Saety and Relablty o Dstrbuted Embedded Systems Techncal Report ESL 04-01 Smulaton o Vehcle Longtudnal Dynamcs Mchael Short Mchael J. Pont and Qang Huang Embedded Systems Laboratory Unversty o Lecester [ESL04-01A - 11 OCTOBER 2004]
Project summary Ths techncal report s one o a seres (lsted n ull below). Together these reports descrbe a complete hardware-n-the-loop (HIL) smulaton that reproduces the behavour o a passenger car travellng down a motorway. In the smulaton, the speed and poston o the car are determned by an adaptve cruse control system mplemented usng one or more embedded mcrocontrollers. The test bed s ntended to be used to assess and compare derent sotware archtectures or use n dstrbuted embedded systems, partcularly those or whch hgh relablty s a key desgn consderaton. Full lst o reports n ths seres Avalable now: ESL04/01 Smulaton o Vehcle Longtudnal Dynamcs ESL04/02 Smulaton o Motorway Trac Flows ESL04/03 Development o a Hardware-n-the-Loop Test Faclty or Automotve ACC Implementatons Forthcomng: ESL04/04 Control Technologes For Automotve Drve-By-Wre Applcatons ESL04/05 10-Node Dstrbuted ACC System: Co-Operatve Implementaton ESL04/06 10-Node Dstrbuted ACC System: Pre-Emptve Implementaton Acknowledgements The work descrbed n ths report was supported by the Leverhulme Trust (F / 00212 / D) [ESL04-01A.DOC - 11 OCTOBER 2004]
Contents 1. Introducton...1 1.1 The Two-Wheel Tracton Model...1 1.2 System Block Dagram...2 2. Equatons O moton...3 2.1 Vehcle load dstrbuton...3 2.2 Drag orces...4 2.3 Tractve propertes o the tyre/road nterace...4 2.4 Wheel dynamc equatons...6 3. Powertran Model...8 3.1 Engne torque curve...8 3.2 Engne dynamcs...10 3.3 Gearbox model...11 4. Brake System Model...12 4.1 Servo actuated brake system...12 4.2 Brake rcton characterstc...13 5. Parameter Selecton...14 6. Smulaton Results...15 6.1 Acceleraton Perormance...15 6.2 Brakng Perormance...16 7. Conclusons...17 [ESL04-01A.DOC - 11 OCTOBER 2004]
1. Introducton Ths document descrbes the development o a model o the longtudnal dynamcs o a passenger car. The model descrbed here provdes a sutably detaled descrpton o a host vehcle, whch s controlled by a dstrbuted embedded system n a hardware-n-the-loop realtme test aclty. The motorway smulaton envronment n whch the testng s to take place s detaled n an assocated techncal report (ESL04/02), whle the development o the nterace to the embedded system and the overall system ntegraton can be ound n the report ESL04/03. 1.1 The Two-Wheel Tracton Model For a smulaton o vehcle dynamc perormance, the gross vehcle dynamcs and the tyre/wheel dynamcs must both be consdered. These can both be captured by smpled lumped mass models, and may consst o sngle-wheel versons, two-wheel versons, or ull our-wheel models or cornerng as well as acceleraton/brakng analyss [Gllespe 1992]. In order to smulate the dynamcs o a passenger vehcle whlst drvng down a motorway, the lateral orces actng upon the vehcle may be, n general, neglected. Ths s because the motorway systems n many countres worldwde have been desgned to be as straght as possble, and the orces nvolved durng steerng to change lanes have a relatvely small eect, and act or only small perods o tme, when compared to the much larger longtudnal orces nvolved when crusng at hgh speed. Ths s n stark comparson to, or example, a racecar smulaton where the acceleraton, brakng and cornerng dynamcs are much more coupled and essental to produce a realstc model. For ths reason, a ull our-wheel model s consdered unnecessary or the purposes o ths smulaton. Passenger vehcles o the type under smulaton here are generally bult to be as stable as possble wth a centre o gravty (COG) as close to the centre o the car as possble. However, the eects o longtudnal load transer are stll present, and or ths reason a smpled two-wheel dynamc model s best suted to descrbe the dynamcs. Fgure 1 shows a schematc o the model. It can be seen rom the gure that car s represented by a lumped mass m, whch has a orward velocty V n the X-drecton. The vehcle s wheels have a rotaton rate ω, a rollng radus o R, and a polar moment o nerta J, where the subscrpt =,r descrbes ether the ront or rear wheel. A coecent o rcton µ exsts between the wheel s tyre and the contact surace. The dmensons B and C represent the dstance between the vehcles centre o mass and the ront and rear axles respectvely, wth the wheelbase represented by the dmenson L. The road gradent θ ndcates the road s nclnaton rom the normal. [ESL04-01A.DOC - 11 OCTOBER 2004] PAGE 1
