Trajectory Tracking Control of Driverless Racing Car under Extreme Conditions

Aiming at the problem that it is difficult to ensure the trajectory tracking accuracy and driving stability of driverless racing car under the extreme conditions of high-speed turning with different road adhesion coefficients, a trajectory tracking control strategy is proposed. Firstly, the road adhesion coefficient is estimated using the extended Kalman filter algorithm. Draw the phase plane diagram of the vehicle’s centroid sideslip angle - centroid sideslip angular velocity, use the two-line method to determine the phase plane stability area, and obtain the expected limit vehicle speed under different road adhesion coefficients and different front wheel steering angle. Tracking of the desired limit travel speed is achieved through drive and brake control. Secondly, a predictive control algorithm based on adaptive prediction horizon model is designed as a lateral motion control strategy to improve the trajectory tracking accuracy. Finally, using MATLAB/Simulink and CarSim co-simulation, the results show that the proposed control strategy can ensure the driving stability of the driverless racing car and improve the trajectory tracking accuracy under extreme conditions.


I. INTRODUCTION
In order to meet the current requirements for safe, intelligent, and efficient traffic development, and to solve problems such as environmental pollution, traffic safety and urban traffic congestion, driverless has become a hot research direction for major car companies and universities [1]. As one of the core technologies of driverless car, the motion control technology of driverless car determines all the actions performed by driverless car according to the instructions. A controller with excellent performance is the foundation of driverless technology. Since the development of driverless car control technology, the current motion control research mainly focuses on medium and low speed normal operating conditions, and trajectory tracking under extreme operating conditions is a difficult problem to be solved [2]. For driverless racing car, due to their special working conditions, driving in a track surrounded by cones of different colors, it is necessary to drive at a higher speed as much as possible without rushing out of the track limit to obtain better results. If the speed is too fast, it may be driven in the extreme working conditions of high-speed turning. Currently, it is necessary to ensure the stability and carry out the trajectory tracking control.
The common method of early research on vehicle extreme conditions control is to integrate path planning and tracking. Its advantage is that it can plan the optimal path of the vehicle in real time. However, this method has a large amount of calculation, and it is difficult to ensure real-time performance. In obtaining optimal trajectories, one approach is to generate driving trajectories through optimization algorithms. For instance, Gerdts et al. [3] introduced the method of optimal control and moving vision to obtain the best driving trajectory of the vehicle, so that the vehicle can track the obtained optimal trajectory. Velenis et al. [4] used the optimal control method to constrain the acceleration to obtain the optimal speed, thereby increasing the lap speed of the car. Cardamone et al. [5] decomposed the track to obtain the best cornering trajectory and used a genetic algorithm to find a balance between track length and track curvature to improve the speed of the car through corners. Some scholars VOLUME XX, 2017 3 also use the method of simulating the technology of professional racing drivers to solve the problem of extreme conditions of vehicles. Theodosis et al. [6] used a set of simple curves to simulate the optimal trajectory of the vehicle during the turning process, and analyzed professional driving techniques, and concluded that the trajectory is a longer clothoid curve can improve the lap time. Casanova et al. [7] used a sequential quadratic programming (SQP) algorithm that minimizes functions of restricted variables to solve the resulting nonlinear programming problem to obtain the minimum vehicle maneuvering time.
In recent years, with the development of high-precision positioning technology, the simple trajectory tracking control method has gradually become the mainstream method to study the extreme conditions control of vehicles. Its characteristic is that the tracking effect has nothing to do with the path, and the control algorithm only needs to make the vehicle give full play to its own performance to obtain the optimal control effect. Therefore, many scholars focus on the control algorithm research. Bobier et al. [8] used the sliding mode variable structure algorithm to control the vehicle when the vehicle reached the control limit, so that the vehicle can track the trajectory quickly and smoothly. Kapania et al. [9] improved the trajectory tracking performance by making the direction of the vehicle's center of mass velocity the same as the tangent direction of the desired path by means of feedforward control. Erlien et al. [10] consider local path planning and path tracking, designed a model predictive controller for trajectory tracking control. Kapania et al. [11] introduced an iterative learning control algorithm to improve iterative information as transient driver input. Novi et al. [12] proposed a hierarchical control method based on nonlinear model predictive control (NMPC), using high-level MPC to calculate the optimal speed curve, and low-level NMPC to constrain the curve, improving the real-time performance of the control algorithm. Wang et al. [13] proposed a lateral control algorithm combining comprehensive feedforwardfeedback and active disturbance rejection control compensation, which improved the trajectory tracking accuracy of high-speed driverless car and the robustness of the controller. Nevertheless, the control effect of the algorithm is also affected by the performance of the vehicle itself in extreme conditions. Therefore, some scholars impose constraints on vehicles to improve the control effect of the algorithm. Jin et al. [14] determined the vehicle limit stable car speed on the road with different adhesion coefficients and different front wheel rotation angles according to the phase plane diagram of the car's center of mass sideslip angle -the center of mass sideslip angle. Li et al. [15] studied the nonlinear characteristics of tires and combined tire nonlinearity with model predictive control to improve vehicle stability under extreme conditions. Chen et al. [16] planned the limit speed of the vehicle through the tire friction limit circle, designed a horizontal and vertical coordinated controller for trajectory tracking, and designed a stability controller to improve the driving stability of the vehicle. Xin et al. [17] improved the stability of the vehicle under extreme conditions by constraining the tire slip angle of the vehicle and constraining the vehicle within the stability boundary. Sun et al. [18] took the direction angle deviation of the velocity vector as the control reference, replaced the actual center of mass side-slip angle with the ideal center-of-mass side-slip angle, used the affine approximation method for the steering angle of the rear wheels to improve the trajectory tracking accuracy. For the vehicle extreme conditions control after the constraints are imposed, it can be regarded as solving the problem of high-order nonlinear systems with constraints, and advanced control methods can also be used. Such as BLF-based method is used to solve problems [19]- [20]. A second-order sliding mode control method [21] and an adaptive SOSM controller design by using a slidingmode-based filter [22]. The advanced control method has better control effect, considering that the driverless racing car is a very complex nonlinear system, there are different working conditions in the process of trajectory tracking. Therefore, to ensure the stability of the driverless racing car during the trajectory tracking control process as much as possible, this paper has not yet used the advanced control method in the system.
In summary, scholars have done a lot of work on the trajectory tracking control of driverless vehicle under extreme conditions. However, at present, most of the literatures focus on driverless vehicle, and there are few studies on driverless racing car. Driverless racing car have the characteristic of high-speed turning. Therefore, for the extreme conditions of high-speed turning on road with different adhesion coefficients of driverless racing car, we propose a trajectory tracking control strategy. Compared with the existing research, the main contributions include the following two aspects. 1) By plotting the phase plane of centroid sideslip angle and centroid sideslip angle velocity, the expected limit speed of the vehicle under different road adhesion coefficient and different front wheel steering angle is obtained. Compared with a fixed speed, it can better improve the stability of the vehicle under extreme working conditions. 2) Based on the three degree of freedoms model of the vehicle, a linear model predictive controller is designed. According to PGC and the lateral position error, the fuzzy controller is used to adaptively control the prediction horizon of the linear model predictive controller. Compared with the fixed prediction time domain, the adaptive prediction time domain can significantly improve the trajectory tracking accuracy of the vehicle.

