Proceedings of the 7th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-, 28 Detection of Critical Driing Situations using Phase Plane Method for Vehicle Lateral Dynamics Control by Rear Wheel Steering Anne on Vietinghoff Haiyan Lu Uwe Kiencke Institute of Industrial Information Technology, Uniersität Karlsruhe (TH) Abstract: A method for detecting laterally critical driing situations based on the ehicle body side slip angle () is inestigated. Based on a nonlinear ehicle model, the ehicle stability is analyzed graphically on the β β plane (or phase plane). The stable area can be determined depending on the wheel turn angle, the elocity and the road friction coefficient µ. The detection of critical situations ia the phase plane method is integrated into a gain scheduling control for the stabilization of the lateral ehicle dynamics. The control concept focuses on minimizing the by means of rearwheel steering. Since the cannotbe measured in series productionehicles it is estimated using an ExtendedKalmanFilter that combines the lateral and longitudinal ehicle dynamics. The oerall control concept including the EKF and the phase plane method are alidated with the ehicle simulation software CarMaker r. Besides open-loop maneuers, additional close-loopmaneuers are conducted in order to inestigate the drier s influence on the controller performance and it s reaction to the controller actiity. Keywords: Vehicle dynamic systems; Nonlinear system control; Gain scheduling; Nonlinear obsererand filter design; Phase plane. INTRODUCTION Electronic Stability Control systems that aid the drier in maintaining control of the ehicle is an intense research area for many manufacturers. It is of great importance for these systems to detect the actually critical driing situations. Principally there are two approaches to analyze the ehicle stability. The first approach assumes that the ehicle behaior is stable as long as the ehicle remains within the linear region. Nonlinear dynamics are an indicator that the tire forces are saturated since they come close tothe limitofadhesion. Correspondentdetectionmethods analyze the deiation of measurable characteristic alues from a linear reference model, e.g. the yaw rate or the required steering wheel angle. A detailed oeriew oer these methods is gien in Hiemer (2). The second approach analyzes the formal stability of a ehicle model. This caneither be a linearbicycle model (Isermann (24)) or a nonlinear model (Chung and Kyongsu (26)). The latter approach will be inestigated in this paper. Loss-of-control accidents where a ehicle spins out are particularly seere because of the lateral collision that follows. Before this kind of crash the usually increases. The is hence an important key ariable for judging the stability ofa ehicle. Consequently, the controlaim ofmost four wheel steering concepts is the reduction of the (see e.g. Shibahata et al. (986)). In this paper a gain scheduling controller with actie rear wheel steering will be presented. The controller design is based on a nonlinear double track model. Since state feedback is utilized, all state ariables are to be known. Howeer, sensors for the are too sensitie or expensie for series production ehicles. Instead, the will be estimated with an Extended Kalman Filter. The process model, the phase plane method and the oerall controller are alidated using CarMaker r. This simulation tool proides a detailed and flexible ehicle model including a drier as well as a ariety of preimplemented test runs. 2. PHASE PLANE METHOD 2. Nonlinear Single Track Model For the analysis of the ehicle dynamics on the β β plane the nonlinear bicycle model is used. The two wheels on the same axle are merged to one resulting wheel. Fig. shows the bicycle model including the most important forces and ehicle parameters. For simplification the longitudinal tire forces are neglected (F LF = F LR = ). The angles β and are assumed to be small and the elocity is regarded to be constant. Then, the lateral ehicle dynamics can be represented as: Simulation tool for irtual driing, deeloped by IPG Automotie GmbH, Karlsruhe, Germany 978--234-789-2/8/$2. 28 IFAC 694.382/2876--KR-.8
