Electric Vehicle Lateral Dynamics Control based on Instantaneous Cornering Stiffness Estimation and an Efficient Allocation Scheme

Transcription

1 Electric Vehicle Lateral Dynamics Control based on Instantaneous Cornering Stiffness Estimation and an Efficient Allocation Scheme A. Viehweider Y. Hori The University of Tokyo, Department of Advanced Energy, Kashiwa, Japan ( The University of Tokyo, Kashiwa, Japan. Abstract: Beyond the establishment of a polution free mobility, electric vehicles show attractive features as far as vehicle motion control is concerned. In this work an electric vehicle equipped with four independent In-Wheel-Motors and active front and rear steering system is considered. The 6 actuators should be controlled in a way to guarantee safe, comfortable and low energy consumption drive. Generally, Linear Time Invariant (LTI) vehicle models are appropriate if the motion of the vehicle does not approach the physical limits of the tire force saturation and road conditions remain constant. Otherwise, they fail to predict the vehicle dynamics. On the other hand the control design approach based on LTI models is straightforward. The approach taken in this contribution is the following: use a robust sliding mode controller based on a slowly linear time varying model, together with an efficient estimation scheme for the instantaneous cornering stiffness in order to prevent the vehice to reach lateral tire force saturation by restricting the possible actuator outputs in a control allocation scheme. Computer simulation runs with the professional vehicle dynamics environment CarSim show good performance of the control scheme. Further work will be the validation with real vehicle experiments and modelling of consumed energy for consideration of energy efficency in the control scheme. Keywords: Vehicle Modelling, Electric Vehicle Control, In Wheel Motor, Cornering Stiffness, Lateral Dynamics, Roll Motion, Control Allocation. 1. INTRODUCTION Electric Vehicle (EV) research draws a lot of attention these days. EVs seems to be the right answer to depleting oil resources and a mandatory reduction of carbon dioxide emissions due to the greenhouse effect. But the EV technology offers also advantages for the vehicle dynamics control. As particularly advantageous for the purpose of motion control can the integration of In Wheel Motors (IWMs) in each wheel be considered. Together with Active Front and Rear Steering (AFS and ARS) six degrees of freedom are available in order to determine an advantageous vehicle behaviour as far as longitudinal and lateral dynamics, as well as roll and pitch motion is considered. To implement a sophisticated comprehensive control concept a valid vehicle model must be established. This contribution focuses on the modelling issues and implications for the controller design of a 4 IWMs driven EV with AFS and ARS with regard to lateral dynamics and roll motion control during constant or nearly constant velocity. At the core of this approach is a vehicle model that takes into account the particularities of the 4 IWMs driven EVs with consideration of the anti lift and anti dive forces (vertical forces generated by the IWMs) as introduced in Kat- This research has been supported by the kakenhi grant Nr during a fellowship program of the Japanese Society for the Promotion of Science. suyama (2011). A linear slowly parameter variant model is used for the controller design. Nonlinear effects due to reaching the lateral tire force saturation (e.g. by heavy steering command and/or low µ road) are counteracted by a robust control approach and the determination of the lateral tire force saturation which is detected with an Instantaneous Cornering Stiffness (ICS) estimator which differs from Sienel (1997) as far as no particular mass distribution of the vehicle is assumed but lateral tire force sensors (nowadays available at a reasonable price) are used. Simulation results show the effectiveness of the modelling and control approach. The contribution is structured in the following way: In section 2 the basics of EV dynamic modelling are shortly given: At constant velocity v the lateral and roll dynamics can be described by a 4th order linear state space model with two slowly time varying road dependent parameters. The peculiarities of In-Wheel motor traction can be summarized by a linear mapping between the four driving forces generated by the motor torques and resulting moments and forces acting on the vehicle. The non linearity due to lateral tire force saturation can be grasped with the ICS. Section 3 describes an efficient estimation schemes for the determination of the ICS based on lateral tire force sensors. Section 4 develops a comprehensive control and allocation scheme based on the introduced modelling and

