Research Report. Eiichi Ono, Yoshikazu Hattori, Yuji Muragishi. Abstract. Special Issue Estimation and Control of Vehicle Dynamics for Active Safety

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1 Specal Issue Estmaton and Control of Vehcle Dynamcs for Actve Safety 7 Research Report Estmaton of Tre Frcton Crcle and Vehcle Dynamcs Integrated Control for Four-wheel Dstrbuted Steerng and Four-wheel Dstrbuted Tracton/Brakng Systems Ech Ono, Yoshkazu Hattor, Yuj Muragsh Abstract To mprove the performance of vehcle dynamcs control systems, t s mportant to be able to estmate the frcton force characterstcs between the tres and the road In ths paper, we estmate the radus of the tre frcton crcle by usng the relatonshp between the Self-Algnng Torque (SAT), and the lateral and longtudnal forces actng on each tre Then, we propose a vehcle dynamcs control system for four-wheel dstrbuted tracton/brakng and four-wheel dstrbuted steerng that s based on an on-lne nonlnear optmzaton algorthm that mnmzes the maxmum µ rate of the four tres by usng the estmaton of the radus of the tre frcton crcle Keywords Vehcle dynamcs control, Self algnng torque, Vehcle dynamcs ntegrated management, Tre frcton crcle R&D Revew of Toyota CRDL Vol No

2 8 Introducton In turns wth low lateral acceleratons, the characterstcs of Self-Algnng Torque (SAT) become more saturated than those of the lateral force, such that an algorthm for estmatng the tre characterstcs based on the relatonshp between, ) SAT and the lateral force has been developed Whle such methods can be appled to pure lateral slp moton, we clarfed the relatonshp between the tre grp margn and SAT wth combned lateral and longtudnal slp by usng a brush model 3, ) and are proposng a method for estmatng the radus of the tre frcton crcle wth combned lateral and longtudnal slp 5) Furthermore, we are proposng a vehcle dynamcs control system for four-wheel dstrbuted steerng and four-wheel dstrbuted tracton/brakng systems, that ams to optmze the tre frcton crcle of each wheel Then, the effects of the vehcle dynamcs control system on fourwheel dstrbuted steerng and four-wheel dstrbuted tracton/brakng systems are shown by smulatons Estmaton of tre frcton crcle Each tre grp margn s estmated from the SAT, longtudnal and lateral force of each ndvdual tre 5) The tre grp margn ε, normal SAT T SAT and SAT model rate γ SAT are defned as ε T SAT γ T T SAT l 6 SAT F F F x y, () l F x K F β 3, (), (3) l : ground contact length, K β : cornerng stffness, F : radus of tre frcton crcle, F x, F y : longtudnal and lateral force, and T SAT : self-algnng torque (SAT) By usng the brush model, the relatonshp between the SAT model rate, longtudnal force, and tre grp margn can be derved as F x 3 3 γ 6 3 SAT ( ε ε ) Kβ y Fx 3 3 = ε( ε ε ) ( ε 3ε ε) 5 K () Fgure shows the tre grp margn ε as a functon of the SAT model rate γ SAT and normalzed longtudnal force F x /K β, as calculated from Eq () Ths fgure shows that the tre grp margn can be descrbed by a monotonous functon of the SAT model rate and the normalzed longtudnal force Then, the tre grp margn can be estmated from γ SAT and F x /K β by usng a three-dmensonal map as shown n Fg Furthermore, the radus of the tre frcton crcle F can be estmated as: F = Fx F ε y β (5) 3 Vehcle dynamcs control for four-wheel dstrbuted steerng and tracton/brakng system 3 Herarchcal control structure The herarchcal control structure shown n Fg s adopted for vehcle dynamcs control 6) [Vehcle Dynamcs Control] Ths layer calculates the target force and the moment of the vehcle n order to acheve the desred vehcle moton correspondng to the drver's pedal nput and the steerng wheel angle The desgned target force and moment also satsfy the robust stablty condton 7) [Force & Moment Dstrbuton] The target force and moment of the vehcle moton are dstrbuted to the target forces of each wheel, based on the tre grp margn n ths layer Tre grp margn 5 - Fx/K [rad] -5 5 SAT model rate Fg Relatonshp between γ, F x /K β and e R&D Revew of Toyota CRDL Vol No

