Dynamic Decoupling and Trajectory Tracking for Automated Vehicles Based on the Inverse System
Dynamic Decoupling and Trajectory Tracking for Automated Vehicles Based on the Inverse System
Yu, Yinghong;Li, Yinong;Liang, Yixiao;Zheng, Ling;Yang, Wei
applied sciences Article Dynamic Decoupling and Trajectory Tracking for Automated Vehicles Based on the Inverse System 1 , 2 1 , 2 , 1 , 2 1 , 2 1 , 2 Yinghong Yu , Yinong Li *, Yixiao Liang , Ling Zheng and Wei Yang The State Key Lab of Mechanical Transmission, Chongqing University, Chongqing 400044, China; email@example.com (Y.Y.); firstname.lastname@example.org (Y.L.); email@example.com (L.Z.); firstname.lastname@example.org (W.Y.) School of Automotive Engineering, Chongqing University, Chongqing 400044, China * Correspondence: email@example.com; Tel.: +86-130-7543-1806 Received: 15 September 2020; Accepted: 19 October 2020; Published: 22 October 2020 Abstract: A simultaneous trajectory tracking and stability control method is present for the four-wheel independent drive (4WID) automated vehicles to handle dynamic coupling maneuvers. To conquer the disadvantage that attendant disturbances caused by the dynamic coupling of traditional decentralized control methods degenerate the trajectory tracking accuracy, the proposed method takes advantage of the idea of decoupling to optimize the tracking performance. After establishing the dynamic model of the 4WID automated vehicles, the coupling mechanism of the vehicle dynamic control and its negative eect on trajectory tracking were studied at ﬁrst. The inverse system model was then determined by machine learning and connected in series with the controlled object to form a pseudo linear system to realize dynamic decoupling. Finally, diering from previous tracking methods following the apparent lateral position and longitudinal velocity references, the pseudo linear system tracks the ideal intermediate targets transferred from the target trajectory, that is, the accelerations of vehicle in longitudinal, lateral and yaw directions, to indirectly achieve trajectory tracking and validly restrain the vehicle motion. The eectiveness of the proposed method, i.e., the high tracking accuracy and the stable driving performance, is veriﬁed through three coupling driving scenarios in the CarSim-Simulink co-simulations platform. Keywords: trajectory tracking; the four-wheel independent drive vehicles; automated vehicles; the inverse system; dynamic decoupling 1. Introduction Automated vehicles (AVs) provide safe, cheap, and ecient travel as well as attracting widespread research interest in industry and academia [1,2]. The trajectory tracking module manipulates vehicle chassis actuators to reach the target position at the right time, which is a core part of AVs and directly aects driving safety and comfort . According to the dierent control structures, the existing trajectory tracking control can be divided into a decentralized control method and centralized control method. The decentralized control method decomposes the trajectory tracking problem into longitudinal velocity control issue and lateral position tracking issue, and the corresponding control laws of these subsystems need to be designed, respectively. In the past few decades, the problem of lateral position tracking has always been the core issue. In order to improve tracking accuracy or control stability, many lateral position tracking methods have been proposed, such as preview , pure-pursuit (PP) , Stanley , linear quadratic regulator (LQR)  and model predictive control (MPC) . After being integrated with longitudinal controllers, these methods could realize accurate tracking under most scenarios [9–12]. However, Appl. Sci. 2020, 10, 7394; doi:10.3390/app10217394 