Fgure 1: Schematc O The Two-Wheel Tracton Model 1.2 System Block Dagram The dagram below shows a hgh-level system block dagram outlnng the man elements o the vehcle model, and the system varable dependences between them. The man nputs to the model are the throttle and brake settngs, and the man outputs are the vehcles velocty, and the ront and rear wheel veloctes. Addtonally, there are three other parameters o nterest: the road gradent, road condtons and wnd speed. Fgure 2: Vehcle System Block Dagram It can be seen rom the model that the many system elements are tghtly coupled and hghly nteractve, and many are also non-lnear. It can also be seen n the model that the vehcle s ront-wheel drve. Ths report wll rst consder the gross vehcle dynamcs that govern the generaton o vehcle speed, and consder each block n turn backward toward the systems nputs. Once each element had been descrbed, sutable values are then determned or each system parameter. Followng ths, a sutable methodology or solvng the equatons n real-tme s then developed. [ESL04-01A.DOC - 11 OCTOBER 2004] PAGE 2
2. Equatons O moton The prmary orces o nterest n the smulaton o vehcle longtudnal dynamcs are the orces actng about the X-axs, actng on the vehcle through the tyre/road nterace. These orces act to both accelerate and brake the car, and are proportonal to the normal orce Z at the tyre/road contact ponts: FX = µ FZ, = r, Equaton 04/01/A The total orce actng upon the vehcle body due to these orces s smply the summaton o the reacton orces at the ront and rear contact ponts: F X T = 2 F + 2 F X X r Equaton 04/01/B It s straghtorward to choose the orward velocty V and tyre rotatonal veloctes ω r and ω as the man dynamc states to be smulated. Consderng the orward velocty V, summng the orces actng on the car s mass m: V FX T + g sn( θ ) Fd ( V ) = m. Equaton 04/01/C Where F XT s obtaned rom equaton B, g sn(θ ) s the gradent term and F d (V) s the orce due to drag, dscussed n secton 2.2. In order to evaluate equaton B, we must apply equaton A and calculate the normal orces actng at each wheel; n order to do so we requre knowledge o the load dstrbuton at each wheel. 2.1 Vehcle load dstrbuton The statc load dstrbuton s smply a uncton o vehcle geometry and the grade angle, and s obtaned by summng the orces at each contact pont. There s, however, a dynamc load dstrbuton that can transer load between the ront and rear wheels as the vehcle accelerates and brakes. Examnng the geometry, the ollowng two equatons may be ormulated to descrbe the ront and rear normal orces: FZ C H. = mg cos( θ ) + sn( θ ) mv L L H L Equaton 04/01/D [ESL04-01A.DOC - 11 OCTOBER 2004] PAGE 3
FZ r B H. = mg cos ( θ ) sn( θ ) + mv L L H L Equaton 04/01/E The rst terms on the rght o these equatons are the statc load terms, whlst the second, acceleraton dependent terms, are the dynamc loadng terms. In agreement wth ntuton, whlst the vehcle s acceleratng the load s transerred to the rear wheels, and durng brakng t s transerred to the ront wheels. 2.2 Drag orces The equatons o moton contan a drag term F d (V) that s both velocty dependent and acts to lmt the vehcle lnear maxmum speed. The drag term s a combnaton o both aerodynamc resstance orces and rollng resstance orces, whch are both unctons o the orward velocty V [Gllespe 1992]: 1 2 Fa = ρacd ( V ), Fr = mgc 2 F = F + F d a r r ( V ) Equaton 04/01/F Where C r s the rollng resstance coecent, A s the rontal area o the vehcle and C d s the aerodynamc drag coecent. The prevalng wnd speed may be added to the vehcle speed beore evaluaton o the aerodynamc drag. The densty o ar, ρ, can be taken to be equal to 1.23 Kg/m 3. Now that the gross orces actng on the vehcle body have been examned, the nteracton between the vehcle tyre and the road surace that produces the tractve orces must be consdered. 2.3 Tractve propertes o the tyre/road nterace Evdence has shown that the eectve coecent o orce transer µ s a consequence o the derence between the orward velocty o the car V and the rollng speed o the tyre, ωr. Although many derent parameters can and do aect the rcton couplng, or a model such as ths t s best descrbed n terms o the wheel slp rato λ, whch s dened below [Olsen et al. 2003]: V ωr λ =, = r, max( V, ω R ) Equaton 04/01/G [ESL04-01A.DOC - 11 OCTOBER 2004] PAGE 4