7th IFAC World Congress (IFAC'8) Seoul, Korea, July 6-, 28 F SR F LR CoG β F SF F LF equilibrium point saddle point limiting point l R Fig.. Bicycle model m m 2 ρ l F m( β + ) F SF F SR = () J Z ψ FSF l F + F SR l R =. (2) Therein F SF and F SR are the front and rear lateral tire forces respectiely: F SF = F SFL + F SFR, F SR = F SRL + F SRR. (3) The lateral force of each tire is computed using the Magic Formula tire model with pure side slip as described in Pacejka (22). The Magic Formula tire model is alidated according to Fig. 2. testrun CarMaker r F Zij α ij µ MF y x F Sij (reference) F Sij (MF) Fig. 2. Flow diagram for alidation of the Magic Formula tire model Fig. 3 shows the results for an elk test on a wet road surface with µ =.8, where the ehicle speed is set to 6 km /h. FLF L/[N] FLRL/[N] 2 2 4 6 2 4 6 8 2 2 CarMaker Magic Formula 4 2 4 6 8 FLF R/[N] FLRR/[N] 6 4 2 2 2 4 6 8 4 2 2 2 4 6 8 Fig. 3. Validation of the Magic formula tire model The lateral tire forces gien by the Magic Formula tire model are almost identical to the reference from CarMaker r. A ariety of further testruns confirmed that the Magic Formula tire model proides a ery high accuracy for the lateral tire forces. 2.2 β β Phase Plane Analysis In this paper a typical compact passenger car is used to analyze the stability of the nonlinear single track model on the β β plane. Fig. 4 shows the state trajectory under arious initial conditions on the phase plane for straightahead driing on a dry road surface (µ =) with =3 m /s. compare Fig. 4. β β plane for = rad, = 3 m /s and µ = The state trajectories are symmetric around the point of origin which coincides with the equilibrium point in this case. The two saddle points are unstable balance points and represent the maximum at which the ehicle can be stabilized statically. Since these points are equilibrium points they are always located on the β axis ( β = ). Fig. shows the state trajectory on the phase plane with the wheel turn angle increased to.2 rad. The ehicle Fig.. β β plane for =.2 rad, = 3 m /s and µ = speed and the road friction coefficient remain unchanged. The grey points represent those equialent in Fig. 4. The comparison of Fig. with Fig. 4 shows that the stable limit becomes smallerinthe steereddirection, while it becomes larger in the opposite direction. An excessie steering operation will lead to a disappearance of the equilibrium point, which means, the ehicle cannot be stabilized any more if the excessie steering remains uncorrected. With the ehicle speed reduced from 3 m /s to 8 m /s the stability limit is expanded as shown in Fig. 6. If the ehicle dries on a surface with low friction coefficient, it is contracted in the steered as well as in die opposite direction as shown in Fig. 7. Reason for this is that the maximum transmittable tire force decreases with increasing ehicle speed or decreasing road friction coefficient. According to the analysis on the phase plane the main reasons for an unstable ehicle behaior are primarily an excessie steering operation, a ehicle speed that is too 69
7th IFAC World Congress (IFAC'8) Seoul, Korea, July 6-, 28 Fig. 6. β β plane for = rad, = 8 m /s and µ = Fig. 7. β β plane for = rad, = 3 m /s and µ =.6 high or driing on a low-µ road surface. Based on the aboe knowledge an algorithm was deeloped to determine the stable area under arious driing conditions. A diamond form as suggested in Chung and Kyongsu (26) is chosen as the stable area (see Fig. 8). Thus, the implementation Fig. 8. Determination of the stable area on the β- β plane of the algorithm in Simulink r is eidently simplified. The two saddle points and the two limiting points are calculated offline depending on the wheel turn angle, the ehicle speed and the road friction coefficient µ. These points are stored in a 3D lookup table. The resulting ideal stability limit is scaled downwith a factor a ( < a ) for aerage driers, because they usually already feel unsafe far before the ideal stability limit is reached and therefore misjudge the driing situation. During driing, the algorithm continuously obseres the location of (β, β) on the phase plane. As soon as the current stable region is left, the driing situation is regarded to be critical and an actiation flag (=) is passed to the lateral dynamics controller. 