Research on Intelligent Vehicle Trajectory Planning and Control Based on an Improved Terminal Sliding Mode

Abstract

Aiming at precisely tracking an intelligent vehicle on a desired trajectory, this paper proposes an intelligent vehicle trajectory planning and control strategy based on an improved terminal sliding mold. Firstly, the traditional RRT algorithm is improved by using the target bias strategy and the separation axis theorem to improve the algorithm search efficiency. Secondly, an improved terminal sliding mode controller is designed. The controller comprehensively considers the lateral error and heading error of the tracking control, and the stability of the control system is proven by the Lyapunov function. Finally, the performance of the designed controller is verified by the Matlab-Carsim HIL simulation platform. The test results of the Matlab-Carsim HIL simulation platform show that, compared with the general terminal sliding mode controller, the improved terminal sliding mode controller designed in this paper has higher control accuracy and better robustness.

1. Introduction

With the development of society and technology, intelligent driving has become more and more widely used to reduces incidences of traffic accidents and improve driving safety [1]. The Apollo autonomous driving platform is already in mass production in 2021, and Tencent released the newly upgraded TAI4.0 smart cockpit solution also in this year. In 2021, SAIC Audi and Alibaba Cloud officially signed a memorandum of cooperation, covering areas such as the internet of vehicles and digital ecology. Therefore, intelligent driving has become one of the directions of future automobile development.

The intelligent vehicle is an advanced system that mainly integrates technologies such as environment perception, trajectory planning, and tracking control [2]. Among them, trajectory planning and tracking control are the key components of intelligent vehicles. They are also important guarantees for the safe driving of intelligent vehicles. Therefore, intelligent vehicles have attracted the increasing attention of researchers domestically and internationally.

Most trajectory planning algorithms for intelligent vehicles originated in the field of mobile robots [3]. They are applied to complex traffic road environments through continuous improvement and optimization. In order to deal with complex dynamic environments, researchers have proposed global planning algorithms and local planning algorithms [4]. In order to solve the real-time obstacle avoidance problem of mobile robots, Song et al. proposed a global dynamic path planning method for mobile robots based on the improved a* algorithm [5]. Moreover, Tim et al. proposed a far-sighted two-step, multi-layered graph-based trajectory planner [6]. The planner was designed to generate action sets for multiple drivable trajectories, allowing adjacent behavior planners to choose the most appropriate actions for the global state in the scene. Jonatan et al. proposed a 3D online path planning method with kinematic constraints to solve the path planning problem of mobile robots in underwater unknown environments [7]. This algorithm can avoid obstacles in real time, but the computational complexity is large, resulting in low efficiency. In order to reduce the computational complexity of the algorithm, Zhang et al. proposed an optimal path planning method based on a genetic algorithm and a simulated annealing algorithm [8]. Bi et al. proposed a trajectory optimization method for a cable shovel robot based on a multi-objective genetic algorithm [9]. The method has strong adaptability and reduces the energy consumption of robot operations. In order to further improve the planning efficiency, the RRT algorithm is widely used because of its probabilistic completeness and the advantages of considering multiple constraints [10]. The early RRT algorithm has a slow convergence speed. It is difficult to apply to unmanned vehicles with high real-time performance [11]. For this, researchers propose bidirectional expanded random tree to further improve the convergence speed of the algorithm [12]. Niu et al. proposed an intelligent vehicle path planning method based on an improved RRT algorithm [13]. Liu et al. proposed a target-biased bidirectional fast exploration random tree algorithm based on curve smoothing [14]. The algorithm reduces the number of search nodes and improves the search efficiency. Tahir et al. proposed a potentially guided intelligent bidirectional RRT* algorithm [15]. The new concept of the potential-guided bidirectional tree was introduced into the algorithm. The results showed that the algorithm greatly improved the convergence speed and the memory utilization. Velocity planning is also an important part of trajectory planning. Thomas et al. proposed an optimization-based velocity plan [16]. The algorithm is capable of real-time adaptive velocity planning. It is suitable for racing cars operating at their handling limit and at speeds in excess of 200 km/h.

