Model Predictive Control for Vehicle Yaw Stability

Transcription

1 EngOpt d Intenational Confeence on Engineeing Optimization Rio de Janeio, Bazil, 1-5 July 212. Model Pedictive Contol fo Vehicle Yaw Stability Mooyong Choi and Seibum Choi Depatment of Mechanical Engineeing, KAIST, 335 Gwahagno, Yuseong-gu, Daejeon, 35-71, Koea 1. Abstact Yaw stability of an automotive vehicle in a tun is citical to the oveall stability of the vehicle. In this pape, we pesent a method of vehicle stability contol (VSC) based on Model Pedictive Contol (MPC). Conventional VSCs wok passively as they detect ecessive yaw ate o slip angle of a vehicle. Howeve, in many cases, when ecessive yaw ate o slip angle of a vehicle is detected, the vehicle is aleady in unstable states. Using the MPC scheme, the poposed contolle can actuate bakes to geneate coection moment in advance of vehicle being unstable by pedicting vehicle movement of seveal hunded milliseconds ahead. The diffeences of desied vehicle states fom the bicycle model and the estimated vehicle state ae minimized by applying the MPC scheme. The pefomance of the poposed method is evaluated using the vehicle dynamics softwae CARSIM. 2. Keywods: Vehicle Dynamics, Yaw Stability, Model Pedictive Contol (MPC), Optimization 3. Intoduction Vaious types of Electonic Stability Contol (ESC) have been developed by many eseaches [1][2][3]. Basically, these ESC algoithms ae activated to eet coection moment when ecessive diffeences of actual and desied yaw ates o immodeate side slip angles ae detected. Howeve, when ecessive side slip angles o ecessive diffeences of actual and desied yaw ates ae detected, in many cases, vehicles ae aleady in unstable states. Since vehicle in unstable states tend to quickly spin out o bounce out fom its desied tajectoy, small delay of ESC actuation can be followed by fatal accidents. To ovecome this dawback of conventional ESC, this pape focuses on development of a new ESC algoithm based on MPC scheme. ist, bicycle model is designed to captue the chaacteistics of vehicle lateal dynamics. The initial coneing stiffness is used to geneate desied yaw ate. On-line coneing stiffness adaptation is pefomed using the tie model [4]. The MPC scheme geneates the coective yaw moment to minimize the coective yaw moment and the diffeence between the desied yaw ate and actual yaw ate. 4. Development of Yaw Stability Contolle based on MPC 4.1. Vehicle Lateal Dynamics Modeling In this section, a simple bicycle model with dynamic tie model is used to geneate desied yaw ate. This bicycle model as shown in ig.1 descibes vehicle lateal dynamics. igue 1. Bicycle Model The equations of motion fo the bicycle model ae as follows: mv ( ) yf y I l l z f yf y (1) (2) Whee C yf f f C y l f f f v l v

2 Dynamic tie model as defined in [5] can be obtained as follows: C, y _ lag y _ lag y KV The lagged lateal tie foces can be defined as follows:, y _ lag y _ lag yf y _ lag y _ lag y Using the fist ode dynamic tie model and laplace tansfom, font and ea lateal tie foce equations can be eplaced by the follows: mv ( s ) y C, K, L 1.5m y K yf _ lag y _ lag I s l l z f yf _ lag y _ lag, y As a esult, the following state space fom equation can be obtained X AX B (3) X 1 C C 1 C l C l 1 1 f f f 2 mv mv A C l C l C l C l 1 f f f f I I v z z C f mv B Cl f f Iz By using the dynamic tie model, the bicycle model can be efomed to have the fou states including deivatives of yaw ate and side slip angle and lagged chaacteistics of tie foces to bette descibe vehicle lateal dynamics. 4.2 Tie Model In a egula bicycle model, the coneing stiffnessesc f, C ae set to be constants. Howeve, as slip angles incease o tie longitudinal foces ae applied, those coneing stiffnesses decease. To take these phenomenon into account, the combined longitudinal and lateal bushed tie model [t] which is simple enough to be un in eal-time while complicated enough to captue the tie nonlinea chaacteistics is used. The equations of the tie model is as follow: T (4) (5)

3 Whee The following paametes ae identified in eal-time using the lineaized ecusive least squaes method [] in eal-time.,, 4.3. Lineaized Tie Model Since we have the nonlinea tie model which is identified in eal-time, the nonlinea tie chaacteistics can be eflected to the bicycle model by lineaizing the tie foce cuve at cuent slip angle. igue 2. Geneic Lateal Tie oce vs Slip Angle Cuve y C f (6) C y f y (7) The tie foce can be epessed as in (7) by lineaizng the tie model at the cuent slip angle. Using (7) fo the font and ea tie foces, the bicycle model with dynamics tie model can be modified as follow: X A X B C D M c (8) A 1 C C 1 C l C l 1 1 mv mv 1 I I v z z f f f C l C l C l C l 1 f f f f