3. NONLINEAR DOUBLE TRACK MODEL The controller design is based on a nonlinear model considering all four wheels. Fig. 9 shows the ehicle model with an additional rear wheel steering input. The b R F SRL n R F SRR n R F LRL F LRR F WX CoG l R m β m 2 ρ l F F SF L n F F SF R n F Fig. 9. Four wheel ehicle model with 3 DOF F LF L F LF R center of graity is assumed to be at road leel. The roll and ertical motions are neglected. By applying Newton s laws a nonlinear state space model for the motion in the longitudinal, lateral and ertical direction can be deried: ẋ(t) = f(x(t), u(t)), y(t) = Cx(t) (4) with three state ariables and six input ariables x(t) = [ β ] T, u(t) = [F LFL F LFR F LRL F LRR ] T. Additionally it depends on the lateral wheel forces F Sij. 3. Estimation Since the β cannot be measured directly, it is estimated using an extended Kalman-Filter (EKF). The filter design is based on the nonlinear double track model (4). In on Vietinghoff et al. (27) an Extended Kalman Filter (EKF) was presented based on the same model unless the rear wheel turn angle was not included. Additionally, in on Vietinghoff et al. (27) the lateral wheel forces were approximated ia a compact model that did not consider different friction coefficients. Now, a simplified magic tire formula is incorporated: ( ) α ij F Sij = F Zmax,ij sin C ij arctan(b ij ) () µ where ( F Zmax,ij = µ F Zij + k ) Fz,ij(F Z,ij F Zij ). (6) F Z,ij F Zij can be deried from the static normal forces as well as the longitudinal and lateral acceleration. The road friction coefficient µ is assumed to be known. The remaining parameters C ij, B ij, k Fz,ij and F Z,ij are identified through a nonlinear least squares estimator. Besides the lateral wheel forces the nonlinear double track model (4) additionally requires the knowledge of the y b F x 696
7th IFAC World Congress (IFAC'8) Seoul, Korea, July 6-, 28 longitudinal wheel forces. Thereto a model of the drietrain is included. Input ariables of this longitudinal model are the brake pressure P, the clutch position α c and the throttle angle α g. Utilizing the indiidual wheel speeds, the longitudinal and lateral acceleration as well as the yaw rate as measurement ariables, the Extended Kalman Filter proides estimates for the and the elocity. Details about the Kalman Filter design and the drie-train model can be found in (on Vietinghoff et al., 27). Fig. shows the estimated for an elk test on a wet road surface (µ =.8). The ehicle speed is set to 2 reference estimated 2 2 3 4 6 7 8 9 Fig.. Estimated for an elk test at 4 km /h on wet surface with µ =.8 4 km /h. The estimated follows the reference from CarMaker r ery well. In the following the estimated VB- SSA will be utilized for a state feedback control reducing the. 4. GAIN SCHEDULING CONTROL BY ACTIVE REAR WHEEL STEERING In on Vietinghoff et al. (2) a gain scheduling control concept is proposed, which minimizes the by means of combining braking force control of indiidual wheels and an actie front wheel steering. Here, a controller will be deried, that stabilizes the ehicle in critical driing situations by rear wheel steering only. 4. Controller Design Since the four-wheel ehicle model (4) is nonlinear, a nonlinear control concept is required. A most widely used nonlinear control design approach is the gain scheduling control, where the nonlinear system is linearized about a family of equilibrium operating points. Then, linear controllers can be designed for each equilibrium. Online, the appropriate controller can be selected ia a scheduling ariable depending on the current operating point. Since the ehicle dynamics are essentially influenced by the ehicle speed, it is chosen to be the scheduling ariable. Assuming straight-ahead driing, the equilibrium operating points can be determined with the following condition: ẋ eq = f(x eq, u eq ) = (7) and take the form: x eq,i = [ i ] T, i = i. m/s, i =,...,6 (8) u eq,i =,eq,i =. (9) Now the nonlinear system can be linearized around the 6 sets of equilibrium points. The state feedback control gain for each linearized system is determined with the linearquadratic regulator (LQR) algorithm. Since the primary aim of the controller is the minimization of the, the weighting factor for the is chosen significantly higher than that for the other two states. The control logic of the gain scheduling