Cost Functions for Assessment of Vehicle Dynamics

Transcription

1 Cost Functions for Assessment of Vehicle Dynmics Dzmitry Svitski Automotive Engineering Deprtment Ilmenu University of Technology Ilmenu, Germny Pvel Nedom, Jroslv Mchn Skod Auto Mld Boleslv, Czech Republic Jiri Plihl Institute of Informtion Theory nd Automtion Acdemy of Sciences of the Czech Republic Prgue, Czech Republic Vlentin Ivnov, Klus Augsburg Automotive Engineering Deprtment Ilmenu University of Technology Ilmenu, Germny Abstrct The pper ddresses methodology of the ssessment of vehicle dynmics on the bsis of set of cost functions. The proposed concept covers different domins like longitudinl, lterl nd verticl dynmics with possibility to deduce complex cost function hving three min ttributes: (i) dimensionless form, (ii) rnge from 0 to 1, nd (iii) weighting fctors for ech individul domin. The ttributes (i) nd (ii) re subjected to the comprison of bseline, reference, nd ctul vlues of corresponding prmeters of vehicle dynmics like ccelertion, yw rte, slip nd others. The ttribute (iii) links to the choice of weighting fctors from the nlysis of vehicle mneuvers. Appliction fields of the developed individul nd complex cost functions re ssessment of vehicle performnce nd stbility, optimiztion of relevnt control systems, nd choice of proper control strtegies / tuning of control gins by utonomous or coopertive opertion of severl systems of vehicle dynmics control. The pper introduces the implementtion of the proposed methodology by the exmple of Electronic Stbility Progrm (ESP) system. Cse studies illustrte results of the clcultion of the cost functions for the voidnce nd sllom mneuvers. The corresponding computtionl procedures re bsed on full vehicle softwre simultor, vlidted with experiments on the rel vehicle. Keywords vehicle dynmics; cost function; weighting fctors; electronic stbility control; modeling I. INTRODUCTION Recent developments in utomotive control rise number of issues tht ddress the complex ssessment of vehicle dynmics for the system benchmrking nd the optimiztion of control lgorithms. The reson is tht typicl stte-of-the-rt ground vehicle belongs to the clss of multi-ctuted objects. Such vehicle involves different subsystems, which hve functions of dynmics control nd cn operte both utonomously nd in integrtion with one nother. For instnce, the longitudinl nd lterl vehicle dynmics cn be simultneously controlled through nti-lock brking (ABS), trction slip control, electronic stbility progrm (ESP), ctive differentil, torque vectoring, nd so on. The resulting integrtion of subsystems clls for the development of n The reserch leding to these results hs received funding from the Europen Union Seventh Frmework Progrm FP7/ under grnt greement no pproprite nlyticl method tht cn ssess the performnce of the controlled vehicle mneuvers from viewpoint of longitudinl, lterl nd verticl dynmics, driver comfort, energy efficiency nd other fctors. The resonble solution for such method cn be the implementtion of set of cost functions tking into ccount vehicle mneuver conditions, rod prmeters, nd opertionl chrcteristics of the vehicle. The compiltion of cost functions cn use conventionl indictors of vehicle stbility, efficiency, comfort nd so on. The relevnt vehicle dynmic prmeters re dequtely described in vrious stte-of-the-rt reference books [1-3]. Within this context, n pproprite solution cn include the ppliction of normlized prmeters tht is of specil interest by the ssessment of combined lterl nd longitudinl mneuvers. Exmples of normlized nd dimensionless description of tire forces nd moments, cmber nd slip ngles, nd wheel slip rtios were proposed in works [4, 5]. Other studies re lso known where more specified methods hve been introduced to describe the cost functions. For instnce, the work [6] proposes the combined ssessment of lterl nd ride dynmics using the frequency-dependent weighting index of lterl ccelertion. The cost functions presented in the ctul study re bsed on n originl methodology tht ws firstly introduced in [7]. The methodology uses set of globl functions derived from the typicl prmeters dopted for the evlution of vehicle response chrcteristics, such s yw rte, ccelertion, sideslip ngle, nd so on. The individul components of the functions re connected through weighting fctors, which cn be vrible depending on the kind of vehicle mneuver s well s on the driving conditions. The detils of the methodology re given in the subsequent sections of the pper. The procedure of clcultion nd the ppliction of cost functions will be illustrted for two cse studies covering the complex vehicle dynmics during voidnce nd sllom mneuvers. II. GENERALIZED CONCEPT OF COST FUNCTIONS The performnce nd effectiveness of vehicle mneuver cn be estimted through prmeters relted to different domins. Domins under discussion re longitudinl, lterl /13/$31.00 c 2013 IEEE 48