II. LONGITUDINAL MOTION CONTROL STRATEGY
In this article, for the trajectory tracking control of the driverless racing car under extreme conditions, it has good VOLUME XX, 2017 3 trajectory tracking accuracy whereas ensuring the stability of its trajectory tracking process. A trajectory tracking control strategy is proposed, as shown in Fig. 1. It mainly includes longitudinal motion control and lateral motion control. Longitudinal motion control mainly includes road adhesion coefficient identification, stable limit vehicle speed generation, driving and braking control. Firstly, the vehicle three-degree-of-freedom model and the Dugoff tire model are built, and the force analysis of the vehicle model and tires is carried out to obtain the state space expression of the system, and the Kalman filter algorithm is used to estimate the road adhesion coefficient. Secondly, a simplified two-degree-of-freedom model is established, the phase plane diagram of centroid sideslip angle-centroid sideslip angular velocity is drawn, and the two-line method is used to determine the stable area of the phase plane, and the stable driving limit speed of different road adhesion coefficients and front wheel steering angle is obtained.
According to the estimated current road adhesion coefficient and the steering angle of the front wheels of the vehicle, the stable driving limit speed of the vehicle in the current state is obtained. Drive and brake control is performed with the steady running vehicle speed as the desired vehicle speed. Lateral motion control is mainly based on the principle of model predictive control (MPC), and a linear model predictive controller is designed. According to the average of the second-order differential quotient of the road and the lateral position error, the adaptive control of the fuzzy controller is adopted for the prediction horizon of the model predictive controller. According to the difference between the current state vector of the vehicle and the reference state vector, the front wheel steering angle is solved, so that the vehicle can track the reference trajectory.

