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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 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; 20173201003@cqu.edu.cn (Y.Y.); liangyixiao@cqu.edu.cn (Y.L.); zling@cqu.edu.cn (L.Z.); yangwei0705@gmail.com (W.Y.) School of Automotive Engineering, Chongqing University, Chongqing 400044, China * Correspondence: ynli@cqu.edu.cn; 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 e ect 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, di ering 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 e ectiveness 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 a ects driving safety and comfort [3]. According to the di erent 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 [4], pure-pursuit (PP) [5], Stanley [6], linear quadratic regulator (LQR) [7] and model predictive control (MPC) [8]. 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 [13] designed the lateral controller considering the time-varying velocity, which can eliminate the lateral disturbance caused by longitudinal control. Attia et al. [14] proposed the nonlinear model predictive control (NMPC) considering the characteristics of vehicle body motion coupling and tire force coupling, which can e ectively solve the problem of motion interference between di erent directions. Besides the lateral control compensation, Kanayama [15] applied the Lyapunov method to solve the integrated longitudinal and lateral tracking problem. Menour [16] used the di erential 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 e ect 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 e ective method to eliminate the interactions of various vehicle motion directions. To decouple the lateral and the yaw motion of the vehicle, Marino [19] 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 [20] 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 [21] 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 [22] and guarantee the accuracy motion tracking [23]. The input number of 4WID electric vehicle system is equal to the output number, which is a positive system [24] 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 e ectiveness 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 e ects 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 [25]. The planar motion of the vehicle can be front tires are equivalent to steering angle  [25]. 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 di erential algorithm is adopted, the longitudinal forces are synthetically considered as the total longitudinal force F and the additional yaw moment M [26]. 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 [26]. (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 di erence 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 [27]. . . . Then, introducing the small angle hypothesis, the tire lateral force is proportional to its slip angle [27]. =   =  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' 2 2 + , (11) . . m m m x x . . 2 2 C l C l C l + C l l (1 k )F  2C l .. M f f r r y f f r r ' f r t f f 𝐶 𝑙 −𝐶 𝑙 𝐶 𝑙 +𝐶 𝑙 𝑙 1−𝑘 𝐹 𝛿 2𝐶 𝑙 𝛿 𝑀 𝑦 𝜑 ' = 2 2 + + , (12) . . (12) 𝜑 = −2 −2 + + , I I I I I z z x z x z z 𝐼 𝐼 𝑥 𝐼 𝑥 𝐼 𝐼 . . . The The dyn dynamics amicscan can be be r rewr ewritten itten in the in the state-space state-space form, form, with with the the st states ates be beinging X = 𝑿=x, y 𝑥,' ,𝑦, ,𝜑 and , an the d the control inputs being 𝑼= 𝐹 ,𝛿, 𝑀 ,i.e., control inputs being U = (F ,, M ) ,i.e., t z 𝑦 𝜑 . . −𝑘 𝑥 2 3 2 . 3 y +𝑙' −𝛿 . ⎡ 1 ⎤ k x ⎡ D 𝜑 0 ⎤ 𝑥 𝑥 6 . +l .  7 6 7 f ' 0 6 1 2𝐶 70 6 7 x x m 6⎢ 7 ⎥ 6 7 2C 0 ⎢ ⎥ 6 7 𝑚 f 𝑚 6 7 6 m m 7 . 6 7 . C +C C l C l 𝐶 f+𝐶 r 𝐶 f 𝑙 f −𝐶 r r 𝑙 6 7 6 7 ⎢ ⎥ ⎢ ⎥ 6 2C 7 6 7 (1k ) . . 6 1− r𝑘 𝛿 f2𝐶 7 X = ' 2 2 X + U, (13) 6 −𝜑 −2 −2 7 𝑿 = 𝑿+ 6 7 𝑼, 6 7 (13) mx mx 0 ⎢ ⎥ 6⎢ 7 ⎥ 6 7 0 𝑚𝑥 𝑚𝑥 6 m m 7 2 2 6 7 6 7 C l C l C l +C l 6 r r r r 7 𝑚 𝑚 f f f f 6 7 ⎢4 5 ⎥ l (1k ) 2C l ⎢ r ⎥ 4 f f f 5 0 2 . 2 . 𝐶 𝑙 −𝐶 𝑙 𝐶 𝑙 +𝐶 𝑙 𝑙 1− 𝑘 𝛿 2𝐶 𝑙 ⎢ ⎥ 1 I x I x z z ⎢ I I I ⎥ z z z 0−2 −2 ⎣ 𝐼 𝑥 𝐼 𝑥 ⎦ ⎣ ⎦ 𝐼 𝐼 𝐼 In Equation (13), the input matrix is a non-diagonal matrix, which means each state of the system In Equation (13), the input matrix is a non-diagonal matrix, which means each state of the system is a ected by multiple inputs. The input matrix is not a constant coecient matrix, but contains input is affected by multiple inputs. The input matrix is not a constant coefficient matrix, but contains input elements and state elements, which indicates that there are complex interactions among three directions elements and state elements, which indicates that there are complex interactions among three and causes the attendant disturbances when controlling one of the directions, such as the2C  /m is directions and causes the attendant disturbances when controlling one of the directions, such as the a longitudinal resistance caused by lateral control input. −2𝐶 𝛿 𝑚 is a longitudinal resistance caused by lateral control input. Once ignoring the attendant disturbance, the control errors appear and deteriorate tracking [28]: Once ignoring the attendant disturbance, the control errors appear and deteriorate tracking [28]: . . . e = e cos e + e sin e , (14) 𝑒 =𝑒 cos 𝑒 ' +𝑒 sin ' 𝑒 , x y (14) 𝑒 =𝑒 cos 𝑒 −𝑒 sin 𝑒 , . . . (15) e = e cos e e sin e , (15) ' ' y x 3. Methodology 3. Methodology Once an inverse system whose inputs and outputs are strictly opposed to the origin system Once an inverse system whose inputs and outputs are strictly opposed to the origin system exists, the decoupling method based on the inverse system will be valid, as shown in Figure 2; that exists, the decoupling method based on the inverse system will be valid, as shown in Figure 2; that is, the inverse system and the control object are connected in series to form an equivalent multiple is, the inverse system and the control object are connected in series to form an equivalent multiple single-input single-output combined system without interactions. single-input single-output combined system without interactions. Figure 2. The decoupling principle of the inverse system. Figure 2. The decoupling principle of the inverse system. 3.1. 3.1. Decoupling Decoupling of of D Dynamics ynamics 3.1.1. Proof of Reversibility 3.1.1. Proof of Reversibility In order to apply the decoupling algorithm, it is necessary to prove the reversibility of the vehicle In order to apply the decoupling algorithm, it is necessary to prove the reversibility of the vehicle dynamic. Here, the proof is derived based on the Interactor Algorithm 1 [29,30]. dynamic. Here, the proof is derived based on the Interactor Algorithm [29,30]. For a multiple-input and multiple-output (MIMO) nonlinear system, it can be written as For a multiple-input and multiple-output (MIMO) nonlinear system, it can be written as (16) 𝑿 = 𝑓 𝑿, 𝑼 , X = f(X, U), (16) 𝒀=ℎ 𝑿 , (17) where the input vector is 𝑼=𝑢 ,𝑢 …,𝑢 , the state vector is 𝑿= 𝑥 ,𝑥 …, 𝑥 and the output vector is 𝒀=𝑦 ,𝑦 …,𝑦 =ℎ 𝑿 ,ℎ 𝑿 …,ℎ 𝑿 . Appl. Sci. 2020, 10, 7394 5 of 17 Y = h(X), (17) h i where the input vector is U = u , u ::: , u , the state vector is X = [x , x ::: , x ] and the output 1 2 p 1 2 n h i h i T T vector is Y = y , y ::: , y = h (X), h (X) ::: , h (X) . q q 1 2 1 2 Algorithm 1. Interactor Algorithm 1: Defining the superscript r , which means taking the derivative for k-th component y , r is the order when k k k h i (r ) control input U = u , u ::: , u firstly appears, and the derivative is noted as y 1 2 p 2: For k = 1, q 2 3 6 G 7 6 k1 7 6 7 3: Defining a criterion vector G = 6 7, and G is null when k = 1 k (r ) 0 4 k 5 4: Calculating the rank of the Jacobian matrix R = rank(@G /@U) k k 5: If R = k 6: = r k k 7: End If 8: End For 9: If  n i=1 10: The system is reversible. 11: End If . . . As for the proposed vehicle dynamics, the output vector is Y = x, y,' . Setting y = x, which does not include any input variable, G needs to be rewritten with respect 1 1 . . . 2 2 . . 2C  C  C l (1) y ' F k x f f f f t D . . to its derivative [21]: that is, G = y = y' + + 2 + 2 , the rank of the m m m m m x x Jacobian matrix of G to U is . . y ' . +l . R = rank(@G /@U) = 1 x x = 1, (18) 1 1 2C 0 m m So that, = r = 1. And y = y, which also does not include any input variable. Hence, 1 1 2 2 3 " # .. 6 G 7 6 7 6 7 setting G = 6 7 = .. , 2 (1) 4 5 2 3 0 0 6 7 6 7 6 7 ( ) R = rank @G /@U = 6 7 = 2, (19) 2 2 2C 4 (1k ) f 5 m m 2 3 .. 2 3 6 7 6 7 6 G 7 6 7 .. 6 7 6 7 6 7 6 7 we can get that = r = 1. At last, setting G = 6 7 = 6 y 7, 2 2 3 (1) 4 5 6 7 6 7 .. y 4 5 . . 2 3 y ' . . 6 +l  7 6 1 x x 7 6 7 2C 0 6 7 6 m m 7 6 7 6 7 2C (1k ) f 6 r 7 R = rank(@G /@G) = = 3, (20) 3 3 6 7 6 0 7 6 m m 7 6 7 6 ( ) 7 l 1k  2C l 4 f r f f 5 I I I z z z and this yields = r = 1, then 3 3 = 3, (21) i=1 Appl. Sci. 2020, 10, 7394 6 of 17 According to the reversibility judgment [31], there is an inverse system for the 4WID vehicle without (1) (1) (1) hidden dynamics [32], and the input and output of the inverse system are U = y , y , y = inv 2 3 .. .. .. x, y,' and Y = (F ,, M ) . inv t z As the 3DOF vehicle dynamic system is reversible and based on the Equation (13), the inverse system can be written as Equation (22): . . 2 3 12 . 3 y ' k x 6 . +l .  7 f 6 7 6 1 7 ' 0 x x 6 7 6 7 m 2C 0 6 7 6 7 6 7 6 m m 7 6 7 . C +C C l C l 6 7 f r f f r r 6 7 6 2C 7 (1k ) 6 7 6 r f 7 ' 2 . 2 . Y = X 6 7 inv 6 7 6 7 0 mx mx 6 7 6 7 6 m m 7 2 2 6 7 6 7 C l C l C l +C l 6 r r r r 7 6 7 f f f f l (1k ) 2C l 4 5 4 f f f 5 0 2 . 2 . I x I x I I I z z z z z (22) . . 2 3 y ' . +l . 