5 Ameican Contol Confeence June 8-, 5 Potland, O, USA WeA43 Yaw Stability Contol of an Automotive ehicle via Genealied Pedictive Algoithm Sohel Anwa Abstact Yaw stability of an automotive vehicle in a steeing maneuve is citical to the oveall safety of the vehicle n this pape we pesent a theoetical development and expeimental esults of a vehicle yaw stability contol system based on genealied pedictive contol (GPC method he contolle ties to pedict the futue yaw ate of the vehicle and then takes contol action at pesent time based on futue yaw ate eo he poposed contolle utilies the insight into the yaw ate eo gowth when the automobile is in an undestee o ovestee condition on a low fiction coefficient suface in a handling maneuve Expeimental esults show that the pedictive featue of the poposed contolle povides an effective way to contol the yaw stability of a vehicle Y NODUCON AW Stability Contol (YSC systems have been established in the automotive industy as a safety/pefomance featue YSC geneally pevents the vehicle fom unde-steeing o ove-steeing in a handling maneuve (eg lane change, slalom, etc, paticulaly on a low fiction coefficient suface t also helps the dive maintain yaw stability of the vehicle in a high G handling maneuve A pedictive yaw stability contolle based on GPC [] is pesented in this pape he pedictive natue of the contol algoithm would povide an insight into the incipient yaw instability that can be contolled with appopiate actuation system his featue may impove the pefomance of the vehicle yaw stability (ie esponse, etc, paticulaly on low fiction coefficient sufaces he contol algoithm compaes the vehicle yaw ate fom a poduction gade yaw ate senso with a desied value (which is computed based on vehicle speed and steeing wheel angle f the yaw ate eo (the diffeence between the desied and measued yaw ate exceeds a cetain theshold, a contolling yaw moment is calculated based on a pedictive contol stategy his yaw toque command is then tanslated into actuato command(s Manuscipt eceived Septembe 4, 4 his wok was suppoted by the Chassis Advanced echnology Depatment of isteon Copoation S Anwa was with the Chassis Advanced echnology Depatment, isteon Copoation, Deabon, M 486 USA He is cuently with the Mechanical Engineeing Depatment, Pudue School of Engineeing and echnology, UPU, ndianapolis, N 46 USA (phone: 37-74-764; fax: 37-74-9744; e-mail: soanwa@ iupuiedu Sato et al [] investigated a fou wheel steeing system with the use of yaw ate feedback and steeing angle feedfowad contol When the vehicle deflects due to a sudden side wind, oad suface distubance, o abupt baking, steeing is automatically coected though the ea wheel to significantly impove fowad stability Shibahata et al [3] discussed a chassis contol stategy fo impoving the limit pefomance of vehicle motion hey studied the effects of baking foce distibution on a vehicle s lateal and longitudinal diections t was claimed that contolling the lateal distibution of the baking foce on the font wheels was effective fo the impovement of the vehicle stability while that on the ea wheels was effective fo extending the limit of vehicle motion Wang et al [4] pesented a method to impove the handling and stability of vehicles by contolling yaw moment geneated by diving / baking foces Yaw moment was contolled by the feedfowad compensation of steeing angle and velocity to minimie the side slip angle at the vehicle cente of gavity hey povided simulation esults and scaled expeimental esults to veify thei claims Wang and Nagai [5] discussed an integated contol system poviding high pefomance within ties stong nonlinea aeas with adaptability to the changing oad and othe conditions, by optimally contolling the font and ea steeing angles and the yaw moment, based on the infomation of system paametes Simulation esults wee povided to pove the claims Savkoo and Chou [6] investigated the application of active aeodynamic devices fo suppessing paasitic motion and fo impoving the esponse of vehicles to steeing maneuves, within the scope of the linea dynamic behavio he impovements in the pefomance of the base-line vehicle that wee achievable by the application of diect yaw and oll moments by applying eithe an open loop contol pe-filte o a state feedback contol law based on LQ design hey obseved that the contol stategy yielded a supeio pefomance but demanded uneasonably lage moments fom the actuatos in the context of available aeodynamic foces hey also obseved that the demand on diect yaw and oll moment of actuatos is modest when the actuatos ae contolled using the LQ feedback only and if the contol design was used to tack a desied yaw ate tajectoy and simultaneously to educe the paasitic olling motion Nagai et al [7] pesented an integated contol system of active ea wheel steeing and yaw moment -783-998-9/5/$5 5 AACC 435
