A Neural Network Combined Inverse Controller for a Two-Rear-Wheel Independently Driven Electric Vehicle

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1 Energies 24, 7, ; doi:.339/en77464 Article OPEN ACCESS energies ISSN A Neural Network Combined Inverse Controller or a wo-rear-wheel Independently Driven Electric Vehicle Duo Zhang, Guohai Liu *, Wenxiang Zhao, Penghu Miao, Yan Jiang and Huawei Zhou School o Electrical and Inormation Engineering, University o Jiangsu, Zhenjiang 223, China; s: zhangduo@mail.ujs.edu.cn (D.Z.); zwx@mail.ujs.edu.cn (W.Z.); miaopenghu@63.com (P.M.); jyan@mail.ujs.edu.cn (Y.J.); zhouhuawei@mail.ujs.edu.cn (H.Z.) * Author to whom correspondence should be addressed; ghliu@mail.ujs.edu.cn; el.: Received: 4 April 24; in revised orm: 2 June 24 / Accepted: 8 July 24 / Published: 22 July 24 Abstract: Vehicle active saety control is attracting ever increasing attention in the attempt to improve the stability and the maneuverability o electric vehicles. In this paper, a neural network combined inverse (NNCI) controller is proposed, incorporating the merits o let-inversion and right-inversion. As the let-inversion sot-sensor can estimate the sideslip angle, while the right-inversion is utilized to decouple control. hen, the proposed NNCI controller not only linearizes and decouples the original nonlinear system, but also directly obtains immeasurable state eedback in constructing the right-inversion. Hence, the proposed controller is very practical in engineering applications. he proposed system is co-simulated based on the vehicle simulation package CarSim in connection with Matlab/Simulink. he results veriy the eectiveness o the proposed control strategy. Keywords: neural network combined inverse; sot-sensor; decoupling control; electric vehicles; two-rear-wheel independently driven. Introduction Due to the drastic issues o environmental pollution and the energy-consumption crisis, many kinds o electric vehicles (EVs) are quickly being developed. Among them, the multi-wheel independently-driven EVs driven by either two or our motors are considered as a uture trend. hese motors are integrated into the drive wheels and can be controlled independently [ 4]. In this coniguration, the motors

2 Energies 24, directly drive the wheels so that the power train is shortened and EV eiciency is raised. Furthermore, since the driving/braking orce o each wheel can be independently controlled, some vehicle saety system such as anti-brake system and direct yaw-moment control (DYC) become more lexible and easible [5 9]. Moreover, the driving/braking orce can be estimated in real time so that the observer or estimate technique can be applied easily. hereore, the perormances o the EVs are greatly improved. It is well known that the vehicle dynamics are not completely independent in both directions. his is due to three major types o the coupling eects, namely the kinetic coupling, the tire orce coupling and the weight shit coupling. hese coupling eects become increasingly signiicant as maneuvers involve higher accelerations, larger tire orces, or reduced road riction []. It also has been veriied that the dynamic coupling compensation does signiicantly improve the control perormance o the vehicles []. However, the majority o the existing research has ocused on either pure longitudinal or pure lateral control. In act, the integration o both control problems is one o the most challenging issues. Only a handul o works have been reported on merging both control tasks into one goal [2 4]. Furthermore, as a typical nonlinear, time-varying and coupled system together with dynamic uncertainties, the vehicle precise dynamic model cannot be obtained in practices [5]. In act, the perormance o a vehicle active saety system depends not only on the control algorithm, but also on the accurate measurement o the vehicle motion states. In the DYC system, it is necessary to accurately measure yaw rate and sideslip angle. he yaw rate inormation is easily obtained by using low-cost gyro sensors, but unortunately, the direct measurement o the sideslip angle is only provided by special devices (optical sensors or global positioning system inertial sensors combinations), which are nowadays very expensive and unsuitable or ordinary vehicles [6,7]. hus, the sideslip angle must be estimated in real-time. For this reason, a variety o estimation methods have been studied extensively, such as Kalman ilter/extended Kalman ilter observers [8,9], recursive least square algorithms [2], adaptive observers [2] and else observers [22 25]. Most o these state observers and related algorithms rely heavily on an accurate tire model or a vehicle model. hey are able to provide a satisactory estimate results only these models are accurately known. However, the tire model and the vehicle model will vary during the vehicle operation. Since the neural network (NN) has certain sel-learning ability to adapt to environment and system characteristics, it has been used to estimate the sideslip angle [3,4,26,27]. In these literatures, the sideslip angle was considered as a map or unction o the vehicle dynamics usually measured on board such as the lateral acceleration and the yaw rate. he results obtained seem to be promising, but these methods need a large amount o experimental data and the results depend on the data accuracy. Furthermore, these NN-based methods were seldom theoretically proved or its selection o training parameters. In addition, these methods have poor adaptability or the complex vehicle operating conditions. hereore, these methods are limited in practice. he main contribution o this paper is proposing a new control method, named NN combined inverse (NNCI) controller or the two-rear-motor independently driven EV to address two issues: () A new NN let inversion (NNLI) sot-sensor which combines the NN and the inverse system is employed to accurately estimate the sideslip angle; (2) he NN right-inversion (NNRI) controller is used to decouple the sideslip angle and the yaw rate. he remainder o this paper is organized as ollows: Section 2 briely describes the model o the vehicle, and Section 3 shows the design o the NNCI

