evluted numerclly, wthn the tte updte loop, by re-evlutng f nd h for mll chnge n ech of the fve tte. 3.3 Adptng Model Prmeter Gven the generl form for f ( x(t) ), the model cn redly be pecfed n term of n extended tte vector χ (t), to nclude ny number of the prmeter : ( x (t), η, u(t) ) + ( ) ω(t) x (t), η, u(t) ω(t) x& (t) f χ& ( t) = = () η& (t) f If the orgnl model vld, ll the prmeter re tme nvrnt, o f ( x(t), η, u(t) ) = nd the rte of chnge of thee prmeter tte gven by ω (t). he new model error covrnce nd model/enor cro covrnce mtrce become Q Qχ = E( ωω ) E( ωω ), S E( ωω ) χ S = E( ωυ 4. Smulton Experment ) (3) 4. Degn Proce or ll the multon experment, error covrnce mtrce Q, R nd S re etmted from ngle reference multon, degned to excte the ytem t hgh mpltude, over rnge of frequence. he throttle nd teer nput for the ource model re llutrted for th 5 econd tet n gure, long wth the wheel velocty output from the ource model, whch ued n nput to the etmtor (w). he tet degned to be chevble n rel vehcle, o the rndom teer nd throttle ecton re generted ung whte noe whch bndlmted wthn phyclly relble rnge, to 5Hz. 4 teer ngle, δ (deg) normled throttle wheel peed, w (m/)/ gure : et nput for noe mtrx dentfcton Equton () nd (7) re ppled to the multon reult to etblh tme htore for ω nd υ, nd Q, R nd S re then formed eprtely for the lner nd extended Klmn flter, n n obvou wy, from equton (6). he oberver re teted on ndependent tet nput, wth mplng ntervl =. ec. 4. Non-dptve Oberver Performnce frt exmned wth no prmeter dptton ( η = ), for δ = 4 tep teer, wth zero throttle nput, t 8m/ (gure 3). he plot how ource

7 model tme htore gnt reult for both degn of Klmn flter nd lo trce from the Klmn flter model multed wthout enor feedbc, for th fve econd tet. he lner Klmn flter etmte re poor for th frly evere teer nput. he umpton tht both peed orwrd peed, u (m/) 5 () ource model; ll of the EK reult mght be mproved by utble prmeter dentfcton n t model. One mmedte beneft of EK over the lner flter llutrted n plot (g); vertcl lod n ntermedte vrble n the EK model. Although tedy-tte error re gn een n th predcton, the blty of EK to etmte vlue uch th led to more complete dynmc chrcterton of the vehcle. Converely, the lgorthm lo preent poor reult n t recontructon of roll rte plot (c). h problem, lo een n erler tude (ee Bet et l, ) cued by the tructurl dfference between ource nd EK roll model, cung lrge expected model error for p n the Q mtrx, nd hence llowng greter trnmon of enor noe nto the tte etmte. Sdelp, v (m/) Roll rte, p (rd/) (b) Source model Extended Klmn flter Lner Klmn flter EK model (c) Long. enor, (m/) Lt. enor, (m/) (e) (f) Yw rte, r (rd/) (d) (g) gure 3 : Non-dptve hee oberver error reult cn be reduced by ncreng the EK nd cornerng tffne re contnt, cue lrge error model order further, but gnfcnt mprovement cn lo whch re prtculrly evdent n plot (b) for delp, nd be cheved through the ddton of roll rte enor, n the trcng of lterl ccelerton plot (f). we wll ee n Secton 4.4. Wth the nonlner model, the EK more ccurte n, r nd for much of the tet, n v. hee reult re 4.3 Introducng Prmeter Adptton good n pte of the very hgh, lthough reltc, noe A gnfcnt error from the EK n gure 3 the level whch hve been mpoed on the enor. Alo overetmton of pe delp. h cued by error note the dotted lne tme htore, whch how tht lrge n the lp to tyre force reltonhp, o could be tedy-tte dfference ext between the EK nd mproved by modfyng the tyre model; n obvou Vertcl lod, z, (N) 6