3 9 [Wheel Control] Ths layer controls the moton of each wheel so as to acheve the target force Ths paper descrbes a force & moment dstrbuton algorthm n detal 3 Nonlnear optmal control A vehcle model s descrbed usng the coordnates shown n Fg 3, wth the -axs beng the drecton of the target resultant force and the Y-axs beng the drecton perpendcular to the -axs The µ rate γ ( =,, 3, ) of each wheel s defned as Fx Fy γ ε =, (6) F F x, F y : longtudnal and lateral force of the -th wheel, F : the radus of the frcton crcle of the -th wheel In ths study, the µ rate of each wheel s Informaton Drver nput Vehcle Dynamcs dynamcs control Control Force & moment Moment dstrbuton Dstrbuton Wheel Control control Actuator (Result) Force & moment of vehcle Lateral & longtudnal forces of tres (Target) Drvng torque, Brake pressure, Steer angle controlled so as to have the same value as γ = γ = γ, (, j =,, 3, ) (7) Furthermore, the angle between the drecton of the frcton force and the -axs s expressed as q ( =,, 3, ), as shown n Fg In the second layer of herarchcal control, the value of q that mnmzes γ by satsfyng the followng constrants s calculated by usng Sequental Quadratc Programmng (SQP) γ Fcos q = F γ Fsn q = F γ F( dcos q lsn q)= M x (8) y (9) z () Eq (8) represents a constrant ndcatng that the resultant force n the -axs drecton s the target value F x Eq (9) represents a constrant ndcatng that the resultant force n the Y-axs drecton s target value F y Eq () represents a constrant ndcatng that the moment around the center of gravty of the vehcle s the desred yaw moment M By elmnatng γ from Eqs (8)-(), the constrants of q can be obtaned F{ ( Fx d Mz ) cos q Fx lsn q}= () F{ Fy dcos q ( Fy l Mz ) sn q}= () The performance functon J that mnmzes γ s defned as Fg Herarchcal vehcle dynamcs management algorthm -axs J (3) γ d, l : constants of a typcal moment arm In ths paper, we use the followng constants d T = x y z df lf M = f Tr () d -d F F F x l l Y-axs F y Target resultant Force & moment -l- 3 M -l Tre force γ q F -axs d 3 d F 3 F Frcton crcle Frcton crcle F Y-axs Fg 3 Coordnate system correspondng to resultant force Fg Relatonshp between drecton of target resultant force and drecton of frcton force R&D Revew of Toyota CRDL Vol No

4 Lf Lr l = (5) Snce the numerator on the rght sde of Eq (3) s a constant, maxmzng J mples mnmzng γ By substtutng Eqs (8)-() nto Eq (3), J can be rewrtten as, J = d F F cos q l F F sn q = F d Fx dmz cos q l Fy lm z sn q (6) Then, the optmzaton problem can be formulated as follows Problem : We need to fnd q ( =,, 3, ), that maxmzes the performance functon (6) wth constrant equatons () and () Problem can be solved by usng SQP Frst, snq and cosq can be lnearzed around q as: sn q = sn q cos q (q - q ) (7) cos q = cos q sn q (q - q ) (8) Then, constrant equatons () and () can be expressed as follows F ( Fxd Mz) sn q Fxlcos q q {( ) ( ) = F F d M q sn q cos q x z Fxl( qcos q sn q)} { } F F d sn q F l M cos q q y y z { = F F d q sn q cos q x y y ( )} (9) Fyl Mz qcos q sn q Next, snq and cosq can be approxmated around q by a nd order Taylor expanson as: sn q sn q = sn q cos q( q q) q q () () cos q cos q = cos q sn q( q q) q q () Then, performance functon (6) can be expressed as J = F {( d Fx dmz ) q ( l Fy lm z ) q } q cos sn = {( ) d F d M q cos q sn q M F d cos q l sn q z { ( ) } x z = F D( q ) Y, (3) N =, () D N = (d F x - d M z )(q cosq - snq ) (l F y l M z )(q snq cosq ), (5) = d F d M cos q l F lm sn q Y = d F d M x z (6) ( l F lm q ) q q q y z sn cos (7) Assumng that D >, the problem of maxmzng J can be rewrtten as the problem mnmzng K =, (8) by the varable transformaton p F q (9) = D ( ) Further, lnearzed constrant equatons (9) and () can be expressed as A A A3 A B, A A A A = (3) 3 p B F A = {( Fxd Mz) sn q Fxlcos q}, (3) A ( ) D x z y z p ~ F = Fy dsn q ( Fy l Mz ) cos q D ( y z) ( )} l F lm q sn q cos q q ( x z) d F d M ( y z) l F lm D q q sn q qcos q cos q qsn q B = F ( Fx d Mz ) ( q ) sn q cos q q cos q qsn q, N D, (3) F, xl{ ( q ) cos q sn q} (33) R&D Revew of Toyota CRDL Vol No