The maxmum uncton n the denomnator o the above equaton allows ts use or both acceleraton and brakng models. A slp rato equal to zero means that the orward velocty and tyre rollng speed are equal, whch mples an absence o ether engne or brake torque. A postve slp rato mples that the tyre has a postve nte rollng velocty, and the car posses a greater nte orward velocty. A negatve slp rato mples that the car has a nte orward velocty and the tyre has a greater equvalent postve rollng velocty. At each extreme,.e. +1 and 1, the wheel s ether locked at zero speed, or spnnng wth the vehcle at zero speed. When both tyre and car velocty are equal to zero, the slp s mathematcally undened, and s taken to be zero or smulaton purposes. Expermental studes have produced several clearly dened rcton/slp characterstcs between the tyre and road surace or a varety o derent drvng suraces and condtons [Stchn 1984]. For the purposes o smulaton, our types o road condton are to be modelled: Normal: The road s dry and maxmum tracton s theoretcally possble. Wet/Ranng: Overall tracton s reduced by about 20 %. Snow: Un-packed snow les on the road surace. Maxmum tracton reduced by 65 %. Ice: Packed rozen snow and black ce le on the road surace. Hghly dangerous maxmum tracton reduced by 85 %. Graphcally, these our condtons are shown n gure 3 below: Slp/Frcton Graph 1 Tyre Frcton (u) 0.5 0-0.5 Normal Wet Snow Ice -1-1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 Slp Rato Fgure 3: Plot O Frcton/Slp Characterstcs For a gven set o tyre test data, several analytcal models exst to analyse and smulate these relatonshps, the most popular o whch s the Pacejka magc model [Bakker et al. 1989]. The Pacejka model s dened mathematcally as ollows: [ESL04-01A.DOC - 11 OCTOBER 2004] PAGE 5
µ ( λ ) = D sn( C arctan( Bλ E( Bλ arctan( Bλ )))), = r, Equaton 04/01/H The model denes n excess o 40 constants that are determned rom the gven set o expermental data, and the overall model coecents B, C, D and E are then calculated rom a combnaton o these constants. The requred model coecents to produce the slp/rcton relatonshps as shown n the graph above were determned to be as shown n Fgure 4: Pacejka Coecents Surace B C D E Dry Tarmac 10 1.9 1 0.97 Wet Tarmac 12 2.3 0.82 1 Snow 5 2 0.3 1 Ice 4 2 0.1 1 Fgure 4: Equvalent Model Parameters In general, the model produces a good approxmaton o the tyre/road rcton nterace, wth the only notable anomaly occurrng when the road surace s covered n snow. Under these condtons, t has been noted that the peak rcton coecent occurs when the slp s at ether extreme o the range (.e. 1,1). Ths s due to the snowplough eect and s not well captured by ths model [Olsen et al. 2003]. The computatonal complexty nvolved n usng a more accurate snow model s not requred or the purposes o ths smulaton. In order to calculate the slp, and thus determne the eectve tractve orces, the equatons o moton that allow calculaton o wheel velocty wll be examned. 2.4 Wheel dynamc equatons In a smlar manner to developng the gross equatons o moton or the vehcle body, summng the torques about each wheel enables the ollowng equaton or wheel acceleraton to be wrtten:. τ e + τ r τ b τ d ( ω ) ω =, = r, J Equaton 04/01/I Where τ e s the torque delvered by the engne to each wheel and τ b s the torque appled to each wheel due to the brakes. τ r s the reacton torque on each wheel due to the tyre tractve orce: [ESL04-01A.DOC - 11 OCTOBER 2004] PAGE 6
