2007-01-3642 A Novel Integrated Vehicle Chassis Controller Coordinating Direct Yaw Moment Control and Active Steering Daofei Li and Fan Yu Shanghai Jiaotong University Copyright © 2007 SAE International ABSTRACT To optimally allocate control authorities to co-existing active actuators, an integrated chassis controller with main/servo-loop structure is designed to coordinate direct yaw moment control and active steering Firstly, a sliding mode controller in the main-loop calculates the desired stabilizing forces Then in the servo-loop, by directly considering limits of road friction and actuators, a quadratic programming based control allocation approach is adopted to reasonably and optimally distribute these deisired forces to tire actuator actions Co-simulation of Matlab/Simulink and Carsim clarifies that the proposed controller could significantly improve vehicle handling performances INTRODUCTION During recent decades, many advanced chassis control systems, e.g active steering, direct yaw moment control (DYC) and active roll control, have been extensively researched and developed Among them, active steering and DYC are two basic approaches to vehicle lateral motion control In order to generate proper tire lateral forces, active steering system could provide additional steering angles to driver’s command DYC could effectively improve vehicle lateral stability through differential braking/traction, i.e an active stabilizing yaw moment can be generated from the difference of left and right tire longitudinal forces However, these individually designed chassis control systems still have their own limits and drawbacks, e.g active steering can not generate more tire lateral forces in near saturation region Under certain circumstances, unnecessary interferences between subsystems will deteriorate vehicle stability and ride comfort generation is definitely a key factor that determines whether the intention of driver or chassis controller could be realized Therefore, to design an accurate and practical chassis controller, it is necessary to consider tire constraints, e.g tire-road friction limits and steering limits However, among these existing publications on IVCC design, only a few of them considered the real constraints For instance, to get a greater stability margin, references [5] and [6] used different optimization methods to minimize tire workload and thus to keep the tire control forces as small as possible On the other hand, control allocation (CA) approaches, which have been widely adopted in the dynamic control of over-actuated aircrafts [7-10] and marine vessels [11], have recently been introduced into automotive control area[12-14] One of its advantages is that different actuator limits can be directly considered and the formulated optimal control problem is solved through linear or nonlinear programming methods Based on the foregoing studies, in order to further consider different constraints and thus to improve tire force generation accuracy, this paper proposes a topdown integrated controller for a future vehicle, in which the braking and steering of four wheels can be independently maneuvered (e.g by drive-by-wire technology) The remainder of this paper is organized as follows: section describes the nonlinear vehicle model considering tire nonlinearities; section 3, and describe the proposed chassis control system, including detailed main-loop and servo-loop controller design; section presents the results of co-simulation using Matlab/Simulink and Carsim; and finally, section gives the conclusion VEHICLE MODEL Therefore, during the past few years, aiming to compensate the drawbacks through coordination, integrated vehicle chassis control (IVCC) system has become a hot topic of vehicle dynamics research, [1-6] Through subsystem coordination, IVCC tries to optimally allocate the control authorities of different actuators Since vehicle motion control forces are mostly generated from tire-road contact patches, the accuracy of tire force For main-loop controller design, a simplified vehicle model (Figure 1) with three-DOF (i.e longitudinal, lateral and yaw motion) is used The vehicle dynamics can be represented in Eq ªmv (v x v y r )º «m ( v r ) ằ x ô v v y ằ Izz r ơô ¼» (1) Fu In above equation, the generalized force Fu for vehicle motion control can be calculated from tire control force vector ucf [Fx1 Fy " Fx Fy ]T , i.e Fu [Fx Fy Fz ]T reference model, a sliding mode controller is designed to obtain the desired stabilizing forces, i.e Fud [Fxd Fyd M zd ]T 2) (2) Mf ucf where Mf is a constant u matrix As for tire modeling, a simple but accurate approach of Burckhardt [15] is adopted, which could describe both tire nonlinearities and combined slip characteristics Fy dr Fy G4 Fy1 xu b Fx df Fy D2 V2 Fx V4 Fx1 Gf Mz D4 D1 V1 Fx yu V3 G3 Fy D3 G2 Fx In the servo-loop, the stabilizing forces Fud would be optimally allocated to tire actuator