In order to effectively track the desired trajectory, researchers have proposed a series of tracking control algorithms. The current motion control algorithms mainly include PID control, SMC control and MPC control. Among them, sliding mode variable structure control has a strong robustness to nonlinear systems and unknown disturbances, so researchers have made different improvements to it. Cen et al. used the enhanced variable structure based on sliding mode to realize the trajectory tracking control of the robot [17]. Moreover, he designed the shear control and sliding mode control of the non-holonomic wheeled mobile robot. Gao et al. constructed a new variable-structure sliding mode by designing the update law of the sliding mode parameters [18]. This improved the convergence speed of the system. The chattering of the sliding mode controller seriously affects the accuracy of the controller. Therefore, Sun et al. proposed a composite sliding mode controller [19]. The controller consists of a novel sliding mode control based on hybrid reaching law and an extended sliding mode disturbance observer. Experiments show that this method can effectively suppress chattering and shorten the approach time. Wu et al. proposed a method using a fuzzy sliding mode variable structure [20]. This method can suppress the chattering of the actuated joints and minimize the pose error of the robot end effector. In order to make the sliding mode controller approach the sliding mode surface quickly in a limited time, and improve the convergence speed, Truong et al. proposed a backstepping global fast terminal sliding mode control method for the trajectory tracking control problem of industrial robots [21]. This method makes the control error of the controller converge in a short time. Wu et al. designed a hierarchical controller based on a model predictive control and sliding mode control that could simultaneously track the reference trajectory and the desired vehicle speed [22]. The method adopted a new reaching law, which further reduced the chattering of the system while ensuring that the system converged quickly to the sliding mode surface.

Aiming at the accuracy of intelligent vehicle tracking the desired trajectory, this paper designs a trajectory plan based on improved RRT algorithm and an improved terminal sliding mode intelligent vehicle tracking control strategy. Improved motion planning and motion control methods have many advantages. Firstly, the traditional RRT algorithm was improved by using the goal bias strategy and the separation axis theorem to improve the planning efficiency of the algorithm. Secondly, an improved terminal sliding mode controller was designed, which comprehensively considered the lateral error and heading error of the tracking control to improve the accuracy of the system tracking control. The rest of the paper is organized as follows: Section 2 discusses trajectory planning based on the improved RRT algorithm; Section 3 discusses building the controller model; Section 4 discusses the design of the improved terminal sliding mode controller; Section 5 discusses the trajectory planning simulation experiments and Matlab-Carsim HIL simulation platform experiments; Section 6 discusses the conclusions of this study.

2. Improved RRT Algorithm

This chapter makes improvements to the basic RRT algorithm. Firstly, the sampling area is restricted. When the algorithm performs random point sampling, the target bias strategy is used [23]. When the random tree expands, it will drive the random point X_rand to expand in the direction of the target point X_goal. Then, according to the vehicle′s nonholonomic constraints, the sampling of random points is restricted to the feasible region of the forward direction. The feasible region takes the longitudinal axis of the vehicle as the middle line. According to the steering range of the vehicle, a forward sampling range is generated. When the random tree is expanded, the size of the vehicle and the obstacle is considered, and the separation axis theorem is used for collision detection to ensure that the planned path will not collide with the surrounding obstacles. The algorithm flow chart of this paper is shown in Figure 1.

2.1. Extended Target Bias

The RRT algorithm is characterized by the random search of the space. Such search path planning brings huge advantages, but it also brings the disadvantages of the RRT algorithm [24]. Before the algorithm obtains the path solution, the random tree search is completely random. The random tree often expands to some places far away from the target, and the algorithm efficiency is not high. In order to improve the efficiency of the algorithm, this paper adopts the extended goal bias strategy. By artificially guiding the generation of random points, the target point X_goal is selected with a certain probability as the random point X_rand in the loop. In order to maintain the ability of random trees to expand the unknown space, the probability P is usually not selected as too large, usually 0.05–0.1. P = 0.1 is used in this paper. The comparison of the path generated by the basic RRT algorithm and the target-biased RRT algorithm with probability P = 0.1 is shown in Figure 2. As shown in Figure 2a, the basic RRT algorithm has many sampling points, and the path distance is long. As shown in Figure 2b, the improved algorithm has fewer sampling nodes and the path distance is short. Therefore, the improved algorithm reduces the path length and the number of sampling points, which improves the efficiency of the algorithm.