τ r = R FX, = r, Equaton 04/01/J I C s the vscous rcton co-ecent o the th wheel, the vscous rcton torque τ d(ω) can be wrtten as: τ d ( ω ) = ω, = r, C Equaton 04/01/K Solvng these equatons and ntegratng them wrt tme allows a smulaton o the gross longtudnal moton o the vehcle and each o ts wheels. In order to determne the torque delvered to each wheel n equaton I, by the engne and the brakng system, t s necessary to develop urther model equatons that descrbe the vehcle s Powertran and brakng systems. [ESL04-01A.DOC - 11 OCTOBER 2004] PAGE 7
3. Powertran Model The vehcle Powertran ncludes the engne tsel, the torque converter, gearbox and nal drve derental to delver the torque to the ront wheels. When the car s n moton, t s assumed that the engne speed s equal to the wheel speed scaled by the current gear and nal axle drve transmsson system. However, when the car s statonary or at very low speeds n 1 st gear, t must be assumed that the torque converter s n operaton and a nte amount o lud slp s occurrng. Under these condtons the actual torque delvered s determned by a combnaton o the converter slp, the throttle settng and wheel velocty. Snce the smulaton o the vehcle at low speed s generally not requred, t s assumed that the torque converter s locked at all tmes and no lud couplng takes place, and the engne RPM s saturated at some mnmum value. In addton, snce we are assumng the vehcle s drve-by-wre, t s assumed that small servomotors actvate both the throttle and brakes. Fgure 5 shows a schematc block dagram o the Powertran model. Fgure 5: Powertran Block Dagram Beore consderng the engne and servo dynamcs, the amount o torque that s avalable or a gven engne RPM wll be consdered. 3.1 Engne torque curve A typcal modern, hgh-end passenger vehcle such as that whch s under smulaton wll have a large capacty engne wth a maxmum RPM o around 6000. For example, a typcal Mercedes-Benz V8 engne s capable o delverng approxmately 800 NM o torque at around 3700 RPM. A torque/rpm plot o a V8 engne such as ths s shown n gure 6 below: [ESL04-01A.DOC - 11 OCTOBER 2004] PAGE 8
Typcal V8 Torque Data Engne Torque (Nm) 900 800 700 600 500 400 300 200 100 0 Typcal 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 Engne Speed (RPM) Fgure 6: Typcal Engne Torque Data The standard method o modellng a torque/rpm relatonshp such as ths s to employ a look-up-table o data values and use nterpolaton or ntermedate values. However, n order to provde or an eectve, ecent smulaton, a second-order polynomal equaton o the orm shown below was employed to capture the relatonshp: y = a + bx + 2 cx Equaton 04/01/L The use o a polynomal also has the advantage that a nte torque s avalable at low RPM wthout the torque converter model. Ater applyng curvlnear regresson on the avalable data to determne sutable coecents, the ollowng equaton can be used to provde a good approxmaton o the maxmum avalable engne torque (T Max ) at a gven RPM (R) or a typcal V8 engne: T Max = 528.7 + 0.152R 0.0000217R 2 Equaton 04/01/M Hgher-order polynomals may be used to produce an mproved t, yet as shown n gure 7 ths s not explctly necessary. I the current gear rato s η g and the nal drve rato s η, the engne RPM R s determned as ollows: R = η η g ω Equaton 04/01/N [ESL04-01A.DOC - 11 OCTOBER 2004] PAGE 9
Typcal V8 Torque Data 900 800 700 600 500 400 300 200 100 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 Engne Torque (Nm) Typcal Model Engne Speed (RPM) Fgure 7: Model vs. Typcal Torque Data. Ths torque/rpm characterstc denes the maxmum torque that the engne can delver at a gven RPM; however the actual torque developed at the crankshat s a uncton o the chemcal energy delvered nto the engne by the current throttle settng, and the dynamcs o the engne tsel. 3.2 Engne dynamcs I we assume that the converson o chemcal energy nto output torque by the engne may be descrbed by a rst order tme lag, and the throttle s actuated by a servo wth an assocated tme lag, these lags may be lumped together nto a sngle equvalent lag τ es. Denng an energy transer co-ecent µ e governng the actual amount o torque developed as a uncton o the maxmum, T Max, the ollowng equatons may be used to descrbe the wheel torque τ e to the throttle settng u t : µ e = 0.01u t τ es µ e τ = µ T e e Max η η g. Equaton 04/01/O Equaton 04/01/P Where η g and η are the current gear rato and nal drve rato respectvely, and the nput throttle settng u t s constraned to a value between 0-100 %. The nal element o the Powertran that requres a descrpton s the automatc gearbox, whch decdes the current gear and consequently the value o η g. [ESL04-01A.DOC - 11 OCTOBER 2004] PAGE 10
