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DGPR‐MPC: Learning‐based model predictive controller for autonomous vehicle path following

DGPR‐MPC: Learning‐based model predictive controller for autonomous vehicle path following INTRODUCTIONIn recent decades, autonomous driving technologies have had the potential to significantly improve driving safety, reduce traffic congestion, and increase mobility [1]. Perception, motion planning modules, policy‐making, and accurate and stable path following are crucial for a fully autonomous driving system, as they directly affect passenger safety and comfort.As a crucial enabling technology for autonomous driving, path following primarily employs the control algorithm to calculate the front wheel angle in order to ensure stability and safety while following the reference path. The vehicle dynamics model significantly impacts the success of the path following the mission. In the middle of the 20th century, the linear two‐degrees‐of‐freedom (2DOF) steering model was first investigated. Sahoo et al. [2] modelled the vehicle using the 2DOF model, analyzed the transient response, and designed the steering controller using the 2DOF model. Segel et al. [3] proposed a three‐degrees‐of‐freedom (3DOF) steering model with three degrees of freedom. The vehicle was considered a linear dynamic system when the mathematical model was developed. Yaw, lateral, and roll motion were determined to describe the vehicle's transient response. With the development of analytical mechanics, more complex vehicle models have been created, such as 7DOF [4], 17DOF [5], and even more degrees of freedom models, which are widely used in path tracking due to their simple forms. However, most current lateral control systems use a simplified modelling method for the mechanism, which limits the enhancement of control performance under wide‐range driving conditions.In recent years, data‐driven technologies have rapidly evolved and been widely implemented in autonomous vehicles [6–8]. The learning‐based approach does not require specific information about the physical structure and parameters of the system; rather, it requires only input–output mapping data. Ghazizadeh et al. [9] studied the neural network vehicle model and proposed the ‘Neuro‐Vehicle' model in 1996. James et al. [10] compared the performance of the standard linear state space model and the neural network model under large‐scale real‐world driving conditions using longitudinal driving data. The results demonstrated that the neural network model significantly enhanced the accuracy of longitudinal state prediction. Based on the gated recurrent unit (GRU), Sieberg et al. [11] established the vehicle side‐slip dynamic model. They fused the prior knowledge of the Kalman filter in the network to improve the modelling accuracy. Mauro Da Lio et al. [12] conducted layered modelling of longitudinal vehicle dynamics using a convolutional neural network (CNN) and recurrent neural network (RNN) and compared the impact of different topological structures on the modelling accuracy and network robustness. The aforementioned methods demonstrate that neural networks can model dynamic systems. However, these modelling methods are typically employed for vehicle state estimation and cannot be combined with the control method effectively.Significant research has been conducted on the path following control of autonomous vehicles in the existing literature. Pure pursuit (PP) [13] based on geometric models is widely used in autonomous systems in the real world. Fuzzifying the control variables enables the fuzzy control method [14] to simulate human driving behaviour. However, the optimal performance of the control system cannot be achieved because the parameters are obtained primarily through trial‐and‐error. Recently, methods based on optimization theory and dynamic models, such as linear quadratic regulator (LQR) and model predictive control (MPC) [15–17], have been widely used for the path‐following control of autonomous vehicles. At each control step, these methods require the repeated solution of an optimization problem. Rafaila et al. [18] used a 2DOF model and a magic tire formula to develop a non‐linear model predictive control (NMPC) method to accommodate the highly dynamic non‐linear characteristics of the vehicle turning. Onieva et al. [19] utilized the self‐learning capability of the legacy algorithm to learn the behaviour of human‐driven vehicles and improved the tolerance to uncertainty and imprecision of vehicle parameters under high‐speed road conditions, thereby achieving high‐precision control effects. Chen et al. [20] proposed a novel data‐driven non‐linear model control method based on the high‐order neural network (RHONN) modelling method. Teng et al. [21] developed an echo state model (ESM) with a rational data organization and an efficient neural network structure for high‐precision four‐step‐ahead prediction in both dry and slippery road conditions. Kim et al. [22] combine deep learning with a fully differentiable physics model to endow the neural network with available prior knowledge. The reinforcement learning control algorithm [23] enables agents to explore unknown environments, but successful training is typically challenging. Overall, the existing control algorithms meet the requirements for all most scenarios, for example, urban roads, highways, and parking. However, conventional control algorithms are susceptible to instability when the vehicle is operating in restricted conditions, such as high‐speed obstacle avoidance. Since the vehicle is a typical non‐linear system, it is worthwhile investigating new control methods employing learning‐based tools in order to further enhance the vehicle's dynamic performance.Motivated by the above discussions, a novel non‐linear predictive control method enabled by deep Gaussian process regression (DGPR) modelling is proposed in this paper. The main contributions of this study are shown as follows:The vehicle is a complex dynamic system whose path‐following accuracy and stability depend on the precision of its model. This paper proposes a deep learning approach to model vehicle dynamics, which can simulate potential system states that are hard to analyse or estimate, enhance the modelling accuracy, and predict the state of autonomous vehicles in uncertain environments.A novel vehicle dynamic modelling method that integrates traditional machine learning and deep learning to address the issue of errors caused by model mismatch in model‐based controllers. The method incorporates a branch in the model structure that estimates the uncertainty of model input.A DGPR‐based non‐linear model predictive controller is designed to improve the performance of the autonomous vehicle's path following and lateral stability. Then, simulation‐based validations of the proposed algorithms are performed on an autonomous vehicle in a variety of dynamic driving scenarios.The remainder of this paper is organized as follows. Section 2 describes the dynamics of two‐wheeled vehicles and introduces the DGPR vehicle dynamic model design. Section 3 introduces the model predictive controller design. Section 4 describes the procedures for data collection, model training, and simulation validation. Section 5 provides the results of the simulation. Finally, conclusions are provided in Section 6.VEHICLE DYNAMIC MODELLINGVehicle lateral dynamicsThe vehicle dynamic system is a non‐linear and time‐varying system. We neglect vehicle roll, pitch, and motions. Then, the 2DOF signal‐track model, as shown in Figure 1 and Equation (1).1U̇y=Fyr+Fyfcosδf+Fxfsinδfm−rUxṙ=aFyfcosδf+aFxfsinδf−bFyrIz$$\begin{equation} \def\eqcellsep{&}\begin{array}{l} {{\dot U}_y} = \dfrac{{{F_{yr}} + {F_{yf}}\cos {\delta _f} + {F_{xf}}\sin {\delta _f}}}{m} - r{U_x}\\[12pt] \dot r = \dfrac{{a{F_{yf}}\cos {\delta _f} + a{F_{xf}}\sin {\delta _f} - b{F_{yr}}}}{{{I_z}}} \end{array} \end{equation}$$1FIGUREPhysical model.where U denotes the velocity of the vehicle's centre of mass; Ux,Uy${U_x},{U_y}$ are the velocities at the centre of mass of the vehicle along the x and y directions of the vehicle body coordinate system, respectively. αf,αr${\alpha _f},{\alpha _r}$ represent the side‐slip angles of the front and rear wheels, respectively; β denotes the sideslip angle of the centre of mass; r represents the vehicle yaw rate; L=a+b$L = a + b$ is the wheelbase; m is the vehicle weight; Iz${I_z}$ is the moment of inertia of the vehicle around t he z‐axis of the centre of mass; Fyf,Fyr${F_{yf}},{F_{yr}}$ are the resultant lateral forces on the front and rear tires, respectively; Fxf${F_{xf}}$ is the resultant longitudinal force on the front axle tires and δf${\delta _f}$ is the front wheel steering angle.Considering the relationship between the vehicle coordinate system and the inertial coordinate system2ė=Uxsin(Δφ)+Uycos(Δφ)Δφ̇=r−ṡρṡ=Uxcos(Δφ)+Uysin(Δφ)1−ρe$$\begin{equation} \def\eqcellsep{&}\begin{array}{l} \dot e = {U_x}\sin (\Delta \varphi ) + {U_y}\cos (\Delta \varphi )\\[6pt] \Delta \dot \varphi = r - \dot s\rho \\[8pt] \dot s = \dfrac{{{U_x}\cos (\Delta \varphi ) + {U_y}\sin (\Delta \varphi )}}{{1 - \rho e}} \end{array} \end{equation}$$where e$e\;$represents the lateral error, Δφ$\Delta \varphi \;$is the heading error, andρ$\;\rho \;$represents the curvature of the desired path.Brush tire modelThe brush tire model [24] has been utilized extensively in parameter estimation and lateral control studies. The tire lateral force is calculated as follows:3Fy=−Cαtanα+Cα23μFz|tanα|tanα−Cα327μ2Fz2tan3α,|α|<αsat−μFzsgnα,otherwise$$\begin{equation} {F_y} = \left\{ \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} \def\eqcellsep{&}\begin{array}{l} - {C_\alpha }\tan \alpha + \dfrac{{C_\alpha ^2}}{{3\mu {F_z}}}|\tan \alpha |\tan \alpha \\[11pt] - \dfrac{{C_\alpha ^3}}{{27{\mu ^2}F_z^2}}{\tan ^3}\alpha ,|\alpha | &lt; {\alpha _{{\rm{sat }}}}\\ \end{array} \\[5pt] { - \mu {F_z}{\mathop{\rm sgn}} \alpha ,{\rm{otherwise}}} \end{array} \right.