2.2. Random Point Expansion Optimization

When using RRT to plan the vehicle path, considering the vehicle kinematics constraints and limiting the sampling area can better meet the driving requirements of the vehicle steering mechanism and the vehicle following control requirements. The two-degree-of-freedom vehicle motion model is established with the center of the vehicle′s rear axle as the reference point. The vehicle kinematics model is shown in Figure 3, (x, y) are the coordinates of the vehicle′s rear axle axis, which is in the inertial coordinate system OXY, and μ is the heading angle of the vehicle. δ_f is the front wheel angle; β is the center of mass slip angle; B is the vehicle yaw angle; v is the velocity of the vehicle center of mass; L_r is the distance between the center of mass and the rear axle; and L_f is the distance between the center of mass and the front axle.

The differential equation formulas of the vehicle model are derived:

$$\dot{x}=\nu cos\mu \Rightarrow x_{t+dt}=x_t+\nu \mathit{cos}\mu ·dt$$

(1)

$$\dot{y}=\nu \mathit{sin}\mu \Rightarrow y_{t+dt}=y_t+\nu \mathit{sin}\mu ·dt$$

(2)

$$\dot{\mu }=\omega \Rightarrow {\mu }_{t+dt}={\mu }_t+\left(\nu /L\right)·\mathit{tan}{\delta }_f·dt$$

(3)

Based on the above vehicle kinematics model, the feasible path of the vehicle in path planning is shown in Figure 4. According to Figure 4, the feasible path of the vehicle restricts the selection of the random point X_rand within the feasible area of the vehicle. As shown in Figure 5, the random point X_rand is generated in the feasible area of the vehicle. The positive direction of the X-axis of the centerline of the sampling node area is equal to the direction of the vehicle. The boundaries of the feasible area are L₁ and L₂. The vehicle′s heading angle range is set to −40° to 40°, and the distance between the random point and the tree node is greater than a search step d; that is, the restricted search area is within the feasible sector in front of the vehicle and greater than the step d.

To determine whether the random point X_rand is valid, it can be obtained based on the slope of the line between the nearest tree node X_near and the parent node (here, the path starting point X_int) and the line between the nearest tree node X_near and the random point X_rand. In Figure 5, the coordinates of the parent node of the tree are (x_i1, y_i1), the coordinates of the nearest tree node X_near are (x_n1, y_n1), and the coordinates of the random point X_rand are (x_r1, y_r2). k_l is the slope of the line between the nearest tree node and the parent node. k₂ is the slope of the line connecting the nearest tree node to a random point. The formulas of k₁ and k₂ are as follows:

$$k_1=\left(y_{n1}-y_{i1}\right)/\left(x_{n1}-x_{i1}\right)$$

(4)

$$k_2=\left(y_{r1}-y_{n1}\right)/\left(x_{r1}-x_{n1}\right)$$

(5)

$$\mu =\mathit{arctan}\left(\left(k_1-k_2\right)/\left(1+k_1·k_2\right)\right)$$

(6)

If the value of |μ| is less than μ_max, the generated random point is valid. The extended constraint formula is as follows:

$$\left|\mu \right|<{\mu }_{max}={45}^{°}$$

(7)

2.3. Collision Detection

In the path planning of the improved RRT method, the geometric dimensions of vehicles and obstacles are considered. In this paper, the geometric dimensions of vehicles and obstacles were abstracted into rectangles. The separation axis theorem was used to transform the collision detection problem between vehicles and obstacles into the intersection test problem of different rectangles [25]. The principle of the separation axis is to project the convex polygon onto a vector separation axis to see if the projections of the two polygons overlap. If they do not overlap, the projections of the objects on the separation axis do not overlap each other, and there is no collision between these two objects [26].

The schematic diagram of the separating axis theorem is shown in Figure 6: the two rectangles are in the global coordinate xoy, and rectangles A and B represent vehicles and obstacles, respectively. The unit vectors of rectangle A in the local coordinates with A_o as the origin are A_x and A_y. The unit vectors of rectangle B in the local coordinates with B_o as the origin are B_x and B_y. The length of rectangle A is L_a, the width is W_a, the length of rectangle B is L_b, the width is W_b, T is the distance between the center point of rectangle A and the center point of rectangle B. μ is the heading angle of the vehicle in the global coordinate system. φ₁ is the angle between the center line of the two rectangles and the X axis of the global coordinate system.

There are four separation axes L, which are the x and y axes of the local coordinates of the A and B rectangles. To detect whether the two rectangles collide, you only need to determine whether the projections of the two rectangles on all the separation axes meet all the following conditions.