3.3 Gearbox model In practce, many commercally avalable passenger vehcles now have automatc gearboxes that are controlled by an electronc control unt (ECU). Snce ths element o the Powertran s not under consderaton n the development o the embedded control system, t s assumed that the automatc gearbox s explctly modelled as part o the smulaton. Many o the modern gearbox ECU s are connected to the vehcle communcatons network, and run sophstcated adaptve algorthms nvolvng many nstrumented varables such as RPM, vehcle speed, drvng wheel speed and throttle settng. An automatc gearbox s best descrbed n terms o a sht-map, that relates the threshold or changng each gear up or down as a uncton o throttle settng and wheel speed [Gllespe 1992]. Fgure 8 below shows the smpled sht map or the 5-speed gearbox that s to be used n ths smulaton: Fgure 8: Automatc Gearbox Sht Map It can be seen n gure that there are two corner ponts n the sht prole o each gear one at 30% throttle and the other at 80%. Ths enables the engne to operate n the desrable regon o ts torque curve at most tmes. In order to allow or better acceleraton perormance, the throttle s ncreased the engne RPM s allowed to ncrease n proporton beore changng up a gear. In addton, the sht map allows the gearbox to drop nto a lower gear sudden acceleraton s desred by sharply ncreasng the throttle. The ollowng secton descrbes the nal element o the model, the brakng system. [ESL04-01A.DOC - 11 OCTOBER 2004] PAGE 11
4. Brake System Model The brake system model used n ths smulaton s based around a servo-valve actuated system. Although some systems under development allow or a purely servo-actuated brake or each wheel, a more common current mplementaton s to be used. In ths system, an engnemounted hydraulc pump generates a hydraulc pressure o 150 Bar, whch s ed to one o our servo-actuated proportonal valves to modulate the brake lne pressure delvered to each wheel. Snce the model under development here s only a two wheel tracton model, only two valves and brakng systems are under consderaton: the ront and rear. Fgure 9 below shows a schematc o ths brakng system: Fgure 9: Brake System Block Dagram 4.1 Servo actuated brake system Although somewhat o a smplcaton, the dynamcs o the servo valve and the hydraulc systems can be modelled as smple lags n the tme doman [Gerdes et al. 1993]. These two lags can be ncorporated nto a sngle lag τ bs or the purposes o our smulaton. I the ront and rear callpers are modelled as a smple pressure gan K c, then the pressure appled to the brake dsk p b can be modelled as ollows: pb = 1.5Kc ub τ bs p, = r,. b Equaton 04/01/Q In ths equaton, the nput brake settng u b s constraned to a value between 0-100 %, representng the amount o pressure to apply. The constant o 1.5 relates u b to the ntermedate pressure n the brake lne (0-150 Bar). [ESL04-01A.DOC - 11 OCTOBER 2004] PAGE 12
4.2 Brake rcton characterstc Ths brake lne pressure s then converted nto a brakng torque va a rcton relatonshp that vares wth vehcle speed, temperature and several other parameters [Gerdes et al. 1993]. However, a quas-lnear relatonshp may be assumed and a smpled rcton characterstc s utlsed or ths smulaton: ω τ b = pb Kb mn( 1, ), = r, α Equaton 04/01/R Where K b s a pressure/torque converson constant or each brake system. It can be seen that as the wheel velocty ω approaches zero, the eectve brake torque also approaches zero. Above a wheel velocty o α, the steady state brake torque wll be equal to p b multpled by K b. In order to smply thngs urther, n the smulaton K c wll be taken to be unty and ts value lumped nto K b. In a typcal passenger vehcle, the ront brakes tend to have a larger capacty than the rear brakes due to the eects o dynamc load transer and suspenson. Now that sutable equatons have been developed to descrbe each element o the system, some parameters that represent a modern passenger vehcle wll be suggested. [ESL04-01A.DOC - 11 OCTOBER 2004] PAGE 13