\end{equation}$$where Cα${C_\alpha }$ and μ$\mu \;$denote the tire corner stiffness and road friction coefficient, respectively,Fz$\;{F_z}\;$represents the tire vertical force; α$\alpha \;$denotes the tire side‐slip angle, and αsat${\alpha _{sat}}$ denotes the tire saturation side‐slip angle. The side‐slip angles of the front and rear tires are respectively calculated by4αf=arctanUy+arUx−δfαr=arctanUy−brUx$$\begin{equation} \def\eqcellsep{&}\begin{array}{l} {\alpha _f} = \arctan \left(\dfrac{{{U_y} + ar}}{{{U_x}}}\right) - {\delta _f}\\[14pt] {\alpha _r} = \arctan \left(\dfrac{{{U_y} - br}}{{{U_x}}}\right) \end{array} \end{equation}$$The amount of normal force experienced on each tire is respectively calculated by5Fzf=bLmg−hCGLmaxFzr=aLmg+hCGLmax$$\begin{equation} \def\eqcellsep{&}\begin{array}{l} {F_{zf}} = \dfrac{b}{L}mg - \dfrac{{{h_{CG}}}}{L}m{a_x}\\[20pt] {F_{zr}} = \dfrac{a}{L}mg + \dfrac{{{h_{CG}}}}{L}m{a_x} \end{array} \end{equation}$$where hCG${h_{CG}}\;$represents the height to the vehicle centre of gravity, g denotes the gravity acceleration, L is the vehicle wheelbase, and ax${a_x}$represents the vehicle longitudinal acceleration.The proposed bicycle model is employed to design the nominal model predictive controller and compare its performance to that of the DGPR‐MPC.Deep Gaussian process regressionDue to the need for real‐time performance by autonomous vehicles, the nominal vehicle dynamic model is simplified to reduce system complexity. However, we use a leaning‐based method to build the vehicle dynamic model to improve the performance and enable the automatic model adaptation. For instance, it must be able to model complex and difficult‐to‐model forces that change as a function of the coefficient of road‐tire friction, while remaining computationally efficient enough for real‐time control.The structure of DGPR is shown in Figure 2. The model's hidden layer consists of two layers with 100 neurons per layer. The activation function is the Softplus activation function. Motivated by states and controls considered in the physical‐based model, the DGPR's input is the velocity, steering angle, and front longitudinal force, are formed into a signal state as shown in Equation (6). At a current time, the current model state is concatenated with delayed model states to form the input to the model as shown in Equation (7). The output layer of the model has four neurons. The output of the first and second neurons is the next time step's mean lateral velocity and yaw rate. Furthermore, the output of the third and fourth neurons is the variance of the next time step's lateral velocity and yaw rate.6xt=[r,Uy,Ux,δf,Fxf]$$\begin{equation}{x_{t = }}[r,{U_y},{U_x},{\delta _f},{F_{xf}}]\end{equation}$$7ht=[xt,xt−1,…,xt−N]$$\begin{equation}{h_t} = [{x_t},{x_{t - 1}}, \ldots ,{x_{t - N}}]\end{equation}$$The forward calculation of DGPR is shown in Equation (8)2FIGUREThe structure of DGPR. DGPR, deep Gaussian process regression.8z1=w1Tht+b1a1=log(1+ez1)z2=w2Ta1+b2a2=log(1+ez2)z3=w3Ta2+b3θ=(w1,b1,w2,b2,w3,b3)μUy,t+1rt+1=z3[1:2]σUy,t+1σr,t+1=actELU+1(z3[3:4])$$\begin{equation} \left\{ \def\eqcellsep{&}\begin{array}{l} {z_1} = w_1^T{h_t} + {b_1}\\[5pt] {a_1} = \log (1 + {e^{{z_1}}})\\[5pt] {z_2} = w_2^T{a_1} + {b_2}\\[5pt] {a_2} = \log (1 + {e^{{z_2}}})\\[5pt] {z_3} = w_3^T{a_2} + {b_3}\\[5pt] \theta = ({w_1},{b_1},{w_2},{b_2},{w_3},{b_3})\\[5pt] \left[ \def\eqcellsep{&}\begin{array}{l} {\mu _{{U_{y,t + 1}}}}\\[5pt] {r_{t + 1}} \end{array} \right] = {z_3}[1:2]\\[15pt] \left[ \def\eqcellsep{&}\begin{array}{l} {\sigma _{{U_{y,t + 1}}}}\\[5pt] {\sigma _{r,t + 1}} \end{array} \right] = ac{t_{ELU + 1}}{\rm{ (}}{z_3}[3:4]{\rm{)}} \end{array} \right.\end{equation}$$where N − 1 is the number of delay states.a1 and a2 is are the expression of the Softplus. θ is the parameter learned by the DGPR. To ensure the positive of the variance, we use a modified Exponential Linear Unit (ELU) [25] activation, which is shown in Equation (9) and Figure 3.9actELU+1=x+1,x≥0ex,otherwise$$\begin{equation} ac{t_{ELU + 1}} = \left\{ \def\eqcellsep{&}\begin{array}{ll} x + 1, &\quad x \ge 0\\[5pt] {e^x},&\quad {\rm{ otherwise}} \end{array} \right.\end{equation}$$3FIGUREThe modified exponential linear unit (ELU) activation.DESIGN OF THE DGPR‐MPCThe model predictive controller is a technique for controlling linear and non‐linear systems that relies on optimization. This process involves minimizing the cost function in order to select the most suitable values for the plant model, which then provides optimal performance for following the reference output. To achieve this, the MPC controller predicts the vehicle's path (for tasks such as vehicle path following) for the next p time steps (prediction horizon) by utilizing the plant model, allowing it to anticipate and adjust for any potential errors. By utilizing different scenarios, the MPC can minimize the error between the predicted and reference trajectories. In order to maximize performance, the accuracy of the vehicle model must be high, accounting for all vehicle dynamics, including non‐linearity and uncertainty, while keeping computational power requirements low. To facilitate comparison with the physical model‐based path following controller using the same control logic, the DGPR is employed to design the control algorithm, as shown in Figure 4.4FIGUREThe general model predictive controller model.Here, the dynamic model of vehicle is rewritten as10ξt+1=DGPR(ξt,ξt−1,…,ξt−n,ut,ut−1,…,ut−n)$$\begin{equation}{\xi _{t + 1}} = DGPR({\xi _t},{\xi _{t - 1}}, \ldots ,{\xi _{t - n}},{u_t},{u_{t - 1}}, \ldots ,{u_{t - n}})\end{equation}$$where T represents the sampling time for the discrete state‐space model.The output variable is defined as11ηref(t)=[Yref(t),φref(t)]T$$\begin{equation}{\eta _{ref}}(t) = {[{Y_{ref}}(t),{\varphi _{ref}}(t)]^T}\end{equation}$$and the reference trajectory is written as12η(t)=[Y(t),φ(t)]T$$\begin{equation}\eta (t) = {[Y(t),\varphi (t)]^T}\end{equation}$$The vehicle position is shown in Figure 1. It determines the lateral error and the heading error of the autonomous vehicle relative to the reference path. Moreover, the physical model and the DGPR use kinematics equations to predict the lateral position of the vehicle. The kinematics of the vehicle does not need to be studied, and it can be calculated by Equation (4).The optimization objective function is defined as13argminJ=∑t=1Np(QY(et)2+Qψ(Δψt)2)+∑t=1Nc(Qδ(Δδt)2)+∑t=1Np((σUy)2+(σr)2)$$\begin{equation} \def\eqcellsep{&}\begin{array}{l} \arg \min J = \displaystyle\sum_{t = 1}^{{N_p}} {({Q_Y}{{({e^t})}^2} + {Q_\psi }{{(\Delta {\psi ^t})}^2})} \\[5pt] + \displaystyle\sum_{t = 1}^{{N_c}} {({Q_\delta }{{(\Delta {\delta ^t})}^2})} + \displaystyle\sum_{t = 1}^{{N_p}} {({{({\sigma _{Uy}})}^2} + {{({\sigma _r})}^2})} \end{array} \end{equation}$$14subjectto,xt+k+1=DGPR(xt+k|t,ut+k|t,θ)uk=uk−1+Δukx(0|t)=x(t)u(k)∈Uk∈[t,t+Nc]x̂∈Xk∈[t,t+Np]$$\begin{equation} \def\eqcellsep{&}\begin{array}{l} {\rm{subject to,}}\\[5pt] {x_{t + k + 1}} = DGPR({x_{t + k|t}},{u_{t + k|t}},\theta )\\[5pt] {u_k} = {u_{k - 1}} + \Delta {u_k}\\[5pt] x(0|t) = x(t)\\[5pt] u(k) \in Uk \in [t,t + {N_c}]\\[5pt] \hat x \in Xk \in [t,t + {N_p}] \end{array} \end{equation}$$where Np${N_p}$ is prediction horizon and Nc${N_c}$ is prediction horizon. J is the cost function, the state and input constraints are expressed by X and U,respectively. QY${Q_Y}$ and Qφ${Q_\varphi }$ are the weight of lateral and heading error. Qδ${Q_\delta }$ is the weight of front wheel angle. x̂$\hat x$ is the future state predicted based on the current measurement and the historical state. The first item indicates the ability of the system to follow the reference trajectory, requiring the vehicle to track the lateral error and heading deviation of the reference trajectory as small as possible. The second item indicates the requirement for controlling incremental constraints, ensuring that the control variables change quickly