A. DUGOFF TIRE MODEL
The Dugoff tire model is used for the tire model of the whole vehicle. Compared with other models, the Dugoff tire model can calculate the longitudinal force and lateral force through the slip rate and tire slip angle, and the road adhesion coefficient can be independent as a separate variable in the tire force, which meets the require of road adhesion coefficient estimation. The formula is as follows: where Fx_ij is the tire longitudinal force, Fy_ij is the tire lateral force, , such as: Fx_fl is the longitudinal force of the left front wheel, μ_ij is the road adhesion coefficient, Fz_ij is the tire vertical load, Cx is the longitudinal stiffness of the tire, Cy is the cornering stiffness of the tire, λ_ij is the slip rate, αij is the side slip angle of tire.
The formula for calculating slip rate is: where ωij is the tire angular velocity; Rij is the tire radius.
Normalize the tire force to get a new form: where o x F is the normalized tire longitudinal force, 0 y F is the normalized tire lateral force.

B. VEHICLE ESTIMATION MODEL
The vehicle dynamics model is the basis for designing the road adhesion coefficient estimator. This section uses a threedegree-of-freedom dual-track dynamic model that comprehensively considers longitudinal, lateral, yaw, and longitudinal and lateral vertical load transfer, as shown in Fig where ax is the longitudinal acceleration, ay is the lateral acceleration, vx is the longitudinal speed, vy is the lateral speed,  is the yaw angular acceleration, m is the vehicle mass, Iz is the moment of inertia about the z-axis, Tf is the wheelbase of the front axle, Tr is the wheelbase of the rear axle, a and b are the distances between the center of mass and the front and rear axles.
The formula for calculating the vertical load of the tire is as follows: where hg is the height of the gravity center from the ground.

C. ROAD ADHESION COEFFICIENT ESTIMATION ALGORITHM
Based on the establishment of the vehicle model and the Dugoff normalized tire model, the extended Kalman filter algorithm is used to design a road adhesion coefficient estimator, which finally realizes the estimation of the road adhesion coefficient. The principle of the extended Kalman filter algorithm is as follows: where x is the state vector, z is the observed vector, A is the state matrix, B is the control matrix, u is the control vector, Q is the process noise covariance matrix, R is the measurement noise covariance matrix.

D. EXPECTED LIMIT DRIVING SPEED GENERATION
The complex nonlinear dynamic system of the vehicle is regarded as a controlled system, and the phase plane method is used to analyze its stability. The three degrees of freedom dynamic model of the vehicle proposed above has many parameters, so it is simplified to get nonlinear two degrees of freedom model: where Ff is the longitudinal force of the front tire, Fr is the longitudinal force of the rear tire.
For the above two degrees of freedom vehicle model. In a certain initial state  and , the solution of the equation is a phase trajectories start from the initial state. Choosing different initial states, multiple phase trajectories can be formed to form the phase plane of the centroid sideslip angleyaw rate (  − ). Similarly, for different initial states  and  , can get the phase plane of the centroid sideslip angle-the centroid sideslip angular velocity (  − ). Reference [23] shows that the  − -phase plane is more suitable for vehicle stability analysis than the  − -phase plane. Therefore, this This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. article selects the  − -phase plane to obtain the expected limit driving speed [14]. The phase plane region is composed of multiple phase trajectories. The region of convergence and equilibrium is called the stable region, and the divergent region is called the unstable region. When the vehicle travels under different working conditions, the difference in road adhesion coefficient, front wheel angle, and driving speed will have an impact on the stable area of the  − -phase plane. In this article, the two-line method is used to divide the stable region of the phase plane [24]. A tangent is drawn at the equilibrium points at both ends of the phase plane, and the boundary equation of the stable region is: Setting the road adhesion coefficient as 0.85 and the front wheel angle as 3 degrees to obtain the stability boundary at different vehicle speeds, as shown in Fig. 3. It can be seen from Fig. 3 that with the change of vehicle speed, the stable boundary of the phase plane changes, and the stable area of the phase plane decrease continuously. Therefore, to obtain the limit speed of the vehicle for stable driving, by setting the road adhesion coefficient and the front wheel angle in a certain state, the vehicle speed is gradually increased to reach the boundary of the stable area. Then the vehicle speed is the expected limit vehicle speed under the current road adhesion coefficient and front wheel steering angle. Fig. 4 shows the relationship between the vehicle parameter curve and the stability boundary when the road adhesion coefficient is 0.3 and the front wheel steering angle is 3 degrees.  It can be seen from Fig. 4 (a) that when the vehicle speed is 50km/h, the vehicle parameter curve exceeds the stability boundary, and the vehicle is in an unstable state. It can be seen from Fig. 4 (b) that when the vehicle speed is reduced to 47km/h, the vehicle parameter curve is very close to the stability boundary. Therefore, under the conditions of road adhesion coefficient of 0.3 and front wheel steering angle of 3 degrees, the expected limit driving speed is 47km/h. This method is used to obtain the expected limit driving speed under different road adhesion coefficients and front wheel steering angle. The range of the road adhesion coefficient is: 0.2 to 0.9, the range of the front wheel steering angle is: 3° to 10°. The desired limit driving speed MAP is obtained as shown in Fig. 5.