2 a 11 = C f + C r m eq v a 13 = m sh(k x m s gh) J x m eq v and a 21 = l f C f l r C r a 41 = m sh(c f + C r ) J x,eq m a 43 = K x m s gh J x,eq a 12 = 1 l f C f l r C r m eq v 2 a 14 = m shc x J x m eq v a 22 = l2 f C f + lrc 2 r (2) v a 42 = m sh(l f C f l r C r ) J x,eq mv a 44 = C x J x,eq Fig. 1. Electric vehicle model for the purpose of modelling Table 1. Vehicle describing parameters m Vehicle mass m eq Equivalent mass m s Vehicle sprung mass C f Front cornering stiffness C r Rear cornering stiffness h Distance RC-COG Moment of inertia z-axis J x Moment of inertia x-axis J x,eq Equivalent moment of inertia x-axis C x Roll damping coefficient K x Roll stiffness coefficient v Longitudinal velocity l f Distance front axle from COG l r Distance rear axle from COG r Wheel radius d Width of the axle tread (d 2 = d 2 ) section 5 shows simulation results of this scheme. In section 6 some conclusions are given. 2. ELECTRIC VEHICLE MODELLING 2.1 Linear vehicle lateral and roll dynamics model Under the assumption of moderate steering angles and constant (or only slow time varying) longitudinal velocity v the EV can be described as linear state space model with the state vector x = [β, γ, φ, φ] T, where β vy v x, γ, φ, φ are the body slip angle, yaw rate, roll angle and roll rate respectively and the measurement vector y = [a y, γ, φ] T according to Fig. 1. As inputs u of the EV we consider the front steering angle δ f, the rear steering angle δ r and an additional yaw and roll moment M z, M x that will be defined later: with ẋ = A(C f, C r )x + B(C f, C r )u = (1) ) ) = ( a11 a 12 a 13 a 14 a 21 a a 41 a 42 a 43 a 44 x + ( va11 v(a ) va 13 va ( b11 b b 21 b 22 b b 41 b 42 0 b 44 y = C(C f, C r )x + D(C f, C r )u = ) = x + u ( ) vb11 vb u and b 11 = C f m eq v b 12 = C r m eq v b 21 = l f C f b 22 = l rc r b 23 = 1 b 41 = m shc f J x,eq m b 42 = m shc r J x,eq m b 44 = 1 J x m eq = m m2 sh 2 J x J x,eq = J x m2 sh 2 (3) m. (4) The meaning of the single parameters that determine the entries of state description matrices A and B is explained in Tab. 1. Equation (1) can be derived by applying the kinematic and dynamic relationships for the bicycle model with roll dynamics. It is assumed that the front and rear lateral forces F yf, F yr depend linearly on the respective front and rear tyre slip angles α f, α r : F yf = C f α f F yr = C r α r α f = γl f v β + δ f α r = γl r v β + δ. (5) r In this model approach - apart from the cornering stiffness values in the linear region C f, C r - all parameters are assumed to be constant. The cornering stiffness parameters of the linear model can be estimated by use of interlaced observer schemes as described in Fujimoto et al. (2007). For the purpose of control design the model in (1) is reduced to a 2 nd order model (x red = [β, γ] T, u red = [δ f, δ r, M z ] T ) where the roll dynamics is not explicitly described but only the effect on the lateral dynamics described as an additional time varying uncertainty (.): ẋ red = A red (C f, C r )x red + B red (C f, C r )u red + (x red, t) ( ) ( ) a = 11 a 12 b a 21 a 22 x red + 11 b 12 0 b 21 b 22 b 23 u red + (.). (6) The uncertainty vector (.) accounts also for non linear phenomena when approaching lateral tire force saturation. A controller based on this vehicle model must be able to perform regardless of this uncertainty. 2.2 Modelling the effect of In-Wheel-Motors on the vehicle The effect of 4 IWMs on the vehicle dynamics can be very compactly described. As opposed to a combustion engine driven vehicle with conventional power train, the IWM generated torque at a wheel leads not only to a force in longitudinal direction (driving force) but also to a remarkable force in vertical direction. This is a