and smoothly. The third item indicates a constraint on the uncertainty of the model's output.In each control step, the optimal control sequenceU∗=[u1∗,u2∗,u3∗,…,uNp∗]$\;{U^*} = \;[ {u_1^*,u_2^*,u_3^*, \ldots ,u_{{N_p}}^*} ]$ is solved by the optimization algorithm. The first value for dynamic system control is selected and then the entire optimization process is repeated in the next time step.DATE COLLECTION AND MODEL TRAININGDate collectionCarSim was used to collect data with a higher degree of freedom in order to validate the modelling capability of the DGPR for a dynamic system with a high degree of freedom. The CarSim software has 27 degrees of freedom, and the tire non‐linear characteristics in the model were obtained by fitting real data. Therefore, it can be concluded that the vehicle state data output by CarSim is reliable and reflective of the actual characteristics. The established vehicle model was controlled by a Logitech G29 steering wheel and pedal, and the data was collected with MATLAB/Simulink 2019a. During the data collection, the sampling time was set to = 20 ms, as shown in Figure 5. For each episode, the initial state of the vehicle was set to 0, the engine throttle was initialized to 0, the steering wheel angle was initialized to 0, and the road friction coefficient is set 0.5 to 1.0. The range of the throttle opening and brake pedal pressure was limited to [0 0.2] and [0 9] MPa.5FIGURESimulation data collection system based on Matlab/CarSim.Model training and validationThe dataset was divided into 70% training dataset, 15% validation dataset, and 15% test dataset. A DGPR was developed using the deep learning framework PyTorch [26]. The learning rate was 0.0005 and the optimization algorithm is Adam [27]. The loss function is defined as follows:15L=12logmaxσUy,t+1σr,t+1,epseps+μUy,t+1μr,t+1−Uy,t+1rt+12maxσUy,t+1σr,t+1,epseps$$\begin{eqnarray} L &=& \frac{1}{2} \left(\log \left(\max \left(\left[ \def\eqcellsep{&}\begin{array}{l} {\sigma _{{U_{y,t + 1}}}}\\[5pt] {\sigma _{r,t + 1}} \end{array} \right],\quad \left[ \def\eqcellsep{&}\begin{array}{l} eps\\[5pt] eps \end{array} \right]\right)\right.\right.\nonumber\\ &&\left.\left. +\, \frac{{{{\left(\left[ \def\eqcellsep{&}\begin{array}{l} {\mu _{{U_{y,t + 1}}}}\\[5pt] {\mu _{r,t + 1}} \end{array} \right] - \left[ \def\eqcellsep{&}\begin{array}{l} {U_{_{y,t + 1}}}\\[5pt] {r_{t + 1}} \end{array} \right]\right)}^2}}} {{\max \left( \left[ \def\eqcellsep{&}\begin{array}{l} {\sigma _{{U_{y,t + 1}}}}\\[5pt] {\sigma _{r,t + 1}} \end{array} \right], \left[ \def\eqcellsep{&}\begin{array}{l} eps\\[5pt] eps \end{array} \right]\right)}}\right)\right) \end{eqnarray}$$where eps$eps$ is used for stability and it is set as 1×10−6$1 \times {10^{ - 6}}$. The training loss and validation loss are shown in Figure 66FIGURETraining and validation loss using simulation data.Figure 7 compares the prediction results of the DGPR and the single‐track model on the test data. In the next step, the Euler integral method is used to calculate the yaw rate and the lateral velocity using the physical model prediction value and the 20‐ms time step. At the next time step, the prediction result is compared to the vehicle data measured at that time. The test error based on the physical model is large because the physical model is only described around a single operating point and cannot capture the influence of different friction conditions on physical parameters. The DGPR can identify dynamic effects not accounted for by the physical model. For example, the elastic hysteresis of the tire when the vehicle is running at a low speed, and the weight transfer effect when the vehicle is accelerating, decelerating, or turning rapidly are learned by DGPR. In addition, it should be noted that the DGPR can make accurate predictions about the road on which the vehicle is travelling without requiring a precise estimate of road friction. In contrast, the physical model can only calculate the current dynamic changes of the vehicle using the current control input and state data, which has not only a low level of accuracy, but also inaccurate physical parameters and inadequate modelling.7FIGUREState prediction values using actual data based on DGPR and physical model. DGPR, deep Gaussian process regression.MPC CONTROL PERFORMANCE VALIDATIONThe model of the simulation system was developed on the MATLAB (Simulink)‐CarSim platform. Two scenarios based on the CarSim D‐Class vehicle model were simulated: the eight‐shaped path manoeuvre and the double lane‐change manoeuvre. The objective was to control a vehicle so that it could follow the reference path. The parameters of the vehicle and controller are shown in Tables 1 and 2. To demonstrate the advantage of the DGPR‐MPC, it was compared to the nominal MPC method, which is an optimal control method with a small steady‐state error and a quick response.1TABLEVehicle parameters.SymbolDescriptionValue (Unit)MVehicle total mass1770 (kg)lf${l_f}$Distance of the front wheel axle from the CG1.20 (m)lr${l_r}$Distance of the rear wheel axle from the CG1.43 (m)Iz${I_z}$Vehicle moment of inertia about yaw axis2760 (kg · m2)cf${c_f}$Cornering stiffness of front tires150,000 (N/rad$N/{\rm{rad}}$)cr${c_r}$Cornering stiffness of rear tires170,000 (N/rad$N/{\rm{rad}}$)2TABLEControl parameters.ParameterValuePrediction horizonNp$\;{N_p}$= 11Control horizonNc=11$\;\;{N_c} = \;11$Sample time/sT = 0.02Steer angle limit/rad−0.174to0.174$\; - 0.174\;{\rm{to\;}}0.174$Steer rate limit/rad−0.014to0.014$\; - 0.014\;{\rm{to\;}}0.014$Lateral error weightQY$\;{Q_Y}$1.0Yaw error weightQφ$\;{Q_\varphi }$0.5Control effort weight Qδ${Q_\delta }$0.3Eight‐shaped path simulationAs shown in Figure 8, the time‐based eight‐shaped path was used as a reference path. The eight‐shaped path consisted of two straight lines and two smooth curves. The vehicle ran at high speed (vx=80km/h${v_x} = 80\,{\rm{km/h}}$μ=0.85${\mu } = 0.85$).8FIGUREThe path following trajectory results of the eight‐shape simulation.In our study, we employed the interior point optimizer (IPOPT) to determine the optimal control variables. IPOPT is a powerful solver for non‐linear optimization problems and is integrated into the MATLAB Optimization Toolbox. This allowed us to efficiently solve for the control variables in our problem. Tables 1 and  2Figure 8 depicts the actual vehicle trajectories and their zoom versions. Both the DGPR‐MPC and the nominal MPC were capable of path following. The DGPR‐MPC path following was smoother and had more minor overshoot and steady‐state errors than the nominal MPC. The numerical simulation result of the absolute value of lateral offset is summarized in Table 3, which intuitionally verifies the same conclusion.3TABLEComparison results of lateral offset eCG${e_{CG}}$.Nominal MPCDGPR‐MPC80 km/heCG${e_{CG}}$Min (m)−0.2824−0.1232Max (m)0.30600.1835Mean (m)0.12300.5090Standard deviation (m)0.10700.0496DGPR, deep Gaussian process regression; MPC, model predictive control.The lateral offset is shown in Figure 9. Both approaches could stabilize the path following errors. In contrast, the DGPR‐MPC could converge to the steady‐state error in a transient period. In addition, the DGPR‐MPC could almost eliminate steady‐state error.9FIGUREThe lateral offset results of the eight‐shaped simulation.The results of the control input, front wheel angle, lateral acceleration, and yaw rate are shown in Figure 10, where the front wheel steering angles of both methods were maintained within the acceptable magnitude range. DGPR‐MPC reduces the peak value of dynamic state quantity and improves stability.10FIGUREThe steering angle, lateral accelerations, and yaw rates of the eight‐shape simulation.Figure 11 shows the calculation time of the MPC controller during each control cycle in the simulation of the eight‐shape. It can be found that the calculation time of each step of DGPR‐MPC is basically between 20 and 35 ms, which is already within an acceptable range. In the future, we will use C++ for programming. And efficient solver casadi will be used to solve the non‐linear optimization problems and reduce solving time [28].11FIGURECalculation time of the MPC controller during each control cycle in the simulation of the eight‐shape. MPC, model predictive control.Double lane‐change simulationIn this simulation, the vehicle ran at a high speed of 100 km/h for a double lane change on the road with a low road friction of μ=0.5${\rm{\;}}\mu = 0.5$. The actual trajectories of the vehicle are shown in Figure 12. The path following based on the DGPR‐MPC can navigate the autonomous vehicle closer to the reference path compared to the nominal MPC.12FIGUREThe path following trajectory results of the double line‐change simulation.The simulation results for the lateral offset of the CG are shown in Figure 13. It can be seen that the DGPR‐MPC controller changed over a greater range than the nominal MPC as the curvature varied. The numerical simulation result of the absolute