E. DRIVE CONTROL
The driving force control uses the difference between the expected vehicle speed and the actual vehicle speed as the input of the PID controller, and outputs the motor torque to the motor model to accelerate the vehicle. The control principle is shown in Fig. 6.

F. BRAKE CONTROL
Calculate the braking deceleration according to the difference between the expected vehicle speed and the actual vehicle speed, convert the braking deceleration into braking torque, and output the braking oil pressure to decelerate the vehicle according to the relationship between the braking torque and the braking oil pressure. The control principle is shown in Fig. 7.

FIGURE 7. Brake Control Principle
According to the difference between the desired vehicle speed and the current vehicle speed, the required braking deceleration is calculated as: where a is the braking deceleration, t is the braking time, and vx_ref is the desired vehicle speed. The total braking force is evenly distributed to the four wheels, and the braking torque of the four wheels is calculated as: 4 ma TR = (21) where T is the braking torque, R is the wheel radius. According to the proportional relationship between the braking torque and the brake oil pressure, the braking pressure applied to the four-wheel cylinders is obtained as: where Pf is the braking pressure of the front wheel, Pr is the braking pressure of the rear wheel, k1=95 is the proportional relationship of the front wheel, k2=50 is the proportional relationship of the rear wheel.

A. VEHICLE PREDICTION MODEL
Based on the vehicle three degrees of freedom dual-track model proposed above, there are certain requirements for the real-time performance of the algorithm in the process of designing the lateral motion controller. Therefore, the model is simplified to a three degrees of freedom singletrack model, as shown in Fig. 8.
where Flf and Flr is the longitudinal force of the front tire and rear tire, Clf and Clr is the longitudinal stiffness of the front tire and rear tire, Sf and Sr is the slip rate of the front tire and rear tire, Fcf and Fcr is the lateral force of the front tire and rear tire, Ccf and Ccr is the lateral stiffness of the front tire and rear tire, δf is the steering angle of front wheels, αf and αr is the tire slip angle of front and rear tire, respectively. x and y is the speed of the vehicle in the x and y axis directions, respectively.
According to Newton's second law, and considering the conversion between the vehicle coordinate system and the inertial coordinate system, the vehicle dynamics equation is: sin cos cos sin where x and y is the acceleration of the vehicle in the x and y axis directions, respectively. A and B is the speed of the vehicle in the X and Y axis directions of the inertial coordinate system, respectively.

B. MPC CONTROLLER DESIGN
The linear model predictive controller is used as the lateral motion controller. Compared with the nonlinear This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. ( 1) ( ) ( ) ( ) ( ) (25) where A(k)=I+TA(t), B(k)=TB(t), T is the sample period, Assuming that the reference point is (ξdyn,r , udyn,r), derive the prediction equation of MPC and convert (25) into the form of augmented matrix.
( (26) where A new equation can be written as  where η(k+i|k) is the actual system state, ηref(k+i|k) is the referenced system state, Δu(k+i|k) is the control increment of the front wheel steering angle, Q is the state weight coefficient matrix, R is the control increment weight coefficient matrix, ρ is the relaxation factor weight coefficient, ε is the relaxation factor.
The constraint of the front wheel steering angle and the variation are designed as 20 20 0.85 0.85 Solving the following constrained optimization problem to obtain the optimal control increment acting on the system.  Solving (24), the optimal front wheel steering angle increment sequence is obtained as follows: , , , The first item of the increment sequence is used as the control increment input of the system.