6 7 6 1 x x 7 6 7 2C 0 6 7 6 m m 7 6 7 6 7 2C (1k ) f 6 r 7 + U 6 7 inv 6 0 7 6 m m 7 6 7 6 ( ) 7 l 1k  2C l 4 f r f f 5 I I I z z z 3.1.2. Definition of the Inverse System As shown in Equation (22), the output can be calculated using the Taylor series expansion. However, to avoid the approximation error caused by the omission of high-order terms in the Taylor formula, a machine learning method is applied to identify the relationship between the input and output of the system in this paper. Because the identification of the inverse system is a MIMO regression, .. .. the back propagation neural network (BPNN) model is adopted. Note that x and y are replaced by the measurable a and a : x y .. . . x = a + y', (23) .. . . y = a x', (24) .. . . . Hence, the input and output of the BPNN model are U = a , a ,', x, y,' and Y = nn x y nn (F ,, M ) . t z 3.1.3. Neural Networks Training n o .. . . . The D-class sedan model of CarSim is utilized to obtain the vehicle responses a , a ,', x, y,' x y under stochastic inputsfF ,, M g. To simulate the 4WID vehicle, the total longitudinal force and yaw t z moment are converted to driven torques of each wheel based on Equations (2) and (4). dw (1 k )F  M /d(1 + cos) f l, f r r t z T = + I , (25) f l, f r r dt dw k F  M /d(1 + cos) f l, f r r t z T = + I , (26) rl,rr r r dt where T , T , T , T are the driven torques of the front-left, front-right, rear-left and rear-right tire, f l f l f l f l respectively. r is the radius of each wheel. I is the wheel inertia moment. w and w are the f l, f r f l, f r rotation velocities of each wheel. The longitudinal and lateral acceleration were constrained considering the ride comfort and anti-sideslip [33]. g  a  g, (27) g  a   g, (28) s y s Appl. Sci. 2020, 10, 7394 7 of 17 Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 16 where 𝜂 , 𝜂 and 𝜇 respectively represent the maximum coefficient of longitudinal deceleration, where , and respectively represent the maximum coefficient of longitudinal deceleration, longitudinal d s longitudinal acceleration and lateral acceleration. According to the vehicle dynamics and the acceleration and lateral acceleration. According to the vehicle dynamics and the kinematic model: kinematic model: F 𝐹 a = , (2(29) 9) 𝑎 x = , L L 𝐿 𝐿 R =  , (30) 𝑅= , (30) sinjj jj | | | | sin 𝛿 𝛿 where R is the turning radius of the vehicle, and L is the distance between the front and rear axles. where 𝑅 is the turning radius of the vehicle, and 𝐿 is the distance between the front and rear axles. Then, substituting Equations (29) and (30) into Equations (27) and (28) yields: Then, substituting Equations (29) and (30) into Equations (27) and (28) yields: 𝜇 gL (31) |𝛿 | ≤ , jj  , (31) mg  F  mg, (32) −𝜇𝑚𝑔 ≤ 𝐹 ≤𝜇𝑚𝑔, (32) The interval between two sampling points of longitudinal force, steering angle and additional The interval between two sampling points of longitudinal force, steering angle and additional yaw moment is 2 s. The values of sampling points are stochastically generated, obeying the yaw moment is 2 s. The values of sampling points are stochastically generated, obeying the uniform uniform distribution in the range of (5000, 5000)N, (0.1, 0.1)rad and (500, 500)Nm, respectively. distribution in the range of −5000,5000 𝑁 , −0.1,0.1 and −500,500 , respectively. The The sampling data are interpolated by the Hermite method for smoothness, as shown in Figure 3. sampling data are interpolated by the Hermite method for smoothness, as shown in Figure 3. Figure 3. One group of vehicle stochastic inputs. Figure 3. One group of vehicle stochastic inputs. By setting di erent initial speeds, multiple groups of input signals act on the vehicle model, By setting different initial speeds, multiple groups of input signals act on the vehicle model, then then the inputs and the responses of the vehicle are recorded as the training dataset. To ensure the the inputs and the responses of the vehicle are recorded as the training dataset. To ensure the validity validity of the collected data set, the following two constraints need to be guaranteed. of the collected data set, the following two constraints need to be guaranteed. • The equivalent steering angle is constrained according to the real-time speed. The equivalent steering angle is constrained according to the real-time speed. • The vehicle speed is always positive. The vehicle speed is always positive. 3.1.4. Design of the Trajectory Tracking Controller 3.1.4. Design of the Trajectory Tracking Controller Based on the inverse system, the control framework is built in Figure 4. The target motion Based on the inverse system, the control framework is built in Figure 4. The target motion trajectory transformed into the vehicle dynamic state references—that is, the longitudinal trajectory transformed into the vehicle dynamic state references—that is, the longitudinal acceleration, acceleration, the lateral acceleration, and the yaw acceleration. By following the target vehicle the lateral acceleration, and the yaw acceleration. By following the target vehicle dynamic state, dynamic state, the inverse system exports the vehicle inputs and drives the vehicle tracking desired the inverse system exports the vehicle inputs and drives the vehicle tracking desired trajectory. trajectory. 𝑁𝑚 𝑟𝑎𝑑 𝑔𝐿 Appl. Sci. 2020, 10, 7394 8 of 17 Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 16 Figure 4. Scheme of the designed trajectory tracking controller with decoupling. Figure 4. Scheme of the designed trajectory tracking controller with decoupling. 3.2. The desired Vehicle States 3.2. The Desired Vehicle States Based on the kinematic model, the desired vehicle states, i.e., the references of the inverse system Based on the kinematic model, the desired vehicle states, i.e., the references of the inverse system .. 𝑼 =𝑎 ,𝑎 ,𝜑 , are calculated from the target motion trajectory. U = a , a ,' , are calculated from the target motion trajectory. re f x y Supposing that the references from the planning module are the sequences of resultant speed Supposing that the references from the planning module are the sequences of resultant speed V 𝑉 (t 𝑡 ) an andd trajectory trajectoryY 𝑌(X 𝑋 ), , t is 𝑡 is the time, the time, X and 𝑋 and Y ar 𝑌 e the are the veh vehicle ilongitudinal cle longitudinal andand later lateral position al position in the in the inertial coordinate system. inertial coordinate system. The The spat spatial–temporal ial–temporal rel relationship ationship is is un unified ified by by using using the the travel travel distance distance o offvehicles vehicles [34]. [34]. 𝑉 𝑑𝑡 = ∆𝑋 𝑖 2 +∆𝑌 𝑖 2 , (33) Vdt = DX(i) + DY(i) , (33) where ∆𝑋 𝑖 and ∆𝑌 𝑖 are the longitudinal and lateral spacing between the 𝑖+ 1 th point and the where DX(i) and DY(i) are the longitudinal and lateral spacing between the i + 1 th point and the i th 𝑖 th point in the inertial coordinate system. point in the inertial coordinate system. The desired yaw can be calculated as [35]: The desired yaw can be calculated as [35]: ∆𝑌 𝑖 𝜑 𝑖 =tan , (34) DY( i) ∆𝑋 𝑖 '(i) = tan , (34) DX(i) The vehicle velocity along the 𝑋 and 𝑌 directions in the inertial coordinate system are The vehicle velocity along the X and Y dir ∆𝑋 ections 𝑖 in the ∆𝑌inertial 𝑖 coordinate system are (35) 𝑉 𝑖 = ,𝑉 𝑖 = , ∆𝑡 𝑖 ∆𝑡 𝑖 ( ) ( ) DX i DY i V (i) = , V (i) = , (35) Then, the vehicle longitudinal and lateral velocities are [28] Dt(i) Dt(i) 𝑥 𝑖 =𝑉 𝑖 cos 𝜑 𝑖 +∆𝑌 𝑖 sin 𝜑 𝑖 , (36) Then, the vehicle longitudinal and lateral velocities are [28] 𝑦 𝑖 =−𝑉 𝑖 sin 𝜑 𝑖 +∆𝑌 𝑖 cos 𝜑 𝑖 , (37) x(i) = V (i) cos'(i) + DY(i) sin'(i), (36) The longitudinal, lateral and yaw accelerations are yields: y(i) = V (i) sin'(i) + DY(i) cos'(i), (37) 𝑥 𝑖 ∆𝜑 𝑖 𝑎 𝑖 = −𝑦 𝑘 , (38) The longitudinal, lateral and yaw accelerations are yields: ∆𝑡 𝑖 ∆𝑡 𝑖 x(i) D'(i) 𝑦 𝑖 ∆𝜑 𝑖 a (i) = y(k) , (38) 𝑎 𝑖 = +𝑥 𝑘 , (39) Dt(i) Dt(i) ∆𝑡 𝑖 ∆𝑡 𝑖 y(i) D'(i) ∆𝜑 𝑖 a (i) = + x(k) , (39) Dt(i) Dt(i) ∆𝑡 𝑖 (40) 𝜑 𝑖 = , ∆𝑡 𝑖 D'(i) .. Dt(i) '(i) = , (40) So far, the input of the inverse system, the target motion states of the vehicle 𝑼 =𝑎 ,𝑎 ,𝜑 , Dt(i) has been obtained. Appl. Sci. 2020, 10, 7394 9 of 17 h i .. T So far, the input of the inverse system, the target motion states of the vehicle U = a , a ,' , re f x y Appl. has Sci. been 2020 obtained. , 10, x FOR PEER REVIEW 9 of 16 4. Simulation Results and Analysis 4. Simulation Results and Analysis The simulations are conducted to verify the correctness of the inverse system identification and The simulations are conducted to verify the correctness of the inverse system identification and assess the tracking performance of the proposed method. A D-Class CarSim vehicle model is adopted, assess the tracking performance of the proposed method. A D-Class CarSim vehicle model is and the parameters are shown in Table A2. The proposed trajectory tracking controller is developed in adopted, and the parameters are shown in Table A2. The proposed trajectory tracking controller is Simulink. The Simulink–CarSim interface is shown in Figure 4. developed in Simulink. The Simulink–CarSim interface is shown in Figure 4. 4.1. Verification of the Inverse System Models 4.1. Verification of the Inverse System Models The inverse system is identified by a BPNN model whose structure is 6-50-50-3, i.e., the model The inverse system is identified by a BPNN model whose structure is 6-50-50-3, i.e., the model has six inputs, three outputs and two hidden layers with 50 nodes. The mean square error between has six inputs, three outputs and two hidden layers with 50 nodes. The mean square error between the training data and the prediction of the BPNN is 0.000213 and the regression factor of the model is the training data and the prediction of the BPNN is 0.000213 and the regression factor of the model 0.99941, which indicates that the model has identified the input–output mapping relationship between is 0.99941, which indicates that the model has identified the input–output mapping relationship the of the inverse system. A novel test dataset was collected and applied to evaluate the fitting between the of the inverse system. A novel test dataset was collected and applied to evaluate the performance of the inverse system as Figure 5. fitting performance of the inverse system as Figure 5. Figure Figure 5. 5. T The he re responses sponses o of f t the he back back p pr rop opagation agation ne neural ural netw network ork (B (BPNN) PNN) mo model. del. As shown in Figure 5, three responses of the BPNN model highly coincide with the actual As shown in Figure 5, three responses of the BPNN model highly coincide with the actual value, value, illustrating that the identification of the inverse system is correct and the 4WID vehicle system illustrating that the identification of the inverse system is correct and the 4WID vehicle system is is reversible. reversible. 4.2. Verification of Tracking Performance 4.2. Verification of Tracking Performance Three coupling scenarios were designed to assess the tracking performance, and pure-pursuit Three coupling scenarios were designed to assess the tracking performance, and pure-pursuit and MPC are implemented as the benchmark to compare the improvement of the proposed method. and MPC are implemented as the benchmark to compare the improvement of the proposed method. Note that, to filter out the varying-velocity disturbance, the lateral preview reference of MPC and Note that, to filter out the varying-velocity disturbance, the lateral preview reference of MPC and pure-pursuit is based on time, and a speed preview controller supplemented in parallel. pure-pursuit is based on time, and a speed preview controller supplemented in parallel. 4.2.1. Scenario 1: Lane Change with Deceleration on Dry Road Surface In this scenario, the vehicle drives at the initial speed of 90 km/h on a straight and dry road with the road adhesion coefficient being 0.8, then the decelerating lane change is completed within 58 m, which is a classic collision avoidance scenario. Based on the proposed method, the desired vehicle motion states 𝑼 =𝑎 𝑡 ,𝑎 𝑡 ,𝜑 𝑡 were calculated and the vehicle dynamic states tracking results are shown as Figure 6. Appl. Sci. 2020, 10, 7394 10 of 17 4.2.1. Scenario 1: Lane Change with Deceleration on Dry Road Surface In this scenario, the vehicle drives at the initial speed of 90 km/h on a straight and dry road with the road adhesion coecient being 0.8, then the decelerating lane change is completed within 58 m, which is a classic collision avoidance scenario. h i .. Based on the proposed method, the desired vehicle motion states U = a (t), a (t),'(t) were x y re f calculated and the vehicle dynamic states tracking results are shown as Figure 6. Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 16 Figure 6. The vehicle dynamic states tracking results in scenario 1. Figure 6. The vehicle dynamic states tracking results in scenario 1. Figure 6 shows that the longitudinal acceleration, lateral acceleration of the vehicle coincide with Figure 6 shows that the longitudinal acceleration, lateral acceleration of the vehicle coincide with the desired value and the change in each direction does not a ect the other directions, indicating the desired value and the change in each direction does not affect the other directions, indicating that that the proposed method has decoupled dynamics and the vehicle dynamic states can be accurately the proposed method has decoupled dynamics and the vehicle dynamic states can be accurately tracked in close loop. tracked in close loop. As shown in Figure 7, the real vehicle velocity and lateral position accurately follow the references. As shown in Figure 7, the real vehicle velocity and lateral position accurately follow the references. The velocity tracking error of the proposed method is smallest compared with pure-pursuit control The velocity tracking error of the proposed method is smallest compared with pure-pursuit control and MPC. With the same longitudinal controller, MPC and pure-pursuit control still contribute to and MPC. With the same longitudinal controller, MPC and pure-pursuit control still contribute to di erent speed errors, which means the longitudinal control is a ected by control in other directions. different speed errors, which means the longitudinal control is affected by control in other directions. Even though the lateral control value was correctly calculated by MPC, the longitudinal tracking error Even though the lateral control value was correctly calculated by MPC, the longitudinal tracking results in a delay in the lateral tracking owning to the lateral reference related to the time and does error results in a delay in the lateral tracking owning to the lateral reference related to the time and not consider the longitudinal error within a planning cycle, which is another deterioration of tracking does not consider the longitudinal error within a planning cycle, which is another deterioration of accuracy caused by dynamic coupling. tracking accuracy caused by dynamic coupling. To quantitatively evaluate the tracking performance of the three methods, the results were statistically analyzed. The statistical values of lateral error are shown in Table 1. Compared with pure-pursuit and MPC, the mean square error of velocity (MSE ) of the proposed method is the smallest, and its mean square error of lateral tracking (MSE ) and yaw tracking (MSE ) rank in the middle. Y yaw Figure 7. Trajectory tracking results in scenario 1. To quantitatively evaluate the tracking performance of the three methods, the results were statistically analyzed. Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 16 Figure 6. The vehicle dynamic states tracking results in scenario 1. Figure 6 shows that the longitudinal acceleration, lateral acceleration of the vehicle coincide with the desired value and the change in each direction does not affect the other directions, indicating that the proposed method has decoupled dynamics and the vehicle dynamic states can be accurately Appl. Sci. 2020, 10, 7394 11 of 17 tracked in close loop. As shown in Figure 7, the real vehicle velocity and lateral position accurately follow the references. Table 1. The tracking performance statistics. The velocity tracking error of the proposed method is smallest compared with pure-pursuit control and MPC. With the same longitudinal controller, MPC and pure-pursuit control still contribute to Unit Decoupling Control MPC PP different speed errors, which means the longitudinal control is affected by control in other directions. ( ) max e m 0.1054 0.0113 0.0296 Even though the lateral control value was correctly calculated by MPC, the longitudinal tracking min(e ) m 0.0090 0.1795 0.0399 2 2 error results in a de MSE lay in the lateral trac m /s king own 0.0033 ing to the latera0.0110 l reference rel 0.1164 ated to the time and MSE m 0.0011 0.0036 0.0002 does not consider the lon Y gitudinal error within a planning cycle, which is another deterioration of MSE 0.0382 0.0127 0.0386 yaw tracking accuracy caused by dynamic coupling. Figure 7. Figure 7. Trajectory tracking re Trajectory tracking results sults in s in scenario cenario 1. 1. 4.2.2. Scenario 2: Turn Left with Deceleration at Crossing To quantitatively evaluate the tracking performance of the three methods, the results were statistically analyzed. In urban trac, the intersection is a common scene. The vehicle is required to slow down through a right-angle bend with a radius of 50 m. On a dry road with a road adhesion coecient of 0.8, the vehicle speed decelerates to 9 m/s from 12 m/s within 5 s. Figure 8 shows that there are two big pulses in the target w which are caused by the noncontinuous curvature. As the pulses with large rates of change exceeded the range of the training set, the target w was dicult to track and resulted in a chain reaction in the longitudinal and lateral directions. However, the longitudinal and lateral fluctuations accompanied by yaw pulses do not mean decoupling failure, because besides the fluctuating part, the other targets in the three motion directions are accurately tracked without interferences. This gives us two inspirations; first, the target vehicle motion states should be smooth and remain within the range of the training set; the other is that the curvature of the target position curve designed by the planning level should be as continuous as possible. Figure 9 shows that the proposed method realizes the velocity, position and yaw tracking simultaneously. The fluctuations in a and a are amplified and accumulated, resulting in a stable x y velocity and lateral error of 0.193 km/h and 0.36 m after a 150 m trip. As the lateral error is seriously related to driving safety, the tracking performance of pure-pursuit is the best in this scenario. The MPC failed to reduce the error in the X direction of the Cartesian coordinate system without coordinate transformation, which leads to a larger lateral error in the vehicle coordinate system. Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 16 The statistical values of lateral error are shown in Table 1. Compared with pure-pursuit and MPC, the mean square error of velocity ( ) of the proposed method is the smallest, and its mean square error of lateral tracking ( ) and yaw tracking ( ) rank in the middle. Table 1. The tracking performance statistics. Unit Decoupling Control MPC PP 𝑚𝑎𝑥 𝑒 m 0.1054 0.0113 0.0296 𝑚𝑖𝑛 𝑒 m −0.0090 −0.1795 −0.0399 m ⁄s 0.0033 0.0110 0.1164 m 0.0011 0.0036 0.0002 ° 0.0382 0.0127 0.0386 4.2.2. Scenario 2: Turn Left with Deceleration at Crossing In urban traffic, the intersection is a common scene. The vehicle is required to slow down through a right-angle bend with a radius of 50 m. On a dry road with a road adhesion coefficient of 0.8, the vehicle speed decelerates to 9 m/s from 12 m/s within 5 s. Figure 8 shows that there are two big pulses in the target 𝑤 which are caused by the noncontinuous curvature. As the pulses with large rates of change exceeded the range of the training set, the target 𝑤 was difficult to track and resulted in a chain reaction in the longitudinal and lateral directions. However, the longitudinal and lateral fluctuations accompanied by yaw pulses do not mean decoupling failure, because besides the fluctuating part, the other targets in the three motion directions are accurately tracked without interferences. This gives us two inspirations; first, the target vehicle motion states should be smooth and remain within the range of the training set; the other is that the curvature of the target position curve designed by the planning level should be as continuous as possible. Figure 9 shows that the proposed method realizes the velocity, position and yaw tracking simultaneously. The fluctuations in 𝑎 and 𝑎 are amplified and accumulated, resulting in a stable velocity and lateral error of 0.193 km h and 0.36 m after a 150 m trip. As the lateral error is seriously related to driving safety, the tracking performance of pure-pursuit is the best in this scenario. The MPC failed to reduce the error in the X direction of the Cartesian coordinate system without coordinate Appl. Sci. 2020, 10, 7394 12 of 17 transformation, which leads to a larger lateral error in the vehicle coordinate system. Figure 8. The vehicle dynamic states tracking results in scenario 2. Figure 8. The vehicle dynamic states tracking results in scenario 2. Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 16 Figure 9. Trajectory tracking results in scenario 2. Figure 9. Trajectory tracking results in scenario 2. 4. 4.2.3. 2.3. Scen Scenario ario 3: 3: L Lane ane C Change hange wit with h De Deceleration celeration//A Acceleration cceleration on on a a Wet Wet R Road oad S Surface urface In scen In scenario ario 3 3, , t the he vehicle vehicle de decelerates celerates or or a accelerates ccelerates at at t the he in initial itial spee speed d of of 2 20 0 m/ m/s s w when hen chan changing ging lane on a wet lane on a wet road whose road whoseroad roadadhesion adhesioncoefficient coecient is 0.35. is 0.35. Figure 10a Figure 10a sh shows ows th that at th te three met he three methods hods all st allill still achi achieve eve traject trajectory ory tracking on a lo tracking on w-adhesion road. a low-adhesion Compared wi road. Compar th Figure 7, even though ed with Figure 7, even though the road condi the roadtconditions ions are worse, are wor the lateral and se, the lateral yaw tracking and yaw tracking errors of the proposed method decrease with speed red errors of the proposed method decrease with speed ru eduction. ction. However, t Howeverh , e lat the lateral eral and y and yaw aw t tracking racking p performance erformance o of f t the he M MPC PC g get et w worse. orse. I In n F Figur igur ee10 10 b, b,the the pure- pure-pursuit pursuiand t and MPC MPC fail fail to to tra trackcthe k the ta target rget in in longitudinal, longitudinal lateral , latera and l and yaw yaw mot motion. ionHowever . However, , the thepr proposed oposed method method st still ill fo follows llows t the he const constrained rained vehicle vehicle st states ates at at t the he lowe lower r- -adhesion adhesion roa road d condition, condition, gu guaranteeing aranteeing t the he driv driving ing stability stability .. Figure 10. Trajectory tracking results on low-adhesion road: (a) steering with deceleration, (b) steering with acceleration. As we can see from Figure 11a, the proposed method always keeps the vehicle in the safe area of the sideslip phase-plane like the pure-pursuit and MPC; however, in Figure 11b, since the tracking performance becomes worse, the phase curve of the pure-pursuit and MPC are over the stability 𝑀𝑆𝐸 𝑀𝑆𝐸 𝑀𝑆𝐸 𝑀𝑆𝐸 𝑀𝑆𝐸 𝑀𝑆𝐸 Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 16 Figure 9. Trajectory tracking results in scenario 2. 