value of lateral offset is summarized in Table 4, intuitionally indicating that the DGPR‐MPC‐based method can reduce the magnitude and frequency of the overshoot and obtain a smaller steady‐state error in path following compared to the conventional MPC method.13FIGUREThe lateral offset results of the double line‐change simulation.4TABLEComparison results of lateral offset eCG${e_{CG}}$.Nominal MPCDGPR‐MPC100 km/heCG${e_{CG}}$Min (m)−0.1108−0.0249Max (m)0.11100.0320Mean (m)0.04680.0117Standard deviation (m)0.04290.0106DGPR, deep Gaussian process regression; MPC, model predictive control.Figure 14 depicts the steering angle of the front wheel, lateral acceleration, and yaw rate. The steering angle of the DGPR‐MPC‐based method was relatively smaller than that of the nominal MPC method. The DGPR‐MPC of the path following had a smaller region of lateral acceleration compared to the nominal MPC. In addition, the phenomenon of overshoot for lateral accelerations was significantly mitigated. The path following results demonstrated that the DGPR‐MPC‐based method completed the path more satisfactorily than the nominal MPC method. This result indicates that the possibility of a vehicle surpassing the safe driving area has been dramatically reduced. Figure 15 shows the calculation time of the MPC controller during each control cycle in the simulation of the double‐lane change at constant speed.14FIGUREThe steering angles, lateral accelerations, and yaw rates of the double line‐change simulation.15FIGURECalculation time of the MPC controller during each control cycle in the simulation of the double lane‐change at constant speed. MPC, model predictive control.To verify the robustness of the model, the vehicle ran at a variable speed for a double lane change on the road with a low road friction of μ=0.85$\mu \; = \;0.85$.The reference path model is described in terms of lateral position Yref${Y_{ref}}$ and yaw angle Yawref$Ya{w_{ref}}$ as a function of the longitudinal position X. And the reference speed is shown in Figure 16. The path following based on the DGPR‐MPC can navigate the autonomous vehicle closer to the reference path compared to the nominal MPC. 16Yref=dy12[1+tanh(z1)]−dy22[1+tanh(z2)]$$\begin{equation}{Y_{ref}} = \frac{{{d_{{y_1}}}}}{2}[1 + \tanh ({z_1})] - \frac{{{d_{y2}}}}{2}[1 + \tanh ({z_2})]\end{equation}$$17φref=arctandy11cosh(z1)21.2dx1−dy21cosh(z2)21.2dx2$$\begin{equation} {\varphi _{ref}} = \arctan \left\{ {\left. \def\eqcellsep{&}\begin{array}{l} {d_{y1}}{\left[ {\dfrac{1}{{\cosh ({z_1})}}} \right]^2} \left(\dfrac{{1.2}}{{{d_{x1}}}}\right)\\[11pt] - {d_{y2}}{\left[ {\dfrac{1}{{\cosh ({z_2})}}} \right]^2}\left(\dfrac{{1.2}}{{{d_{x2}}}}\right) \end{array} \right\}} \right.\end{equation}$$wherez1=2.525X−68−1.2or2.525X−180−1.2orz2=2.525X−133−1.2or2.525X−245−1.2,dx1=25,dx2=25anddy1=3.76,dy2=3.76$$\begin{eqnarray} {{z}_{1}} &=&\left( \frac{2.5}{25} \right)\left( \text{X}-68 \right)-1.2\text{ or }\left( \frac{2.5}{25} \right)\left( \text{X}-180 \right)\nonumber\\ &&-\,1.2\text{ or }{{z}_{2}}=\left( \frac{2.5}{25} \right)\left( \text{X}-133 \right)-1.2 \nonumber\\ && \text{or }\left( \frac{2.5}{25} \right)\left( \text{X}-245 \right)- 1.2,{{d}_{x1}}=25,{{d}_{x2}}\nonumber\\ &=& 25\text{and}{{d}_{y1}}=3.76,{{d}_{y2}}=3.76 \end{eqnarray}$$16FIGUREThe reference speed.17FIGUREThe path following trajectory results of the double line‐change simulation.The simulation results for the lateral offset of the CG are shown in Figures 17 and 18. Under the same operating conditions, the control performance of DGPR‐MPC has significantly improved compared with nominal MPC. The absolute value of the lateral offset was numerically simulated and the results are summarized in Table 5. These results suggest that the DGPR‐MPC‐based method can reduce the magnitude and frequency of overshoot and achieve a smaller steady‐state error in path following compared to the conventional MPC method.18FIGUREThe path following trajectory results of the double line‐change simulation.Figure 19 shows the steering angle of the front wheel, lateral acceleration, and yaw rate. Figure 20 shows the calculation time of the MPC controller during each control cycle in the simulation of a double‐lane change at variable speed. The yaw rate of the DGPR‐MPC‐based method was relatively smaller than that of the nominal MPC method. This result indicates that the DGPR‐MPC‐based method completed the path more satisfactorily than the nominal MPC method.19FIGUREThe steering angles, lateral accelerations, and yaw rates of the double line‐change simulation.20FIGURECalculation time of the MPC controller during each control cycle in the simulation of the double lane‐change at variable speed. MPC, model predictive control.5TABLEComparison results of lateral offset eCG${e_{CG}}$.Nominal MPCDGPR‐MPCeCG${e_{CG}}$Min (m)−0.0758−0.0377Max (m)0.07810.0661Mean (m)0.01690.0135Standard deviation (m)0.02630.0219DGPR, deep Gaussian process regression; MPC, model predictive control.CONCLUSIONIn this paper, a learning‐based vehicle dynamics model for path following is developed in order to overcome the problems of low precision and poor stability of path following control when autonomous vehicles operate under conditions of large curvature and time‐varying curvature at high speed. Moreover, the model predictive controller is designed based on the learned model. The feasibility of the proposed algorithm is verified using the high‐fidelity simulation platform CarSim/Simulink. The proposed control strategy can achieve greater path‐following accuracy and superior lateral stability than the path‐following controller based on a physical model.However, the collected data cannot comprehensively cover the limited area of the vehicle, resulting in errors that are unknowable. In future research, the authors intend to combine the learned‐based vehicle dynamic model with the physical model in order to improve prediction performance in the training data coverage area and eliminate the learning‐based model's unknown error.AUTHOR CONTRIBUTIONSXuekai Yu: Methodology; Writing—original draft. Hai Wang: Conceptualization; Supervision. Chenglong Teng: Data curation; Visualization. Long Chen: Funding acquisition; Supervision. Yingfeng Cai: Funding acquisition; Supervision; Writing—review & editing.ACKNOWLEDGEMENTThis work was supported in part by the National Natural Science Foundation of China (52225212,U20A20333).CONFLICT OF INTEREST STATEMENTThe authors declare no conflict of interest.FUNDING INFORMATIONNational Natural Science Foundation of China (52225212, U20A20333)CREDIT CONTRIBUTION STATEMENTXuekai Yu: Provide ideaYingfeng Cai: Methodology, Writing – original draft, Writing – review & editingHai Wang: Funding acquisition, SupervisionLong Chen: Funding acquisition, SupervisionXiaoqing Sun: Data curation, ValidationChenglong Teng: Data curation, ValidationCRediT TAXONOMYXuekai Yu: Methodology, Writing – original draft; Hai Wang: Conceptualization, Supervision; Chenglong Teng: Data curation, Visualization; Xiaoqiang Sun: Investigation, Resources; Long Chen: Funding acquisition, Supervision; Yingfeng Cai: Funding acquisition, Supervision, Writing – review & editingDATA AVAILABILITY STATEMENTThe data that support the findings of this study are available from the corresponding author upon reasonable request.REFERENCESChen, Y., Li, L.: Advances in Intelligent Vehicles. 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DGPR‐MPC: Learning‐based model predictive controller for autonomous vehicle path following

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Abstract

INTRODUCTIONIn recent decades, autonomous driving technologies have had the potential to significantly improve driving safety, reduce traffic congestion, and increase mobility [1]. Perception, motion planning modules, policy‐making, and accurate and stable path following are crucial for a fully autonomous driving system, as they directly affect passenger safety and comfort.As a crucial enabling technology for autonomous driving, path following primarily employs the control algorithm to calculate the front wheel angle in order to ensure stability and safety while following the reference path. The vehicle dynamics model significantly impacts the success of the path following the mission. In the middle of the 20th century, the linear two‐degrees‐of‐freedom (2DOF) steering model was first investigated. Sahoo et al. [2] modelled the vehicle using the 2DOF model, analyzed the transient response, and designed the steering controller using the 2DOF model. Segel et al. [3] proposed a three‐degrees‐of‐freedom (3DOF) steering model with three degrees of freedom. The vehicle was considered a linear dynamic system when the mathematical model was developed. Yaw, lateral, and roll motion were determined to describe the vehicle's transient response. With the development of analytical mechanics, more complex vehicle models have been created, such as 7DOF [4], 17DOF [5], and even more degrees of