4 C f mv B Cl f f Iz f mv C l l f f Iz D 1z I z M c is the coective moment. C is the additional tem due to the lineaization. (3) is used to geneate the desied yaw ate, wheeas (8) is used to calculate the coective yaw moment in the MPC scheme Calculation of Coection Moment based on Model Pedictive Contol M 1 J M W ( ) n1 2 2 n l, n p, n subject to g ( ) is coective yaw moment.,, and ae desied yaw ate fom (3), yaw ate developed in (8), weighting, state constaints, espectively. State constaints can be defined as follow: l W ( ) p g( ) g : y _ ma ma ma yf _ ma b (1 ) a b yf _ ma mv a a (1 ) b b yf _ ma mv a y _ ma y _ ma (9) : b v ma ma _ ma (1) is added to the equation. (1) is detemined fo the vehicle not to eceed the slip angle which geneate the maimum lateal tie foce. o the bounday, a look-up table is used fo pactical eason. (9) is defined as in [6] but 4.4. Distibution of the coective yaw moment ig. 3. shows a vehicle with bake foce applied on the font ight wheel. To geneate the negative coective yaw moment, font ight and font ea wheels could be the places that bake foces to be applied. In case of applying bake foce on the ea ight wheel on a vehicle in a left tun, lateal foce of ea ight wheel deceases. Diminished lateal foces of ea wheels ae followed by ceating positive moment which is undesiable in this case. Wheeas, in case of applying bake foce on the ea ight wheel on a vehicle in a left tun, the lateal foce of the font wheel deceases which is followed by incease of the negative coective yaw moment. Consideing fiction ellipse effect of the ties, wheels that bake foces to be applied ae chosen. The following equations calculate the coective yaw moments as functions of changes of longitudinal tie foces. The change of lateal foce due to additional longitudinal foce applied by bake is calculated using the tie model [4]. M sin l cos tb c _ f f y (11) M tb l c _ f y (12)

5 igue 3. Applying bake foce to geneate yaw moment 35 combined tie foces at 5 degee slip angle fom model 5 coection moment of font wheel 1 coection moment of ea wheel oce (N) 2 15 y Moment (Nm) 3 2 Moment (Nm) slip atio slip atio slip atio (a) Lateal and longitudinal foce vs slip atio (b) Moment when applying bake on R (c) Moment when applying bake on RR igue 4. Applying bake foce to geneate yaw moment In ig. 4., (a) shows longitudinal and lateal tie foces along slip atio at a constant slip angle. (b),(c) show geneated coective moments when applying bake foces on the font ight and ea ight espectively. As shown in (c), the ecessive bake foce on ea ight wheel can geneate coective moment in an opposite diection due to diminished lateal foce. The MPC scheme intoduces in 4.3 calculates the needed coective yaw moment. Then, Eq. (11) o Eq. (12) is used to calculate the equied bake foce using the Newton-Raphson method. The equied bake foces ae geneated by applying pope bake pessue to the vehicle bake system. igue 5. Schematic of the system 5. Simulation Results.6 Yawates.4.2 adian/sec desied -.6 actual without contol actual with contol

6 Maste Cylinde Pessue (Mp Longitudinal Speed (m/s) Maste Cylinde Pessue Longitudinal Speed Coective yaw Moment 1 5 Coective Moment (Nm) Applied Pessues L R RL RR 2 Mpa igue 6. Simulation esult A simulation esult using CARSIM is pesented in ig.6. Befoe the bake pedal is pushed between 4 and 5 second, the vehicle with the contol follows the desied yaw ate with less eo compaing to the vehicle without any contol. Afte the bake pedal is stepped while tuning, the vehicle without contol loses it stability and spins out. Wheeas, the vehicle with contol tack the desied yaw ate even when the dive steps on bake pedal while tuning. 6. Conclusion A method of vehicle stability contol (VSC) based on MPC is pesented. Using the MPC scheme, the poposed contolle can actuate bakes to geneate coection moment in advance of vehicle being unstable by pedicting vehicle movement of seveal hunded milliseconds ahead. The diffeences of desied vehicle states fom the bicycle model and the estimated vehicle state ae minimized by applying the MPC scheme. The simulation esult using CARSIM shows that poposed algoithm successfully stabilize the vehicle even bake in a tun is applied to the vehicle.

7 Acknowledgement This wok was suppoted by the National Reseach oundation of Koea (NR) gant funded by the Koea govenment(mest) (No ) Refeence 1. B.A. Guvenc, T. Acaman, and G. Levent, Coodination of steeing and individual wheel baking actuated vehicle yaw stability contol, Intelligent Vehicle Symposium, Poceedings of IEEE, 23, pp W.J. Manning, M. Selby, D.A. Colla, and M.D. Bown, IVMC: Intelligent vehicle motion contol, SAE Tans (22). 3. M. Nagai, M. Shino, and. Gao, Study on integated contol of active font steeing angle and diect yaw moment, JSAE (22), pp Y.-H.J. Hsu, S.M. Laws, and J.C. Gedes. Estimation of tie slip angle and fiction limits using steeing toque. Contol Systems Technology, IEEE Tans-actions on, 18(4):896{97, July R. Rajamani, Vehicle Dynamics and Contol. New Yok: SpingeVelag, Caig E. Beal, Applications of Model Pedictive Contol to Vehicle Dynamics fo Active Safety and Stability. PhD thesis, Stanfod Univesity, 211.