4.2.3. Scenario 3: Lane Change with Deceleration/Acceleration on a Wet Road Surface In scenario 3, the vehicle decelerates or accelerates at the initial speed of 20 m/s when changing lane on a wet road whose road adhesion coefficient is 0.35. Figure 10a shows that the three methods all still achieve trajectory tracking on a low-adhesion road. Compared with Figure 7, even though the road conditions are worse, the lateral and yaw tracking errors of the proposed method decrease with speed reduction. However, the lateral and yaw tracking performance of the MPC get worse. In Figure 10b, the pure-pursuit and MPC fail to track the target in longitudinal, lateral and yaw motion. However, the proposed method still follows the constrained Appl. Sci. 2020, 10, 7394 13 of 17 vehicle states at the lower-adhesion road condition, guaranteeing the driving stability. Figure 10. Figure 10. Trajectory Trajectory track tracking ing resu results lts on low-a on low-adhesion dhesion roa road: d: (a (a ) steering wi ) steering with th dec deceleration, eleration, (b (b ) st ) steering eering with acceleration. with acceleration. As we can see from Figure 11a, the proposed method always keeps the vehicle in the safe area of As we can see from Figure 11a, the proposed method always keeps the vehicle in the safe area the sideslip phase-plane like the pure-pursuit and MPC; however, in Figure 11b, since the tracking of the sideslip phase-plane like the pure-pursuit and MPC; however, in Figure 11b, since the tracking Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 16 performance becomes worse, the phase curve of the pure-pursuit and MPC are over the stability performance becomes worse, the phase curve of the pure-pursuit and MPC are over the stability boundaries, which means that the proposed method significantly improves the handling stability and boundaries, which means that the proposed method significantly improves the handling stability and has fewer tracking errors compared with the pure-pursuit and MPC algorithms. has fewer tracking errors compared with the pure-pursuit and MPC algorithms. Figure Figure 11. 11. T The si he sidedeslip pha slip phase-psle-plane d ane diagrai m agram: ( : (a) steea ri) ngsteeri withndge with dece celeration; (lbe)ration; ( steeringb w) st itheering with acceleration. acceleration. Figure 12 shows the error statistics results of the longitudinal, lateral and yaw directions of the three tracking controllers under four working conditions, as well as the trajectory tracking error index Figure 12 shows the error statistics results of the longitudinal, lateral and yaw directions of the P of the planar motion weighted by the three directions. three tracking controllers under four working conditions, as well as the trajectory tracking error index 𝑃 of the planar motion weighted by the three directions. P = w MSE + w MSE + w MSE , (41) 1 V 2 Y 3 yaw 𝑃= 𝑤 +𝑤 +𝑤 , (41) Figure 12. The tracking performance index: (a) the longitudinal direction; (b) the lateral direction; (c) the yaw direction; (d) the planar motion. Considering that three directions are equally important in planar motion tracking, the three weights should be the same. However, due to the unit, the mean square error of yaw tracking is very large, so it is reduced by a certain proportion and the weighted vector is 𝒘= 1,1, /180 . As shown in Figure 12a, except for scenario 2, the of the proposed method is the smallest. The accurate velocity tracking illustrates that the dynamic decoupling of the proposed tracking method can effectively reduce the longitudinal interference from other motion directions. Moreover, compared with the other two methods, the proposed method still performs well in the lateral and 𝑀𝑆𝐸 𝑝𝑖 𝑀𝑆𝐸 𝑀𝑆𝐸 𝑀𝑆𝐸 Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 16 boundaries, which means that the proposed method significantly improves the handling stability and has fewer tracking errors compared with the pure-pursuit and MPC algorithms. Figure 11. The sideslip phase-plane diagram: (a) steering with deceleration; (b) steering with acceleration. Figure 12 shows the error statistics results of the longitudinal, lateral and yaw directions of the three tracking controllers under four working conditions, as well as the trajectory tracking error index 𝑃 of the planar motion weighted by the three directions. Appl. Sci. 2020, 10, 7394 14 of 17 𝑃= 𝑤 +𝑤 +𝑤 , (41) Figure 12. The tracking performance index: (a) the longitudinal direction; (b) the lateral direction; Figure 12. The tracking performance index: (a) the longitudinal direction; (b) the lateral direction; (c) (c) the yaw direction; (d) the planar motion. the yaw direction; (d) the planar motion. Considering that three directions are equally important in planar motion tracking, the three Considering that three directions are equally important in planar motion tracking, the three weights should be the same. However, due to the unit, the mean square error of yaw tracking is very weights should be the same. However, due to the unit, the mean square error of yaw tracking is very large, so it is reduced by a certain proportion and the weighted vector is w = (1, 1, pi/180) . large, so it is reduced by a certain proportion and the weighted vector is 𝒘= 1,1, /180 . As shown in Figure 12a, except for scenario 2, the MSE of the proposed method is the smallest. As shown in Figure 12a, except for scenario 2, the of the proposed method is the smallest. The accurate velocity tracking illustrates that the dynamic decoupling of the proposed tracking method The accurate velocity tracking illustrates that the dynamic decoupling of the proposed tracking can e ectively reduce the longitudinal interference from other motion directions. Moreover, compared method can effectively reduce the longitudinal interference from other motion directions. Moreover, with the other two methods, the proposed method still performs well in the lateral and yaw tracking. compared with the other two methods, the proposed method still performs well in the lateral and In summary, as shown in Figure 12d, the trajectory tracking error index of the proposed method proposed is always the lowest, indicating that the proposed tracking method is more suitable for performing trajectory tracking tasks under the coupled conditions than the other two methods. 5. Conclusions To achieve the high trajectory tracking precision in the dynamic coupling scenarios, a simultaneous trajectory tracking and stability control method for 4WID automated electric vehicles is present in this paper based on the inverse system theorem. To reveal the coupling mechanism and to prove the reversibility of 4WID vehicles, the 3DOF vehicle dynamic model is constructed. The inverse system learned by a BPNN model shows e ectiveness to realize dynamic decoupling. The pseudo linear system composed of the inverse system and the controlled object follows the desired vehicle dynamic states to indirectly achieve trajectory tracking. Three typical and common coupled driving conditions are designed to verify the trajectory tracking accuracy under the simultaneous control of vehicles’ longitudinal, lateral and yaw motion. Compared with the pure-pursuit algorithm and the MPC algorithm, the proposed method reduces interactions among vehicle motion directions and reveals better tracking performance. Moreover, since the target states of the vehicle have been constrained within a reasonable range, the decoupling method not only maintains the accurate trajectory tracking but also guarantees the stable vehicle driving under low-adhesion road conditions. Even though the proposed method theoretically shows better control performance, further verifications could be implemented on real vehicles to realize the engineering applications. Author Contributions: Conceptualization, Y.Y. and Y.L. (Yinong Li); methodology, Y.Y. and Y.L. (Yixiao Liang); software and validation, Y.Y. and Y.L. (Yixiao Liang); formal analysis, Y.Y. and W.Y.; writing—review and editing, 𝑀𝑆𝐸 𝑝𝑖 𝑀𝑆𝐸 𝑀𝑆𝐸 𝑀𝑆𝐸 Appl. Sci. 2020, 10, 7394 15 of 17 Y.Y., Y.L. (Yinong Li) and W.Y.; supervision and funding acquisition, Y.L. (Yinong Li) and L.Z. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by Key Research Program of the Ministry of Science and Technology ([Grant No. 2017YFB0102603-3, 2016YFB0100900), Chongqing Science and Technology Program Project Basic Science and Frontier Technology (Grant No. cstc2018jcyjAX0630), Chongqing Technology Innovation and Application Development Major Theme Special Project (Grant No. cstc2019jscx-zdztzxX0032), Graduate Scientific Research and Innovation Foundation of Chongqing (Grant No. CYB19063). Acknowledgments: In this section you can acknowledge any support given which is not covered by the author contribution or funding sections. This may include administrative and technical support, or donations in kind (e.g., materials used for experiments). Conflicts of Interest: We declare that there is no conflict of interests in connection with the paper submitted. Appendix A Table A1. Symbols and definitions of the dynamics model cited. Definition Symbol Unit Vehicle mass m kg Vehicle inertia on yaw direction I kgm . .. Longitudinal speed/acceleration (in xoy) x/x m/s . .. Lateral speed/acceleration (in xoy) y/y m/s Longitudinal force vector on tire (in xoy) F N Lateral force vector on tire (in xoy) F N Distance from c.g. to the front/rear axle l /l m Length of wheelbase d m Longitudinal force on each tire (in tire coordinate) F N xi Lateral force on each tire (in tire coordinate) F N yi Front wheel steering angle on the left/right  / rad f l f r Equivalent steering angle  rad Slip angle of the front/rear wheel / rad Speed angle of the front/rear wheel rad f r Cornering sti ness of the front/rear wheel C /C N/rad Yaw angle of vehicle body (in XOY) ' rad Total input longitudinal force on vehicle c.g. F N Total input yaw moment on vehicle c.g. M Nm Longitudinal force distribute rate on front/rear wheel k /k - Aerodynamic drag coecient k N/(m/s) . . e /e /e Control error on longitudinal/lateral/yaw directions ' m/s/m/s/rad x y . . e /e Tracking error on longitudinal/lateral m/m X Y Table A2. Symbols and definitions of the dynamics model cited. Parameters Definition Value m Vehicle mass 1370 kg Horizontal distance from c.g. to front tires 1.11 m l Horizontal distance from c.g. to rear tires 1.67 m k The resistance coecient of air 0.35 cornering sti ness of front tires 67553 Nm/rad C cornering sti ness of rear tires 49506 Nm/rad I Yaw inertia 2315.3 kg/m References 1. Xu, S.; Peng, H. Design, Analysis, and Experiments of Preview Path Tracking Control for Autonomous Vehicles. IEEE Trans. Intell. Transp. Syst. 2020, 21, 48–58. [CrossRef] 2. Wang, Y.; Shao, Q.; Zhou, J. Longitudinal and lateral control of autonomous vehicles in multi-vehicle driving environments. IET Intell. Transp. Syst. 2020, 14, 924–935. [CrossRef] Appl. Sci. 2020, 10, 7394 16 of 17 3. Chen, T.; Chen, L.; Xu, X.; Cai, Y. Simultaneous path following and lateral stability control of 4WD-4WS autonomous electric vehicles with actuator saturation. Adv. Eng. Softw. 2019, 128, 46–54. [CrossRef] 4. Samuel, M.; Hussein, M.; Mohamad, M. 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Sliding mode direct yaw-moment control design for in-wheel electric vehicles. IEEE Trans. Ind. Electron. 2017, 64, 6752–6762. [CrossRef] Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional aliations. © 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Sciences Multidisciplinary Digital Publishing Institute