5 B = F Fyd ( q ) sn q cos q J F F d q F l M q Fy ( q)= y cos y z sn Fyl Mz {( q ) cos q sn q} (3) Snce mnmzng K mples mnmzng the Eucldan norm of p = [p p p 3 p ] T, the optmal soluton can be derved by usng pseudo-nverse matrx calculaton as p = A, (35) A A3 A B A A A3 A B A represents the pseudo-nverse of matrx A Then, q = dag F F F F D D 3 D3 D A A A3 A B A A A A B 3, (36) q = [q q q 3 q ] T Then, recurson formula solvng of optmal q based on SQP s shown as follows STEP To satsfy the condton D >, the ntal value of q s calculated as l F lm y z q = tan (37) d Fx dmz By substtutng Eq (37), D can be rewrtten as ( ) = d F d M cos q l F lm sn q D x z y z ( x z) ( y z) = d F d M l F lm cos q tan ( x z) ( y z) > = d F d M l F lm STEP q s calculated from Eq (36) STEP 3 We defne a penalty functon as P( q)= ρ JFx ( q) JFy ( q) J ( q ) ρ >, (38), (39) Fx { x z x }, () J ( q)= F ( F d M ) cos q F l sn q l F d F lm d M y z x z 3 () J Fx (q) corresponds to the left sde of Eq (), and J Fy (q) corresponds to the left sde of Eq () If P(q) < P(q ), then q = q, () and we can go to STEP Otherwse, ^q whch satsfes P( ^q)< P( q ) s a straght lne searched n [q, q ] Then, q = q^, (3) and we can go to STEP We know that SQP s one of the most effectve methods of nonlnear optmzaton 8) Further, optmal γ can be obtaned as γ = ( df x) ( lf y) Mz ( ) cos ( ) { sn q } F d F d M q l F lm x z y z () The magntude and drecton of each tre force are translated to the steer angle and longtudnal force of each wheel, and these are controlled n the last layer of the herarchcal control The lateral and longtudnal forces can be obtaned as F x = γ F cos q, (5) F y = γ F sn q (6) Further, by usng a brush model, the slp angle of each wheel can be calculated as Ks κ sn q α = tan, (7) Kα κcos q 3F κ = ( γ) 3, (8) Ks K s : brakng stffness, K α : cornerng stffness Smulaton results { } Fgure 5 shows the smulaton results for fourwheel dstrbuted steerng and tracton/brakng control n the case of a sngle-shot sne-wave steerng maneuver at ν = m/s A lnear model of the vehcle s used for the reference of control The µ rate s controlled such that t s the same for each wheel, and s restrcted to a maxmum value of 95, so that the grp margn s unform, whle the lmt on the frcton crcles s satsfed Ths mples that the R&D Revew of Toyota CRDL Vol No