Condition 1: the projection axis is the x axis of the A rectangular local coordinate system, which is:

$$T\left|\mathit{cos}{\phi }_1\right|>\frac{L_a}{2}\left|\mathit{cos}\mu \right|+\frac{W_a}{2}\left|\mathit{sin}\mu \right|$$

(8)

Condition 2: the projection axis is the y axis of the A rectangular local coordinate system, which is:

$$T\left|\mathit{sin}{\phi }_1\right|>\frac{L_a}{2}\left|\mathit{sin}\mu \right|+\frac{W_a}{2}\left|\mathit{cos}\mu \right|$$

(9)

Condition 3: the projection axis is the x axis of the B rectangular local coordinate system, which is:

$$T\left|\mathit{cos}\left({\phi }_1-\mu \right)\right|>\frac{L_b}{2}\left|\mathit{cos}\mu \right|+\frac{W_b}{2}\left|\mathit{sin}\mu \right|$$

(10)

Condition 4: the projection axis is the y axis of the B rectangular local coordinate system, which is:

$$T\left|\mathit{sin}\left({\phi }_1-\mu \right)\right|>\frac{L_b}{2}\left|\mathit{sin}\mu \right|+\frac{W_b}{2}\left|\mathit{cos}\mu \right|$$

(11)

If the above four determination conditions are met at the same time, it can be determined that the vehicle and the obstacle will not collide.

3. Build the Controller Model

In order to improve the trajectory tracking accuracy of the sliding mode controller, a model that comprehensively considers the lateral error and the heading error was designed. The sliding mode controller was designed based on the model in this section.

3.1. Vehicle Dynamics Model

For the motion control of intelligent vehicles, it is necessary to analyze the dynamic model of the lateral motion of the vehicle. The dynamic model of the vehicle is considered comprehensively [27], as shown in Figure 7. β is the vehicle center of mass sideslip angle. v_x is the longitudinal velocity of the vehicle at the center of mass, and v_y is the lateral velocity of the vehicle at the center of mass.

In the case of a small tire slip angle, the lateral force of the tire is proportional to the slip angle.

The slip angles of the front wheel are as follows:

$${\alpha }_f={\delta }_f-{\theta }_{vf}$$

(12)

The formula for the slip angle of the rear wheel is as follows:

$${\alpha }_r=-{\theta }_{vr}$$

(13)

The lateral force of a single front wheel of the vehicle is as follows:

$$F_{yf}=C_{\alpha f}\left({\delta }_f-{\theta }_{vf}\right)$$

(14)

where δ_vf is the angle between the tire velocity vector and the longitudinal axis of the vehicle, ${\delta }_f$ is the steering angle of the front wheel, and $C_{\alpha f}$ is the proportional coefficient, which is called the cornering stiffness of the front wheel.

Similarly, the lateral force of a single rear wheel of the vehicle is as follows:

$$F_{yr}=C_{\alpha r}\left(-{\theta }_{vr}\right)$$

(15)

where $C_{\alpha r}$ is the proportional coefficient, which is called the cornering stiffness of the rear wheel, and θ_vr is the speed angle of the rear wheel of the vehicle.

According to Formulas (16) and (17), the front wheel speed angle θ_vf and rear wheel speed angle θ_vr of the vehicle are calculated as follows:

$$tan\left({\theta }_{vf}\right)=\left(v_y+L_f\dot{B}\right)/v_x$$

(16)

$$tan\left({\theta }_{vr}\right)=\left(v_y-L_r\dot{B}\right)/v_x$$

(17)

where L_f is the distance between the center of mass and the front axle of the intelligent vehicle, and L_r is the distance between the center of mass and the rear axle of the intelligent vehicle.

The formula obtained by approximating the small angle is as follows:

$${\theta }_{vf}=\left(v_y+L_f\dot{B}\right)/v_x$$

(18)

$${\theta }_{vr}=\left(v_y-L_r\dot{B}\right)/v_x$$

(19)

The yaw dynamics equation obtained by the torque balance around the Z axis is as follows:

$$I_z\ddot{B}=2L_f{F}_{yf}-2L_r{F}_{yr}$$

(20)

where $I_z$ is the yaw moment of inertia of the vehicle body.