2 nd verticl dynmics, driver comfort, hndling, gility nd so on. As result, diverse concepts of cost functions cn be potentilly derived. Within the frmework of the presented study, the requirements to the concept in question re proposed s follows: Ech domin hs n individul cost function. Arguments of the individul cost functions re bsed on corresponding prmeters tht cn be either observed by conventionl vehicle sensors or estimted from models embedded into on-bord processors. Weighting fctors re ssigned to ech rgument of the individul cost functions. The sum of weighting fctors within certin domin is equl to 1. Arguments of the individul cost functions hve numericl form tht is normlized to the rnge [0; 1]. To define complex cost function for severl domins of vehicle dynmics, dditionl weighting fctors hve to be ssigned for ech domin. The sum of weighting fctors for ll domins nlyzed by the complex cost function is equl to 1. Fig. 1 illustrtes the process of composition of individul nd complex cost functions. This process cn be described through the following computtion sequence. 1) Choice of set of prmeters of vehicle dynmics for ech domin. For exmple, the lterl ccelertion, the yw rte nd the side slip ngle cn be ttributed to the domin Lterl dynmics. 2) Trnsformtion of the prmeters to the dimensionless form in the rnge 0 1. The methodology under discussion proposes the following procedure for normliztion (bsed on root men squre function): 1 n f( x) = mx n ref 2 ( xi xi ) i= 1 ( X ) min ( X ), (1) where x is the prmeter (e.g., ccelertion); X(n) is the vector of x-vlues for the vehicle mneuver under ssessment. The indexes ref nd re designted for the reference nd ctul vlues correspondingly. Ech prmeter of the cost function hs n individul procedure of the definition of reference chrcteristics. It should be pointed out tht the function f(x) will be in the rnge [0; 1]. The vlue "0" should be considered s "the best cse": the ctul vlues of vrible coincide with the reference vlues. The vlue "1" should be considered s "the worst cse". 3) Individul weighting fctors re ssigned to ech of prmeters in ccordnce with conditions like m l k w = 1; w = 1;...; w = 1. (2) Ai Bi Ni i= 1 i= 1 i= 1 Fig. 1. Computing of cost functions The composition of weighting fctors depends on the type of mneuver tht will be explined in next section. 4) Clcultion of the individul cost functions: m EA = Ai wai; EB = Bi wbi;...; i= 1 i= 1 k. (3) E = N w N i Ni i= 1 l 2013 IEEE Symposium on Computtionl Intelligence for Engineering Solutions (CIES) 49