contol using baking foces Consideing the tie fiction cicle, the contol system was designed using model matching contol theoy to make the vehicle pefomance follow a desied dynamics model even duing lage deceleations o lateal acceleations hey povided simulation esults to veify the claims made Pak and Ahn [8] descibed an H-infinity yaw moment contol scheme using bake toque fo impoving vehicle pefomance and stability specially in high speed diving he contolle was designed to minimie the diffeence between the pefomance of the actual vehicle behavio and that of its model behavio unde distubance input An eight DO vehicle model was used to veify the enhancement claims on vehicle pefomance and stability Dakunov et al [9] investigated the application of sliding mode contol on the yaw stability contol fo an automobile he contol law was based on optimum seach fo minimum yaw ate via sliding mode contol he developed algoithm detemined the level of vehicle stability though the used of measued vehicle states and then intevened if necessay though individual wheel baking to povide added stability and handling pedictability Hac and Bodie [] discussed a method of impoving vehicle stability and emegency handling using electonically contol chassis systems hey analyed a simple nonlinea vehicle model in the yaw plane to show that the vehicles can become unstable duing potions of handling maneuves pefomed at o close to the limit of adhesion hey also showed that small changes in the balance of tie foces between font and ea axles may affect vehicle yaw moment and stability hey pesented peliminay test esults fo a vehicle with integated closed loop contol of bakes and suspension, pefoming typical handling maneuves he pesent wok utilies the pedictive chaacteistics of the GPC to deive a yaw stability contol algoithm he contol algoithm is based on a lineaied vehicle model his model is then discetied via a bilinea tansfomation he contol algoithm has been validated on a test vehicle Expeimental esults show that the pedictive contolle is effective in minimiing the undestee and ovestee conditions LNEAZED EHCLE MODEL Yaw Stability Contol (YSC System has been aound on the high-end cas fo a numbe of yeas he effectiveness of these systems vaies widely depending on the system design, oad conditions and dive's esponse Most of these systems ae based on empiical data and heavily dependent on testing n the pesent investigation, a moe systematic appoach is taken to develop an YSC system based on a lineaied vehicle model and a pedictive contol algoithm igue illustates the foces acting on the tie contact patches fo a vehicle duing a handling maneuve he yaw dynamics fo the vehicle in such a maneuve can be descibed with following equation []: d = a( yl + y cos + b( y + dt yl + ( c yl sin d y sin + M Whee, ZZ = ehicle yaw inetia; M = Contol yaw moment; xl, yl, x, y, xl, y, y = ie contact patch foces in x- and y-diections as illustated in igue ; = oad wheel angle fo the font wheels; a, b, c, d = Contact patch locations fom the vehicle CG; o simplification, it was assumed in equation ( that: (a oad wheel angle fo the font left tie is equal to the oad wheel angle fo the font ight tie, and (b he foce in x-diection is vey small in a non-baking situation xl α L igue Schematic epesenting vehicle yaw dynamics t is futhe assumed that the nomal foce on the left and ight side of the vehicle is same, ie nomal foce on the font left contact patch is the same as that on font ight contact patch, etc Howeve, the fiction coefficient is assumed to the diffeent fo each contact patch Also, the lateal fiction foces ae assumed to linealy vay with the slip angle [] α L = α = α ; α L = α = α = c α ; = c α ; = c α ; ( yl y wl xl = c wl α L L α yl yl bl 5 c in 5 d in y ont Eddy Cuent Machines M ea Eddy Cuent Machines bl yl whee c L, c, c L, & c β x ae the coneing coefficients fom a two tack vehicle model α L, α, α L, L w α b b x w α y 69 in 44 in a b y 436