The two-degree-of-freedom dynamic model of the intelligent vehicle is as follows:

$$\{\begin{array}{c}\dot{\beta }=\frac{2F_{yf}+2F_{yr}}{m{v}_x}-\dot{B}\\ \ddot{B}=\frac{2L_f{F}_{yf}-2L_r{F}_{yr}}{I_z}\end{array}$$

(21)

3.2. The Driver Model Based on Preview

In the process of trajectory tracking of intelligent vehicles, the lateral error of the vehicle and the heading error of the vehicle need to be considered. If only the front wheel angle is used as a variable, only one error can be controlled. Therefore, for the heading error of the intelligent vehicle when it drives to the preview point, this paper introduces a total error y_m to integrate the two errors, as shown in Figure 8.

When the vehicle is driving, the longitudinal speed v_x is much greater than the lateral speed v_y. Therefore, it is assumed that in the time Δt, the yaw rate of the vehicle remains unchanged, the velocity of the center of mass remains unchanged, and the vehicle moves in a circle. In order to drive to the desired trajectory, the driver previews point D. The actual trajectory follows the arc $\widehat{EC}$, and runs to point D within Δt time. Because the longitudinal velocity v_x is much larger than the lateral velocity v_y, the point C coincides with the point D within Δt time. So, AC = AD = y_m. Therefore, a preview-based vehicle trajectory error model is established [28], as shown in Figure 8. x_m is the preview distance, y_m is the total error, y_m1 is the lateral error, μ₁ is the actual track heading angle of the intelligent vehicle, and μ₂ is the expected track heading angle of the intelligent vehicle.

According to the vehicle trajectory error model based on preview in Figure 8, the formula obtained is as follows:

$$\{\begin{array}{l}{y}_m=y_{m1}+x_{m}sin\left(\Delta \mu \right)\\ {\dot{y}}_{m1}=v_y\mathit{cos}\left(\Delta \mu \right)-v_{x}sin\left(\Delta \mu \right)\\ \begin{array}{l}{\dot{x}}_{m1}=v_y\mathit{sin}\left(\Delta \mu \right)+v_x\mathit{cos}\left(\Delta \mu \right)\hfill \\ \Delta \mu ={\mu }_1-{\mu }_2\hfill \end{array}\end{array}$$

(22)

The heading angular velocity ${\dot{\mu }}_2$ of the desired trajectory is as follows:

$${\dot{\mu }}_2=\frac{{\dot{x}}_{m1}}{R}=\rho {\dot{x}}_{m1}$$

(23)

where $R$ is the radius of the actual trajectory within $\mathsf{\Delta }t$, $\rho$ is the radius of curvature of the actual trajectory, and x_m1 is the distance traveled by the intelligent vehicle along the reference trajectory arc $\widehat{EC}$ within $\mathsf{\Delta }t$ time.

At time Δt, since the longitudinal velocity v_x is much larger than the lateral velocity v_y, Δμ is very small, according to the small-angle approximation:

$$\{\begin{array}{c}{y}_m=y_{m1}+x_m\Delta \mu \\ {\dot{y}}_{m1}=v_y-v_x\Delta \mu \\ \begin{array}{l}{\dot{x}}_{m1}=v_y\Delta \mu +v_x\hfill \\ \begin{array}{l}\Delta \mu ={\mu }_1-{\mu }_2\hfill \\ {\dot{\mu }}_1\approx \dot{B}\hfill \\ {\dot{\mu }}_2=\frac{{\dot{x}}_{m1}}{R}=\rho {\dot{x}}_{m1}\hfill \end{array}\hfill \end{array}\end{array}$$

(24)

where $\dot{B}$ is the yaw rate.

From Formulas (21) and (24), the following formula is obtained:

$$\{\begin{array}{l}{\ddot{y}}_m={\ddot{y}}_{m1}+x_m\Delta \ddot{\mu }\\ {\ddot{y}}_{m1}={\dot{v}}_y-{\dot{v}}_x\Delta \mu -v_x\Delta \dot{\mu }\\ \Delta \ddot{\mu }=\ddot{B}-\rho {\ddot{x}}_{m1}\\ {\ddot{x}}_{m1}={\dot{v}}_y\Delta \mu +v_y\Delta \dot{\mu }+{\dot{v}}_x\end{array}$$

(25)