3 In ccordnce with previous llownces, vlues of individul cost functions belong to the numericl intervl [0; 1]. 5) To deduce the complex cost function covering severl domins of vehicle dynmics, the weighting fctors re designted to individul cost functions (see fctors w A, w B nd w N on Fig. 1). The sum of the domin-bsed weighting fctors hs to be equl to 1. 6) Computtion of the complex (globl) cost: E = E w + E w +... E w. (4) globl A A B B N N ymx = 9.81 μ, (8) ymx where μ ymx is the mximl lterl friction coefficient t given level of longitudinl ccelertion / decelertion. The prmeter y * in Eq. (7) identifies the reference lterl ccelertion tht does not exceed the friction limits nd cn be derived from the look-up-tble (LUT) given for the vehicle s fmily of Steering wheel ngle Lterl ccelertion - functions composed for different longitudinl ccelertions x, Fig. 2. The procedure of the LUT composition is introduced in detils in [7]. The vlues of the resulting function E globl will be within the rnge [0; 1]. Next section of the pper introduces n exmple of the composition of cost functions for domins of lterl, longitudinl nd verticl dynmics. III. EXAMPLE OF COMPOSITION OF COST FUNCTIONS A. Cost function of lterl dynmics The proposed individul cost function for the lterl dynmics E lt uses the prmeters of the lterl ccelertion y, the sideslip ngle, nd the yw rte d/dt. These prmeters cn be mesured by conventionl vehicle sensors or derived from vehicle models relized in on-bord utomotive control systems like ESC. The formultion of the function E lt is s follows: lt y y ( ) β ( β) ψ ( ψ) E = w f Δ + w f Δ + w f Δ, (5) where w i re the weighting fctors, nd ref ref Δ =, Δ β = β β, y y y Δ = ref ψ ψ ψ, (6) Fig. 2. Exmple of definition reference dt for lterl ccelertion The clcultion of reference yw rte (d/dt) ref follows the condition: ψ = ψ, otherwise * * ref ψ, ψ ψmx mx. (9) The function f in Eq. (5) s well s in the subsequent formultions is defined by Eq. (1). The reference lterl ccelertion is clculted s ref y * * y, y ymx = y mx, otherwise. (7) The mximl lterl ccelertion y * cn be defined from the tire friction ellipse s By nlogy with y, the prmeter (d/dt)* in Eq. (9) cn be derived from look-up tbles given for the vehicle s fmily of Steering wheel ngle Yw rte -dependencies composed for different longitudinl ccelertions. The mximum vlue of yw rte cn be in ddition controlled s [8] y mx v x sin βref mx =. (10) vx cos βref ψ The reference sideslip ngle is clculted in ccordnce with the following conditions IEEE Symposium on Computtionl Intelligence for Engineering Solutions (CIES)

4 β ref * β, v vss = (11) β _ ss, otherwise mx The vlue of sideslip ngle for stedy-stte conditions cn be chosen s: 2 * v β = β, (12) mx_ ss 2 vss where v is the ctul bsolute vehicle velocity, mx_ss nd v ss re correspondingly the mximl sideslip ngle nd bsolute vehicle velocity given for the point where the influence of velocity on yw rte becomes negligible (for reference, mx_ss =3 nd v ss =40 km/h were chosen in the presented study bsed on sttisticl dt). The ctul vehicle sideslip ngle cn be computed s follows [3]: β ( vy vx) 1 = tn, (13) where v x is the ctul longitudinl velocity, v y is the ctul lterl velocity. It should be mentioned tht the comprison of nd ref tkes plce only in the cse > ref. B. Cost function of longitudinl dynmics As in the cse of the lterl dynmics, the individul cost functions for the ssessment of the longitudinl dynmics is bsed on the prmeters tht cn be mesured by conventionl on-bord sensors or derived from relevnt sensor informtion. Two following prmeters re proposed in this context: longitudinl ccelertion (decelertion) x, nd slip power losses of the vehicle P sσ. In such cse the bsic formultion of the cost function cn be proposed s ( ) ( ) Elong wx f x wp f Ps Σ = Δ + Δ (14) where w i re the weighting fctors nd Δ =, (15) ref x x x Δ P = P P. (16) ref sσ sσ sσ The longitudinl ccelertion is clculted s, =. (17), otherwise ref * x * x x mx x x mx The mximl longitudinl ccelertion cn be defined from the tire friction ellipse s xmx = 9,81 μ, (18) xmx where μ xmx is the mximl longitudinl friction coefficient t given level of lterl ccelertion / decelertion. The prmeter x * in Eq. (17) identifies the reference longitudinl ccelertion tht does not exceed the friction limits. The vlue of ctul longitudinl ccelertion x is obtined either (i) with conventionl ccelerometer being component of vehicle dynmics control system or (ii) from the vehicle simultor. The slip power losses of the vehicle cn be clculted s: n sσ = wi ωi i i= 1 P T s (19) where n is the number of drive wheels, T w is the wheel torque, ω is the rottionl wheel velocity, s is the wheel slip coefficient. To clculte the reference slip power losses from Eq. (19), certin threshold of wheel slip coefficient cn be proposed. Bsed on typicl friction-slip-curves, the threshold cn correspond to the re of mximl friction coefficient, i.e. s= To clculte the ctul slip power losses from Eq. (19), the wheel slip computed from the informtion of wheel rottionl velocity sensors cn be used. C. Cost function of verticl dynmics Within the frmework of the presented study, the oneprmeter individul cost function is used for the verticl dynmics. This function opertes with the roll ngle θ: vert ( θ ) E = w f Δ, (20) θ where the weighting fctor w θ =1 in the cse under considertion nd ref θ θ θ Δ =. (21) The reference roll ngle cn be chosen s the threshold θ ref =1. As for the sideslip ngle, the comprison of θ nd θ ref tkes plce only in the cse θ > θ ref. D. Globl cost function The globl cost function in ccordnce with Eq. (4) cn be computed s follows: Eglobl = Elong wlong + Elt wlt + Evert wvert. (22) 2013 IEEE Symposium on Computtionl Intelligence for Engineering Solutions (CIES) 51