8 cnddte for dptton the be cornerng tffne on whch the Pcej model depend : C α. In extendng the EK to nclude prmeter tte however, we frt need to provde n etmte for the ddtonl error mtrx component n equton (3). h done by ddng bnd-lmted whte noe to the prmeter η durng the reference multon decrbed n Secton 4., uch tht η t = η + ε t () () where for the th prmeter tte, ε N λη ( ) ( ), ( ) An etmte for ω (t) cn then be found by utbly ccurte numercl dfferentton, ω (t) = ε& ( t) from equton (); th expln the need for ome bndlmtton on ε, nd for thee experment lmt of 5Hz w ued. Mtrce Qχ nd S χ re contructed gn from equton (6), now ung e = [ ω, ω, υ ]. gure 4 how the dpted cornerng tffne, long wth the new EK predcton for v when the tep teer tet repeted. he dptton of C α cued by, nd ct to reduce the ner tedy-tte error n whch Sdelp, v (m/) Adpted C, (N/rd) Source model Extended Klmn flter Lner Klmn flter gure 4 : EK reult for n oberver wth Cα etmton cn be een fter the tep between.5 nd econd n gure 3(f). he reult lrge mprovement n etmton of v. h n excellent reult, but the ccurcy of v turn out to be entve to the choce of λ, whch w et rbtrrly to. here. Alo, the dptton trggered here by n mpulve nput whch cue lttle dynmc exctton n the enor tht drve the dptton proce. he concluon tht dptton would be more robut under contnuou dynmc exctton. 4.4 Prmeter Identfcton Wth dynmc nput, EK cpble of multneouly dptng multple prmeter. Here we conder etmton of the x nd z x locton of the vehcle centre of grvty, o η = [ h, b]. A 5Hz bnd-lmted Gun gnl gn ued for the teer nput here wth n RMS of.5 nd contnt 75Nm engne torque ppled. η then ntled to npproprte vlue, [.7,.], nd gure 5 how the dptton for rnge of choce of λ. Adpted h, (m) Adpted b, (m) Roll rte, p (rd/) Source model EK, λ =.5 EK, λ =. EK, λ =.5 (b) Source model EK, λ =.5 () (c) 8 9 gure 5 : EK reult for C of G dentfcton he concluon tht dptton wor very well, nd rpdly under thee condton. he proce lo robut to vrton of λ, wth potve correlton between peed of repone nd λ, nd lo between tndrd devton of the dpted prmeter, nd λ. h mght reonbly be expected, gven the nfluence tht λ h ettng the reltve blnce between expected enor nd model ( f = ) ccurcy n the oberver degn proce; hgher λ promote greter mgntude of enor feedbc.

9 gure 5(c) gve n exmple of how the tuned prmeter lo mprove tte predcton howng the frt nd lt two econd of the etmton of p. In th tet we hve ncluded n ddtonl roll enor lthough th w not necery to cheve correct dptton of h nd b. he ddtonl enor generlly mprove roll predcton compred wth the gure 3 reult, nd we lo ee roll ccurcy mprove wth tme, h nd b dpt. 4.5 Noe Mtrx Dependence A t h been decrbed nd ued o fr, the dptve EK could be ppled to predct the repone of rel vehcle (lbet not n rel-tme), except n tht we hve umed detled pror nowledge of the error correlton n Q, R nd S. In prctce thee re very dffcult to obtn, o EK performnce under nomnl noe mtrce lo of nteret. he centre of grvty experment therefore repeted, th tme wth Q nd R determned dgonl mtrce mply proportonl to expected tte nd enor RMS mpltude on the noe dentfcton tet, nd wth S =. Q, mod = λ ω, =,5 Q R, j, j = ω ( 5), = λmod υ, > 5 j =,5 Conderng two lterntve for the new nomnl error fctor λ mod, nd gn ung λ =., the dptton reult compred wth the tndrd Q,R,S reult n gure 6. (or ll three reult we hve reverted here to the tndrd four ccelerometer enor et.) Remrbly, the nomnl noe ettng lo nduce good dptton of centre of grvty, wth no