& α ae slip angles associated with each wheel µ L, µ, µ L, & µ ae the fiction coefficients associated with each oad-tie contact patch With above simplification, the following yaw dynamics equation is obtained: a( c L + c α cos + b( c L + c α + & = (3 ( c * c L d * c α sin + M Now, the slip angles can be elated to the body side slip angle, oad wheel angle, and the yaw angle by the following elationship (figue α = ( β a; α = ( β + b (4 Substituting the above elationship (4 in equation (3, and assuming the c L = c = c and c L = c = c, the following equation is obtained: {ac cos + ( c d c sin } { a b c & = {ac + cos ( c d c sin } } (5 {ac cos + ( c d c sin + bc } β + M Now, the side-slip and state equation is obtained as follows [] ( & β + = m ( xl + x sin ( yl + y cos ( yl + y cos β + ( xl + x cos + ( yl + y sin + (6 sin β m ( xl + x Again, assuming that the foces in x-diection in a nonbaking situation ae vey small, and substituting the elationship between slip angle & lateal foces, and the slip angle equation & assuming that vaiation of β is vey small about the opeating value, and using the assumption that C L = C = C and C L = C = C, we obtain: C ( β cos C β & β = C a C b m cos + (7 Combining the equations (5 and (7, we obtain: C cos + C ac cos bc m m & β ac c d ac c d = + {{ cos + ( * β { cos ( * + & c sin bc } c sin} a b c } (8 C cos m + M ac cos + ( c d c sin Now fo the sake of simplicity, let us lineaie the above equation about = he following equation of obtained: ( C + C ( ac bc m m M ac bc a c b c & β β = + (9 & ( ( heefoe the plant dynamics (vehicle yaw dynamics can be epesented by the following set of equations: x& a a x b x = + u; y = [ c c ] ( x& a a x b x x = β; x = ; b = ; b = ; c = ; c = a a ( C = + C m ( ac bc = ; a ; a ( ac = m ( a c = bc b c A tansfe function epesentation of the above statespace system is given by the following equation: ( s ( s a / = ( M ( s s ( a + a s + ( aa aa he above tansfe function can be discetied in ode to obtain a discete time tansfe function A bilinea tansfomation is utilied fo this pupose ( ( n + n + n = ; n = ( a ; M ( ( d + d + d a n = ; n = ( + a d = ( aa aa ( a + a + 4 ( d = ( aa aa 8 d = ( aa aa + ( a + a + 4 n the above set of equations, epesents the sample time PEDCE CONOL LAW Like most of the YSC algoithm, the poposed contol algoithm also equies the knowledge of desied vehicle yaw ate, given the steeing angle and vehicle speed he objective of the contolle is to tack the desied yaw ate by minimiing the sum of futue yaw ate eos N J = [ des ( t + j ( t + j] (3 j= J = Yaw ate pefomance index fo the vehicle N = Pediction hoion des ( t + j = ied yaw ate at time (t+j ( t + j = Pedicted yaw ate at time (t+j 437
Genealied pedictive contol (GPC utilies Diophantine type discete mathematical identities to obtain pedicted plant output in the futue n addition to its pedictive capabilities, GPC has been shown to be obust against modeling eos and extenal distubances [] n the following section, a discete vesion of the GPC (Genealied Pedictive Contol is deived he tansfe function in equation (3 can be ewitten as: ( d + d + d ( = ( n + n + n M ( (4 Now the Diophantine pediction equation (j-step ahead pedicto is given by, j E j ( ( d + d + d + j ( = (5 E j ( - = A polynomial in - with ode (j- j ( - = A polynomial in - of degee Multiplying both sides of equation (6 by ( t + j, ( t + j = j ( t + Ej( n + n + n M ( t + j (6 he objective function can now be ewitten in matix fomat as, J = [ ] [ ] (7 = [ ( t + ( t + ( t + N] = [ ( t + ( t + ( t + N] t + = ( t + G M ( t ( ( t + N = G ( j ( t + G M ( t + N N N = E j ( ( n + n + n he pedicted slip equations can be e-witten in a matix fomat as follows: = G * U + f G = U = [ M f ( t + = [ G ( G ( g g g N = g ( t M i g g N + g ( t + M g i ] M + g f = [ f ( t + f ( t + f ( t + N] i ( t + N ] ( t + ( t (8 he objective function can now be ewitten as follows: J = [ f GU ] [ f GU ] (9 Minimiation of the objective function yield the following pedictive contol law: U = [ G G] G ( f ( n the above equation, U is a vecto o obtain the contol law at pesent time, only the fist element of U is used heefoe the contol law is given by: M ( t = M ( t + g ( f ( whee g is the fist ow of Equation ( is the pedictive contol law fo the yaw stability contol system Now, the contol moment can be geneated via a numbe of actuation systems n this paticula