freedom models, which are widely used in path tracking due to their simple forms. However, most current lateral control systems use a simplified modelling method for the mechanism, which limits the enhancement of control performance under wide‐range driving conditions.In recent years, data‐driven technologies have rapidly evolved and been widely implemented in autonomous vehicles [6–8]. The learning‐based approach does not require specific information about the physical structure and parameters of the system; rather, it requires only input–output mapping data. Ghazizadeh et al. [9] studied the neural network vehicle model and proposed the ‘Neuro‐Vehicle' model in 1996. James et al. [10] compared the performance of the standard linear state space model and the neural network model under large‐scale real‐world driving conditions using longitudinal driving data. The results demonstrated that the neural network model significantly enhanced the accuracy of longitudinal state prediction. Based on the gated recurrent unit (GRU), Sieberg et al. [11] established the vehicle side‐slip dynamic model. They fused the prior knowledge of the Kalman filter in the network to improve the modelling accuracy. Mauro Da Lio et al. [12] conducted layered modelling of longitudinal vehicle dynamics using a convolutional neural network (CNN) and recurrent neural network (RNN) and compared the impact of different topological structures on the modelling accuracy and network robustness. The aforementioned methods demonstrate that neural networks can model dynamic systems. However, these modelling methods are typically employed for vehicle state estimation and cannot be combined with the control method effectively.Significant research has been conducted on the path following control of autonomous vehicles in the existing literature. Pure pursuit (PP) [13] based on geometric models is widely used in autonomous systems in the real world. Fuzzifying the control variables enables the fuzzy control method [14] to simulate human driving behaviour. However, the optimal performance of the control system cannot be achieved because the parameters are obtained primarily through trial‐and‐error. Recently, methods based on optimization theory and dynamic models, such as linear quadratic regulator (LQR) and model predictive control (MPC) [15–17], have been widely used for the path‐following control of autonomous vehicles. At each control step, these methods require the repeated solution of an optimization problem. Rafaila et al. [18] used a 2DOF model and a magic tire formula to develop a non‐linear model predictive control (NMPC) method to accommodate the highly dynamic non‐linear characteristics of the vehicle turning. Onieva et al. [19] utilized the self‐learning capability of the legacy algorithm to learn the behaviour of human‐driven vehicles and improved the tolerance to uncertainty and imprecision of vehicle parameters under high‐speed road conditions, thereby achieving high‐precision control effects. Chen et al. [20] proposed a novel data‐driven non‐linear model control method based on the high‐order neural network (RHONN) modelling method. Teng et al. [21] developed an echo state model (ESM) with a rational data organization and an efficient neural network structure for high‐precision four‐step‐ahead prediction in both dry and slippery road conditions. Kim et al. [22] combine deep learning with a fully differentiable physics model to endow the neural network with available prior knowledge. The reinforcement learning control algorithm [23] enables agents to explore unknown environments, but successful training is typically challenging. Overall, the existing control algorithms meet the requirements for all most scenarios, for example, urban roads, highways, and parking. However, conventional control algorithms are susceptible to instability when the vehicle is operating in restricted conditions, such as high‐speed obstacle avoidance. Since the vehicle is a typical non‐linear system, it is worthwhile investigating new control methods employing learning‐based tools in order to further enhance the vehicle's dynamic performance.Motivated by the above discussions, a novel non‐linear predictive control method enabled by deep Gaussian process regression (DGPR) modelling is proposed in this paper. The main contributions of this study are shown as follows:The vehicle is a complex dynamic system whose path‐following accuracy and stability depend on the precision of its model. This paper proposes a deep learning approach to model vehicle dynamics, which can simulate potential system states that are hard to analyse or estimate, enhance the modelling accuracy, and predict the state of autonomous vehicles in uncertain environments.A novel vehicle dynamic modelling method that integrates traditional machine learning and deep learning to address the issue of errors caused by model mismatch in model‐based controllers. The method incorporates a branch in the model structure that estimates the uncertainty of model input.A DGPR‐based non‐linear model predictive controller is designed to improve the performance of the autonomous vehicle's path following and lateral stability. Then, simulation‐based validations of the proposed algorithms are performed on an autonomous vehicle in a variety of dynamic driving scenarios.The remainder of this paper is organized as follows. Section 2 describes the dynamics of two‐wheeled vehicles and introduces the DGPR vehicle dynamic model design. Section 3 introduces the model predictive controller design. Section 4 describes the procedures for data collection, model training, and simulation validation. Section 5 provides the results of the simulation. Finally, conclusions are provided in Section 6.VEHICLE DYNAMIC MODELLINGVehicle lateral dynamicsThe vehicle dynamic system is a non‐linear and time‐varying system. We neglect vehicle roll, pitch, and motions. Then, the 2DOF signal‐track model, as shown in Figure 1 and Equation (1).1U̇y=Fyr+Fyfcosδf+Fxfsinδfm−rUxṙ=aFyfcosδf+aFxfsinδf−bFyrIz$$\begin{equation} \def\eqcellsep{&}\begin{array}{l} {{\dot U}_y} = \dfrac{{{F_{yr}} + {F_{yf}}\cos {\delta _f} + {F_{xf}}\sin {\delta _f}}}{m} - r{U_x}\\[12pt] \dot r = \dfrac{{a{F_{yf}}\cos {\delta _f} + a{F_{xf}}\sin {\delta _f} - b{F_{yr}}}}{{{I_z}}} \end{array} \end{equation}$$1FIGUREPhysical model.where U denotes the velocity of the vehicle's centre of mass; Ux,Uy${U_x},{U_y}$ are the velocities at the centre of mass of the vehicle along the x and y directions of the vehicle body coordinate system, respectively. αf,αr${\alpha _f},{\alpha _r}$ represent the side‐slip angles of the front and rear wheels, respectively; β denotes the sideslip angle of the centre of mass; r represents the vehicle yaw rate; L=a+b$L = a + b$ is the wheelbase; m is the vehicle weight; Iz${I_z}$ is the moment of inertia of the vehicle around t he z‐axis of the centre of mass; Fyf,Fyr${F_{yf}},{F_{yr}}$ are the resultant lateral forces on the front and rear tires, respectively; Fxf${F_{xf}}$ is the resultant longitudinal force on the front axle tires and δf${\delta _f}$ is the front wheel steering angle.Considering the relationship between the vehicle coordinate system and the inertial coordinate system2ė=Uxsin(Δφ)+Uycos(Δφ)Δφ̇=r−ṡρṡ=Uxcos(Δφ)+Uysin(Δφ)1−ρe$$\begin{equation} \def\eqcellsep{&}\begin{array}{l} \dot e = {U_x}\sin (\Delta \varphi ) + {U_y}\cos (\Delta \varphi )\\[6pt] \Delta \dot \varphi = r - \dot s\rho \\[8pt] \dot s = \dfrac{{{U_x}\cos (\Delta \varphi ) + {U_y}\sin (\Delta \varphi )}}{{1 - \rho e}} \end{array} \end{equation}$$where e$e\;$represents the lateral error, Δφ$\Delta \varphi \;$is the heading error, andρ$\;\rho \;$represents the curvature of the desired path.Brush tire modelThe brush tire model [24] has been utilized extensively in parameter estimation and lateral control studies. The tire lateral force is calculated as follows:3Fy=−Cαtanα+Cα23μFz|tanα|tanα−Cα327μ2Fz2tan3α,|α|<αsat−μFzsgnα,otherwise$$\begin{equation} {F_y} = \left\{ \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} \def\eqcellsep{&}\begin{array}{l} - {C_\alpha }\tan \alpha + \dfrac{{C_\alpha ^2}}{{3\mu {F_z}}}|\tan \alpha |\tan \alpha \\[11pt] - \dfrac{{C_\alpha ^3}}{{27{\mu ^2}F_z^2}}{\tan ^3}\alpha ,|\alpha | &lt; {\alpha _{{\rm{sat }}}}\\ \end{array} \\[5pt] { - \mu {F_z}{\mathop{\rm sgn}} \alpha ,{\rm{otherwise}}} \end{array} \right.