MODELLING AND CONTROL OF A VEHICLE WITH SINGLE-WHEEL CHASSIS ACTUATORS Ralf Orend Lehrstuhl für Regelungstechnik Universität Erlangen-Nürnberg Cauerstraße 7, 958 Erlangen, German ralf.orend@rt.eei.uni-erlangen.de Abstract: For a vehicle equipped with active single-wheel steering, brake, drive and suspension sstems a nonlinear vehicle model is presented. On the basis of this model an integrated vehicle dnamics control is developed comprising all of the mentioned chassis actuators to control the plane vehicle motion. The basic control strateg consists in two parts. One part is a flatness based tracking controller for the vehicle motion, delivering a aw moment and forces in longitudinal and lateral direction as control commands. The other part is an analtical allocation of these control commands into adequate commands for the chassis actuators, i.e. steer angles, wheel speeds and a wheel load intervention. Copright c IFAC 5 Kewords: automotive control, vehicle dnamics, chassis control, nonlinear control, model-based control. INTRODUCTION This contribution deals with the longitudinal and lateral dnamics of a vehicle equipped with active single-wheel chassis actuators, i.e. single-wheel steering, brake, drive and suspension sstems. Thus the steering angle and the brake resp. drive torque at each single wheel are controlled individuall. Moreover the wheel load distribution is influenced b active wheel load interventions. All in all this provides the possibilit to control the tre forces at each single wheel independentl up to the limit of adhesion with the view to achieve a desired plane vehicle motion. With regard to this constellation an appropriate vehicle model is developed. Its main feature is that it can be inverted analticall, i.e. one can calculate the steering angles and the wheel speeds from the desired vehicle motion. On the basis of this model a strateg is derived to control the plane vehicle motion comprising all of the mentioned single-wheel chassis actuators. The aim of this contribution is to present a structured procedure to design a model-based integrated vehicle dnamics control for a vehicle equipped with active single-wheel steering, brake, drive and suspension sstems. The paper is organied as follows: The nonlinear vehicle model is presented in Sec.. In Sec. 3 the control strateg is derived and an appropriate controller design is carried out. In Sec. 4 the efficienc of the designed integrated vehicle dnamics control is finall tested in computer simulations.. VEHICLE MODEL The longitudinal and lateral vehicle dnamics are described b a nonlinear twotrack vehicle model. The vehicle is assumed to behave like a rigid bod moving on a plane road. Vertical dnamic
effects, like roll, pitch and heave movements will not be considered; the are supposed to be either compensated b the active suspension sstem or are suppressed b an appropriate mechanical design of the wheel suspension. u = s l l v s r l v s l l h s r l h F, G F T = [ ] F F F F F 3 F 3 F 4 F 4. (). Dnamics of the plane vehicle motion The plane vehicle motion is specified b the aw rate ψ, the sideslip angle β and the vehicle velocit v. These three variables of motion are determined b the resulting aw moment M, the resulting longitudinal force F and the resulting lateral force F acting on the vehicle with its mass m and its aw moment of inertia J. F F 3 (a) F F 3 s l F F F 4 s r F F 4 l v F M l h v v 3 (b) v v 3 v s l β v v v v 4 ψ s r v v 4 Fig.. Schematic depiction of a plane vehicle motion: (a) forces and moments, (b) velocities Regarding the vehicle-fied coordinate sstem (which is rotating with the angular speed ψ relativ to a road-fied inertial sstem) the basic equations of the plane vehicle motion can be written as follows (cf. Fig. ): d ψ J v = v ψ + dt v v ψ m M F, F f() }{{ m u } B () ψ ψ β = arctan v ψ ψ v, v = v cosβ v v + v v v sin β h () h() with the longitudinal and lateral velocit v > resp. v of the center of gravit of the vehicle. Neglecting eternal disturbances, e.g. air resistance, the resulting aw moment and the resulting longitudinal and lateral force u = [ M F F are generated b longitudinal and lateral tre forces F i resp. F i (cf. Fig. a): l v l h The longitudinal and lateral velocities at the wheel centers