actions, i.e active steering angles and wheel control torques The control goal of the servo-loop is to minimize Fud tracking error E , control input magnitudes u and increments 'u The constraints that need to be considered include tire-road adhesion limits (approximately “friction circle”), rate/magnitude limits of active steering and wheel torque MAIN-LOOP: SLIDING MODE CONTROLLER To get enough robustness against uncertainties (e.g variation of vehicle parameters and sensor information noises), sliding mode method is employed in the mainloop to control this nonlinear system with crosscoupling[16-17] The simplified vehicle dynamics (Eq 1) can be rewritten as follows: X i where X U Fud êơv x v y r ẳ T Figure Three-DOF vehicle model CHASSIS CONTROLLER STRUCTURE The entire controller is functionally realized through a main-loop and servo-loop structure, as shown Figure , and the control inputs are Si Tw i ud = arg min{E ,u ,'u} ei ȁ i ȟ i (4) X i X id , ȁ i is positive and wheret state error is ei ȟ i ³ ei (IJ )dIJ By following the procedures of sliding mode controller design, the sliding control law is designed as Ui 1 gˆ i [uˆ i k1i Si k2i sgn(Si )], (5) where fˆ i and gˆ i are the nominal values of fi and g i , respectively; uˆ i fˆ i X id ȁ i ei ; k1i and k 2i are carefully-selected control parameters [18] Through straightforward derivation, it could be verified that the designed control law can guarantee the reachability for the sliding surface defined above To avoid chattering of control inputs, the sign function is replaced with saturation function, which is defined as sat(Si ĭi ) 1) (3) In order to suppress the tracking error of motion variables, the sliding surface is chosen as a G fi ( X , t ) g i ( X , t ) U i ° ° ® ° °¯ sgn(Si ) Si t ĭi Si ĭi Si ĭi Figure Scheme of integrated vehicle chassis controller Finally, the desired stabilizing force Fud summarized as In the main-loop, the driver intention is firstly translated into desired motion variables v xd ,v yd , rd by a reference model In order to track the Fud ªmv (v xd v y r ȁ1e1 k11S1 k 21sat(S1ĭ1 ))º « m (v r ȁ e k S k sat(S ĭ )) » v 2 12 22 2 x « ằ ôơ ằẳ Izz ( e3 k13S3 k 23 sat(S3 ĭ3 )) (6) can be (7) SERVO-LOOP: FORCE ALLOCATION In our previous studies [18-19], tire constraints are treated indirectly using Sequential Quadratic Programming (SQP) approach (Figure 3b) In order to establish a dynamic trade-off between control inputs uc and Fud tracking error E , the desired increments of tire control inputs 'uc are achieved by minimizing a properly-weighted cost function JSQP , i.e by solving Eq JSQP T T E T WE E 'uc Wǻuc 'uc uc Wuc uc wJSQP w'uc (8) Tire force allocation based on CA; Inverse tire model: calculating desired tire variables; Actuator regulation: calculating final actuator inputs J CA ucf ucf U c Đ Fxi à ă â Fyi Fud Di (k ) max d įti (k 1) d įDi (k ) max Đ sLdi à ă â D di ¹ (11) where įti (k 1) is total wheel steering angle in the (k 1)th sampling step, į Di is equivalent driver steering angle As for steering rate limits, there approximately exists į | (įai (k 1) įai (k ))Ts , where Ts is the sampling time So the maximum active steering rate į max can be considered through Eq 12 įti (k ) Ts į max d įti (k 1) d įti (k ) Ts į max (9) However, the calculated final control inputs using this approach will still break these limits in some driving situation To directly consider these limits, a control allocation (CA) approach is adopted to allocate Fud to tire longitudinal forces Fxi and lateral forces Fyi The servoloop controller is thus realized through three steps (Figure 3a), i.e 1) 2) 3) Both rate and magnitude limits of actuator can be converted to bounds of Fxi and Fyi Considering the maximum active steering angle į max , there is (12) Since tire slip angle can be calculated as Į i įti ı i , combining Eq 11 and Eq 12, the bounds of i th tire slip angle in the (k 1)th sampling step can be determined by Eq 13 They could be further translated into tire lateral force bounds, Fyi and Fyi max , by using the linearized cornering stiffness CĮ (k ) , i.e 'Fy CĮ (k )'Į °° ® ° °¯ Į i max(įti (k ) Ts į max įDi (k ) įai max ) ı i Įi max min(įti (k )Ts į maxįDi (k )įai max )ıi (13) Similarly, as for powertrain and brake limits, when the wheel is driven or braked, the bounds of Fxi can also be determined By combining above limits, the feasible region U c of tire force vector can be further approximated using lower and upper bounds, i.e § Twi à ă â G Uc {ucf | ucf d ucf d ucf max } (14) CA problem with equality constraint Considering tire constraints, the optimization problem of Fud allocation can be formulated as follows wJ SQP Fud w'uc Đ 'sLdi à ă â 'Ddi ¹ CA problem: Given U c as the feasible region, ucf ucf (k ) as the initial value, find a feasible and optimal ucf that minimizes the cost function JCA , i.e Đ Twi à ă â G ¹ [ucfT Wuucf (ucf ucf )T Wǻu (ucf ucf )] Figure Scheme of servo-loop