\end{equation}$$where Cα${C_\alpha }$ and μ$\mu \;$denote the tire corner stiffness and road friction coefficient, respectively,Fz$\;{F_z}\;$represents the tire vertical force; α$\alpha \;$denotes the tire side‐slip angle, and αsat${\alpha _{sat}}$ denotes the tire saturation side‐slip angle. The side‐slip angles of the front and rear tires are respectively calculated by4αf=arctanUy+arUx−δfαr=arctanUy−brUx$$\begin{equation} \def\eqcellsep{&}\begin{array}{l} {\alpha _f} = \arctan \left(\dfrac{{{U_y} + ar}}{{{U_x}}}\right) - {\delta _f}\\[14pt] {\alpha _r} = \arctan \left(\dfrac{{{U_y} - br}}{{{U_x}}}\right) \end{array} \end{equation}$$The amount of normal force experienced on each tire is respectively calculated by5Fzf=bLmg−hCGLmaxFzr=aLmg+hCGLmax$$\begin{equation} \def\eqcellsep{&}\begin{array}{l} {F_{zf}} = \dfrac{b}{L}mg - \dfrac{{{h_{CG}}}}{L}m{a_x}\\[20pt] {F_{zr}} = \dfrac{a}{L}mg + \dfrac{{{h_{CG}}}}{L}m{a_x} \end{array} \end{equation}$$where hCG${h_{CG}}\;$represents the height to the vehicle centre of gravity, g denotes the gravity acceleration, L is the vehicle wheelbase, and ax${a_x}$represents the vehicle longitudinal acceleration.The proposed bicycle model is employed to design the nominal model predictive controller and compare its performance to that of the DGPR‐MPC.Deep Gaussian process regressionDue to the need for real‐time performance by autonomous vehicles, the nominal vehicle dynamic model is simplified to reduce system complexity. However, we use a leaning‐based method to build the vehicle dynamic model to improve the performance and enable the automatic model adaptation. For instance, it must be able to model complex and difficult‐to‐model forces that change as a function of the coefficient of road‐tire friction, while remaining computationally efficient enough for real‐time control.The structure of DGPR is shown in Figure 2. The model's hidden layer consists of two layers with 100 neurons per layer. The activation function is the Softplus activation function. Motivated by states and controls considered in the physical‐based model, the DGPR's input is the velocity, steering angle, and front longitudinal force, are formed into a signal state as shown in Equation (6). At a current time, the current model state is concatenated with delayed model states to form the input to the model as shown in Equation (7). The output layer of the model has four neurons. The output of the first and second neurons is the next time step's mean lateral velocity and yaw rate. Furthermore, the output of the third and fourth neurons is the variance of the next time step's lateral velocity and yaw rate.6xt=[r,Uy,Ux,δf,Fxf]$$\begin{equation}{x_{t = }}[r,{U_y},{U_x},{\delta _f},{F_{xf}}]\end{equation}$$7ht=[xt,xt−1,…,xt−N]$$\begin{equation}{h_t} = [{x_t},{x_{t - 1}}, \ldots ,{x_{t - N}}]\end{equation}$$The forward calculation of DGPR is shown in Equation (8)2FIGUREThe structure of DGPR. DGPR, deep Gaussian process regression.8z1=w1Tht+b1a1=log(1+ez1)z2=w2Ta1+b2a2=log(1+ez2)z3=w3Ta2+b3θ=(w1,b1,w2,b2,w3,b3)μUy,t+1rt+1=z3[1:2]σUy,t+1σr,t+1=actELU+1(z3[3:4])$$\begin{equation} \left\{ \def\eqcellsep{&}\begin{array}{l} {z_1} = w_1^T{h_t} + {b_1}\\[5pt] {a_1} = \log (1 + {e^{{z_1}}})\\[5pt] {z_2} = w_2^T{a_1} + {b_2}\\[5pt] {a_2} = \log (1 + {e^{{z_2}}})\\[5pt] {z_3} = w_3^T{a_2} + {b_3}\\[5pt] \theta = ({w_1},{b_1},{w_2},{b_2},{w_3},{b_3})\\[5pt] \left[ \def\eqcellsep{&}\begin{array}{l} {\mu _{{U_{y,t + 1}}}}\\[5pt] {r_{t + 1}} \end{array} \right] = {z_3}[1:2]\\[15pt] \left[ \def\eqcellsep{&}\begin{array}{l} {\sigma _{{U_{y,t + 1}}}}\\[5pt] {\sigma _{r,t + 1}} \end{array} \right] = ac{t_{ELU + 1}}{\rm{ (}}{z_3}[3:4]{\rm{)}} \end{array} \right.\end{equation}$$where N − 1 is the number of delay states.a1 and a2 is are the expression of the Softplus. θ is the parameter learned by the DGPR. To ensure the positive of the variance, we use a modified Exponential Linear Unit (ELU) [25] activation, which is shown in Equation (9) and Figure 3.9actELU+1=x+1,x≥0ex,otherwise$$\begin{equation} ac{t_{ELU + 1}} = \left\{ \def\eqcellsep{&}\begin{array}{ll} x + 1, &\quad x \ge 0\\[5pt] {e^x},&\quad {\rm{ otherwise}} \end{array} \right.\end{equation}$$3FIGUREThe modified exponential linear unit (ELU) activation.DESIGN OF THE DGPR‐MPCThe model predictive controller is a technique for controlling linear and non‐linear systems that relies on optimization. This process involves minimizing the cost function in order to select the most suitable values for the plant model, which then provides optimal performance for following the reference output. To achieve this, the MPC controller predicts the vehicle's path (for tasks such as vehicle path following) for the next p time steps (prediction horizon) by utilizing the plant model, allowing it to anticipate and adjust for any potential errors. By utilizing different scenarios, the MPC can minimize the error between the predicted and reference trajectories. In order to maximize performance, the accuracy of the vehicle model must be high, accounting for all vehicle dynamics, including non‐linearity and uncertainty, while keeping computational power requirements low. To facilitate comparison with the physical model‐based path following controller using the same control logic, the DGPR is employed to design the control algorithm, as shown in Figure 4.4FIGUREThe general model predictive controller model.Here, the dynamic model of vehicle is rewritten as10ξt+1=DGPR(ξt,ξt−1,…,ξt−n,ut,ut−1,…,ut−n)$$\begin{equation}{\xi _{t + 1}} = DGPR({\xi _t},{\xi _{t - 1}}, \ldots ,{\xi _{t - n}},{u_t},{u_{t - 1}}, \ldots ,{u_{t - n}})\end{equation}$$where T represents the sampling time for the discrete state‐space model.The output variable is defined as11ηref(t)=[Yref(t),φref(t)]T$$\begin{equation}{\eta _{ref}}(t) = {[{Y_{ref}}(t),{\varphi _{ref}}(t)]^T}\end{equation}$$and the reference trajectory is written as12η(t)=[Y(t),φ(t)]T$$\begin{equation}\eta (t) = {[Y(t),\varphi (t)]^T}\end{equation}$$The vehicle position is shown in Figure 1. It determines the lateral error and the heading error of the autonomous vehicle relative to the reference path. Moreover, the physical model and the DGPR use kinematics equations to predict the lateral position of the vehicle. The kinematics of the vehicle does not need to be studied, and it can be calculated by Equation (4).The optimization objective function is defined as13argminJ=∑t=1Np(QY(et)2+Qψ(Δψt)2)+∑t=1Nc(Qδ(Δδt)2)+∑t=1Np((σUy)2+(σr)2)$$\begin{equation} \def\eqcellsep{&}\begin{array}{l} \arg \min J = \displaystyle\sum_{t = 1}^{{N_p}} {({Q_Y}{{({e^t})}^2} + {Q_\psi }{{(\Delta {\psi ^t})}^2})} \\[5pt] + \displaystyle\sum_{t = 1}^{{N_c}} {({Q_\delta }{{(\Delta {\delta ^t})}^2})} + \displaystyle\sum_{t = 1}^{{N_p}} {({{({\sigma _{Uy}})}^2} + {{({\sigma _r})}^2})} \end{array} \end{equation}$$14subjectto,xt+k+1=DGPR(xt+k|t,ut+k|t,θ)uk=uk−1+Δukx(0|t)=x(t)u(k)∈Uk∈[t,t+Nc]x̂∈Xk∈[t,t+Np]$$\begin{equation} \def\eqcellsep{&}\begin{array}{l} {\rm{subject to,}}\\[5pt] {x_{t + k + 1}} = DGPR({x_{t + k|t}},{u_{t + k|t}},\theta )\\[5pt] {u_k} = {u_{k - 1}} + \Delta {u_k}\\[5pt] x(0|t) = x(t)\\[5pt] u(k) \in Uk \in [t,t + {N_c}]\\[5pt] \hat x \in Xk \in [t,t + {N_p}] \end{array} \end{equation}$$where Np${N_p}$ is prediction horizon and Nc${N_c}$ is prediction horizon. J is the cost function, the state and input constraints are expressed by X and U,respectively. QY${Q_Y}$ and Qφ${Q_\varphi }$ are the weight of lateral and heading error. Qδ${Q_\delta }$ is the weight of front wheel angle. x̂$\hat x$ is the future state predicted based on the current measurement and the historical state. The first item indicates the ability of the system to follow the reference trajectory, requiring the vehicle to track the lateral error and heading deviation of the reference trajectory as small as possible. The second item indicates the requirement for controlling incremental constraints, ensuring that the control variables change quickly and smoothly. The third item indicates a constraint on the uncertainty of the model's output.In each control step, the optimal control sequenceU∗=[u1∗,u2∗,u3∗,…,uNp∗]$\;{U^*} = \;[ {u_1^*,u_2^*,u_3^*, \ldots ,u_{{N_p}}^*} ]$ is solved by the optimization algorithm. The first value for dynamic system control is selected and then the entire optimization process is repeated in the next time step.DATE COLLECTION AND MODEL TRAININGDate collectionCarSim was used to collect data with a higher degree of freedom in order to validate the modelling capability of the DGPR for a dynamic system with a high degree of freedom. The CarSim software has 27 degrees of freedom, and the tire non‐linear characteristics in the model were obtained by fitting real data. Therefore, it can be concluded that the vehicle state data output by CarSim is reliable and reflective of the actual characteristics. The established vehicle model was controlled by a Logitech G29 steering wheel and pedal, and the data was collected with MATLAB/Simulink 2019a. During the data collection, the sampling time was set to = 20 ms, as shown in Figure 5. For each episode, the initial state of the vehicle was set to 0, the engine throttle was initialized to 0, the steering wheel angle was initialized to 0, and the road friction coefficient is set 0.5 to 1.0. The range of the throttle opening and brake pedal pressure was limited to [0 0.2] and [0 9] MPa.5FIGURESimulation data collection system based on Matlab/CarSim.Model training and validationThe dataset was divided into 70% training dataset, 15% validation dataset, and 15% test dataset. A DGPR was developed using the deep learning framework PyTorch [26]. The learning rate was 0.0005 and the optimization algorithm is Adam [27]. The loss function is defined as follows:15L=12logmaxσUy,t+1σr,t+1,epseps+μUy,t+1μr,t+1−Uy,t+1rt+12maxσUy,t+1σr,t+1,epseps$$\begin{eqnarray} L &=& \frac{1}{2} \left(\log \left(\max \left(\left[ \def\eqcellsep{&}\begin{array}{l} {\sigma _{{U_{y,t + 1}}}}\\[5pt] {\sigma _{r,t + 1}} \end{array} \right],\quad \left[ \def\eqcellsep{&}\begin{array}{l} eps\\[5pt] eps \end{array} \right]\right)\right.