3 Energies 24, controller in detail. he simulated results are presented in Section 4. Finally, some conclusions and uture works are presented in Section wo-rear-wheel Independently Driven EV In this section, the vehicle with the ront-wheel-steering and two-rear-wheel independently driven is assumed to be a constant velocity. By ignoring body roll, the vehicle model includes our degrees-o-reedom (4DOF), namely the lateral, the yaw motions and the two drive wheels. Figure shows the two-wheel independently driven EV. he lateral and yaw motions o the vehicle can be expressed as ollows: m( v y+v xγ)= ( F yl+ Fyr ) cos δ + F yrl+ Fyrr d d I Z γ=l( F yl +Fyr ) cosδ -l( r F yrl +F yrr ) + ( Fyl -Fyr ) sinδ - ( Fxrl -Fxrr ) 2 2 () where v x is the longitudinal vehicle velocity; v y is the lateral vehicle velocity; γ is the yaw rate and β is the sideslip angle; F x and F y are the longitudinal and lateral tire orces, respectively; he abbreviations l, r, rl and rr are ront let, ront right, rear let and rear right, respectively; δ is the ront steering angle input; l and l r are the distances rom the center o gravity to ront and rear axle; d is the width o the vehicle; m represents the vehicle mass; I Z is the moment o inertia about the yaw axis. Figure. wo-rear-wheel independently driven EV. he lateral tire orces o the ront and rear tires are determined by using a linear approximation as ollows: F y = C α (2) F y = C r α r (3) where C and C r are the lateral cornering stiness o the ront and rear tires; α and α r are the slip angles o the ront and rear tires. Furthermore, they can be expressed as:

4 Energies 24, he wheel rotational dynamics is represented by: l γ α β+ -δ (4) vx lγ r αr β- v (5) x w i i xi J ω F R (6) where ù i is the tire rotational speed; R is the tire eective rolling radius; J w is the wheel rotational inertia; i is the torque o the motor i rr,rl. he additional yaw moment M Z is: d M Z ( Fxrl Fxrr ) (7) 2 By applying small angle assumption cosδ» and sinδ» δ, the state equation o the 4DOF vehicle can be written as: é 2C + 2Cr 2C l -2Crl ù é r 2C ù é ù ú β mvx mv x éβù mv úé x δ ù = úê 2 2 2C γ 2C l γ l 2Crlr 2C l 2Crl + ê ú r úêm ú (8) - + Z ê ë û ë û ë úû - I I Z IZvx Z I êë úû êë Z úû 3. Neural Network Combined Inverse Control he NN inverse method combines the merits o the inverse system theory and the NN technology together [28 3]. It solves the problems that the analytical inverse system method excessively depends on the exact system model and the expression o the inverse system. 3.. Neural Network Right Inversion Controller For the system in Equation (8), the system state variable is: x = ( x,x) = ( β,γ) (9) 2 he system control variable is: and the system output variable is: ( u,u) ( δ,m ) u = = () 2 Z y = ( y,y) = ( β,γ) () 2 he state equation o the 4DOF vehicle can be rewritten as: (, ) x= xu (2) where:

5 Energies 24, æ2c + 2Cr 2C l - 2Crlr 2C ö x+ ( -) x 2 2- u mv mv mx ç 2C l - 2Cl 2C l + 2Cl 2C l - + ç è ø x x ( xu, ) = ç (3) 2 2 r r r r x + x 2 u u2 IZ IZvx IZ IZ Using the Interactor algorithm [3], the Jacobi matrix is: Axu (, ) é ù y -2C y é ù u u mx ê ú ê ú (4) 2 = = 2C y ê 2 y ê 2ú - ú u IZ I Z u ê ú ëê ë û 2 ûú Due to: 2C Det( A( x,u )) = ¹ (5) mi x Z the A( x,u ) is nonsingular. Also, the system relative-order α = [α,α 2] = [,], and the system order () () is α+ α2 = 2. hereore, this system is invertible. By setting y = x= v= γ and y2 = x 2 = v2 = β, the inverse system F can be described as: u= F( x, v, v ) (6) A back-propagation NN is trained oline to approximate the inverse system in Equation (6). When the learning process is over, the NNRI system is constructed. A pseudo-linear composite system can be obtained by cascading this NNRI system beore the original nonlinear system. hen, a complex multi-variable vehicle dynamic system with strong coupling and uncertainties is linearized and decoupled into two linear integral subsystems (ã subsystem and â subsystem), as shown in Figure 2. As a result, the closed-loop control can be realized by using linear control theory. 2 Figure 2. Diagram o pseudo-linear system. v y u y v s y v y 2 2 u 2 x y 2 v 2 s y Neural Network Let Inversion Sideslip Angle Sot-Sensor From Equation (6), it can be known that the input variables o the inverse system are the derivative o output variables o the original vehicle system i.e., β and γ. In other words, to construct the NNRI

6 Energies 24, controller, it is essential to obtain the state variables. As mentioned above, the sot-sensor to estimate the sideslip angle is needed or some reasons such as system cost. Generally speaking, or a non-linear system, in its interior, it can be assumed that there exists an assumed inherent sensor subsystem as shown in Figure 3. he variables xˆ xˆq (can be estimated) are the inputs o the subsystem, while the variables xq+ xi which can be measured directly are the outputs. he variables u i are the input variables o the controlled system. I a let inversion o the assumed inherent sensor can be constructed and cascaded behind the assumed inherent sensor, a so-called composite identity system is obtained, whose outputs would be the identity mapping o its inputs. It means that the outputs o the assumed inherent sensor inversion will reproduce completely the inputs o the assumed inherent sensor. hereore, the non-directly measured variables xˆ xˆq o the controlled system can be estimated rom the variables xq+ xi, which can be measured directly. Figure 3. Measuring principle based on let inversion. For the system in Equation (2), its state variable is: ( ) x = x,x,x,x = ( γ, β,v,a ) (7) x x he input o the system is: he directly measured variable is: ( ) ( ) u= u,u = δ,m (8) 2 Z = ( ) = ( ) (9) z z,z,z γ,v,a 2 3 x x he non-directly measured variable is: x = ( x ) = ( β) (2) According to the modeling algorithm, the Jacobian matrix rank is obtained: æ Z ö æ Z ö æö rank = rank = rank = è ç x ø è ç β ø èø ç (2) Since the rank is not equal to the number o the non-directly measured variables, it can urther derivate as:

7 Energies 24, γ z, z β 2lC β γ (22) J z β rank rank rank hereore, the model o the assumed inherent sensor is let invertible. And the let inversion o the assumed inherent sensor or sideslip angle can be written as: β = ( u,γ,γ ) (23) Since the system is complex in mathematical model, it is diicult to construct the above assumed inherent sensor inversion by analytic means. Hence, another static NN is used to approximate the above nonlinear unction in Equation (23). hen, the NNLI dynamic sot-sensor is inally completed, which is composed o a static NN and a series o dierentiators. It simpliies the construction o the proposed NN-based sot-sensor in practical use, while is strict enough in theory. he structure o the NNLI sideslip angle sot-sensor is shown in Figure 4. Figure 4. NNLI sideslip angle sot-sensor Neural Network Combined Inverse Controller aking ull advantage o the powerul ability o the let-inversion sot-sensor to estimate system state, the combination o NNRI controller with NNLI sot-sensor results in a new nonlinear control method, which is called NNCI control. he NNLI sot-sensor can estimate the sideslip angle, while the NNRI is used to decouple the sideslip angle and the yaw rate. hen, the proposed NNCI controller not only linearizes and decouples the original nonlinear system, but also directly obtains immeasurable state eedback in constructing the right-inversion. Ater being linearized by the proposed NNCI, the ultimate closed vehicle active saety system can be controlled by a traditional Proportional-Integral (PI) controller as shown in Figure 5, where the 2DOF model is used to generate the desired yaw rate and the desired sideslip angle [9]. he additional yaw moment M Z is allocated into the torques o two driving motors through the control allocation. he control allocation method is that one side o the wheel driving torque decreases, while the other side o the wheel driving torque increases.