5 E. Choice of weighting fctors The choice of weighting fctors cn be done using metheuristic nlysis for different mneuvers nd driving situtions. The procedure of the definition of the weighting fctors is not described in detil in the presented study. However Tble I gives exmples of composition of weighting fctors tht were used in the clcultions discussed in subsequent sections of the pper. TABLE I. EXAMPLES F WEIGHTING FACTORS Weighing fctors Mneuver Numericl vlue w y Stedy-stte (e.g. constnt circle) 0.6 Trnsient (e.g. voidnce) 0.2 w Stedy-stte (e.g. constnt circle) 0.05 Trnsient (e.g. voidnce) 0.3 w Stedy-stte (e.g. constnt circle) 0.35 Trnsient (e.g. voidnce) 0.5 w x Stedy-stte (e.g. constnt circle) 0.6 Trnsient (e.g. voidnce) 0.4 w P Stedy-stte (e.g. constnt circle) 0.4 Trnsient (e.g. voidnce) 0.6 w θ Stedy-stte (e.g. constnt circle) 1.0 Trnsient (e.g. voidnce) 1.0 w lt Stedy-stte (e.g. constnt circle) 0.5 Trnsient (e.g. voidnce) 0.5 w long Stedy-stte (e.g. constnt circle) 0.2 Trnsient (e.g. voidnce) 0.15 w vert Stedy-stte (e.g. constnt circle) 0.3 Trnsient (e.g. voidnce) 0.35 IV. CASE STUDY: ASSESSMENT OF TRANSIENT MANEUEVRS OF THE VEHICLE EQUIPPED WITH ESC SYSTEM This section illustrtes the implementtion of the cost functions to ssess the vehicle dynmics during mneuvers. A. Test vehicle nd simultor The Skod Yeti vehicle hs been used in the study s the test object. The min technicl dt of the vehicle re given in Tble II. The test vehicle is equipped with ESP system. The system ESP comprises severl subsystems: ABS (Anti-lock brking system), MSR (Engine Drg Torque Control), ASR (Trction slip control), EDS (Electronic differentil lock), nd own stbility control system ESC. The experimentl results hve confirmed tht the opertion of ESP system improves understeer chrcteristics of the cr, Fig. 3. For numericl ssessment of this nd other effects, the described methodology ws pplied. To compute nd nlyze the cost functions, number of mneuvers were performed using the instrumented vehicle, Fig. 4, nd then investigted with the full vehicle simultor creted in IPG CrMker softwre environment. Fig. 5 presents the results of the vlidtion of the vehicle simultor by the exmple of mneuver Trck keeping circle with rdius 50 m. Next sub-section presents the clcultion of the cost functions for more complex, trnsient mneuvers. TABLE II. VEHICLE TECHNICAL DATA Prmeter Vlue / Description Totl weight 2080 kg Mximum speed 201 kph Accelertion kph 8,4 s Front xle McPherson suspension with lower tringulr links nd trnsverse torsion stbiliser Rer xle Multi-element suspension with longitudinl nd trnsverse links nd trnsverse torsion stbiliser Steering Direct rck-nd-pinion steering with electromechnicl power steering Tyres 215/60 R16 Dimensions 4223 mm x 1793 mm x 1691 mm Wheelbse 2578 mm Outer turning circle dimeter 10,32 m Fig. 3. Influence of ESC opertion on understeer chrcteristics Fig. 4. Test Vehicle IEEE Symposium on Computtionl Intelligence for Engineering Solutions (CIES)