C. ADAPTIVE PREDICTION HORIZON
The prediction horizon Np is one of the main parameters affecting the control effect of the MPC controller. Therefore, this article uses a fuzzy controller to adaptively control the prediction horizon Np to improve the control accuracy of the controller.
Firstly, the average of the absolute value of the second derivative of the road is introduced to define the lateral position change of the reference path [25], as follows:  j is an integer between 1 to Np+1. The prediction horizon Np represents the prediction step size of the system for the future time. If Np is larger, the characteristic of the system is that it cares more about the state of the future moment, and the weight of the state of the current moment is smaller. Therefore, the control effect shown is that the control accuracy of the current state is reduced, but the stability of the control system is better. On the contrary, the characteristic of the system is that the more it cares about the state of the current moment, the less weight it gives to the state of the future moment. Therefore, the control effect shown is that the control accuracy of the current moment is better, but the stability of the control system is deteriorated due to the small prediction of the future moment. Therefore, the reasonable choice of prediction horizon Np has a very important influence on the control results [26]. The vehicle state of the driverless racing car changes during the trajectory tracking process, and the fixed prediction horizon cannot guarantee the optimal control effect in different states.
Based on the above principles, to ensure the stability of the vehicle and at the same time improve the trajectory tracking accuracy in different vehicle states. According to PGC and lateral position error, the prediction horizon Np is adaptively adjusted using a fuzzy controller. Fixed control horizon Nc is adopted. The adaptive prediction time domain adjustment control structure is shown in Fig. 9.  Fig. 10. The fuzzy rule control table is shown in Table. 1.

V. SIMULATION RESULTS
The controller is built in MATLAB/Simulink, the vehicle model is built in CarSim, and the co-simulation is carried out to verify the proposed control strategy. The main parameters of the vehicle are shown in Table. 2. Considering that the path of the driverless racing competition is a combination of straights and corners, it is to verify the trajectory tracking control under the extreme conditions of high-speed turning. Therefore, a simulation path is established by cubic spline curve fitting. Its reference path and reference yaw angle are shown as follows: where Yref is the reference lateral position, X is the longitudinal position, φref is the reference yaw angle, a=0.0001422, b=-0.02751, c=1.553, d=0.
In the simulation environment, the initial position of the vehicle in the geodetic coordinate system is set to (0,0), and the initial yaw angle is 57.3°. Set up four groups of controllers for comparison, namely: controller A, controller B, controller C, and controller D. Controller A is an adaptive prediction horizon model predictive controller. Controller B is a fixed prediction horizon model predictive controller (Np=16). Controller C is a fixed prediction horizon model predictive controller (Np=11). Controller D is a constant longitudinal vehicle speed, fixed prediction horizon model predictive controller (Np=16). The vehicle speed of controllers A, B, and C are controlled by the longitudinal motion control strategy designed in this article. The controller parameters are shown in Table. 3.

A. CASE STUDY 1: HIGH ADHESION COEFFICIENT ROAD
The road adhesion coefficient is set to 0.8 and the initial vehicle speed is 115 km/h. The simulation results are shown in Fig. 11. As can be seen from Fig. 11 (a)-(f), after 28 m, the entry into the curve starts at this time. Controller D cannot accurately track the trajectory because the vehicle speed is too fast, and the vehicle has a large deviation from the reference trajectory. The other three sets of controllers decelerate according to the front wheel steering angle, and the speed is almost the same, which can track the trajectory stably and accurately. The maximum lateral position error of controller B is 0.21 m. The maximum lateral position error of controller A is 0.15 m. The maximum lateral position error of controller C is 0.2 m. Compared with fixed prediction horizon; adaptive prediction horizon control can effectively improve the accuracy of trajectory tracking control. The road adhesion coefficient estimator has small fluctuations in the estimation process, and finally converges around the set value of 0.8. The maximum estimation error of the road adhesion coefficient is 0.03, which has a good estimation effect. The vehicle speed of 95 km/h at the longitudinal position of 35m-50m is selected as the boundary of the centroid sideslip angle -the centroid sideslip angular velocity phase plane. Only a small part of the vehicle's own trajectory curve of the three controllers exceeds the boundary curve, and the vehicle is in a stable state. Therefore, on the road with high adhesion coefficient, the proposed trajectory tracking control strategy has a better trajectory tracking control effect under extreme conditions.