Dynamic Decoupling and Trajectory Tracking for Automated Vehicles Based on the Inverse System

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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; 20173201003@cqu.edu.cn (Y.Y.); liangyixiao@cqu.edu.cn (Y.L.); zling@cqu.edu.cn (L.Z.); yangwei0705@gmail.com (W.Y.) School of Automotive Engineering, Chongqing University, Chongqing 400044, China * Correspondence: ynli@cqu.edu.cn; 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 e ect 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, di ering 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 e ectiveness 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 a ects driving safety and comfort [3]. According to the di erent 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 [4], pure-pursuit (PP) [5], Stanley [6], linear quadratic regulator (LQR) [7] and model predictive control (MPC) [8]. 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 [13] designed the lateral controller considering the time-varying velocity, which can eliminate the lateral disturbance caused by longitudinal control. Attia et al. [14] proposed the nonlinear model predictive control (NMPC) considering the characteristics of vehicle body motion coupling and tire force coupling, which can e ectively solve the problem of motion interference between di erent directions. Besides the lateral control compensation, Kanayama [15] applied the Lyapunov method to solve the integrated longitudinal and lateral tracking problem. Menour [16] used the di erential 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 e ect 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 e ective method to eliminate the interactions of various vehicle motion directions. To decouple the lateral and the yaw motion of the vehicle, Marino [19] 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 [20] 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 [21] 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 [22] and guarantee the accuracy motion tracking [23]. The input number of 4WID electric vehicle system is equal to the output number, which is a positive system [24] 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 e ectiveness 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 e ects 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 [25]. The planar motion of the vehicle can be front tires are equivalent to steering angle  [25]. 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 di erential algorithm is adopted, the longitudinal forces are synthetically considered as the total longitudinal force F and the additional yaw moment M [26]. 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 [26]. (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 di erence 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 [27]. . . . Then, introducing the small angle hypothesis, the tire lateral force is proportional to its slip angle [27]. =   =  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' 2 2 + , (11) . . m m m x x . . 2 2 C l C l C l + C l l (1 k )F  2C l .. M f f r r y f f r r ' f r t f f 𝐶 𝑙 −𝐶 𝑙 𝐶 𝑙 +𝐶 𝑙 𝑙 1−𝑘 𝐹 𝛿 2𝐶 𝑙 𝛿 𝑀 𝑦 𝜑 ' = 2 2 + + , (12) . . (12) 𝜑 = −2 −2 + + , I I I I I z z x z x z z 𝐼 𝐼 𝑥 𝐼 𝑥 𝐼 𝐼 . . . The The dyn dynamics amicscan can be be r rewr ewritten itten in the in the state-space state-space form, form, with with the the st states ates be beinging X = 𝑿=x, y 𝑥,' ,𝑦, ,𝜑 and , an the d the control inputs being 𝑼= 𝐹 ,𝛿, 𝑀 ,i.e., control inputs being U = (F ,, M ) ,i.e., t z 𝑦 𝜑 . . −𝑘 𝑥 2 3 2 . 3 y +𝑙' −𝛿 . ⎡ 1 ⎤ k x ⎡ D 𝜑 0 ⎤ 𝑥 𝑥 6 . +l .  7 6 7 f ' 0 6 1 2𝐶 70 6 7 x x m 6⎢ 7 ⎥ 6 7 2C 0 ⎢ ⎥ 6 7 𝑚 f 𝑚 6 7 6 m m 7 . 6 7 . C +C C l C l 𝐶 f+𝐶 r 𝐶 f 𝑙 f −𝐶 r r 𝑙 6 7 6 7 ⎢ ⎥ ⎢ ⎥ 6 2C 7 6 7 (1k ) . . 6 1− r𝑘 𝛿 f2𝐶 7 X = ' 2 2 X + U, (13) 6 −𝜑 −2 −2 7 𝑿 = 𝑿+ 6 7 𝑼, 6 7 (13) mx mx 0 ⎢ ⎥ 6⎢ 7 ⎥ 6 7 0 𝑚𝑥 𝑚𝑥 6 m m 7 2 2 6 7 6 7 C l C l C l +C l 6 r r r r 7 𝑚 𝑚 f f f f 6 7 ⎢4 5 ⎥ l (1k ) 2C l ⎢ r ⎥ 4 f f f 5 0 2 . 2 . 𝐶 𝑙 −𝐶 𝑙 𝐶 𝑙 +𝐶 𝑙 𝑙 1− 𝑘 𝛿 2𝐶 𝑙 ⎢ ⎥ 1 I x I x z z ⎢ I I I ⎥ z z z 0−2 −2 ⎣ 𝐼 𝑥 𝐼 𝑥 ⎦ ⎣ ⎦ 𝐼 𝐼 𝐼 In Equation (13), the input matrix is a non-diagonal matrix, which means each state of the system In Equation (13), the input matrix is a non-diagonal matrix, which means each state of the system is a ected by multiple inputs. The input matrix is not a constant coecient matrix, but contains input is affected by multiple inputs. The input matrix is not a constant coefficient matrix, but contains input elements and state elements, which indicates that there are complex interactions among three directions elements and state elements, which indicates that there are complex interactions among three and causes the attendant disturbances when controlling one of the directions, such as the2C  /m is directions and causes the attendant disturbances when controlling one of the directions, such as the a longitudinal resistance caused by lateral control input. −2𝐶 𝛿 𝑚 is a longitudinal resistance caused by lateral control input. Once ignoring the attendant disturbance, the control errors appear and deteriorate tracking [28]: Once ignoring the attendant disturbance, the control errors appear and deteriorate tracking [28]: . . . e = e cos e + e sin e , (14) 𝑒 =𝑒 cos 𝑒 ' +𝑒 sin ' 𝑒 , x y (14) 𝑒 =𝑒 cos 𝑒 −𝑒 sin 𝑒 , . . . (15) e = e cos e e sin e , (15) ' ' y x 3. Methodology 3. Methodology Once an inverse system whose inputs and outputs are strictly opposed to the origin system Once an inverse system whose inputs and outputs are strictly opposed to the origin system exists, the decoupling method based on the inverse system will be valid, as shown in Figure 2; that exists, the decoupling method based on the inverse system will be valid, as shown in Figure 2; that is, the inverse system and the control object are connected in series to form an equivalent multiple is, the inverse system and the control object are connected in series to form an equivalent multiple single-input single-output combined system without interactions. single-input single-output combined system without interactions. Figure 2. The decoupling principle of the inverse system. Figure 2. The decoupling principle of the inverse system. 3.1. 3.1. Decoupling Decoupling of of D Dynamics ynamics 3.1.1. Proof of Reversibility 3.1.1. Proof of Reversibility In order to apply the decoupling algorithm, it is necessary to prove the reversibility of the vehicle In order to apply the decoupling algorithm, it is necessary to prove the reversibility of the vehicle dynamic. Here, the proof is derived based on the Interactor Algorithm 1 [29,30]. dynamic. Here, the proof is derived based on the Interactor Algorithm [29,30]. For a multiple-input and multiple-output (MIMO) nonlinear system, it can be written as For a multiple-input and multiple-output (MIMO) nonlinear system, it can be written as (16) 𝑿 = 𝑓 𝑿, 𝑼 , X = f(X, U), (16) 𝒀=ℎ 𝑿 , (17) where the input vector is 𝑼=𝑢 ,𝑢 …,𝑢 , the state vector is 𝑿= 𝑥 ,𝑥 …, 𝑥 and the output vector is 𝒀=𝑦 ,𝑦 …,𝑦 =ℎ 𝑿 ,ℎ 𝑿 …,ℎ 𝑿 . Appl. Sci. 2020, 10, 7394 5 of 17 Y = h(X), (17) h i where the input vector is U = u , u ::: , u , the state vector is X = [x , x ::: , x ] and the output 1 2 p 1 2 n h i h i T T vector is Y = y , y ::: , y = h (X), h (X) ::: , h (X) . q q 1 2 1 2 Algorithm 1. Interactor Algorithm 1: Defining the superscript r , which means taking the derivative for k-th component y , r is the order when k k k h i (r ) control input U = u , u ::: , u firstly appears, and the derivative is noted as y 1 2 p 2: For k = 1, q 2 3 6 G 7 6 k1 7 6 7 3: Defining a criterion vector G = 6 7, and G is null when k = 1 k (r ) 0 4 k 5 4: Calculating the rank of the Jacobian matrix R = rank(@G /@U) k k 5: If R = k 6: = r k k 7: End If 8: End For 9: If  n i=1 10: The system is reversible. 11: End If . . . As for the proposed vehicle dynamics, the output vector is Y = x, y,' . Setting y = x, which does not include any input variable, G needs to be rewritten with respect 1 1 . . . 2 2 . . 2C  C  C l (1) y ' F k x f f f f t D . . to its derivative [21]: that is, G = y = y' + + 2 + 2 , the rank of the m m m m m x x Jacobian matrix of G to U is . . y ' . +l . R = rank(@G /@U) = 1 x x = 1, (18) 1 1 2C 0 m m So that, = r = 1. And y = y, which also does not include any input variable. Hence, 1 1 2 2 3 " # .. 6 G 7 6 7 6 7 setting G = 6 7 = .. , 2 (1) 4 5 2 3 0 0 6 7 6 7 6 7 ( ) R = rank @G /@U = 6 7 = 2, (19) 2 2 2C 4 (1k ) f 5 m m 2 3 .. 2 3 6 7 6 7 6 G 7 6 7 .. 6 7 6 7 6 7 6 7 we can get that = r = 1. At last, setting G = 6 7 = 6 y 7, 2 2 3 (1) 4 5 6 7 6 7 .. y 4 5 . . 2 3 y ' . . 6 +l  7 6 1 x x 7 6 7 2C 0 6 7 6 m m 7 6 7 6 7 2C (1k ) f 6 r 7 R = rank(@G /@G) = = 3, (20) 3 3 6 7 6 0 7 6 m m 7 6 7 6 ( ) 7 l 1k  2C l 4 f r f f 5 I I I z z z and this yields = r = 1, then 3 3 = 3, (21) i=1 Appl. Sci. 2020, 10, 7394 6 of 17 According to the reversibility judgment [31], there is an inverse system for the 4WID vehicle without (1) (1) (1) hidden dynamics [32], and the input and output of the inverse system are U = y , y , y = inv 2 3 .. .. .. x, y,' and Y = (F ,, M ) . inv t z As the 3DOF vehicle dynamic system is reversible and based on the Equation (13), the inverse system can be written as Equation (22): . . 2 3 12 . 3 y ' k x 6 . +l .  7 f 6 7 6 1 7 ' 0 x x 6 7 6 7 m 2C 0 6 7 6 7 6 7 6 m m 7 6 7 . C +C C l C l 6 7 f r f f r r 6 7 6 2C 7 (1k ) 6 7 6 r f 7 ' 2 . 2 . Y = X 6 7 inv 6 7 6 7 0 mx mx 6 7 6 7 6 m m 7 2 2 6 7 6 7 C l C l C l +C l 6 r r r r 7 6 7 f f f f l (1k ) 2C l 4 5 4 f f f 5 0 2 . 2 . I x I x I I I z z z z z (22) . . 2 3 y ' . +l . 