control system including the phase plane method for detecting the critical driing situations is shown in Fig.. testrun CarMaker r µ drier inputs ω ij a x, a y ext. controller.. Kalman- Filter controller 6 3D-lookup table β d/dt x actiation flag = or β x x eq phase plane Fig.. Control logic of the gain scheduling control system The extended Kalman-Filter estimates the and the ehicle speed. The nearest equilibrium point is chosen depending onthe currentehicle speed. The correspondent control gain out of the 6 gains is read out off the lookup table. It is actiated as soon as the actiation flag from the phase plane method is set to one. In order to aoid an excessie rear wheel steering input the controller output is limited to 3 o.. EVALUATION OF THE GAIN SCHEDULING CONTROLLER The gain scheduling control system is alidated with openloop maneuers as well as with closed-loop maneuers.. Open-loop Maneuer Open-loop maneuers are conducted without a drier and the steering wheel is turned through a prescribed motion. The Sine with Dwell maneuer is an extreme seere steering maneuer introduced by NHTSA (National Highway Traffic Safety Administration) causing a seere spin-out (oersteer) of the ehicle. As shown in Fig. 2, the Sine with Dwell maneuer is based on a single cycle of a sinusoidal steering input as described in Garrot (2). The peak magnitudes of the first and second half-cycles δs/[degree] 2 6 6 2 steering wheel angle 2 3 4 6 7 8 9 Fig. 2. Steering wheel angle for Sine with Dwell test are identical and set to 2 in this case. This maneuer 697
7th IFAC World Congress (IFAC'8) Seoul, Korea, July 6-, 28 includes a pause of ms after completion of the third quarter cycle of the sinusoid. The steering frequency is fixed at.7hz. The entrance speed is set to 8 km /h and the throttle is dropped as soon as the maneuer starts. Fig. 3 compares the ehicle speed,, yaw rate and trajectory of the controlled and uncontrolled ehicle. /[degree/s] /[km/h] y/[m] 8 6 4 3 4 6 7 8 9 4 2 2 ehicle speed 3 4 6 7 8 9 yaw rate 3 4 6 7 8 9 trajectory 3 2 x/[m] Fig. 3. Results for Sine with Dwell test, µ = Starting from a straight drie the maneuer is initialized at about 4.3s. After completion of the steering maneuer at about 6.2s the ehicle is supposed to drie straight again (Fig. 2). Without control the yaw rate continues to rise after the pause at about 6 s until the maximum alue of oer o /s is reached, although the steering wheel angle is decreased again after the pause. The ehicle spins out and the keeps growing up to approximately 4 o. A seere lateral crash is often ineitable in this situation. Fig. 3 shows that the ehicle behaior is improed significantly with the aid of the control system. After the pause the yaw rate is reduced with decreasing steering wheel angle and the stays below o. The rear wheel steering input and the actiation flag are shown in Fig. 4. The rear wheels are turned in the direction of the front /W /[degree]. 3 4 6 7 8 front/rear wheel turn angle actiation flag 3 4 6 7 8 Fig. 4. Rear wheel steering angle and actiation flag for Sine with Dwell test, µ = wheels. Shortly after the beginning of the maneuer, the control system is actiated and remains actie until the end of the maneuer except for a short uncritical phase..2 Closed-loop Maneuer While open loop maneuers neglect the influence of the drier in order to hae conditions that can be reproduced exactly, closed loop maneuers explicitly incorporate the drier. Here the drier has to keep the ehicle on a gien course. Thus, it can be ealuated how the control concept can cope with the drier inputs as additional disturbances on the one hand. On the other hand the drier s reaction to the control interention can be examined. Fig. compares the simulation results of the controlled and uncontrolled ehicle for an elk test on a surface with µ = at a speed of 6 km /h. The additional rear wheel δs/[degree] 2 2 3 4 6 7 8 9 y/[m] 2 2 3 4 6 7 8 9 trajectory 6 4 2 steering wheel angle 2 2 2 3 x/[m] Fig.. Results for the elk test, = 6 km /h, µ = steering input from the gain scheduling controller as well as the actiation flag are shown in Fig. 6. Between around /W 2 3 4 6 7 8 9 front/rear wheel turn angle actiation flag 2 3 4 6 7 8 9 Fig. 6. Rear wheel steering angle and actiation flag for elk test, = 6 km /h, µ = 4.8 s and.4 s the phase plane method regards the driing situation to be critical and the control system is actiated accordingly. As a consequence, the peak of the is strongly reduced and the controlled remains within β <.7 o. It can be seen that the control system turns the rear wheels in the same direction as the front wheels to keep the small (Fig. 6). The maximum 698