The trajectory error ${\ddot{y}}_m$ is obtained according to Equations (14), (15) and (21):

$${\ddot{y}}_m={\dot{v}}_y-{\dot{v}}_x\Delta \mu -v_x\Delta \dot{\mu }+x_m\left(\frac{2L_f{C}_{\alpha f}\left({\delta }_f-{\theta }_{vf}\right)+2L_r{C}_{\alpha r}{\theta }_{vr}}{I_z}-\rho {\ddot{x}}_{m1}\right)$$

(26)

From Formulas (18), (19) and (26), the following formula is obtained:

$$\{\begin{array}{l}{\ddot{y}}_m={\phi }_1+{\phi }_2{\delta }_f+d\\ {\phi }_1={\dot{v}}_y-{\dot{v}}_x\Delta \mu -v_x\Delta \dot{\mu }+{\phi }_3\\ {\phi }_2=x_m\frac{2L_f{C}_{\alpha f}}{I_z}\\ {\phi }_3=x_m\left(2L_r{C}_{\alpha r}\left(v_y-L_r\dot{B}\right)-2L_f{C}_{\alpha f}\left(v_y+L_f\dot{B}\right)\right)/v_x{I}_z\end{array}$$

(27)

where d is the disturbance error which represents the external disturbance.

4. Designing a Sliding Mode Controller

The characteristic of a sliding mode variable structure control is that the structure of the controlled system is not static. It can be dynamically controlled with changes in the system, and it has the advantages of quick response, less influence from external disturbances, and simple control action.

4.1. Sliding Surface Design

In this paper, the sliding mode controller is used to control the preview deviation y_m of the intelligent vehicle, and the control error is as follows:

$$\{\begin{array}{c}\dot{k}=x_1\\ {\dot{x}}_1={\ddot{y}}_m\end{array}$$

(28)

where k = y_m.

The sliding surface design of the general terminal sliding mode is as follows:

$$s=\dot{e}+{\gamma }_{1}e+{\gamma }_2{e}^{\raisebox{1ex}{$q$}\!\left/ \!\raisebox{-1ex}{$p$}\right.}=0$$

(29)

where e is the state quantity, ${\gamma }_1$ > 0, ${\mathsf{\gamma }}_2$ > 0, p and q are positive odd numbers, p > q.

The design of this terminal sliding surface has singularity, and the generation of singularity originates from $e^{\raisebox{1ex}{$q$}\!\left/ \!\raisebox{-1ex}{$p$}\right.}$. In this paper, according to the idea of dynamic sliding mode control, the discontinuous control term is placed in the first derivative of the control input. The singularity of terminal sliding modes is resolved. Therefore, the sliding mode function is defined as follows:

$$\{\begin{array}{c}s=\dot{k}+{\gamma }_{3}k+{\gamma }_4{k}_a\hfill \\ {\dot{k}}_a=k^{\frac{q}{p}}\hfill \end{array}$$

(30)

$$\{\begin{array}{c}{\dot{\gamma }}_3=-{\mu }_{1}s\dot{k}\hfill \\ {\dot{\gamma }}_4=-{\mu }_{2}s{\displaystyle {\int }_0^t}{k}^{\frac{q}{p}}d\tau \hfill \end{array}$$

(31)

where ${\mu }_1$ and ${\mu }_2$ are positive gain parameters.

4.2. Control Law Design

The sliding mode function control law is designed as follows:

$${\delta }_f=-\frac{\left({\gamma }_3\dot{k}+{\gamma }_4{\dot{k}}_a+{\phi }_1+{\epsilon }_{1}sign\left(s\right)+{\epsilon }_{2}s\right)}{{\phi }_2}$$

(32)

where ${\epsilon }_1$ and ${\epsilon }_2$ are positive constants, ${\epsilon }_1>{\left|d\right|}_{max}$, ${\gamma }_3$ and ${\gamma }_4$ are derived from Formula (31), and sign(s) is a sign function. According to Equation (27),${\phi }_2=x_m\frac{2L_f{C}_{\alpha f}}{I_z}$, L_f is the distance from the center of mass to the front axle, C_αf is the cornering stiffness of the front wheel, I_z is the yaw moment of inertia of the vehicle body, and x_m is the preview distance. So, φ₂ will not be 0. Therefore, this formula is correct.