3 Fig. 3. Tire slip angle - lateral tire force characteristic However for large tire slip angle as it may occur on a slippery road this relationship does not longer hold true. In Fig. 3 a realistic tire slip angle - lateral tire force characteristic is given. The cornering stiffness of the linear model C {f,r} as introduced in subsection 2.1 is the slope at tire slip angle α = 0 (yellow line). The Instantaneous Cornering Stiffness (ICS) is the slope of this curve at the current tire slip angle (black line). It can be seen that a low value of ICS is an indication that the tire approaches its saturation point. Fig. 2. Longitudinal and vertical forces due to the 4 independent torques acting on the wheel (top) and resulting forces and moments acting on the vehicle (bottom) consequence of the different location of the instantaneous center of rotation. In the case of small longitudinal tire slip λ and constant velocity the driving force of a single wheel can be easily derived from the torque applied at the wheel: F xi = Ti r and the four different driving forces are related to the resulting vehicle forces and moments as shown in Fig. 2 in the following way: M = F x F z M x M y M z = M F x1 F x2 F x3 F x w f w f w r w r d 2 w f d 2 w f d 2 w r d 2 w f l f w f l f w f l r w r l r w r d 2 d 2 d 2 d 2,, (7) where w f = tan(θ f ), w r = tan(θ r ) and Θ f,θ r are the so called anti-dive and anti-lift angle, respectively and can be considered as constant during drive. In this contribution the independent torques are used to influence F x for keeping constant the velocity v, M z to assist in yaw rate tracking and M x to damp the roll motion behaviour during cornering. 2.3 Definition of Instantaneous Cornering Stiffness for Lateral Tire Force Saturation Estimation In the previous sub sections we introduced a linear vehicle dynamics model for moderate steering. As essential for this model development was the linear relationship between lateral tire forces and tire slip angle as expressed in (5). 3. INSTANTANEOUS CORNERING STIFFNESS ESTIMATION FOR LATERAL TIRE FORCE SATURATION DETECTION In Sienel (1997) an estimation scheme for a particular mass distribution assumption and the positioning of a lateral acceleration sensor at the front tread is given. Since this mass assumption may be violated, in this contribution an approach based on the use of lateral tire force sensors is suggested which is generally valid and not only for a particular mass distribution assumption.the ICS can be approximately estimated by the use of time derivative of tire slip angle and tire lateral force. C f,ins (α) = df y,f (α f ) dα f = This estimation is valid if df y,f dt dα f dt dα f 0. (9) dt The use of the time derivative of the tire slip angle is advantageous in the sense that no explicit estimation of the body slip angle is necessary: with (8) α f = δ f β + l f γ (10) v β a y v x γ. (11) Since practically usable and economic viable tire lateral force sensors come out only recently, first approaches of cornering stiffness were based on using lateral acceleration at the front axle together with the assumption of a particular mass distribution of the vehicle (Sienel (1997)).

4 Fig. 4. Lateral force sensor installed in each wheel to measure the lateral tire force generated by each wheel. (NSK, Japan) The use of lateral tire force sensors as depicted in Fig. 4 allows the determination of the ICS via the following equation: F y,f C f,ins δ f Fy mv γ l f v γ = N (12) D In order to avoid the explicit division of two quantities N, D heavily effected by noise the estimation is embedded in a Recursive Least Squares (RLS) estimation scheme (y = Θx) with y = N = F y,f, x = D = α f and Θ = C f,ins : P k 1x k Π k = λ + P k 1 x 2 k Θ k = Θ k 1 + Π k (y k Θ k 1 x k ) P k = 1 λ [P k 1 Π k x k P k 1 ] (13) with Θ 0 = 0 and P 0 adequately chosen. The RLS scheme (for general information confer Madisetti et al. (1997)) avoids the explicit division, although estimation quality during low values of α f may also be poor. Therefore it is advisable to detect low values of α f and not to trust the estimation in this case. In (12) the time derivatives of several signals are needed ( δ f, γ, F y,f ). Since they are noisy, special filtering techniques are needed. As a particular filtering technology it is referred to Algebraic Real Time derivative Estimation (ARTE) as introduced in Zehetner et al. (2007) and Fliess et al. (2005). 