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 7394 2 of 17 due to the interaction between the motion directions, the motion tracking error of the decentralized control method will increase under the coupling condition, i.e., lane changing with varying speed. To solve the problem, some centralized control methods which designed the recompense law between the longitudinal and lateral controller were developed. Turri  designed the lateral controller considering the time-varying velocity, which can eliminate the lateral disturbance caused by longitudinal control. Attia et al.  proposed the nonlinear model predictive control (NMPC) considering the characteristics of vehicle body motion coupling and tire force coupling, which can eectively solve the problem of motion interference between dierent directions. Besides the lateral control compensation, Kanayama  applied the Lyapunov method to solve the integrated longitudinal and lateral tracking problem. Menour  used the dierential ﬂatness theory to design the control laws of longitudinal and lateral directions, which realized the dynamic trade-o. In the architecture of MPC, the longitudinal and lateral control could be transformed into one constrained optimization problem with full consideration of the coupling eect of vehicle motion [17,18]. However, since the weighted optimization is still a compromise rather than a real decoupling, the improvement in tracking accuracy is not obvious. Therefore, in trajectory tracking, dynamic decoupling is an eective method to eliminate the interactions of various vehicle motion directions. To decouple the lateral and the yaw motion of the vehicle, Marino  calculated eigenvalues of the optimal control system by minimizing the weighted sum of the cross-transfer function. Then, referring to the target sideslip angle and yaw rate obtained from the nonlinear vehicle model, the zero-yaw rate maneuver and the zero lateral speed maneuver were guaranteed to improve vehicle handling performance. Zhang  derived an analytical method to decouple the motion control on vehicles’ longitudinal and lateral directions. The responses of the ideal bicycle model were followed to improve the driving safety and handling performance of the vehicle. Wang  adopted the inverse system to decouple lateral, yaw, and roll motions. The decoupling method can transform the coupled vehicle dynamics system into multiple parallel single input single output (SISO) sub-systems. Then, by tracking the ideal vehicle motion states, e.g., yaw rate, longitudinal acceleration and sideslip angle, the vehicle handling performance is enhanced. However, since the desired motion states are the ideal vehicle model responses according to drivers’ actual input, the decoupling method can usually only be applied in the driver-in-loop system to improve the driving stability under satisfying the driver ’s intention. With the development of the 4WID electrical vehicle, the supplementary control of the yaw direction could be implemented to improve the driving performance  and guarantee the accuracy motion tracking . The input number of 4WID electric vehicle system is equal to the output number, which is a positive system  and can easily be decoupled. Hence, focusing on the poor trajectory tracking accuracy problem in dynamic coupled scenarios, the dynamic model of the 4WID vehicle was ﬁrstly built to study its coupling mechanism. To decouple the vehicle dynamic, the inverse system decoupling framework was proposed, where the back propagation neural network (BPNN) was applied to set up the inverse system, and the training dataset was simulated and collected based on stochastic inputs. The desire vehicle motion states obtained by the target lateral position and longitudinal velocity are followed by the inverse system to achieve