eseach wok we pesent an electomagnetic bake-by-wie based yaw contol system he yaw moment is geneated by selectively enegiing these EM bakes which ae located at the fou cones of the vehicle Let's fist develop the contol law in tems of M then we will pesent the electomagnetic means to delive the yaw moment hee ae two situations that accompanies yaw instability: a Undestee condition and b Ovestee condition n an undestee condition the absolute value of the vehicle yaw ate is always smalle than the absolute value of desied vehicle yaw ate des n an ovestee condition, the absolute value of the vehicle yaw ate is always lage than the absolute value of desied vehicle yaw ate des n an undestee condition, the contol moment is geneated by applying baking toque on the inne wheels wheeas in an ovestee condition the contol yaw moment is geneated by applying baking toque on the oute wheels Now the amount of baking toque on the wheels is dictated by the contol yaw toque M n any of these two vehicle dynamic conditions, eithe both wheels o one wheel (on one side can be baked to geneate M n case of baking only one wheel, howeve, it has been obseved that baking the font wheel is moe effective in an ovestee condition wheeas baking the ea wheel has been found to be moe effective in an undestee condition om an optimal contol point of view, it is ecommended to use only one wheel to geneate the contol moment Based on the above analysis, the contol yaw moment can be elated to bake toques as follows (applied baking foces act only in the diection of tie longitudinal axes Assuming counteclockwise positive, M = cxl cos axl sin dx cos ax sin + cl d ( Undestee Condition: ehicle tuning counteclockwise: bake ea left wheel bl M = cxl = c ; bl = M (3 c ehicle tuning clockwise: bake ea ight wheel b M = dx = d ; b = M (4 d Ovestee Condition: ehicle tuning counteclockwise: bake font ight wheel M [ G G] b ( d cos a sin ; b = M ( d cos a sin = (5 ehicle tuning clockwise: bake font left wheel M bl ( c cos a sin ; bl = M ( c cos a sin = (6 G 438
Since the electomagnetic bakes toque is a function of oto speed, in cetain situations these actuatos may satuate n case of actuato satuation, one wheel actuato may not be able to delive the equested yaw moment n this condition, both font and ea wheel actuatos can be used to geneate the equested yaw moment ist we need a toque estimation algoithm fo the eddy cuent machines efeence [] descibes a toque estimation algoithm fo an eddy cuent machine given the oto speed and excitation cuent as follows: est = f ( ω + f( ω* i + f ( ω * i (7 = etading toque; i = etade feedback cuent f i ( ω = ai + ai ω + aiω (8 a i, a i, a i = identified paametes; ω = oto speed Equations (3 though (7 epesent the contol law fo the yaw management system poposed in this pape t is assumed that means of estimating the tie-oad fiction coefficient and nomal foce on the each tie is available Also, the desied yaw ate is also assumed to be known a pioi eithe via expeimental data o data fom pevious developments EXPEMENAL ESULS he above contol law (3-(7 has been implemented on test vehicle equipped with a hybid electomagneticelecto-hydaulic bake-by-wie system Since these equations will povide yaw stability contol functionality based on a pedetemined desied yaw ate, it is necessay to have this data a pioi Also, fo a smooth vehicle ide and handling it is desied to activate the yaw moment contolle based on a theshold value fo the yaw ate eo n the following expeimental esults, this yaw ate eo theshold was set at 5 degees pe second igue Steeing wheel angle in a slalom maneuve on snow without yaw stability contol A vehicle speed estimato, which is not the subject of this pape, is utilied to obtain the vehicle speed Wheel speed is obtained fo poduction gade wheel speed sensos igues though 5 show the baking expeimental esults fo a slalom maneuve on packed snow without any yaw stability contol hese figues show the baseline pefomance of the vehicle t is evident fom figues 4 and 5 that the vehicle undestees heavily on the packed snow suface he measued vehicle yaw ate significantly lags the desied yaw ate fo the given steeing angle and vehicle speed Now expeimental esults with the pedictive yaw stability contolle ae pesented he following contolle paamete values wee detemined expeimentally which esulted in the optimal pefomance of the contolle: Pediction Hoion = 