dfference n the overll etmte for b, nd only lght vrton for h lo the ucce of dptton telf not entve to λ mod. Adpted b, (m).5 Source model.5 Identfed Q,R,S Nomnl Q,R,S, λ mod =.5 Nomnl Q,R,S, λ mod = gure 6 : Comprng nomnl wth correlted noe mtrce for n EK wth C of G dentfcton or th experment, ccurcy n predcton of the tte nd enor ummred n term of RMS error n ble. Here we ee tht error n etmtng the enor re reduced by ncreng the dptton rte. (Under th EK degn, t the lower ettng, λ mod =.5 whch generte fter dptton, becue expected enor modellng error re lower.) Crtclly, n mprovement n tte recontructon lo een the enor model mprove. Note tht the bet nomnl tet provde RMS error whch re even lower thn for the tndrd Q,R nd S. h hppy conequence of low noe expectton, nd we re comprng flter wth qute dfferent umpton, but the reult neverthele encourgng. QRS Nomnl, Nomnl, Identfed tte/enor λ mod =.5 λ mod =.5 u v p r q ble : Comprng tte nd enor RMS error durng dptton under nomnl nd tndrd noe mtrce. Adpted h, (m) Mult-prmeter Optmton A fnl nvetgton of the potentl of the new lgorthm, we conder ncreng the number of dentfed prmeter. h only feble f lrger number of enor lo condered, clerly there cope for the lgorthm to become poorly condtoned, nd hence untble, o here we hve ugmented the et of four ccelerometer wth roll nd yw rte enor. gure 7 demontrte multneou etmton of m, roll nert, yw nert nd longtudnl m centre poton certnly n mbtou venture, wth ll

10 four prmeter ntled to rdculouly low vlue, η ( ) = [,,,. 5] Adpted prmeter (normled) b.3 I xx -3 I zz -3 m -3 gure 7 : An exmple of mult-prmeter dentfcton ung EK he reult re mpreve, nd t nteretng to ee tht the prmeter re dptng t dfferent rte through the ten econd tet. I zz nd b lo pper to temporrly ettle t ntermedte vlue pobly locl mnm up to bout four econd. One dppontng footnote however, tht no utble vlue for λ mod could be found to me th dptton wor wth nomnl noe mtrce pont whch doe erve to emphe the mportnce of error expectton n th mult-dmenonl proce. pplcton n off-lne tte nd prmeter recontructon for exmple to t vehcle degn n the motorport feld. he mot lely future for on-lne development by electve model reducton for exmple by conoldtng the tyre force model to mple generc nonlner lod-dependent functon of mll number of prmeter. Reference. Bet M.C. nd Gordon.J. (999), A Rndomed Integrl Error Crteron for Prmetrc Identfcton of Dynmc Model of Mechncl Sytem IMechE Journl of Sytem nd Control Engneerng (Prt I), Vol 3, pp Kence, U. nd D, A. (997), Obervton of Lterl Vehcle Dynmc Control Engneerng Prctce, Vol 5, No 8, pp Bet M.C. nd Gordon.J. (998), Rel-me Stte Etmton of Vehcle Hndlng Dynmc Ung n Adptve Klmn lter proceedng from the 4 th Interntonl Sympoum on Advnced Vehcle Control (AVEC), Ngoy, Jpn, September 998, pp Mllen W.. nd Mllen D.L. (995), Rce Cr Vehcle Dynmc SAE Interntonl. 5. Gelb, A. (974), Appled Optml Etmton, MI Pre, Cmbrdge M. 6. Bet M.C., Gordon.J. nd Dxon P.J. (), An Extended Adptve Klmn lter for Rel-tme Stte Etmton of Vehcle Hndlng Dynmc Vehcle Sytem Dynmc (ccepted for publcton Dec 999). Concludng Remr he new oberver / dentfer wor well wthn the multon envronment condered here, nd t robutne to degn prmeter vrton me ucce n rel-vehcle pplcton lely. Some good reult wth nomnl noe mtrce lo mprove the lelhood of prctcl vblty, epeclly f thee too re dpted method ext n Gelb (974). Long computer proceng tme me on-lne pplcton mpoble gven the model tructure ued here, but the ytem tll h cope for mmedte