8 Energies 24, Figure 5. Block diagram o proposed NNCI control system. d s v x g * b * b s - s - d M Z F zi u rl u rr rl rr b g b 4. Veriication he perormance o the control algorithm is evaluated by Matlab/Simulink-CarSim cosimulation. A vehicle with two motors at the rear wheels is developed using CarSim. he NNRI controller and NNLI sot-sensor are constructed by using Matlab/Simulink. o veriy proposed controller, the dynamics o the NNCI vehicle are compared with those o the integrated AFS + DYC one. 4.. Ramp Steering Maneuver Firstly, in the ramp steering maneuver, the proposed NNCI vehicle is evaluated as compared with the AFS + DYC one. A ramp steering maneuver is perormed by comprising a ramp steering wheel input shown in Figure 6a. he vehicle travels on a wet road (μ =.4) at a constant speed o km/h. In this case, the NNCI vehicle has smaller slip angle than the AFS + DYC one, as shown in Figure 6b. his is due to the employed NNCI controller, which decouple the yaw rate and the slip angle. While as shown in the Figure 6c,e, the NNCI vehicle has slower yaw rate and smaller lateral acceleration than the AFS + DYC one. As a result, the NNCI vehicle has an increased turning radius as shown in Figure 6g. his is because NNCI controller needs longer time to perorm the control strategy than AFS + DYC one. Figure 6d shows the NNLI sot-sensor s estimated results in the ramp steering maneuver. Furthermore, rom the Figure 6, the longitudinal orces o the two rear wheels o the NNCI vehicle are smaller than those o the AFS + DYC one.

9 Energies 24, Figure 6. Comparison o AFS + DYC vehicle and NNCI vehicle in ramp steering maneuver. (a) Ramp steering; (b) Sideslip angle; (c) Yaw rate; (d) Estimated sideslip angle; (e) Lateral acceleration; () wo-rear-wheel longitudinal orces; (g) Lateral displacement. 7 Steering wheel angle (deg) 5 3 Sideslip angle (deg) DOF AFS+DYC NNCI (a) Yaw rate (deg/s) DOF AFS+DYC NNCI. -.9 Carsim (b) NNLI -.35 AFS+DYC (c) NNCI Sideslip angle (deg) Lateral acceleration (m/s 2 ) Longitudinal orce (N) AFS+DYC Fx_rl NNCI Fx_rl (d) AFS+DYC Fx_rr NNCI Fx_rr () Lateral displacement (m) AFS+DYC (e) NNCI Longitudinal displacement (m) (g)

10 Energies 24, Single Lane Change Steering Maneuver Secondly, they are compared in the single lane change maneuver. A single lane change is perormed comprising a single sine steering wheel input shown in Figure 7a. he vehicle also travels on a wet road (μ =.4) at a constant speed o km/h. he yaw rate and the slip angle responses to the sine steering are presented in Figure 7b,c. he NNCI vehicle traces the desired yaw rate more precisely than the AFS + DYC one. In particular, the NNCI vehicle has much lower sideslip angle than the AFS + DYC one, and the yaw rate and sideslip angle are decoupled very well. he longitudinal orces o the two rear wheels are compared in Figure 7. he lateral displacement o the NNCI vehicle is also smaller than that o the AFS + DYC one as shown in Figure 7g. Figure 7. Comparison o AFS+DYC vehicle and NNCI vehicle in single lane change maneuver. (a) Single lane change steering; (b) Sideslip angle; (c) Yaw rate; (d) Estimated sideslip angle; (e) Lateral acceleration; () wo rear wheels longitudinal orces; (g) Lateral displacement. 6 Steering wheel angle (deg) DOF AFS+DYC NNCI -6 (a) 6 2DOF AFS+DYC NNCI Sideslip angle (deg) Yaw rate (deg/s) 3-3 Sideslip angle (deg) Carsim (b) NNCI -6 Lateral acceleration (m/s 2 ) AFS+DYC (c) NNCI -.2 (d) -.4 (e)