\right.\nonumber\\ &&\left.\left. +\, \frac{{{{\left(\left[ \def\eqcellsep{&}\begin{array}{l} {\mu _{{U_{y,t + 1}}}}\\[5pt] {\mu _{r,t + 1}} \end{array} \right] - \left[ \def\eqcellsep{&}\begin{array}{l} {U_{_{y,t + 1}}}\\[5pt] {r_{t + 1}} \end{array} \right]\right)}^2}}} {{\max \left( \left[ \def\eqcellsep{&}\begin{array}{l} {\sigma _{{U_{y,t + 1}}}}\\[5pt] {\sigma _{r,t + 1}} \end{array} \right], \left[ \def\eqcellsep{&}\begin{array}{l} eps\\[5pt] eps \end{array} \right]\right)}}\right)\right) \end{eqnarray}$$where eps$eps$ is used for stability and it is set as 1×10−6$1 \times {10^{ - 6}}$. The training loss and validation loss are shown in Figure 66FIGURETraining and validation loss using simulation data.Figure 7 compares the prediction results of the DGPR and the single‐track model on the test data. In the next step, the Euler integral method is used to calculate the yaw rate and the lateral velocity using the physical model prediction value and the 20‐ms time step. At the next time step, the prediction result is compared to the vehicle data measured at that time. The test error based on the physical model is large because the physical model is only described around a single operating point and cannot capture the influence of different friction conditions on physical parameters. The DGPR can identify dynamic effects not accounted for by the physical model. For example, the elastic hysteresis of the tire when the vehicle is running at a low speed, and the weight transfer effect when the vehicle is accelerating, decelerating, or turning rapidly are learned by DGPR. In addition, it should be noted that the DGPR can make accurate predictions about the road on which the vehicle is travelling without requiring a precise estimate of road friction. In contrast, the physical model can only calculate the current dynamic changes of the vehicle using the current control input and state data, which has not only a low level of accuracy, but also inaccurate physical parameters and inadequate modelling.7FIGUREState prediction values using actual data based on DGPR and physical model. DGPR, deep Gaussian process regression.MPC CONTROL PERFORMANCE VALIDATIONThe model of the simulation system was developed on the MATLAB (Simulink)‐CarSim platform. Two scenarios based on the CarSim D‐Class vehicle model were simulated: the eight‐shaped path manoeuvre and the double lane‐change manoeuvre. The objective was to control a vehicle so that it could follow the reference path. The parameters of the vehicle and controller are shown in Tables 1 and 2. To demonstrate the advantage of the DGPR‐MPC, it was compared to the nominal MPC method, which is an optimal control method with a small steady‐state error and a quick response.1TABLEVehicle parameters.SymbolDescriptionValue (Unit)MVehicle total mass1770 (kg)lf${l_f}$Distance of the front wheel axle from the CG1.20 (m)lr${l_r}$Distance of the rear wheel axle from the CG1.43 (m)Iz${I_z}$Vehicle moment of inertia about yaw axis2760 (kg · m2)cf${c_f}$Cornering stiffness of front tires150,000 (N/rad$N/{\rm{rad}}$)cr${c_r}$Cornering stiffness of rear tires170,000 (N/rad$N/{\rm{rad}}$)2TABLEControl parameters.ParameterValuePrediction horizonNp$\;{N_p}$= 11Control horizonNc=11$\;\;{N_c} = \;11$Sample time/sT = 0.02Steer angle limit/rad−0.174to0.174$\; - 0.174\;{\rm{to\;}}0.174$Steer rate limit/rad−0.014to0.014$\; - 0.014\;{\rm{to\;}}0.014$Lateral error weightQY$\;{Q_Y}$1.0Yaw error weightQφ$\;{Q_\varphi }$0.5Control effort weight Qδ${Q_\delta }$0.3Eight‐shaped path simulationAs shown in Figure 8, the time‐based eight‐shaped path was used as a reference path. The eight‐shaped path consisted of two straight lines and two smooth curves. The vehicle ran at high speed (vx=80km/h${v_x} = 80\,{\rm{km/h}}$μ=0.85${\mu } = 0.85$).8FIGUREThe path following trajectory results of the eight‐shape simulation.In our study, we employed the interior point optimizer (IPOPT) to determine the optimal control variables. IPOPT is a powerful solver for non‐linear optimization problems and is integrated into the MATLAB Optimization Toolbox. This allowed us to efficiently solve for the control variables in our problem. Tables 1 and  2Figure 8 depicts the actual vehicle trajectories and their zoom versions. Both the DGPR‐MPC and the nominal MPC were capable of path following. The DGPR‐MPC path following was smoother and had more minor overshoot and steady‐state errors than the nominal MPC. The numerical simulation result of the absolute value of lateral offset is summarized in Table 3, which intuitionally verifies the same conclusion.3TABLEComparison results of lateral offset eCG${e_{CG}}$.Nominal MPCDGPR‐MPC80 km/heCG${e_{CG}}$Min (m)−0.2824−0.1232Max (m)0.30600.1835Mean (m)0.12300.5090Standard deviation (m)0.10700.0496DGPR, deep Gaussian process regression; MPC, model predictive control.The lateral offset is shown in Figure 9. Both approaches could stabilize the path following errors. In contrast, the DGPR‐MPC could converge to the steady‐state error in a transient period. In addition, the DGPR‐MPC could almost eliminate steady‐state error.9FIGUREThe lateral offset results of the eight‐shaped simulation.The results of the control input, front wheel angle, lateral acceleration, and yaw rate are shown in Figure 10, where the front wheel steering angles of both methods were maintained within the acceptable magnitude range. DGPR‐MPC reduces the peak value of dynamic state quantity and improves stability.10FIGUREThe steering angle, lateral accelerations, and yaw rates of the eight‐shape simulation.Figure 11 shows the calculation time of the MPC controller during each control cycle in the simulation of the eight‐shape. It can be found that the calculation time of each step of DGPR‐MPC is basically between 20 and 35 ms, which is already within an acceptable range. In the future, we will use C++ for programming. And efficient solver casadi will be used to solve the non‐linear optimization problems and reduce solving time [28].11FIGURECalculation time of the MPC controller during each control cycle in the simulation of the eight‐shape. MPC, model predictive control.Double lane‐change simulationIn this simulation, the vehicle ran at a high speed of 100 km/h for a double lane change on the road with a low road friction of μ=0.5${\rm{\;}}\mu = 0.5$. The actual trajectories of the vehicle are shown in Figure 12. The path following based on the DGPR‐MPC can navigate the autonomous vehicle closer to the reference path compared to the nominal MPC.12FIGUREThe path following trajectory results of the double line‐change simulation.The simulation results for the lateral offset of the CG are shown in Figure 13. It can be seen that the DGPR‐MPC controller changed over a greater range than the nominal MPC as the curvature varied. The numerical simulation result of the absolute value of lateral offset is summarized in Table 4, intuitionally indicating that the DGPR‐MPC‐based method can reduce the magnitude and frequency of the overshoot and obtain a smaller steady‐state error in path following compared to the conventional MPC method.13FIGUREThe lateral offset results of the double line‐change simulation.4TABLEComparison results of lateral offset eCG${e_{CG}}$.Nominal MPCDGPR‐MPC100 km/heCG${e_{CG}}$Min (m)−0.1108−0.0249Max (m)0.11100.0320Mean (m)0.04680.0117Standard deviation (m)0.04290.0106DGPR, deep Gaussian process regression; MPC, model predictive control.Figure 14 depicts the steering angle of the front wheel, lateral acceleration, and yaw rate. The steering angle of the DGPR‐MPC‐based method was relatively smaller than that of the nominal MPC method. The DGPR‐MPC of the path following had a smaller region of lateral acceleration compared to the nominal MPC. In addition, the phenomenon of overshoot for lateral accelerations was significantly mitigated. The path following results demonstrated that the DGPR‐MPC‐based method completed the path more satisfactorily than the nominal MPC method. This result indicates that the possibility of a vehicle surpassing the safe driving area has been dramatically reduced. Figure 15 shows the calculation time of the MPC controller during each control cycle in the simulation of the double‐lane change at constant speed.14FIGUREThe steering angles, lateral accelerations, and yaw rates of the double line‐change simulation.15FIGURECalculation time of the MPC controller during each control cycle in the simulation of the double lane‐change at constant speed. MPC, model predictive control.To verify the robustness of the model, the vehicle ran at a variable speed for a double lane change on the road with a low road friction of μ=0.85$\mu \; = \;0.85$.The reference path model is described in terms of lateral position Yref${Y_{ref}}$ and yaw angle Yawref$Ya{w_{ref}}$ as a function of the longitudinal position X. And the reference speed is shown in Figure 16. The path following based on the DGPR‐MPC can navigate the autonomous vehicle closer to the reference path compared to the nominal MPC. 16Yref=dy12[1+tanh(z1)]−dy22[1+tanh(z2)]$$\begin{equation}{Y_{ref}} = \frac{{{d_{{y_1}}}}}{2}[1 + \tanh ({z_1})] - \frac{{{d_{y2}}}}{2}[1 + \tanh ({z_2})]\end{equation}$$17φref=arctandy11cosh(z1)21.2dx1−dy21cosh(z2)21.2dx2$$\begin{equation} {\varphi _{ref}} = \arctan \left\{ {\left. \def\eqcellsep{&}\begin{array}{l} {d_{y1}}{\left[ {\dfrac{1}{{\cosh ({z_1})}}} \right]^2} \left(\dfrac{{1.2}}{{{d_{x1}}}}\right)\\[11pt] - {d_{y2}}{\left[ {\dfrac{1}{{\cosh ({z_2})}}} \right]^2}\left(\dfrac{{1.2}}{{{d_{x2}}}}\right) \end{array} \right\}} \right.