control: (a) CA approach; (b) SQP approach ucf Uc subject to Eq 16 TIRE FORCE ALLOCATION Mf ucf Tire force limits Tire actuator constraints, including limits on road friction, steering and wheel torque, can be converted into tire force limits as follows For each tire, the friction limit can be described as FHi Fxi2 Fyi2 d P peak Fzi (15) (10) Fud (16) In Eq 15, Wu and Wǻu are weighting matrices corresponding to magnitudes and increments of tire control force inputs However, Eq 15 might have no feasible solution satisfying the equality constraint Actually, Eq 16 could be properly relaxed in some situation For instance, when the vehicle is lightly braked in split-mu road, to suppress the lateral motion, it is more important to keep To take the weighted allocation errors into account, a weighted-least-squares problem can be formulated as Eq 17, where Ȗ ǻu , Ȗ e and We are weighting parameters Wuucf ucf Uc 2 Ȗ ǻu Wǻu (ucf ucf 2 Ȗ e We (Fud Mf ucf ) (17) SPQP formulation By using the sign preserving method [10], Eq 15 is replaced with a new equality constraint Eq 18, where the three scaling factors Ȝi are all positive and less than one By doing this, the signs of Fud will be preserved and with different weightings on Ȝi , the allocation accuracy of Fxd , Fyd and M zd can be emphasized to different degrees Mf ucf diag (Ox Oy Oz )Fud (18) Therefore, the linearly constrained quadratic programming problem is finally formulated as in Eq 19 and will be solved in each sample time step min[(ucf ucf )T Wǻu (ucf ucf ) ucfT Wuucf ucf Ȝi i ¦ Wi (1 Ȝi )2 ] x , y ,z (19) subject to Eq 18 and Eq 20 êucf ô ằ ô ằd ô ằ ô ằ ơô ẳằ ê ô cf » « » « » « x » « » « » « y » « » « » ôơ z ằẳ u êucf max « » » d« « » » « ơô ằẳ (20) INVERSE TIRE MODEL & ACTUATOR REGULATION Given tire forces ucf [Fxi ;Fyi ] and other sensor information, the desired tire variables ( sLdi , Į di ) can be calculated through an analytical inverse tire model or a look-up table Finally, using a Proportional-Integral(PI) regulator [18], the slip controller would further converted desired sLdi into wheel control torque Twi , as shown in Eq 21 As for tire slip angle regulation, the increment of steering angle FWL reff K p (sLd sL ) (sLd sL )K i s Tw (21) SIMULATION RESULTS The proposed integrated vehicle chassis controller is evaluated through co-simulation of Matlab/Simulink and MSC Carsim MSC Carsim is a professional software tool to simulate the dynamic behaviour of ground vehicles In the simulation, the controller designed in the Simulink module outputs steering angles and wheel torques to Carsim module; while in Carsim module, a sports car with four fully nonlinear independent suspensions is established Additionally, aerodynamics is also considered The simulation scenarios include step steer, split-mu braking and driving under crosswind disturbance The responses of vehicles with different controllers (i.e SPQP-based CA controller and SQP controller) will be compared STEP STEER MANEUVER The vehicle initial speed is 120 Km/h, the peak tire-road friction coefficient is 1.2 (dry asphalt), and the front wheels will be steered to degrees in 1.5 seconds As shown in Figure 4, the conventional vehicle (‘PSV’) loses stability quickly, with both yaw rate and sideslip far from the desired responses (‘Ref’) Evidently, through control intervention, both IVCC controllers could track the reference yaw rate and suppress the vehicle sideslip motion, while the yaw rate tracking error of CA based vehicle (‘CA’) is slightly lower than SQP based vehicle (‘SQP’) Yaw rate (deg/sec) WLS formulation 'į i is assumed to be the same as the desired slip angle increment 'Į di , i.e 'įi 'Į di 30 20 10 0 Time (sec) CG sideslip (deg) the allocation errors of Fyd and M zd (rather than Fxd ) as small as possible There are two alternative methods to relax this equality constraint, i.e through WeightedLeast-squares (WLS) formulation [20] and signpreserving quadratic programming (SPQP) formulation Through proper selecting the weighting parameters, WLS and SPQP could be both effective in suppressing allocation errors and keeping tire control force usage as low as possible Ref CA SQP PSV -1 Time (sec) Figure Vehicle responses in step steer maneuver Due to the large driver steering angle at high speed, the resultant lateral acceleration (up to 0.8g) would cause large load transfer Thus outer tires would share more vertical loads than inner tires Since the CA based controller could take the variations of tire vertical load into account, the rear right tire could be steered to a smaller angle, i.e įa is nearly two degrees smaller than įa , as shown in Figure As for wheel control torques shown in Figure 6, with CA based controller, the maximum wheel torque is clearly smaller than that of SQP based controller the CA controller could improve vehicle straight running ability and braking performance, even under extreme driving condition Ga3 (CA) Ga3 , Ga4, (SQP) Ga4 (CA) Ga1 (CA) -2 Figure Trajectories of vehicles in split-mu braking maneuver Ga1 , Ga2, (SQP) Ga2 (CA) -4 Time (sec) F ud Error with CA controller 1000 E Fx E Fy ud Figure Active steering angle in step steer maneuver Error Active steering angle (deg) -1000 Time (sec) with SQP controller Tw (Nm) -2000 400 Time (sec) Tw1 Tw2 Tw3 Tw4 200 -200 F 2000 Error 200 Time (sec) F Error with SQP controller ud Tw1 Tw2 Tw3 Tw4 ud Tw (Nm) 400 -200 E Mz F with CA controller Figure Tracking errors of desired stabilizing force in split-mu braking maneuver Time (sec) Figure Wheel control torque in step steer maneuver SPLIT-MU BRAKING With an initial speed of 50km/h, the vehicle will be braked from t = 1sec (target deceleration í0.3g) The friction coefficient of left side is 1.2 (dry asphalt), while that of the right side is is 0.2 (ice surface) The trajectories of vehicles are shown in Figure For the conventional vehicle, the lateral offset is the largest (up to 3.7m), while its braking distance within seconds also accumulates to 62m (5.1m longer than that of both SQP and CA controller cases) On the other hand, the vehicle sideslip offset of CA controller case is the shortest (limited within 0.15m), which demonstrates that Actually it can be further explained by Fud tracking error as shown in Figure With control allocation (CA) method, Fud could be almost perfectly tracked, with maximum tracking errors 700N, 200N and 300Nm, respectively However, with SQP controller, the tracking error of lateral force is nearly 1000N for a long period and the error of stabilizing yaw moment is up to 1200Nm, which explains the larger lateral offset (up to 1.5m) DRIVING UNDER CROSSWIND DISTURBANCE With an initial speed of 120Km/h, the vehicle is subjected to crosswind disturbance The wind properties and vehicle responses are shown in Figure During the disturbance period 0-5sec, as for the conventional vehicle, a larger lateral offset is caused, with yaw rate and CG sideslip angle up to 2.5 deg/sec and 0.3 deg, respectively On the other hand, by applying active yaw moment and active steering, the undesired lateral offset could be greatly suppressed, i.e about 25 meters less than that of conventional vehicle This open-loop scenario simulation also implies that the proposed controller is quite robust against uncertain disturbances Lateral, Y (m) Sideslip (deg) Yaw rate (deg/s) Crosswind Crosswind speed (Km/h) & heading angle (deg) 100 90deg 50 Speed Heading 60Km/h 25deg 10 CA PSV -5 2 10 10 0.5 -0.5 Time (sec) 40 20 0 100 200 300 400 Longitudinal, X (m) Figure Crosswind definition and vehicle responses CONCLUSION Since more and more advanced active control systems have been equipped in modern vehicles, subsystem coordination becomes an important issue that should be considered Based on the main/servo-loop control structure, this paper proposes a chassis controller coordinating steering, braking and traction Simulation results confirm that the integrated controller is quite robust and it could significantly improve vehicle stability However, by using control allocation approach in stabilizing force distribution, this methodology is also suitable for broader applications in vehicle chassis control For example, if active suspension is also available, tire vertical loads might also be taken as control inputs in force allocation Future work will include further analysis of controller parameter tunning Additional actuators such as active suspension might also be taken into consideration ACKNOWLEDGMENTS The authors would like to thank National Natural Science Foundation of China (NSFC) for its great support REFERENCES Fruechte, R D., Karmel, A M Rillings, J H., Schilke, N A., Boustany, N M, & Repa, B S., “Integrated vehicle control," in IEEE Vehicular Technology Conference, San Francisco, CA, USA, pp 868-877, 1989 Manning, W., Crolla, D., Brown, M & Selby, M., “Coordination of chassis subsystems for vehicle motion control," in Proceeding of AVEC’2000, pp 313-319, 2000 Nagai, M., Shino, M & Gao, F., “Study on integrated control of active front steer angle and direct yaw moment," JSAE Review, Vol 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Integrated Vehicle Dynamic Control Design Based on Optimum Tire Force Distribution, Journal of Shanghai Jiaotong University (Chinese Edition), in press ... paper proposes a chassis controller coordinating steering, braking and traction Simulation results confirm that the integrated controller is quite robust and it could significantly improve vehicle. .. with CA based controller, the maximum wheel torque is clearly smaller than that of SQP based controller the CA controller could improve vehicle straight running ability and braking performance,... On the other hand, by applying active yaw moment and active steering, the undesired lateral offset could be greatly suppressed, i.e about 25 meters less than that of conventional vehicle This