4.3. Proof of Stability

According to the Lyapunov function, the following formula is obtained:

$$V=\frac{1}{2}{s}^2$$

(33)

Take the derivative of V according to the formula:

$$\begin{array}{ll}\dot{V}& =s\dot{s}\\ & =s\left(\ddot{k}+{\dot{\gamma }}_{3}k+{\gamma }_3\dot{k}+{\dot{\gamma }}_4{k}_a+{\gamma }_4{\dot{k}}_a\right)\\ & =s\left({\phi }_1+{\phi }_2{\delta }_f+d+{\dot{\gamma }}_{3}k+{\gamma }_3\dot{k}+{\dot{\gamma }}_4{k}_a+{\gamma }_4{\dot{k}}_a\right)\end{array}$$

(34)

From Formulas (30)–(32), the following formulas are obtained:

$$\begin{array}{ll}\dot{V}& =s\left(-{\mu }_{1}s{\left(k\right)}^2-{\mu }_{2}s{\left({{\displaystyle \int }}_0^t{k}^{\frac{q}{p}}d\tau \right)}^2-{\epsilon }_{1}sign\left(s\right)-{\epsilon }_{2}s+d\right)\\ & =sd-{\epsilon }_1\left|s\right|-{\epsilon }_2{s}^2-{\mu }_2{s}^2{\left({{\displaystyle \int }}_0^t{k}^{\frac{q}{p}}d\tau \right)}^2-{\mu }_1{s}^2{\left(k\right)}^2\end{array}$$

(35)

where $-{\epsilon }_2{s}^2-{\mu }_2{s}^2{\left({{\displaystyle \int }}_0^t{k}^{\frac{q}{p}}d\tau \right)}^2-{\mu }_1{s}^2{\left(k\right)}^2$<0, ${\epsilon }_1>{\left|d\right|}_{max}$, so $\dot{V}<0$. Therefore, for any ${\epsilon }_1>{\left|d\right|}_{max}$, the designed controller is convergent.

5. Experiments

In this paper, the trajectory planning simulation experiment of the improved RRT algorithm and the Matlab-Carsim HIL simulation platform experiment were carried out, respectively. Among them, the trajectory planned by the improved RRT algorithm was used as the expected trajectory. The Matlab-Carsim HIL simulation platform experiment used the state quantity output by Carsim to control the state of the bench. It simulated the motion control of the real intelligent vehicle. Through this experiment, it can be verified that the designed controller can be applied to the intelligent driving platform. In this paper, the result of trajectory planning is magnified ten times as the desired trajectory. The initial speed of the experiment is set to 60 km/h, the initial positions are (20 m, 20 m), and the target positions are (300 m, 300 m).

5.1. Introduction to Simulation Hardware

In order to verify the trajectory tracking performance of the improved terminal sliding mode controller designed in this paper, simulation experiments were carried out using MATLAB 2019b, Carsim2019 and Matlab-Carsim HIL simulation platforms. Matlab-Carsim HIL simulation platform is shown in Figure 9. It includes a servo motor driver, display, industrial computer, rear wheel drive motor, etc.

5.2. Trajectory Planning Simulation Experiment

Figure 10 shows the simulation comparison of different trajectory planning algorithms. Figure 10a is the simulation image of the improved RRT algorithm: blue represents static obstacles of different shapes, and black represents the final trajectory. Figure 10b is the simulation image of the improved A* algorithm: blue represents static obstacles of different shapes, and black represents the final trajectory. The initial positions are (2 m, 2 m), and the target positions are (30 m, 30 m). The improved RRT algorithm takes 0.02 s, and the improved A* algorithm takes 0.05 s.

Experimental results: from Figure 10, it can be found that the trajectory planned by the improved RRT algorithm in this paper can effectively avoid obstacles and the algorithm is extended, taking into account vehicle kinematics and geometric constraints. The planned trajectory is relatively smooth, which is good for vehicle tracking. According to the simulation time, the planning efficiency of the improved RRT algorithm is much higher than that of the improved A* algorithm.

5.3. Sliding Mode Controllers Chattering Analysis

The chattering of the sliding mode controller mainly comes from the sign function. Too much chattering is not conducive to the tracking control, and may cause the controller to crash. Through experiments, it was found that adjusting the size of ${\epsilon }_1/{\epsilon }_2$ can effectively suppress chattering. However, suppressing chattering to a certain extent will affect the accuracy of the controller, resulting in a larger tracking error. Therefore, while ensuring a certain tracking accuracy, chattering with little amplitude variation is acceptable. The test results of ${\epsilon }_1/{\epsilon }_2$ from large to small are shown in Figure 11.