the trajectory tracking and the dynamic decoupling. Finally, the simulation results compared with the pure-pursuit algorithm and MPC algorithm verify the eectiveness of the proposed trajectory tracking method. The paper is organized as follows: a three degrees of freedom (DOF) vehicle model is constructed and the coupling eects are analyzed in Section 2; Section 3 introduces the principle of the proposed decoupling trajectory tracking method; in Section 4, the simulation results are presented and discussed. Conclusions are given in Section 5. Appl. Sci. 2020, 10, 7394 3 of 17 2. Coupling Mechanism of 4WID Vehicle The two-track vehicle dynamic model representing the 4WID vehicle established in the Cartesian coordinate system to study the dynamic characteristics and coupling mechanism of the vehicle, as shown in Figure 1. Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 16 Figure 1. The two-track model with a reference position trajectory. Figure 1. The two-track model with a reference position trajectory. Assuming the small steering turning, and for simplification, the steering angles of the left and Assuming the small steering turning, and for simpliﬁcation, the steering angles of the left and right right front tires are equivalent to steering angle 𝛿 . The planar motion of the vehicle can be front tires are equivalent to steering angle . The planar motion of the vehicle can be expressed as: expressed as: .. . . . mx = my' + F cos F sin + F cos F sin + F + F k x (1) x1 y1 x2 y2 x3 x4 D 𝑚𝑥 =𝑚𝑦 𝜑 +𝐹 cos 𝛿 − 𝐹 sin 𝛿 + 𝐹 cos 𝛿 − 𝐹 sin 𝛿 + 𝐹 +𝐹 −𝑘 𝑥 (1) .. . . my = mx' + F sin + F cos + F sin + F cos + F + F , (2) x1 y1 x2 y2 y3 y4 (2) 𝑚𝑦 =−𝑚𝑥 𝜑 +𝐹 sin 𝛿 + 𝐹 cos 𝛿 + 𝐹 sin 𝛿 + 𝐹 cos 𝛿 + 𝐹 +𝐹 , .. I ' = l (F + F ) sin + F + F cos + d (F F ) cos F F sin + F F l F + F , (3) z f x1 x2 y1 y2 x1 x2 y1 y2 x3 x4 r y3 y4 𝐼 𝜑 =𝑙 𝐹 +𝐹 sin 𝛿 + 𝐹 +𝐹 cos 𝛿 (3) where the deﬁnition of each symbol is showed in Table A1. +𝑑 𝐹 −𝐹 cos 𝛿 − 𝐹 −𝐹 sin 𝛿 + 𝐹 −𝐹 −𝑙 +𝐹 , When the tire force dierential algorithm is adopted, the longitudinal forces are synthetically considered as the total longitudinal force F and the additional yaw moment M . t z where the definition of each symbol is showed in Table A1. When the tire force differential algorithm is adopted, the longitudinal forces are synthetically F = F + F + F + F (4) t x2 x3 x1 x4 considered as the total longitudinal force 𝐹 and the additional yaw moment 𝑀 . (4) 𝐹 =𝐹 ( +𝐹 +𝐹 +𝐹) M = d DF cos + DF , (5) where the longitudinal force distribution ratio of the front and rear axles k is deﬁned as: (5) 𝑀 =𝑑 ∆𝐹 cos 𝛿 + ∆𝐹 , F + F where the longitudinal force distribution ratio of the front and rear x3 axles 𝑘 is defined as: x4 k = , (6) 𝐹 +𝐹 𝑘 = , (6) The additional yaw moment M is caused by the longitudinal force dierence DF between the vehicle’s two sides: The additional yaw moment 𝑀 is caused by the longitudinal force difference ∆𝐹 between the DF = F F = F F , (7) x1 x2 x3 x4 vehicle’s two sides: Then, introducing the small angle hypothesis, the tire lateral force is proportional to its slip (7) ∆𝐹 = 𝐹 −𝐹 =𝐹 −𝐹 , angle . . . . Then, introducing the small angle hypothesis, the tire lateral force is proportional to its slip angle . = = y + l ' /x, (8) f f f 𝛼 = 𝛿 − 𝜃 =𝛿− 𝑦 +𝑙 𝜑 𝑥⁄ , (8) . . . = = y l ' /x, (9) r r r ⁄ (9) 𝛼 = 𝜃 =− 𝑦 −𝑙 𝜑 𝑥 , The two-track model is simplified to a 3DOF model. 2𝐶 𝛿 𝐶 𝛿 𝐶 𝑙 𝛿 𝐹 𝑘 𝑥 𝑦 𝜑 (10) 𝑥 =𝑦 𝜑 + − − +2 +2 , 𝑚 𝑚 𝑚 𝑚 𝑥 𝑚 𝑥 𝐶 +𝐶 𝐶 𝑙 −𝐶 𝑙 2𝐶 + 1−𝑘 𝐹 𝛿 𝑦 𝜑 (11) 𝑦 =−𝑥 𝜑 −2 −2 + , 𝑚 𝑥 𝑚 𝑥 𝑚 Appl. Sci. 2020, 10, 7394 4 of 17 The two-track model is simpliﬁed to a 3DOF model. . . 2 2 2C C C l .. . . F k x f f y f f ' t D x = y' + + 2 + 2 , (10) . . m m m m m x x . . 2C + (1 k )F C + C C l C l r t r y r r ' f .. . . f f f Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 16 y = x'