\end{equation}$$wherez1=2.525X−68−1.2or2.525X−180−1.2orz2=2.525X−133−1.2or2.525X−245−1.2,dx1=25,dx2=25anddy1=3.76,dy2=3.76$$\begin{eqnarray} {{z}_{1}} &=&\left( \frac{2.5}{25} \right)\left( \text{X}-68 \right)-1.2\text{ or }\left( \frac{2.5}{25} \right)\left( \text{X}-180 \right)\nonumber\\ &&-\,1.2\text{ or }{{z}_{2}}=\left( \frac{2.5}{25} \right)\left( \text{X}-133 \right)-1.2 \nonumber\\ && \text{or }\left( \frac{2.5}{25} \right)\left( \text{X}-245 \right)- 1.2,{{d}_{x1}}=25,{{d}_{x2}}\nonumber\\ &=& 25\text{and}{{d}_{y1}}=3.76,{{d}_{y2}}=3.76 \end{eqnarray}$$16FIGUREThe reference speed.17FIGUREThe path following trajectory results of the double line‐change simulation.The simulation results for the lateral offset of the CG are shown in Figures 17 and 18. Under the same operating conditions, the control performance of DGPR‐MPC has significantly improved compared with nominal MPC. The absolute value of the lateral offset was numerically simulated and the results are summarized in Table 5. These results suggest that the DGPR‐MPC‐based method can reduce the magnitude and frequency of overshoot and achieve a smaller steady‐state error in path following compared to the conventional MPC method.18FIGUREThe path following trajectory results of the double line‐change simulation.Figure 19 shows the steering angle of the front wheel, lateral acceleration, and yaw rate. Figure 20 shows the calculation time of the MPC controller during each control cycle in the simulation of a double‐lane change at variable speed. The yaw rate of the DGPR‐MPC‐based method was relatively smaller than that of the nominal MPC method. This result indicates that the DGPR‐MPC‐based method completed the path more satisfactorily than the nominal MPC method.19FIGUREThe steering angles, lateral accelerations, and yaw rates of the double line‐change simulation.20FIGURECalculation time of the MPC controller during each control cycle in the simulation of the double lane‐change at variable speed. MPC, model predictive control.5TABLEComparison results of lateral offset eCG${e_{CG}}$.Nominal MPCDGPR‐MPCeCG${e_{CG}}$Min (m)−0.0758−0.0377Max (m)0.07810.0661Mean (m)0.01690.0135Standard deviation (m)0.02630.0219DGPR, deep Gaussian process regression; MPC, model predictive control.CONCLUSIONIn this paper, a learning‐based vehicle dynamics model for path following is developed in order to overcome the problems of low precision and poor stability of path following control when autonomous vehicles operate under conditions of large curvature and time‐varying curvature at high speed. Moreover, the model predictive controller is designed based on the learned model. The feasibility of the proposed algorithm is verified using the high‐fidelity simulation platform CarSim/Simulink. The proposed control strategy can achieve greater path‐following accuracy and superior lateral stability than the path‐following controller based on a physical model.However, the collected data cannot comprehensively cover the limited area of the vehicle, resulting in errors that are unknowable. In future research, the authors intend to combine the learned‐based vehicle dynamic model with the physical model in order to improve prediction performance in the training data coverage area and eliminate the learning‐based model's unknown error.AUTHOR CONTRIBUTIONSXuekai Yu: Methodology; Writing—original draft. Hai Wang: Conceptualization; Supervision. Chenglong Teng: Data curation; Visualization. Long Chen: Funding acquisition; Supervision. Yingfeng Cai: Funding acquisition; Supervision; Writing—review & editing.ACKNOWLEDGEMENTThis work was supported in part by the National Natural Science Foundation of China (52225212,U20A20333).CONFLICT OF INTEREST STATEMENTThe authors declare no conflict of interest.FUNDING INFORMATIONNational Natural Science Foundation of China (52225212, U20A20333)CREDIT CONTRIBUTION STATEMENTXuekai Yu: Provide ideaYingfeng Cai: Methodology, Writing – original draft, Writing – review & editingHai Wang: Funding acquisition, SupervisionLong Chen: Funding acquisition, SupervisionXiaoqing Sun: Data curation, ValidationChenglong Teng: Data curation, ValidationCRediT TAXONOMYXuekai Yu: Methodology, Writing – original draft; Hai Wang: Conceptualization, Supervision; Chenglong Teng: Data curation, Visualization; Xiaoqiang Sun: Investigation, Resources; Long Chen: Funding acquisition, Supervision; Yingfeng Cai: Funding acquisition, Supervision, Writing – review & editingDATA AVAILABILITY STATEMENTThe data that support the findings of this study are available from the corresponding author upon reasonable request.REFERENCESChen, Y., Li, L.: Advances in Intelligent Vehicles. 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Journal

IET Intelligent Transport Systems – Wiley

Published: Jun 3, 2023

Keywords: autonomous driving; control non‐linearities

References

  • Advances in Intelligent Vehicles
    Chen, Y.; Li, L.
  • Evaluation of the transient response and implementation of a heading‐angle controller for an autonomous ground vehicle
    Sahoo, S.; Subramanian, S.C.; Srivastava, S.
  • Theoretical prediction and experimental substantiation of the response of the automobile to steering control
    Segel, L.
  • Yaw moment control of four wheel steering vehicle by fuzzy approach
    Kazemi, R.; Keshavarz Bahaghighat, M.; Panahi, K.
  • SFNet‐N: An improved SFNet algorithm for semantic segmentation of low‐light autonomous driving road scenes
    Wang, H.
  • CenterNet‐Auto: A multi‐object visual detection algorithm for autonomous driving scenes based on improved centernet
    Wang, H.
  • CenterPoint‐SE: A single‐stage anchor‐free 3‐D object detection algorithm with spatial awareness enhancement
    Wang, H.
  • Neural networks representation of a vehicle model: ‘neuro‐vehicle (NV)’
    Ghazizadeh, A.; Fahim, A.; El‐Gindy, M.
  • Longitudinal vehicle dynamics: A comparison of physical and data‐driven models under large‐scale real‐world driving conditions
    James, S.S.; Anderson, S.R.; Lio, M.D.
  • Hybrid state estimation combining artificial neural network and physical model
    Sieberg, P.M.
  • Modelling longitudinal vehicle dynamics with neural networks
    Da Lio, M.; Bortoluzzi, D.; Papini, G.P.R.
  • Simulation performance evaluation of pure pursuit, stanley, LQR, MPC Controller for Autonomous Vehicles
    Liu, J.
  • Fuzzy observer‐based transitional path‐tracking control for autonomous vehicles
    Hu, C.; Chen, Y.; Wang, J.
  • Trajectory tracking algorithm for autonomous ground vehicle
    Zakaria, M.A.
  • Reference trajectory tuning of model predictive control
    Yamashita, A.S.
  • Nonlinear predictive control of a mobile robot: A solution using metaheuristcs
    Merabti, H.; Belarbi, K.; Bouchemal, B.
  • Nonlinear model predictive control of autonomous vehicle steering
    Rafaila, R.C.; Livint, G.
  • Automatic path following control for unmanned vehicles via genetic algorithms
    Onieva, E.; Naranjo, J.E.; Milanés, V.
  • Rhonn modelling‐enabled nonlinear predictive control for lateral dynamics stabilization of an in‐wheel motor driven vehicle
    Chen, H.; Zhang, J.; Lv, C.
  • Multistep ahead prediction of vehicle lateral dynamics based on echo state model
    Teng, C.
  • Physics embedded neural network vehicle model and applications in risk‐aware autonomous driving using latent features
    Kim, T.; Lee, H.; Lee, W.
  • Path tracking control and identification of tire parameters using on‐line model‐based reinforcement learning
    Kim, T.; Jin Kim, H.
  • Vehicle Dynamics
    Ellis, J.R.
  • Fast and accurate deep network learning by exponential linear units (ELUs)
    Clevert, D.‐A.; Unterthiner, T.; Hochreiter, S.
  • Pytorch: An imperative style, high‐performance deep learning library
    Paszke, A.
  • Real‐time neural MPC: Deep learning model predictive control for quadrotors and agile robotic platforms
    Salzmann, T.