v i resp. v i are given b (cf. Fig. b): v = [ v v v v v 3 v 3 v 4 v 4 = ([ ψ v cosβ v sin β ] G ) T. (3) g() Remark : Throughout the tet the inde i refers to the respective wheel of the vehicle (e.g. v i ): front left: i = rear left: i = 3. Distribution of the wheel loads front right: i = rear right: i = 4 Due to the fact that the tre forces are acting on road level, i.e. with a lever to the center of gravit of the vehicle, roll and pitch moments are generated. As no vertical vehicle motion is assumed to occur, there has to be a static force and moment equilibrium (cf. Fig. ) mg hf = F l v l v l h l h F F hf s l s r s l s 3 r F 4 V F (4) with the wheel loads F i and the sums of the longitudinal and lateral tre forces F resp. F. mg h mg F F F +F 3 F +F 4 F 3 +F 4 F +F s l s r l h l v Fig.. Vertical force and moment equilibrium This leads to the wheel load distribution F = V + + h u +n V F (5) } {{ } (u) with the Moore-Penrose inverse (see e.g. (Ben- Israel and Greville, 3)) and a kernel of V
V + = V T (V V T ), n V = [ and an arbitrar parameter F. (6) For a vehicle with passive suspension, the parameter F is firml specified b the mechanical setup of the chassis, i.e. particularl b anti-roll bars. Regarding a vehicle with an active suspension sstem, one is able to control the wheel load intervention F to achieve a desired wheel load distribution subject to the driving situation (see e.g. (Smakman, ))..3 The tre-road contact The modelling of the tre-road contact is crucial in ever vehicle model, since it is the point where the tre forces are determined. The paper at hand acts on the maime to account for the important aspects, but to keep it as simple as possible. The considered features are force saturation, combined slip, degressive dependenc on the wheel load and dependenc on the friction coefficient. For the sake of simplicit an isotropic tre behaviour is assumed (see e.g. (Burckhardt, 993)), i.e. the tre force F i and the tre slip s i point in the same direction and the magnitude of the tre force is independent of its direction (cf. Fig. 4). Therefore the tre slip is considered as a vector that is caused b a relativ velocit at the contact patch between tre belt and road. It is quantified b the following definition which is a vectorial interpretation of the usual scalar definition: s i = v i v i = v i (v ci + v i ), r si(δ i,ω i;v i) v ci = r i ω i [ cosδi sin δ i (7) with the velocit v ci at the circumference of the wheel, the velocit v i = [ v i v i at the wheel center, the steering angle δ i, the wheel speed ω i and the effective free rolling radius r i (cf. Fig. 3). v i v i v i δ i v i v ci vi v i Fig. 3. Wheel velocit and speed ω i v ci The following approach for the relation between tre force, tre slip and wheel load is motivated b r i the basic equations of the well known Magic tre formula (see e.g. (Pacejka and Besselink, 997)). The influence of the friction coefficient is thereb included as proposed in e.g. (Ammon, 997). The tre force F i = [ F i F i and the adhesion limit, i.e. the peak value of the tre force are specified b: ( ( )) F i = F s i i sin C i arctan B i µ i s i s i, (8) r F i(s i;f i) = µ i F i ( + k F,i F,i F i F,i ) ( ) (9) with the friction coefficient between tre and road µ i and the tre parameters B i >, C i >, k F,i, F,i >. The resulting tre characteristics are schematicall depicted in Fig. 4. s i F i F i F i F i µ i s i µ i F i Fig. 4. Isotropic tre behaviour with its basic characteristics.4 Complete nonlinear vehicle model For a better overview of the complete vehicle model, the prior results are compactl depicted in Fig. 5. Therefore the wheel individual nonlinear scalar functions in (7) resp. (8) and the according scalar variables are merged in vectors: r s = [ r s r s r s3 r s4 r F = [ r F r F r F3 r F4 δ = [ δ δ δ 3 δ 4 ω = [ ω ω ω 3 ω 4 s = [ s s s 3 s 4 3. CONTROL STRATEGY On the basis of the developed sstem description of the vehicle, an appropriate control strateg is derived. Therefor it is useful to take a closer look at the basic nature of the vehicle model. It consists in the dnamic subsstem S dn that is fed back and actuated b the static subsstem S stat (cf. Fig. 5). Starting from this insight the following control strateg is proposed (cf. Fig. 6).