6 7 6 1 x x 7 6 7 2C 0 6 7 6 m m 7 6 7 6 7 2C (1k ) f 6 r 7 + U 6 7 inv 6 0 7 6 m m 7 6 7 6 ( ) 7 l 1k  2C l 4 f r f f 5 I I I z z z 3.1.2. Definition of the Inverse System As shown in Equation (22), the output can be calculated using the Taylor series expansion. However, to avoid the approximation error caused by the omission of high-order terms in the Taylor formula, a machine learning method is applied to identify the relationship between the input and output of the system in this paper. Because the identification of the inverse system is a MIMO regression, .. .. the back propagation neural network (BPNN) model is adopted. Note that x and y are replaced by the measurable a and a : x y .. . . x = a + y', (23) .. . . y = a x', (24) .. . . . Hence, the input and output of the BPNN model are U = a , a ,', x, y,' and Y = nn x y nn (F ,, M ) . t z 3.1.3. Neural Networks Training n o .. . . . The D-class sedan model of CarSim is utilized to obtain the vehicle responses a , a ,', x, y,' x y under stochastic inputsfF ,, M g. To simulate the 4WID vehicle, the total longitudinal force and yaw t z moment are converted to driven torques of each wheel based on Equations (2) and (4). dw (1 k )F  M /d(1 + cos) f l, f r r t z T = + I , (25) f l, f r r dt dw k F  M /d(1 + cos) f l, f r r t z T = + I , (26) rl,rr r r dt where T , T , T , T are the driven torques of the front-left, front-right, rear-left and rear-right tire, f l f l f l f l respectively. r is the radius of each wheel. I is the wheel inertia moment. w and w are the f l, f r f l, f r rotation velocities of each wheel. The longitudinal and lateral acceleration were constrained considering the ride comfort and anti-sideslip [33]. g  a  g, (27) g  a   g, (28) s y s Appl. Sci. 2020, 10, 7394 7 of 17 Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 16 where 𝜂 , 𝜂 and 𝜇 respectively represent the maximum coefficient of longitudinal deceleration, where , and respectively represent the maximum coefficient of longitudinal deceleration, longitudinal d s longitudinal acceleration and lateral acceleration. According to the vehicle dynamics and the acceleration and lateral acceleration. According to the vehicle dynamics and the kinematic model: kinematic model: F 𝐹 a = , (2(29) 9) 𝑎 x = , L L 𝐿 𝐿 R =  , (30) 𝑅= , (30) sinjj jj | | | | sin 𝛿 𝛿 where R is the turning radius of the vehicle, and L is the distance between the front and rear axles. where 𝑅 is the turning radius of the vehicle, and 𝐿 is the distance between the front and rear axles. Then, substituting Equations (29) and (30) into Equations (27) and (28) yields: Then, substituting Equations (29) and (30) into Equations (27) and (28) yields: 𝜇 gL (31) |𝛿 | ≤ , jj  , (31) mg  F  mg, (32) −𝜇𝑚𝑔 ≤ 𝐹 ≤𝜇𝑚𝑔, (32) The interval between two sampling points of longitudinal force, steering angle and additional The interval between two sampling points of longitudinal force, steering angle and additional yaw moment is 2 s. The values of sampling points are stochastically generated, obeying the yaw moment is 2 s. The values of sampling points are stochastically generated, obeying the uniform uniform distribution in the range of (5000, 5000)N, (0.1, 0.1)rad and (500, 500)Nm, respectively. distribution in the range of −5000,5000 𝑁 , −0.1,0.1 and −500,500 , respectively. The The sampling data are interpolated by the Hermite method for smoothness, as shown in Figure 3. sampling data are interpolated by the Hermite method for smoothness, as shown in Figure 3. Figure 3. One group of vehicle stochastic inputs. Figure 3. One group of vehicle stochastic inputs. By setting di erent initial speeds, multiple groups of input signals act on the vehicle model, By setting different initial speeds, multiple groups of input signals act on the vehicle model, then then the inputs and the responses of the vehicle are recorded as the training dataset. To ensure the the inputs and the responses of the vehicle are recorded as the training dataset. To ensure the validity validity of the collected data set, the following two constraints need to be guaranteed. of the collected data set, the following two constraints need to be guaranteed. • The equivalent steering angle is constrained according to the real-time speed. The equivalent steering angle is constrained according to the real-time speed. • The vehicle speed is always positive. The vehicle speed is always positive. 3.1.4. Design of the Trajectory Tracking Controller 3.1.4. Design of the Trajectory Tracking Controller Based on the inverse system, the control framework is built in Figure 4. The target motion Based on the inverse system, the control framework is built in Figure 4. The target motion trajectory transformed into the vehicle dynamic state references—that is, the longitudinal trajectory transformed into the vehicle dynamic state references—that is, the longitudinal acceleration, acceleration, the lateral acceleration, and the yaw acceleration. By following the target vehicle the lateral acceleration, and the yaw acceleration. By following the target vehicle dynamic state, dynamic state, the inverse system exports the vehicle inputs and drives the vehicle tracking desired the inverse system exports the vehicle inputs and drives the vehicle tracking desired trajectory. trajectory. 𝑁𝑚 𝑟𝑎𝑑 𝑔𝐿 Appl. Sci. 2020, 10, 7394 8 of 17 Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 16 Figure 4. Scheme of the designed trajectory tracking controller with decoupling. Figure 4. Scheme of the designed trajectory tracking controller with decoupling. 3.2. The desired Vehicle States 3.2. The Desired Vehicle States Based on the kinematic model, the desired vehicle states, i.e., the references of the inverse system Based on the kinematic model, the desired vehicle states, i.e., the references of the inverse system .. 𝑼 =𝑎 ,𝑎 ,𝜑 , are calculated from the target motion trajectory. U = a , a ,' , are calculated from the target motion trajectory. re f x y Supposing that the references from the planning module are the sequences of resultant speed Supposing that the references from the planning module are the sequences of resultant speed V 𝑉 (t 𝑡 ) an andd trajectory trajectoryY 𝑌(X 𝑋 ), , t is 𝑡 is the time, the time, X and 𝑋 and Y ar 𝑌 e the are the veh vehicle ilongitudinal cle longitudinal andand later lateral position al position in the in the inertial coordinate system. inertial coordinate system. The The spat spatial–temporal ial–temporal rel relationship ationship is is un unified ified by by using using the the travel travel distance distance o offvehicles vehicles [34]. [34]. 𝑉 𝑑𝑡 = ∆𝑋 𝑖 2 +∆𝑌 𝑖 2 , (33) Vdt = DX(i) + DY(i) , (33) where ∆𝑋 𝑖 and ∆𝑌 𝑖 are the longitudinal and lateral spacing between the 𝑖+ 1 th point and the where DX(i) and DY(i) are the longitudinal and lateral spacing between the i + 1 th point and the i th 𝑖 th point in the inertial coordinate system. point in the inertial coordinate system. The desired yaw can be calculated as [35]: The desired yaw can be calculated as [35]: ∆𝑌 𝑖 𝜑 𝑖 =tan , (34) DY( i) ∆𝑋 𝑖 '(i) = tan , (34) DX(i) The vehicle velocity along the 𝑋 and 𝑌 directions in the inertial coordinate system are The vehicle velocity along the X and Y dir ∆𝑋 ections 𝑖 in the ∆𝑌inertial 𝑖 coordinate system are (35) 𝑉 𝑖 = ,𝑉 𝑖 = , ∆𝑡 𝑖 ∆𝑡 𝑖 ( ) ( ) DX i DY i V (i) = , V (i) = , (35) Then, the vehicle longitudinal and lateral velocities are [28] Dt(i) Dt(i) 𝑥 𝑖 =𝑉 𝑖 cos 𝜑 𝑖 +∆𝑌 𝑖 sin 𝜑 𝑖 , (36) Then, the vehicle longitudinal and lateral velocities are [28] 𝑦 𝑖 =−𝑉 𝑖 sin 𝜑 𝑖 +∆𝑌 𝑖 cos 𝜑 𝑖 , (37) x(i) = V (i) cos'(i) + DY(i) sin'(i), (36) The longitudinal, lateral and yaw accelerations are yields: y(i) = V (i) sin'(i) + DY(i) cos'(i), (37) 𝑥 𝑖 ∆𝜑 𝑖 𝑎 𝑖 = −𝑦 𝑘 , (38) The longitudinal, lateral and yaw accelerations are yields: ∆𝑡 𝑖 ∆𝑡 𝑖 x(i) D'(i) 𝑦 𝑖 ∆𝜑 𝑖 a (i) = y(k) , (38) 𝑎 𝑖 = +𝑥 𝑘 , (39) Dt(i) Dt(i) ∆𝑡 𝑖 ∆𝑡 𝑖 y(i) D'(i) ∆𝜑 𝑖 a (i) = + x(k) , (39) Dt(i) Dt(i) ∆𝑡 𝑖 (40) 𝜑 𝑖 = , ∆𝑡 𝑖 D'(i) .. Dt(i) '(i) = , (40) So far, the input of the inverse system, the target motion states of the vehicle 𝑼 =𝑎 ,𝑎 ,𝜑 , Dt(i) has been obtained. Appl. Sci. 2020, 10, 7394 9 of 17 h i .. T So far, the input of the inverse system, the target motion states of the vehicle U = a , a ,' , re f x y Appl. has Sci. been 2020 obtained. , 10, x FOR PEER REVIEW 9 of 16 4. Simulation Results and Analysis 4. Simulation Results and Analysis The simulations are conducted to verify the correctness of the inverse system identification and The simulations are conducted to verify the correctness of the inverse system identification and assess the tracking performance of the proposed method. A D-Class CarSim vehicle model is adopted, assess the tracking performance of the proposed method. A D-Class CarSim vehicle model is and the parameters are shown in Table A2. The proposed trajectory tracking controller is developed in adopted, and the parameters are shown in Table A2. The proposed trajectory tracking controller is Simulink. The Simulink–CarSim interface is shown in Figure 4. developed in Simulink. The Simulink–CarSim interface is shown in Figure 4. 4.1. Verification of the Inverse System Models 4.1. Verification of the Inverse System Models The inverse system is identified by a BPNN model whose structure is 6-50-50-3, i.e., the model The inverse system is identified by a BPNN model whose structure is 6-50-50-3, i.e., the model has six inputs, three outputs and two hidden layers with 50 nodes. The mean square error between has six inputs, three outputs and two hidden layers with 50 nodes. The mean square error between the training data and the prediction of the BPNN is 0.000213 and the regression factor of the model is the training data and the prediction of the BPNN is 0.000213 and the regression factor of the model 0.99941, which indicates that the model has identified the input–output mapping relationship between is 0.99941, which indicates that the model has identified the input–output mapping relationship the of the inverse system. A novel test dataset was collected and applied to evaluate the fitting between the of the inverse system. A novel test dataset was collected and applied to evaluate the performance of the inverse system as Figure 5. fitting performance of the inverse system as Figure 5. Figure Figure 5. 5. T The he re responses sponses o of f t the he back back p pr rop opagation agation ne neural ural netw network ork (B (BPNN) PNN) mo model. del. As shown in Figure 5, three responses of the BPNN model highly coincide with the actual As shown in Figure 5, three responses of the BPNN model highly coincide with the actual value, value, illustrating that the identification of the inverse system is correct and the 4WID vehicle system illustrating that the identification of the inverse system is correct and the 4WID vehicle system is is reversible. reversible. 4.2. Verification of Tracking Performance 4.2. Verification of Tracking Performance Three coupling scenarios were designed to assess the tracking performance, and pure-pursuit Three coupling scenarios were designed to assess the tracking performance, and pure-pursuit and MPC are implemented as the benchmark to compare the improvement of the proposed method. and MPC are implemented as the benchmark to compare the improvement of the proposed method. Note that, to filter out the varying-velocity disturbance, the lateral preview reference of MPC and Note that, to filter out the varying-velocity disturbance, the lateral preview reference of MPC and pure-pursuit is based on time, and a speed preview controller supplemented in parallel. pure-pursuit is based on time, and a speed preview controller supplemented in parallel. 