B. CASE STUDY 2: MEDIUM ADHESION COEFFICIENT ROAD
The road adhesion coefficient is set to 0.6 and the initial vehicle speed is 95 km/h. The simulation results are shown in Fig. 12. As can be seen from Fig. 12 (a)-(f), after 28 m, the entry into the curve starts at this time. Controller D cannot accurately track the trajectory because the vehicle speed is too fast, and the vehicle has a large deviation from the reference trajectory. But after the longitudinal position of 70m, since the curvature of the curve is smaller than that of the first curve, the controller D can continue to track the trajectory. The other three sets of controllers decelerate according to the front wheel steering angle, and the speed is almost the same, which can track the trajectory stably and accurately. The maximum lateral position error of controller B is 0.22 m. The maximum lateral position error of controller A is 0.15 m. The maximum lateral position error of controller C is 0.2 m. However, after 90m, due to the small prediction horizon, the lateral position error of the exit increases, and the maximum is 0.4m. The road adhesion coefficient estimator has small fluctuations in the estimation process, and finally converges around the set value of 0.6. The maximum estimation error of the road adhesion coefficient is 0.02, which has a good estimation effect. The vehicle speed of 85 km/h at the longitudinal position of 35m-50m is selected as the boundary of the centroid sideslip angle -the centroid sideslip angular velocity phase plane. The vehicle trajectory curves of the three groups of controllers are almost all within the boundary curve, and the vehicle is in a stable state. Therefore, the proposed trajectory tracking control strategy also has a good trajectory tracking control effect under the medium adhesion coefficient road surface.

C. CASE STUDY 3: LOW ADHESION COEFFICIENT ROAD
The road adhesion coefficient is set to 0.6 and the initial vehicle speed is 70 km/h. The simulation results are shown in Fig. 13. As can be seen from Fig. 13 (a)-(f), after 28 m, the entry into the curve starts at this time. Controller D cannot accurately track the trajectory because the vehicle speed is too fast, and the vehicle has a large deviation from the reference trajectory. The other three sets of controllers decelerate according to the front wheel steering angle, and the speed is almost the same, which can track the trajectory stably and accurately. Due to the low adhesion coefficient of the road surface, compared with the previous two working conditions, the lateral position error of controller D exceeds 3 m, and the trajectory tracking cannot be continued. The maximum lateral position error of controller B is 0.3m. The maximum lateral position error of controller A is 0.3m. Before the longitudinal position of 90m, the maximum lateral position error of controller C is 0.25m. However, after 90m, due to the small road adhesion coefficient and the small prediction horizon, the lateral position error of the exit increases to 0.3m. The road adhesion coefficient estimator has small fluctuations in the estimation process, and finally converges around the set value of 0.3. The maximum estimation error of the road adhesion coefficient is 0.01, which has a good estimation effect. The vehicle speed of 55 km/h at the longitudinal position of 35m-50m is selected as the boundary of the centroid sideslip angle -the centroid sideslip angular velocity phase plane. The vehicle trajectory curves of the three groups of controllers are almost all within the boundary curve, and the vehicle is in a stable state.
Therefore, under the low adhesion coefficient road surface, if the prediction horizon is too small, the trajectory tracking control effect will be deteriorated. The adaptive prediction horizon adjustment effect is not obvious, and the trajectory tracking control effect of controller A and controller B is close.

VI. CONCLUSION
In this article, the problem of trajectory tracking control of driverless racing car under extreme conditions is addressed, while ensuring its stability and tracking accuracy. A trajectory tracking control strategy is proposed, and the controller is designed based on the proposed control strategy.
Validated by co-simulation with MATLAB/Simulink and CarSim. The results show that the trajectory tracking control strategy proposed in this article can consider the vehicle stability and trajectory tracking accuracy under the extreme conditions of high-speed turning with different road adhesion coefficients.
In future work, we will carry out hardware-in-the-loop experiments to verify the control strategy on the hardwarein-the-loop test bench to test the feasibility of the algorithm on the test bench. In addition, we will improve the road adhesion coefficient estimation algorithm. The estimation algorithm based on tire force used in this article has a certain error in the road adhesion coefficient estimation when the tire force changes greatly. Meanwhile, we will study the use of advanced control methods in this system to improve the control accuracy.