g δ ω v s F u ẋ r s r F G B h F n V F f S stat S dn Fig. 5. Block diagram of the complete nonlinear vehicle model The static subsstem S stat is compensated b its inverse Sstat for the purpose of achieving a desired aw moment and a desired longitudinal and lateral force, i.e. u = u d. The remaining dnamic subsstem S stat is controlled b a feedback tracking controller C T in order to track a desired vehicle motion d. The wheel center velocit v i = [ v i v i follows from the measured vehicle motion according to (3), and the tre slip s i is derived from (8): s i = µ ( i tan arcsin F ) i B i C i F i F i. () Controller ẏ d u d d C T δ ω u stat S stat S dn F S The adhesion limit is determined b the wheel load F i (see (9)) which results from (5) subject to the desired moment and forces u = u d : Vehicle [ F F F 3 F 4 = F = (u)+n V F. (3) Fig. 6. Schematic depiction of the control strateg The basic idea behind this strateg is to design a motion controller C T that uses aw moment and longitudinal and lateral force as control variables which are translated b Sstat into control commands for the chassis actuators. In doing so, the desired moment and forces u d are allocated to steering angles δ, wheel speeds ω and a wheel load intervention F. Remark : This paper acts on the assumption that the wheel speeds are controlled b underling wheel speed controllers that operate active brake and drive sstems. 3. Inversion of the static subsstem The inversion of the static nonlinearit S stat is accomplished graduall b the following steps. The steering angle δ i and the wheel speed ω i at each single wheel are derived from (7): ([ ] ) (si v i + v i ) δ i = arctan [ ], () (si v i + v i ) ω i = r i s i v i + v i. () The tre force F i = [ F i F i depends on the desired moment and forces u = u d and is obtained from the the general solution of (): [ F T F T F 3 T F ] 4 T = F = G + u+n G F (4) with the Moore-Penrose inverse and a kernel of G G + = G T (GG T ), N G = l s l s (l = l v + l h, s = s l + s r ) T, (5) and arbitrar parameters F. These parameters can be interpreted as tre forces that have no influence on the resulting aw moment and the resulting longitudinal and lateral force u, but do affect the distribution of the longitudinal and lateral tre forces F. This is analogous to the wheel load intervention F regarding the wheel load distribution F (see (3)). Remark 3: The ratio between the magnitude of the tre force and its adhesion limit in ()
η i := F i (6) represents the utilisation of the adhesion potential of the tre. B definition the limitations η i hold for each wheel. For this reason a perfect compensation of S stat is not possible, as these limitations remain and have in fact to be taken into account particularl with regard to the feedback controller design. Nevertheless this difficult is factored out in this paper. To determine the et arbitrar parameters F and F this stud embarks on the strateg to achieve the smallest possible utilisations of the adhesion potentials η i at all four wheels, thus keeping all tres as much as possible below their adhesion limit and ensuring an optimal safet reserve in ever driving situation. Hence the parameters F and the wheel load intervention F are determined b an optimisation approach according to (Orend, 5). 3. Tracking control of the dnamic subsstem After compensating the static subsstem S stat b its inverse Sstat (cf. Fig. 6) the remaining dnamic subsstem S dn is considered. It is described b (cf. Fig. 5 and ()) ẋ = f() + Bu, = h(). (7) In the following a flatness based analsis of the sstem (7) is performed. To this end the definition of flatness is recalled (see e.g. (Fliess et al., 995)). A sstem is flat, iff there eists a flat output f with the following properties: I The flat output f is a function of the states, the inputs u and a finit numnber of its time derivatives u, ü,..., u (α). II The dimension of the flat output f equals the dimension of the differential independent components of the inputs u. III All sstem variables, i.e. the states and the inputs u can be epressed b the flat output f and a finit number of its time derivatives ẏ f, ÿ f,..., (γ) f. In the sequel it is shown that the sstem (7) is flat, i.e. that a flat output is given b: f = f ψ f = β = = h(). (8) f3 v Since f is a function of alone condition I is satisfied. Also condition II is met as all components of u are differentiall independent, i.e. ( ) (f() + Bu) rank = rank(b) = 3 (9) u because of det(b) = (J m ) (see ()), and the flat output f has the same dimension as u: dim( f ) = dim() = dim(u) = 3. () It can be shown