4.2.1. Scenario 1: Lane Change with Deceleration on Dry Road Surface In this scenario, the vehicle drives at the initial speed of 90 km/h on a straight and dry road with the road adhesion coefficient being 0.8, then the decelerating lane change is completed within 58 m, which is a classic collision avoidance scenario. Based on the proposed method, the desired vehicle motion states 𝑼 =𝑎 𝑡 ,𝑎 𝑡 ,𝜑 𝑡 were calculated and the vehicle dynamic states tracking results are shown as Figure 6. Appl. Sci. 2020, 10, 7394 10 of 17 4.2.1. Scenario 1: Lane Change with Deceleration on Dry Road Surface In this scenario, the vehicle drives at the initial speed of 90 km/h on a straight and dry road with the road adhesion coecient being 0.8, then the decelerating lane change is completed within 58 m, which is a classic collision avoidance scenario. h i .. Based on the proposed method, the desired vehicle motion states U = a (t), a (t),'(t) were x y re f calculated and the vehicle dynamic states tracking results are shown as Figure 6. Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 16 Figure 6. The vehicle dynamic states tracking results in scenario 1. Figure 6. The vehicle dynamic states tracking results in scenario 1. Figure 6 shows that the longitudinal acceleration, lateral acceleration of the vehicle coincide with Figure 6 shows that the longitudinal acceleration, lateral acceleration of the vehicle coincide with the desired value and the change in each direction does not a ect the other directions, indicating the desired value and the change in each direction does not affect the other directions, indicating that that the proposed method has decoupled dynamics and the vehicle dynamic states can be accurately the proposed method has decoupled dynamics and the vehicle dynamic states can be accurately tracked in close loop. tracked in close loop. As shown in Figure 7, the real vehicle velocity and lateral position accurately follow the references. As shown in Figure 7, the real vehicle velocity and lateral position accurately follow the references. The velocity tracking error of the proposed method is smallest compared with pure-pursuit control The velocity tracking error of the proposed method is smallest compared with pure-pursuit control and MPC. With the same longitudinal controller, MPC and pure-pursuit control still contribute to and MPC. With the same longitudinal controller, MPC and pure-pursuit control still contribute to di erent speed errors, which means the longitudinal control is a ected by control in other directions. different speed errors, which means the longitudinal control is affected by control in other directions. Even though the lateral control value was correctly calculated by MPC, the longitudinal tracking error Even though the lateral control value was correctly calculated by MPC, the longitudinal tracking results in a delay in the lateral tracking owning to the lateral reference related to the time and does error results in a delay in the lateral tracking owning to the lateral reference related to the time and not consider the longitudinal error within a planning cycle, which is another deterioration of tracking does not consider the longitudinal error within a planning cycle, which is another deterioration of accuracy caused by dynamic coupling. tracking accuracy caused by dynamic coupling. To quantitatively evaluate the tracking performance of the three methods, the results were statistically analyzed. The statistical values of lateral error are shown in Table 1. Compared with pure-pursuit and MPC, the mean square error of velocity (MSE ) of the proposed method is the smallest, and its mean square error of lateral tracking (MSE ) and yaw tracking (MSE ) rank in the middle. Y yaw Figure 7. Trajectory tracking results in scenario 1. To quantitatively evaluate the tracking performance of the three methods, the results were statistically analyzed. Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 16 Figure 6. The vehicle dynamic states tracking results in scenario 1. Figure 6 shows that the longitudinal acceleration, lateral acceleration of the vehicle coincide with the desired value and the change in each direction does not affect the other directions, indicating that the proposed method has decoupled dynamics and the vehicle dynamic states can be accurately Appl. Sci. 2020, 10, 7394 11 of 17 tracked in close loop. As shown in Figure 7, the real vehicle velocity and lateral position accurately follow the references. Table 1. The tracking performance statistics. The velocity tracking error of the proposed method is smallest compared with pure-pursuit control and MPC. With the same longitudinal controller, MPC and pure-pursuit control still contribute to Unit Decoupling Control MPC PP different speed errors, which means the longitudinal control is affected by control in other directions. ( ) max e m 0.1054 0.0113 0.0296 Even though the lateral control value was correctly calculated by MPC, the longitudinal tracking min(e ) m 0.0090 0.1795 0.0399 2 2 error results in a de MSE lay in the lateral trac m /s king own 0.0033 ing to the latera0.0110 l reference rel 0.1164 ated to the time and MSE m 0.0011 0.0036 0.0002 does not consider the lon Y gitudinal error within a planning cycle, which is another deterioration of MSE 0.0382 0.0127 0.0386 yaw tracking accuracy caused by dynamic coupling. Figure 7. Figure 7. Trajectory tracking re Trajectory tracking results sults in s in scenario cenario 1. 1. 4.2.2. Scenario 2: Turn Left with Deceleration at Crossing To quantitatively evaluate the tracking performance of the three methods, the results were statistically analyzed. In urban trac, the intersection is a common scene. The vehicle is required to slow down through a right-angle bend with a radius of 50 m. On a dry road with a road adhesion coecient of 0.8, the vehicle speed decelerates to 9 m/s from 12 m/s within 5 s. Figure 8 shows that there are two big pulses in the target w which are caused by the noncontinuous curvature. As the pulses with large rates of change exceeded the range of the training set, the target w was dicult to track and resulted in a chain reaction in the longitudinal and lateral directions. However, the longitudinal and lateral fluctuations accompanied by yaw pulses do not mean decoupling failure, because besides the fluctuating part, the other targets in the three motion directions are accurately tracked without interferences. This gives us two inspirations; first, the target vehicle motion states should be smooth and remain within the range of the training set; the other is that the curvature of the target position curve designed by the planning level should be as continuous as possible. Figure 9 shows that the proposed method realizes the velocity, position and yaw tracking simultaneously. The fluctuations in a and a are amplified and accumulated, resulting in a stable x y velocity and lateral error of 0.193 km/h and 0.36 m after a 150 m trip. As the lateral error is seriously related to driving safety, the tracking performance of pure-pursuit is the best in this scenario. The MPC failed to reduce the error in the X direction of the Cartesian coordinate system without coordinate transformation, which leads to a larger lateral error in the vehicle coordinate system. Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 16 The statistical values of lateral error are shown in Table 1. Compared with pure-pursuit and MPC, the mean square error of velocity ( ) of the proposed method is the smallest, and its mean square error of lateral tracking ( ) and yaw tracking ( ) rank in the middle. Table 1. The tracking performance statistics. Unit Decoupling Control MPC PP 𝑚𝑎𝑥 𝑒 m 0.1054 0.0113 0.0296 𝑚𝑖𝑛 𝑒 m −0.0090 −0.1795 −0.0399 m ⁄s 0.0033 0.0110 0.1164 m 0.0011 0.0036 0.0002 ° 0.0382 0.0127 0.0386 4.2.2. Scenario 2: Turn Left with Deceleration at Crossing In urban traffic, the intersection is a common scene. The vehicle is required to slow down through a right-angle bend with a radius of 50 m. On a dry road with a road adhesion coefficient of 0.8, the vehicle speed decelerates to 9 m/s from 12 m/s within 5 s. Figure 8 shows that there are two big pulses in the target 𝑤 which are caused by the noncontinuous curvature. As the pulses with large rates of change exceeded the range of the training set, the target 𝑤 was difficult to track and resulted in a chain reaction in the longitudinal and lateral directions. However, the longitudinal and lateral fluctuations accompanied by yaw pulses do not mean decoupling failure, because besides the fluctuating part, the other targets in the three motion directions are accurately tracked without interferences. This gives us two inspirations; first, the target vehicle motion states should be smooth and remain within the range of the training set; the other is that the curvature of the target position curve designed by the planning level should be as continuous as possible. Figure 9 shows that the proposed method realizes the velocity, position and yaw tracking simultaneously. The fluctuations in 𝑎 and 𝑎 are amplified and accumulated, resulting in a stable velocity and lateral error of 0.193 km h and 0.36 m after a 150 m trip. As the lateral error is seriously related to driving safety, the tracking performance of pure-pursuit is the best in this scenario. The MPC failed to reduce the error in the X direction of the Cartesian coordinate system without coordinate Appl. Sci. 2020, 10, 7394 12 of 17 transformation, which leads to a larger lateral error in the vehicle coordinate system. Figure 8. The vehicle dynamic states tracking results in scenario 2. Figure 8. The vehicle dynamic states tracking results in scenario 2. Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 16 Figure 9. Trajectory tracking results in scenario 2. Figure 9. Trajectory tracking results in scenario 2. 4. 4.2.3. 2.3. Scen Scenario ario 3: 3: L Lane ane C Change hange wit with h De Deceleration celeration//A Acceleration cceleration on on a a Wet Wet R Road oad S Surface urface In scen In scenario ario 3 3, , t the he vehicle vehicle de decelerates celerates or or a accelerates ccelerates at at t the he in initial itial spee speed d of of 2 20 0 m/ m/s s w when hen chan changing ging lane on a wet lane on a wet road whose road whoseroad roadadhesion adhesioncoefficient coecient is 0.35. is 0.35. Figure 10a Figure 10a sh shows ows th that at th te three met he three methods hods all st allill still achi achieve eve traject trajectory ory tracking on a lo tracking on w-adhesion road. a low-adhesion Compared wi road. Compar th Figure 7, even though ed with Figure 7, even though the road condi the roadtconditions ions are worse, are wor the lateral and se, the lateral yaw tracking and yaw tracking errors of the proposed method decrease with speed red errors of the proposed method decrease with speed ru eduction. ction. However, t Howeverh , e lat the lateral eral and y and yaw aw t tracking racking p performance erformance o of f t the he M MPC PC g get et w worse. orse. I In n F Figur igur ee10 10 b, b,the the pure- pure-pursuit pursuiand t and MPC MPC fail fail to to tra trackcthe k the ta target rget in in longitudinal, longitudinal lateral , latera and l and yaw yaw mot motion. ionHowever . However, , the thepr proposed oposed method method st still ill fo follows llows t the he const constrained rained vehicle vehicle st states ates at at t the he lowe lower r- -adhesion adhesion roa road d condition, condition, gu guaranteeing aranteeing t the he driv driving ing stability stability .. Figure 10. Trajectory tracking results on low-adhesion road: (a) steering with deceleration, (b) steering with acceleration. As we can see from Figure 11a, the proposed method always keeps the vehicle in the safe area of the sideslip phase-plane like the pure-pursuit and MPC; however, in Figure 11b, since the tracking performance becomes worse, the phase curve of the pure-pursuit and MPC are over the stability 𝑀𝑆𝐸 𝑀𝑆𝐸 𝑀𝑆𝐸 𝑀𝑆𝐸 𝑀𝑆𝐸 𝑀𝑆𝐸 Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 16 Figure 9. Trajectory tracking results in scenario 2. 