6 tre forces are used effcently n the frcton crcles Fgure 6 compares the maxmum resultant force of the smulaton result for a sngle-shot sne-wave Slp angle [rad] Yaw velocty [rad/s] Longtudnal force [N] Steer angle [rad] γ ( µ rate) Fg 5 Fg 6 5 Reference (lnear model) Front : sold Rght Rear : dashed Left Wthout control Tme [s] Smulaton result for four-wheel dstrbuted steerng and tracton/brakng control (sngle-shot sne-wave steerng at ν = m/s) 77% Four-wheel -wheel dstrbuted dstrbuted tracton/brakng control control 78% Actve front-rear steerng and four-wheel -wheel dstrbuted tracton/brakng control 5% -wheel Four-wheel dstrbuted dstrbuted steerng steerng and and -wheel four-wheel dstrbuted tracton/brakng control 5 5 Maxmum resultant force [N] Comparson of maxmum resultant force of smulaton result for a sngle-shot sne-wave steerng maneuver at ν = m/s steerng maneuver at ν = m/s Ths fgure shows that the lmted performance (maxmum resultant force) s mproved through the use of ntegrated control The maxmum resultant force s mproved by 78 % as a result of addng actve front-rear steerng control to four-wheel dstrbuted tracton/brakng control, and by a further 5 % as a result of addng an ndependent steerng mechansm (four-wheel dstrbuted steerng) 5 Concluson In ths paper, the radus of each tre frcton crcle s estmated from the relatonshp between the Self- Algnng Torque (SAT) and the lateral and longtudnal forces actng on each tre, and a method of vehcle dynamcs control that uses the tre frcton crcle of each wheel for the optmum s proposed A dstrbuton algorthm usng Sequental Quadratc Programmng (SQP) s used to calculate the magntude and drecton of the tre forces, such that they satsfy the constrants correspondng to the target force and moment of the vehcle moton and mnmze each tre µ rate The proposed algorthm, beng characterzed by pseudo-nverse matrx calculaton, features hgh-speed calculaton and accurate calculaton that satsfy the constrants, so that t can be appled to four-wheel dstrbuted steerng and four-wheel dstrbuted tracton/brakng systems for whch the optmzaton of the eght parameters s necessary Then, the effectveness of the vehcle dynamcs control for four-wheel dstrbuted steerng and four-wheel dstrbuted tracton/brakng systems s shown by smulatons References ) Pasterkamp, W R and Pacejka, H B : "On Lne Estmaton of Tyre Characterstcs for Vehcle Control", AVEC '9, (99), 5-56 ) Fukada, Y : Japanese Unexamned Patent (n Japanese), (995) 3) Bernard, J E, et al : "Tre Shear Force Generaton durng Combned Steerng and Brakng Maneuvers", SAE Tech Pap Ser, No7785 (977) ) Burkard, H and Calame, C : "Rotatng Wheel Dynamometer wth Hgh Frequency Response", Tre Technol Int 998, (998), ) Ono, E, et al : "Estmaton of Tre Grp Margn Usng Electrc Power Steerng System", Proc 8th IAVSD Symp Dyn of Veh on Roads and Tracks, (3) R&D Revew of Toyota CRDL Vol No

7 3 6) Hattor, Y, et al : "Force and Moment Control wth Nonlnear Optmum Dstrbuton for Vehcle Dynamcs", AVEC ', (), ) Ono, E, et al : "Bfurcaton n Vehcle Dynamcs and Robust Front Wheel Steerng Control", IEEE Trans on Control Syst Technol, 6-3(998), - 8) Han, S P : "A Globally Convergent Method for Nonlnear Programmng", J Optm Theory Appl, (978), (Report receved on Sep, 5) Ech Ono Research felds : Vehcle Dynamcs Control Academc degree : Dr Eng Academc socety : Soc Instrum Control Eng, Soc Automot Eng Jpn Awards : SICE Award for Outstandng Paper, 995 SICE Chubu Chapter Award for Outstandng Research, 999 SICE Chubu Chapter Award for Outstandng Technology, IFAC Congress Applcatons Paper Prze, Paper Award of AVEC, Paper Award of AVEC, Yoshkazu Hattor Research felds : Vehcle Dynamcs Control, Vehcle Dynamcs Analyss & Modelng Academc socety : Soc Instrum Control Eng, Inst Syst, Control Inform Eng, Soc Automot Eng Jpn Yuj Muragsh Research felds : Vehcle Dynamcs Control Academc socety : Jpn Soc Mech Eng, Jpn Flud Power Syst Soc R&D Revew of Toyota CRDL Vol No