that condition III is satisfied as well. The parameterisation of the states in f follows from (): f = h ( f ) = f3 cos f =: H ( f ). () f3 sin f Inserting this result in (7), the inputs u can be epressed b f as (see ()): u =B (ẋ f()) ( ) =B H ( f ) ẏ f f(h ( f )) f J ẏ f = m(ẏ f3 cos f f3 ( f + ẏ f )sin f ) m(ẏ f3 sin f + f3 ( f + ẏ f )cos f ) = : H u ( f, ẏ f ). Given a desired trajector for the flat output () f,d = d = [ ψ d β d v d (3) the flatness based tracking controller reads where u = u d = H u ( f, w), w = ẏ f,d Re (4) R = diag(r, R, R 3 ), e = f f,d. (5) This controller achieves the linear tracking error dnamics ė + Re = for the closed loop sstem. Note that the control law (4) consists in a feedforward part ẏ f,d and a proportional feedback part Re. The feedback part is done without integral action in order to leave it to the driver to compensate for stead state tracking errors. 4. SIMULATION RESULTS The efficienc of the designed integrated vehicle dnamics control is tested in computer simulations. As virtual eperimental vehicle a more comple and detailled vehicle model of a mediumclass car is used showing the following features:
ψ [ /s] β [ ] v [km/h] 8 6 4 ψ ψd [ /s] β βd [ ] v vd [km/h] 5 5.5.5 5 5 δ [ ] δ 3 4 [ ] F [N] 5 5 4 3 4 4 ω [/min] ω 3 4 [/min] ηi ma i=...4 8 6 4 8 6 3 4 4.5 Fig. 7. Simulation results: variables of motion (actual - ; desired - -) and tracking error car bod motion with all translational and rotational degrees of freedom, suspension kinematics and elastokinematics, wheel rolling and travel motions, tre dnamics, air resistance, etc. The modelling of the chassis actuators is ideal, i.e. without dnamics and limitations. The wheel speeds are controlled b underling wheel speed controllers that are et not further discussed. Remark 4: In these simulations the active suspension sstem onl takes care of the desired wheel load intervention but does not compensate for vertical movements of the car bod, contrar to the assumption made in Sec.. As driving manoeuvre a fast lane change with medium deceleration is eamined. The maimum absolut lateral acceleration is 8 m/s, and the longitudinal acceleration is kept at a constant value of 5 m/s. The desired aw rate ψ d and the desired sideslip angle β d are provided b a single-track reference model using a single-sine steer input with a frequenc of.5 H. The desired vehicle velocit v d starts at km/h and decreases with a constant rate. The simulation results are depicted in Fig. 7 and Fig. 8. It can be stated that the applied vehicle dnamics control works well since good tracking is achieved. The simulation results show relative small tracking errors with regard to the driving situation: the vehicle is controlled up to its cornering limit, as values of η i = are reached. 5. CONCLUSIONS This contribution showed a structured and analtical approach to handle a multitude of different chassis actuators in order to control the horiontal vehicle motion. Fig. 8. Simulation results: actuator commands and utilisation of the adhesion potentials A nonlinear model-based control strateg for a vehicle with single-wheel steering, brake, drive and suspension sstems was designed. Therefore in the modelling certain simplifications and assumptions were made concerning mainl the tre behaviour. B this means one obtained a pragmatic vehicle model that is not too comple and can be inverted analticall, but nevertheless ehibits the essential nonlinear effects of the horiontal vehicle dnamics. It turned out to be a proper basis for a flatness based tracking controller desing that was carried out in the sequel. The vehicle dnamics control demonstrated its efficienc and performance in computer simulations. REFERENCES Ammon, D. (997). Modellbildung und Sstementwicklung in der Fahreugdnamik. B.G. Teubner. Stuttgart. Ben-Israel, A. and T.N.E. Greville (3). Generalied Inverses. Springer-Verlag. New York. Burckhardt, M. (993). Fahrwerktechnik: Radschlupfregelssteme. Vogel-Verlag. Würburg. Fliess, M., J. Levine, P. Martin and P. Rouchon (995). Flatness and defect of nonlinear sstems: Introductor theor and eamples. Int. J. Control 6 pp. 37 36. Orend, R. (5). Vehicle Dnamics Feedforward Control with Optimal Utilisation of the Adhesion Potentials of all four Tres (in German). at - Automatisierungstechnik 53 pp. 7. Pacejka, H.B. and I.J.M. Besselink (997). Magic formula tre model with transient properties. Vehicle Sstem Dnamics Supplement 7 pp. 34 49. Smakman, H. (). Functional Integration of Slip Control with Active Suspension for Improved Lateral Vehicle Dnamics. Herbert Ut Verlag. München.