3; Contol Hoion = ; Speed igue 3 ehicle speed in a slalom maneuve on snow without yaw stability contol igue 4 ied and measued yaw ate in a slalom maneuve on snow without yaw stability contol igue 5 Yaw ate eo in a slalom maneuve on snow without yaw stability contol theshold fo YC activation = 4 kph; Yaw ate eo theshold fo YC activation = +/- 5 degees igues 6 and 7 show the steeing wheel angle and vehicle speed fo the test vehicle with stability contol he steeing wheel input is simila to the pevious case Howeve, the vehicle speed is a function of the vehicle yaw pefomance, since the dive tends to educe the vehicle speed when the yaw ate eo is lage Hence, the vehicle speed pofile diffes fom the pevious case he nomal foce on each wheel is estimated fom the static weight distibution and dynamic weight tansfe in an acceleation/deceleation event igue 8 shows desied and measued vehicle yaw ates and the yaw ate eo espectively in a slalom maneuve on a packed snow suface t is clea that the yaw ate eo is vey small compae to the pevious case he vehicle yaw ate tacks the desied yaw ate faily well and as a esult the vehicle speed could be kept elatively highe thoughout igue 6 Steeing wheel angle in a slalom maneuve on snow with yaw stability contol 439
the maneuve igue 7 ehicle speed in a slalom maneuve on snow with yaw stability contol igue 8 ied and measued yaw ate in a slalom maneuve on snow with yaw stability contol igue 9 Yaw ate eo in a slalom maneuve on snow with yaw stability contol igue Wheel baking toque command in a slalom maneuve on snow with yaw stability contol igues 9 and show the yaw ate eo fo the vehicle and the wheel baking toque command fom the contolle espectively CONCLUSONS A genealied pedictive contol law has been deived fo a simplified linea vehicle model fo a bake based yaw stability contol system he pedictive natue of the contolle has been utilied to pedict the yaw ate eo gowth which is then utilied to deive the contol law Expeimental esults show that the vehicle can be effectively stabilied in an ovestee/undestee condition on a packed snow suface using the pedictive contolle he vehicle speed could be kept at elatively highe value thoughout the slalom maneuve he measued yaw ate has been found to tack the desied yaw ate well EEENCES [] Clake, DW, Mohtadi, C, uffs, PS, Genealied Pedictive Contol Pat he Basic Algoithm, Automatica, ol 3, No, pp 37-48, 987 [] Sato, H, Kawai, H, and Hitisikoike, M, Development fo fou wheel steeing system using yaw te feedback contol, 99, SAE technical pape seies Passenge Ca Meeting and Exposition, Septembe 6-9, 99, Nashville, N, USA [3] Shibahata, Y, Shimada, AM, Shimada, K, uukawa, Y, mpovement on limit pefomance of vehicle motion by chassis contol, ehicle System Dynamics, v 3, n SUPPL, 994 [4] Wang, Y, Moimoto,, and Nagai, M, Motion Contol of font-wheel-steeing vehicles by yaw moment compensation (Compaison with 4WS Pefomance, ansactions of the Japan Society of Mechanical Enginees, Pat C, v 6, n57, Ma, 994, pp9-97 [5] Wang, Y and Nagai, M, ntegated contol of fouwheel-stee and yaw moment to impove dynamic stability magin, Poceedings of the EEE Confeence on Decision and Contol, v, Decembe -3, 996, Kobe, Japan, pp 783-784 [6] Savkoo, A and Chou, C, Application of aeodynamic actuatos to impove vehicle handling, ehicle System Dynamics, v3, n4, 999, pp345-374 [7] Nagai, M, Yamanaka, S, and Hiano, Y, ntegated contol of active ea wheel steeing and yaw moment contol using baking foces, JSME ntenational Jounal, Seies C, v 4, n, 999, pp 3-38 [8] Pak, JH and Ahn, W S, H-nfinity yaw moment contol with bakes fo impoving diving pefomance and stability, Poceedings of the 999 EEE/ASME ntenational Confeence on Advanced ntelligent Mechatonics, AM, Septembe 9-3, 999, Atlanta, GA, USA, pp 747-75 [9] Dakunov, S, Ashafi, B, and osiglioni, A, Yaw Contol Algoithm via Sliding Mode Contol, Poceedings of the Ameican Contol Confeence, v, Jun 8 3,, Chicago, L, USA, pp 58-583 [] Hac, A and Bodie, M O, mpovements in vehicle handling though integated contol of chassis systems, ntenational Jounal of ehicle ign, v 9, n -,, pp 3-5 [] Anwa, S, A Paametic Model of an Eddy Cuent Electic Machine fo Automotive Baking Applications, EEE ansactions on Contol Systems echnology, ol, ssue 3, May, 4, pp 4-47 [] Kiencke, U and Nielsen, L, "Automotive Contol System fo Engine, Diveline, and ehicle", SAE ntenational, 44