4.2.3. Scenario 3: Lane Change with Deceleration/Acceleration on a Wet Road Surface In scenario 3, the vehicle decelerates or accelerates at the initial speed of 20 m/s when changing lane on a wet road whose road adhesion coefficient is 0.35. Figure 10a shows that the three methods all still achieve trajectory tracking on a low-adhesion road. Compared with Figure 7, even though the road conditions are worse, the lateral and yaw tracking errors of the proposed method decrease with speed reduction. However, the lateral and yaw tracking performance of the MPC get worse. In Figure 10b, the pure-pursuit and MPC fail to track the target in longitudinal, lateral and yaw motion. However, the proposed method still follows the constrained Appl. Sci. 2020, 10, 7394 13 of 17 vehicle states at the lower-adhesion road condition, guaranteeing the driving stability. Figure 10. Figure 10. Trajectory Trajectory track tracking ing resu results lts on low-a on low-adhesion dhesion roa road: d: (a (a ) steering wi ) steering with th dec deceleration, eleration, (b (b ) st ) steering eering with acceleration. with acceleration. As we can see from Figure 11a, the proposed method always keeps the vehicle in the safe area of As we can see from Figure 11a, the proposed method always keeps the vehicle in the safe area the sideslip phase-plane like the pure-pursuit and MPC; however, in Figure 11b, since the tracking of the sideslip phase-plane like the pure-pursuit and MPC; however, in Figure 11b, since the tracking Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 16 performance becomes worse, the phase curve of the pure-pursuit and MPC are over the stability performance becomes worse, the phase curve of the pure-pursuit and MPC are over the stability boundaries, which means that the proposed method significantly improves the handling stability and boundaries, which means that the proposed method significantly improves the handling stability and has fewer tracking errors compared with the pure-pursuit and MPC algorithms. has fewer tracking errors compared with the pure-pursuit and MPC algorithms. Figure Figure 11. 11. T The si he sidedeslip pha slip phase-psle-plane d ane diagrai m agram: ( : (a) steea ri) ngsteeri withndge with dece celeration; (lbe)ration; ( steeringb w) st itheering with acceleration. acceleration. Figure 12 shows the error statistics results of the longitudinal, lateral and yaw directions of the three tracking controllers under four working conditions, as well as the trajectory tracking error index Figure 12 shows the error statistics results of the longitudinal, lateral and yaw directions of the P of the planar motion weighted by the three directions. three tracking controllers under four working conditions, as well as the trajectory tracking error index 𝑃 of the planar motion weighted by the three directions. P = w MSE + w MSE + w MSE , (41) 1 V 2 Y 3 yaw 𝑃= 𝑤 +𝑤 +𝑤 , (41) Figure 12. The tracking performance index: (a) the longitudinal direction; (b) the lateral direction; (c) the yaw direction; (d) the planar motion. Considering that three directions are equally important in planar motion tracking, the three weights should be the same. However, due to the unit, the mean square error of yaw tracking is very large, so it is reduced by a certain proportion and the weighted vector is 𝒘= 1,1, /180 . As shown in Figure 12a, except for scenario 2, the of the proposed method is the smallest. The accurate velocity tracking illustrates that the dynamic decoupling of the proposed tracking method can effectively reduce the longitudinal interference from other motion directions. Moreover, compared with the other two methods, the proposed method still performs well in the lateral and 𝑀𝑆𝐸 𝑝𝑖 𝑀𝑆𝐸 𝑀𝑆𝐸 𝑀𝑆𝐸 Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 16 boundaries, which means that the proposed method significantly improves the handling stability and has fewer tracking errors compared with the pure-pursuit and MPC algorithms. Figure 11. The sideslip phase-plane diagram: (a) steering with deceleration; (b) steering with acceleration. Figure 12 shows the error statistics results of the longitudinal, lateral and yaw directions of the three tracking controllers under four working conditions, as well as the trajectory tracking error index 𝑃 of the planar motion weighted by the three directions. Appl. Sci. 2020, 10, 7394 14 of 17 𝑃= 𝑤 +𝑤 +𝑤 , (41) Figure 12. The tracking performance index: (a) the longitudinal direction; (b) the lateral direction; Figure 12. The tracking performance index: (a) the longitudinal direction; (b) the lateral direction; (c) (c) the yaw direction; (d) the planar motion. the yaw direction; (d) the planar motion. Considering that three directions are equally important in planar motion tracking, the three Considering that three directions are equally important in planar motion tracking, the three weights should be the same. However, due to the unit, the mean square error of yaw tracking is very weights should be the same. However, due to the unit, the mean square error of yaw tracking is very large, so it is reduced by a certain proportion and the weighted vector is w = (1, 1, pi/180) . large, so it is reduced by a certain proportion and the weighted vector is 𝒘= 1,1, /180 . As shown in Figure 12a, except for scenario 2, the MSE of the proposed method is the smallest. As shown in Figure 12a, except for scenario 2, the of the proposed method is the smallest. The accurate velocity tracking illustrates that the dynamic decoupling of the proposed tracking method The accurate velocity tracking illustrates that the dynamic decoupling of the proposed tracking can e ectively reduce the longitudinal interference from other motion directions. Moreover, compared method can effectively reduce the longitudinal interference from other motion directions. Moreover, with the other two methods, the proposed method still performs well in the lateral and yaw tracking. compared with the other two methods, the proposed method still performs well in the lateral and In summary, as shown in Figure 12d, the trajectory tracking error index of the proposed method proposed is always the lowest, indicating that the proposed tracking method is more suitable for performing trajectory tracking tasks under the coupled conditions than the other two methods. 5. Conclusions To achieve the high trajectory tracking precision in the dynamic coupling scenarios, a simultaneous trajectory tracking and stability control method for 4WID automated electric vehicles is present in this paper based on the inverse system theorem. To reveal the coupling mechanism and to prove the reversibility of 4WID vehicles, the 3DOF vehicle dynamic model is constructed. The inverse system learned by a BPNN model shows e ectiveness to realize dynamic decoupling. The pseudo linear system composed of the inverse system and the controlled object follows the desired vehicle dynamic states to indirectly achieve trajectory tracking. Three typical and common coupled driving conditions are designed to verify the trajectory tracking accuracy under the simultaneous control of vehicles’ longitudinal, lateral and yaw motion. Compared with the pure-pursuit algorithm and the MPC algorithm, the proposed method reduces interactions among vehicle motion directions and reveals better tracking performance. Moreover, since the target states of the vehicle have been constrained within a reasonable range, the decoupling method not only maintains the accurate trajectory tracking but also guarantees the stable vehicle driving under low-adhesion road conditions. Even though the proposed method theoretically shows better control performance, further verifications could be implemented on real vehicles to realize the engineering applications. Author Contributions: Conceptualization, Y.Y. and Y.L. (Yinong Li); methodology, Y.Y. and Y.L. (Yixiao Liang); software and validation, Y.Y. and Y.L. (Yixiao Liang); formal analysis, Y.Y. and W.Y.; writing—review and editing, 𝑀𝑆𝐸 𝑝𝑖 𝑀𝑆𝐸 𝑀𝑆𝐸 𝑀𝑆𝐸 Appl. Sci. 2020, 10, 7394 15 of 17 Y.Y., Y.L. (Yinong Li) and W.Y.; supervision and funding acquisition, Y.L. (Yinong Li) and L.Z. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by Key Research Program of the Ministry of Science and Technology ([Grant No. 2017YFB0102603-3, 2016YFB0100900), Chongqing Science and Technology Program Project Basic Science and Frontier Technology (Grant No. cstc2018jcyjAX0630), Chongqing Technology Innovation and Application Development Major Theme Special Project (Grant No. cstc2019jscx-zdztzxX0032), Graduate Scientific Research and Innovation Foundation of Chongqing (Grant No. CYB19063). Acknowledgments: In this section you can acknowledge any support given which is not covered by the author contribution or funding sections. This may include administrative and technical support, or donations in kind (e.g., materials used for experiments). Conflicts of Interest: We declare that there is no conflict of interests in connection with the paper submitted. Appendix A Table A1. Symbols and definitions of the dynamics model cited. Definition Symbol Unit Vehicle mass m kg Vehicle inertia on yaw direction I kgm . .. Longitudinal speed/acceleration (in xoy) x/x m/s . .. Lateral speed/acceleration (in xoy) y/y m/s Longitudinal force vector on tire (in xoy) F N Lateral force vector on tire (in xoy) F N Distance from c.g. to the front/rear axle l /l m Length of wheelbase d m Longitudinal force on each tire (in tire coordinate) F N xi Lateral force on each tire (in tire coordinate) F N yi Front wheel steering angle on the left/right  / rad f l f r Equivalent steering angle  rad Slip angle of the front/rear wheel / rad Speed angle of the front/rear wheel rad f r Cornering sti ness of the front/rear wheel C /C N/rad Yaw angle of vehicle body (in XOY) ' rad Total input longitudinal force on vehicle c.g. F N Total input yaw moment on vehicle c.g. M Nm Longitudinal force distribute rate on front/rear wheel k /k - Aerodynamic drag coecient k N/(m/s) . . e /e /e Control error on longitudinal/lateral/yaw directions ' m/s/m/s/rad x y . . e /e Tracking error on longitudinal/lateral m/m X Y Table A2. Symbols and definitions of the dynamics model cited. 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Published: Oct 22, 2020

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