7th IFAC World Congress (IFAC'8) Seoul, Korea, July 6-, 28 rear wheel steering angle reaches about 2. o at around. s. Fig. 7 shows the simulation results for a slalom test track on a road surface with µ =.6. The ehicle speed is δs/[degree] 2 2 2 3 y/[m] 2 3 trajectory steering wheel angle 2 3 4 6 7 8 9 x/[m] Fig. 7. Results for the Slalom test with µ =.6 set to km /h. Without control the is increasing continuously shortly after the beginning of the maneuer at around s. The front wheels start to slide heaily at around 2 s and the drier is no longer able to keep the ehicle under control. After 8 s he dries with only ery small steering wheel angle in order to keep the ehicle stable on the wet road surface (µ =.6). With the additional rear wheel steering input from the controller the drier can handle the slalom test track successfully. The is reduced significantlyand remains close to zero. The stability of the ehicle is improed accordingly. The rear wheel steering angle generated by the control system and the actiation flag from the phase plane method are shown in Fig. 8. /W /[degree] 2 2 3 front/rear wheel turn angle actiation flag 2 2 3 Fig. 8. Rear wheel steering angle and actiation flag for slalom test, = km /h, µ =.6) 6. CONCLUSION Amodelbasedmethodfor the detectionoflaterallycritical driing situations depending on the ehicle body side slip angle and its angular elocity has been explored. The influence of the steering wheel angle, the ehicle speed and the road friction coefficient on the phase plane has been discussed. A 3D-lookup-table was established to determine the stability limit under arious driing situations. In order to minimize the, a gain scheduling control strategy by means of actie rear wheel steering was deeloped. One open-loop maneuer and two closed-loop maneuers were conducted to ealuate the gain scheduling control system and its actiation according to the integrated phase plane method. The simulation results show that the was successfully minimized by applying additional rear wheel steering on try road surfaces as well as on low-µ road surfaces. As a result, the ehicle behaior was considerably improed in laterally critical driing situations. REFERENCES T. Chung and Y. Kyongsu. Design and Ealuation of Side Slip Angle-Based Vehicle Stability Control Scheme on a Virtual Test Track. In IEEE Transactions on control systems technology, ol. 4, no. 2, 26. W. R. Garrot. The Status of NHTSA s ESC Research. Technical report, NHTSA, 2. M. Hiemer. Model based detection and reconstruction of road traffic accidents. Uniersitätserlag Karlsruhe, 2. R. Isermann. Model-based fault detection and diagnosis - status and applicatons. In 6th Symposium on Automatic Control in Aerospace, 24. H. B. Pacejka. Tyre and ehicle dynamics. Butterworth- Heinemann, Oxford, 22. Y. Shibahata, N. Irie, H. Itoh, and K. Nakamura. The deelopment of an experimental four-wheel-steering ehicle. In SAE transactions, SAE paper 86623, 986. A. on Vietinghoff, A. Feist, and M. Hiemer. Model-based lateral ehicle dynamics control. Reports on Industrial Information Technology, 8:47 6, 2. A. on Vietinghoff, S. Olbrich, and U. Kiencke. ExtendedKalmanFilter forvehicle Dynamics Determination Based on a Nonlinear Model Combining Longitudinal and Lateral Dynamics. SAE paper no. 27--834. Appendix A. NOMENCLATURE The following indices are used as wildcards: i = {F, R} : {Front, Rear} axle j = {L, R} : { Left, Right } side b i F Lij F Sij F Zij J Z l i m n Li wheel track at front/rear axle longitudinal wheel forces lateral wheel forces ertical wheel forces mass moment of inertia about z-axis distance center of graity to front/rear axle ehicle mass in the center of graity longitudinal wheel caster at front/rear wheels elocity in the center of graity β ehicle side slip angle () δ i front/rear wheel turn angle δ S steering wheel angle yaw rate µ road friction coefficient 699