6 Fig. 5. Vlidtion of vehicle simultor on IPG CrMker B. Tests on the vehicle simultor Two mneuvers were chosen for the clcultion of the cost functions for the vehicle simultor: sllom (stndrd ISO 7401) nd voidnce (stndrd ISO 17512). Ech mneuver hs been simulted with three opertion modes: driving without ESP, driving with conventionl ESP configurtion, driving with modified ESP configurtion. The modified ESP configurtion hs been developed fter the nlysis of the cost functions obtined for the conventionl ESP configurtion with the im to minimize the globl cost function E globl. For this purpose, severl prmeters of ESP controller cn be vried. These re the gin of regultor nd the threshold, reching which ESP systems strts operting. Fig. 6-9 present the min test results. The cost functions were computed in ccordnce with equtions nd weighting fctors presented in section III. The reference dt hve been used s for the vehicle without ESP opertion. The nlysis of results llows to drw following conclusions: 1) The complexity of mneuver cn be properly evluted with the proposed methodology. For instnce, the voidnce s less trnsient mneuver s compred with the sllom hs received lower mximum vlues of the individul cost function of lterl dynmics (0.055 vs with conventionl ESP) nd the complex cost function (0.11 vs. 0.15). 2) The use of the individul nd complex cost functions gives the bsis for the tuning of ESP control lgorithms to improve the controlled vehicle opertion in terms of stbility nd performnce. In prticulr, the modified ESP hs reveled during the sllom mneuver the improved stbility performnce (mximum of the individul cost functions ws reduced on points) nd better overll dynmics (mximum of the complex cost functions ws reduced on points). Additionl numericl nlysis of the test results for the vehicle simultor will be ddressed in the extended journl version of this pper. V. CONCLUSIONS The methodology introduced in the pper hs described the procedures of the clcultion of the cost functions for (i) individul ssessment of lterl, longitudinl, nd verticl vehicle dynmics, nd (ii) complex ssessment of globl dynmic behvior of the vehicle. The cost functions re bsed on the comprison of ctul nd reference vlues of lterl nd longitudinl ccelertion, yw rte, sideslip ngle, wheel slip losses, nd roll ngle. The clcultion of cost functions ws illustrted with the cse study for modeling of trnsient mneuvers (sllom nd voidnce) on the vehicle simultor. At tht, the vehicle simultor ws vlidted on the bsis of experiments on the rel vehicle. The developed method for the computtion of the cost functions cn find pplictions not only for the ssessment of vehicle dynmics but lso for optimiztion of lgorithms of vehicle control systems. REFERENCES [1] J. Ellis, Vehicle hndling dynmics, Professionl Engineering Publishing, [2] R. Rjmni, Vehicle dynmics nd control, Springer, [3] J.Y. Wong, Theory of Ground Vehicles, John Wiley & Sons, Inc., New York, [4] W.F. Milliken nd D.L. Milliken, Chssis Design: Principles nd Anlysis, SAE Interntionl, [5] H.S. Rdt nd D.A. Glemming, Normliztion of Tire Force nd Moment Dt, Tire Science nd Technology, Vol. 21, No. 2, pp , [6] H. Peng nd M. Tomizuk, Vehicle Lterl Control for Highwy Automtion, Proc. of Americn Control Conference, pp , [7] V. Ivnov, K. Augsburg, D. Svitski, J. Plihl, P. Nedom nd J. Mchn, Advnced Cost Functions for Evlution of Lterl Vehicle Dynmics,Lecture Notes in Electricl Engineering, Vol. 198, pp , Springer, U. Kiencke nd L. Nielsen, Automotive Control Systems, Springer-Verlg, Berlin-Heidelberg, IEEE Symposium on Computtionl Intelligence for Engineering Solutions (CIES) 53

7 Fig. 6. Dynmic vribles for voidnce mneuver Fig. 7. Cost functions for voidnce mneuver (upper row - conventionl ESP, lower row - modified ESP) IEEE Symposium on Computtionl Intelligence for Engineering Solutions (CIES)

8 Fig. 8. Dynmic vribles for sllom mneuver Fig. 9. Cost functions for sllom mneuver (upper row - conventionl ESP, lower row - modified ESP) 2013 IEEE Symposium on Computtionl Intelligence for Engineering Solutions (CIES) 55