11 Energies 24, Figure 7. Cont. Longitudinal orce (N) AFS+DYC Fx_rl NNCI Fx_rl AFS+DYC Fx_rr NNCI Fx_rr () Lateral displacement (m) AFS+DYC NNCI Longitudinal displacement (m) (g) 4.3. Double Lane Changing Maneuver Finally, in the double lane changing maneuver shown in Figure 8a, the yaw rate and the slip angle responses are compared in Figure 8b,c. he NNLI estimated result is displayed in Figure 8d. he lateral acceleration o the NNCI vehicle is less than.2 g on the slippery road suraces in Figure 8e. It is indicated that controlling the vehicle sideslip angle is beneicial to maintain the vehicle stable in this critical driving conditions. he longitudinal orces o the two rear wheels and the vehicle trajectories are also shown in Figure 8,g. It can be concluded that the AFS + DYC vehicle provide larger lateral displacements than the NNCI one owing to the large yaw rate and lateral acceleration. hese three maneuver results veriy that the actual and estimated values o the sideslip angle agree well. hereore, the NNLI sot-sensor can accurately estimate the sideslip angle. On the other hand, both vehicles can successully track the desired yaw rate as well as minimize the sideslip angle to maintain stable. Meanwhile, by comparing Figures 6 8, it can be seen that the sideslip angle o the NNCI vehicle is smaller than that o the AFS + DYC one. hereore, the stability o the NNCI vehicle is signiicantly improved in critical driving condition. In contrast, the AFS + DYC vehicle has relatively higher yaw rate than the NNCI one, because the proposed decoupling controller needs longer time to calculate the additional yaw moment. hereore, compared with the AFS + DYC vehicle, the NNCI one decreases the maneuverability, but it signiicantly decouples the yaw rate and sideslip angle. Furthermore, in all the cases, the longitudinal orces o the two rear wheels o the NNCI vehicle are smaller than those o the AFS + DYC one. his means that comparing with the integrated AFS + DYC controller, the NNCI one needs lower electric power energy to drive the EV under the same operation condition.

12 Energies 24, Figure 8. Comparison o AFS + DYC vehicle and NNCI vehicle in double lane changing maneuver. (a) Double lane changing steering; (b) Sideslip angle; (c) Yaw rate; (d) Estimated sideslip angle; (e) Lateral acceleration; () wo rear wheels longitudinal orces; (g) Lateral displacement. 4 Steering wheel angle (deg) DOF AFS+DYC NNCI (a) 5 2DOF AFS+DYC NNCI Sideslip angle (deg) Yaw rate (deg/s) Sideslip angle (deg) Longitudinal orce (N) Carsim (b) NNLI AFS+DYC Fx_rl NNCI Fx_rl (d) () AFS+DYC Fx_rr NNCI Fx_rr -5 Lateral acceleration (m/s 2 ) Lateral displacement (m) AFS+DYC (c) NNCI AFS+DYC (e) NNCI Longitudinal displacement (m) (g)

13 Energies 24, Conclusions and Future Works In this paper, a new NNCI decoupling controller has been designed or the two-rear-wheel independently driven EV in order to improve the maneuverability and stability perormance. o estimate the sideslip angle, an assumed inherent sensor let inversion has been treated as the dynamic sot-sensor, which consists o a NN and a series o dierentiators. he yaw rate and the sideslip angle have been decoupled by using the proposed NNCI controller. Also, the NNLI sot-sensor can estimate the sideslip angle accurately. A co-simulation model has been developed and the results have been given, it has been proved that the proposed vehicle oers improved stability and maneuverability. In the uture, these results will be experimentally veriied based on a two-rear-wheel independently driven prototype EV which shown in Figure 9. Figure 9. In-wheel motor, dspace and prototype EV. Acknowledgments his work was supported by the National Natural Science Foundation o China ( and ) and the Specialized Research Fund or the Doctoral Program o Higher Education o China ( ), the Natural Science Foundation o Jiangsu Province (BK23), the innovation plan or PhD candidate o Jiangsu University (CXB_4X), and the Priority Academic Program Development o Jiangsu Higher Education Institutions. Author Contributions In this paper, Guohai Liu and Wenxiang Zhao deined the research theme. Duo Zhang designed the proposed NNCI controller. Penghu Miao and Yan Jiang developed the NNLI sot sensor or the vehicle sideslip angle. Duo Zhang and Penghu Miao developed and implemented the algorithms or the simulation. Duo Zhang and Huawei Zhou analyzed and discussed the results obtained. Conlicts o Interest he authors declare no conlicts o interest.

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