4. CONTROL APPROACH BASED ON ESTIMATION AND EFFICIENT ALLOCATION The modelling considerations in the previous chapters are now used to establish a comprehensive control concept that guarantees good performance even in the non linear domain as shown in Fig. 5. At the core of this concept is a sliding mode controller with adaptive gain, a slight modification of a control concept introduced in Roopaei et al. (2009). It consists of an equivalent control part and a robust control part. The controller relies heavily on the availability of the yaw rate γ and the body slip angle β. Since the body slip angle β cannot be measured at reasonable costs it must reconstructed by use of an observer or an estimation scheme. This is not topic of this contribution. Efficient schemes can be found in Geng et al. (2008), Ngyuen et al. (2011) and Kanghyun et al. (2011). The linear cornering stiffness values (estimated by schemes as presented in Fujimoto et al. (2007)) are used for the equivalent control part of the controller and for the allocation scheme. The controller generates virtual Fig. 5. Comprehensive control approach based on allocation and estimation controller outputs that together with the resulting vehicle longitudinal force F x and an additional roll moment M x that can be used advantageously to damp the roll motion are translated into the demands for the six actuators (two steering angles and 4 driving forces (motor torques)). In order to avoid reaching lateral tire force saturation and beyond appropriate actuator limits (steering angle limits) are detected and the allocation be carried out in a way that these limits are observed. 4.1 Sliding mode controller for Robust Control The lateral vehicle dynamics control concept is based on the vehicle model as described in (6) and the control approach introduced in Roopaei et al. (2009). The control law is composed of a equivalent control part v FF and a sliding mode (robust) part v FF : with v = v FF + v FB (14) v FF = A red (Ĉf, Ĉr)x red (t) + ẋ red,ref v FB = k as sign(s) (15) s = e = x red x ref k as = c 1 s 1 c 2 k as, where the sign function is understood as a smooth approximation of the signum function. Under the assumption of (x red, t) < α (α > 0) and c 1, c 2 > 0 the convergence of this control scheme can be proven (Roopaei et al. (2009)). 4.2 Control allocation The allocation problem for the 6 degrees of freedom electric vehicle (active front and rear steering, 4 independent motor torques) can be described as following: v = Mu, u T Wu = min! u min (t) < u < u max (t), (16) where the allocated variable u = [δ f, δ r, F x1, F x2, F x3, F x4 ] T and the virtual control v = [v 1, v 2, F x, M x ] T. Additionally

5 Fig. 6. Lateral dynamics at v = 50km/h and µ = 0.7with the introduced control, estimation and allocation scheme each allocation equation can be weighted separately by an appropriate weighing matrix T. The problem is solved with Active Set algorithms and reduced to a Weighted Least Square (WLS) problem: min u T Wu + γ w Mu v 2 T subject to u u min (t) <= u <= u max (t). (17) For computational efficient solutions of this allocation problem please confer Härkegård (2003). The matrix W can be used for additional criteria like tire work load minimization as described in Ando et al. (2010). The entries of the matrix M are as follows: M(Ĉf, Ĉr) = Ĉ f mv Ĉ r mv l f Ĉ f 0 0 l r Ĉ r d 2 1 d 2 w f d d 2 w f 0 d 2 1 d 2 w r d d 2 w r T (18) The single actators demand (entries of the vector u) shall be limited: δ f,min (t) δ f δ f,max (t) δ rmin (t) δ r δ rmax (t) F x,1min (t) F F x,2min (t) < x,1 F F < x,1max (t) x,2 F x,2max (t), (19) F x,3min (t) F x,4min (t) F x,3 F x,4 F x,3max (t) F x,4max (t) where in this contribution especially the limits for the steering angles are of interest. By observation of the ICS lateral tire force saturation is detected and the maximal tire slip angle δ {f,r},max is detected where the maximal tire force is achieved. By knowing this bound on the tire slip angle the maximal front and rear steering angle limits can be derived and used in the allocation scheme: Fig. 7. Front and rear steering wheel angles and lower and upper bounds applied in the allocation scheme δ f,max (t) = α f,max l f γ + v ˆβ x δ r,max (t) = α r,max + l rγ + v ˆβ (20) x α {f,r},max is obtained by observing the front and rear ICS, if the ICS falls below a certain threshold (typically between 50% and 70% of the maximal ICS value) the estimated tire slip angle is taken as upper limit for the tire slip. 5. SIMULATION RESULTS The control and allocation scheme of Fig. 5 is tested with the professional vehicle dynamics simulation environment CarSim with a high dimensional non linear realistic vehicle model that takes into account also additional dynamical effects (tire dynamics ecc.). A so called sine steer with a dweel steering wheel command is applied to a vehicle driving with constant velocity v = 50km/h. The road friction µ coefficient of the road is 0.7. The reference yaw rate γ ref and body slip angle β ref are generated from a reference model that takes into account the vehicle velocity v and the driver input by the hand steering wheel. With this velocity v, road condition µ and yaw rate reference γ ref in the case of non inclusion of the actuator limitations