5.4. Matlab-Carsim HIL Simulation Platform Test

Figure 12 shows the comparison experimental image of the Matlab-Carsim HIL simulation platform between the improved terminal sliding mode control and the general terminal sliding mode control. The initial speed was 60 km/h. As shown in Figure 12a, the improved algorithm tracks the desired trajectory image. The black represents the desired trajectory, and the red–dotted line represents the tracking trajectory. Figure 12b shows a partial enlarged image of the improved algorithm tracking the desired trajectory. As shown in Figure 12c, the general algorithm tracks the desired trajectory image. The black represents the desired trajectory, and the blue–dotted line is the tracking trajectory. Figure 12d shows a partial enlarged image of the desired trajectory tracked by the general algorithm. Figure 12e shows the lateral error image when the improved algorithm and the general algorithm track the desired trajectory. The red is the error of the improved algorithm, and the blue is the error of the general algorithm. Moreover, the positional errors at 50 m and 190 m become larger, because a large amount of steering is performed at the corresponding position. Figure 12f shows the image of the front wheel turning angle when the improved algorithm and the general algorithm track the desired trajectory. The red is the improved algorithm, and the blue is the general algorithm. Figure 12g shows the image of the side-slip angle of the centroid when the improved algorithm and the general algorithm track the desired trajectory. The red is the improved algorithm, and the blue is the general algorithm.

Figure 12f,g change the data values greatly at 3 s and 15 s~25 s, because the intelligent vehicle makes a large turn at the corresponding position, especially the continuous turning at 15 s~25 s. Figure 12h is an image of the longitudinal velocity with an acceleration of 1 m/s². The Matlab-Carsim HIL simulation platform test is shown in Figure 13. Figure 13a,b are the images displayed by Carsim in the Matlab-Carsim HIL simulation platform. Figure 13c,d are the test images of the bench in the Matlab-Carsim HIL simulation platform.

Table 1. Maximum lateral error.

Algorithm	50 m	189 m	192 m
Improved(m)	0.194	1.350	−0.797
General(m)	0.846	1.793	−0.849

,

Table 2. Maximum front wheel angle error.

Algorithm	2 s	16 s	20 s
Improved (degree)	12.84	7.99	−2.40
General (degree)	21.98	12.45	−2.36

and

Table 3. Maximum centroid side slip angle error.

Algorithm	2 s	16 s	20 s
Improved (degree)	−2.373	2.335	−1.087
General (degree)	−2.852	2.339	−1.001

show the maximum deviation at the corresponding position. The improved algorithm has a higher tracking accuracy than the general algorithm in discontinuous sharp turns. During continuous steering, the tracking accuracy of the improved algorithm will gradually decrease. However, it can be seen from Figure 12 that the improved algorithm has less chattering and good stability, so it is easier to control.

5.5. Experimental Analysis

In motion planning, the improved RRT trajectory planning algorithm is more efficient than the improved A* algorithm. The trajectory is smoother, which is conducive to intelligent vehicle driving.

In the motion control, the test results of the Matlab-Carsim HIL simulation platform show that, compared with the general terminal sliding mode control, the improved algorithm in this paper has a smaller tracking error in both initial tracking and continuous steering tracking. So, the tracking accuracy is higher. In addition, the front wheel angle and the center of mass slip angle data are smooth, so the body stability is good.

6. Conclusions

Aiming at the motion control problem of intelligent vehicle tracking the desired trajectory effectively, this paper proposes a trajectory planning and control strategy based on an improved terminal sliding mode. This method has the advantages of strong robustness and high tracking accuracy. The specific content of this article is summarized as follows:

• The traditional RRT algorithm is improved by using the target bias strategy and the separation axis theorem, which reduces the sampling area and improves the planning efficiency of the algorithm;
• This paper designs an improved terminal sliding mode controller. The controller comprehensively considers the lateral error and heading error of intelligent vehicle trajectory tracking to improve the accuracy of trajectory tracking;
• In this paper, the simulation experiment of the improved RRT algorithm and the Matlab-Carsim HIL simulation platform experiment are carried out. The experimental results show that the designed controller has high precision and strong robustness. Therefore, the improved algorithm can be applied to the intelligent driving platform and has certain engineering application value.

In future research, the controller needs to be further studied in the continuous steering state so that it can still maintain a high tracking accuracy. Experiments in complex environments will also be required in the future. Therefore, a nonlinear tire model is required.