5. Parameter Selecton Ths secton shall suggest sutable values or the remanng model parameters to acheve perormance smlar to a modern passenger vehcle. The values are not ntended to explctly model a sngle make or model o car, but were determned as average values or ths type o vehcle. Parameters wth the subscrpt should be taken as values or both the ront and rear wheels. L = 3.0 m B = 1.5 m C = 1.5 m H = 0.6 m m = 1626 Kg J R C C C τ τ d r es bs = 4.5 Kg / m = 0.3 m = 0.29 = 0.01 = 0.1 Nm / rads = 0.2 s = 0.2 s 2 1 η = 2.82 :1 η = 3.56 :1 1 η = 2.19 :1 2 η = 1.41:1 3 η = 1:1 4 η = 0.83 :1 5 K K = 13.33 Nm / Bar = 6.666 Nm / Bar α = 0.01 b b r Detaled normaton on specc makes and models o motor vehcles s reely avalable ether drect rom a partcular manuacturer, or va the World Wde Web. [ESL04-01A.DOC - 11 OCTOBER 2004] PAGE 14
6. Smulaton Results Ths secton presents some smple smulaton results that were generated usng the models that have been outlned. Some smple measures o perormance are gleaned rom the smulaton data, n order to very the smulaton parameters gve a realstc perormance. The ollowng secton descrbes the acceleraton perormance o the smulated vehcle. 6.1 Acceleraton Perormance The acceleraton perormance o a passenger vehcle such as ths s normally quoted as the tme taken, n seconds, to reach 60 MPH rom a standng start. Fgure 10 shows the smulaton car velocty n a smlar test, wth tarmac as the road surace. It can be seen that the 0-60 MPH tme s 6.8 seconds, whch s comparable to gures quoted or ths measure o perormance n manuacturers data. The top speed reached was 147.8 MPH, agan whch s a realstc gure or a vehcle wth a V8 engne. Acceleraton Perormance Velocty (MPH) 160 140 120 100 80 60 40 20 0-20 0 5 10 15 20 25 30 35 40 45 50 55 60 Tme (s) Car Velocty Fgure 10: Acceleraton Perormance In addton to observng the car velocty, a plot o each wheel velocty (ront and rear) s gven n gure 11. It can be noted n ths gure that the ront wheels are rotatng somewhat aster than the rear: ths ndcates that there s slp present, creatng the drvng tractve orce. The rear wheels rotate at a value slghtly less than the equvalent rotatng crcumerence o the vehcle, due to the presence o wheel rcton. [ESL04-01A.DOC - 11 OCTOBER 2004] PAGE 15
Acceleraton Perormance Velocty (rads/s) 250 200 150 100 50 0 Front Wheel Velocty Rear Wheel Velocty -50 0 5 10 15 20 25 30 35 40 45 50 55 60 Tme (s) Fgure 11: Wheel Velocty 6.2 Brakng Perormance Another common measure o vehcle perormance s the brakng dstance at varous speeds. Fgure 12 shows the brakng perormance o the vehcle, when braked rom 60 MPH to standstll. It can be seen that the brakng dstance s 47.2m, whch s agan typcal o gures quoted or ths measure o perormance n manuacturers data Brakng Perormance 70 60 Velocty (MPH) 50 40 30 20 10 Car Velocty 0 0 12 22 31 38 43 46 47 Dstance (M) Fgure 12: Brakng Perormance [ESL04-01A.DOC - 11 OCTOBER 2004] PAGE 16
7. Conclusons Ths document has descrbed the development o a sutable model o passenger vehcle longtudnal dynamcs. In order to create a sutable test envronment n whch the project ams may be nvestgated, a realstc motorway smulaton envronment s requred. Ths smulaton envronment s descrbed n detal n techncal report ESL 04/02. [ESL04-01A.DOC - 11 OCTOBER 2004] PAGE 17
Reerences / Bblography Bakker, E., Pacejka, H. and Ldner, L. (1989) A new tre model wth applcaton n vehcle dynamc studes, SAE Paper No. 890087, pp. 101-113. Gerdes, J.C., Macuca, D.B. and Hedrck, J.K. (1993) "Brake System Modelng or IVHS Longtudnal Control, Advances n Robust and N Systems, DSC-Vol. 53, ASME Wnter Annual Meetng, New Orleans, LA. Gllespe, T. (1992) Fundamentals o Vehcle Dynamcs, Socety o Automotve Engneers (SAE), Inc. Olsen, B., Shaw, S.W., and Stepan, G. (2003) Nonlnear Dynamcs o Vehcle Tracton, Vehcle System Dynamcs, Vol. 40, No.6, pp 377-399. Stchn, A. (1984) Acquston o transent tre orce and moment data or dynamc vehcle handlng smulatons, Socety o Automotve Engneers, Vol. 4, No. 831790, pp. 1098-1110. [ESL04-01A.DOC - 11 OCTOBER 2004] PAGE 18