6 Fig. 8. Lateral vehicle dynamics control without (top) M x = 0 and with (bottom) M x = k φ roll damping activated. as expressed in (19) in the control scheme lateral tire force saturation is reached and deterioration of the control performance and even instability may occur. With the introduced control and allocation scheme the lateral tire force saturation region is avoided and the yaw rate tracking achieved as shown in Fig. 6. As can be seen in the lower left and right part of the figure the front and rear ICS are estimated and when the decrease of the ICS reaches a certain level the current tire slip angle is used for the determination of the actuator bounds δ f,max (t), δ r,max (t) (Fig. 7) in order to avoid the tire reaching lateral tire force saturation and beyond. When the front and rear lateral tire force saturation is reached, an additional yaw moment M z (yellow line) is used in order to reach the yaw rate tracking goal. In Fig. 8 the simulation results of the control scheme are compared with roll damping activated M x = k φ, (k > 0) (bottom) and no additional roll moment M x = 0 generated by the IWMs (top). 6. CONCLUSION This contribution has introduced a method to approach EV lateral dynamics control and roll motion damping for EV with IWMs and AFS and ARS. It has been shown how the peculiarities of the IWM driven EV can be modelled and used for improving control performances. The lateral vehicle dynamics model used is quite simple and relies on estimation of the linear cornering stiffness values. Model deficiencies due to non linearity and roll dynamics are summarized in an additional uncertainty vector which is limited in the -norm. Entering the tire force saturation is avoided by an efficient early detection scheme that relies on the estimation of the so called ICS. This detection makes it possible to limit the allowed actuator demand in the control allocation scheme, that maps the virtual controller outputs to the demand values for the six actuators that this kind of vehicle offers. Additional research work will be experimental validation with a test car, and consideration of longitudinal tire force saturation and energy efficiency. REFERENCES Ando, N., and Fujimoto, H.(2010). Yaw-rate Control for Electric Vehicle with Active Front/Rear Steering and Driving/Braking Force Distribution of Rear Wheels. In Proceedings 11th IEEE Int. Workshop on Advanced Motion Control, Nagaoka, Japan, March Fliess M., Mboup M., and Sira-Ramirez, H. (2005). Analyse et representation de signaux transitories: application a la compresion, au debruitage, et a la detection de rupture. In Proceedings of GRETSI 2005, Belgium. Fujimoto H., Fujii K., and Takahashi, N. (2007). Traction and Yaw-rate Control of Electric Vehicle with Slip-ratio and Cornering Stiffness Estimation. In Proc. American Control Conference 2007, Geng, C., and Hori, Y. (2008). Nonlinear Body Slip Angle Observer for Electric Vehicle Stability Control. In Proceedings EVS 23. Härkegård, O. (2003). Backstepping and Control Allocation with Applications to Flight Control. PhD Thesis, Dept. of Electrical Engineering, Linkping University. Kanghyun, N., Oh, S., Fujimoto, H., and Hori, Y. (2011). Vehicle state estimation for advanced vehicle motion control using novel lateral tire force sensors In Proceedings of ACC 2011, Katsuyama, E. (2011). Decoupled 3D Moment Control by In-Wheel Motor. In Proceedings 2011 JSAE Annual Congress (Spring) (Japanese), 1-6. Madisetti, V. K., and Williams, D. B (1997). The Digital Signal Processing Handbook IEEE Press, Chap Ngyuen, B. M., Kanghyun N., Fujimoto H., and Hori, Y. (2011). Modeling of lateral dynamics for motion control of electric vehicle. In Proceedings IIC 2011, OS 12. Roopaei, M., Balas, V. E. (2009). Adaptive Gain Sliding Mode Control in Uncertain MIMO Systems. In Proc. of Int. Workshop on Soft Computing Applications Sienel., W. (1997). Estimation of the tire cornering stiffness and its application to active car steering. Proc. CDC 1997, Zehetner, J., and Horn, M. (2007). Echtzeitableitungsschaetzung am Beispiel automotiver Anwendungen. Proceedings Retzhof Symposium 2007.