Research on Curved Path Tracking Control for Four-Wheel Steering Vehicle considering Road Adhesion Coefficient

Liu, Runqiao; Wei, Minxiang; Sang, Nan; Wei, Jianwei

doi:https://doi.org/10.1155/2020/3108589

Mathematical Problems in Engineering

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Abstract Introduction Conclusions Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 3108589 | https://doi.org/10.1155/2020/3108589

Research on Curved Path Tracking Control for Four-Wheel Steering Vehicle considering Road Adhesion Coefficient

Runqiao Liu,¹Minxiang Wei,¹Nan Sang,²and Jianwei Wei³

Academic Editor: Ramon Sancibrian

Received14 Oct 2019

Revised14 Dec 2019

Accepted26 Dec 2019

Published20 Jan 2020

Abstract

Curved path tracking control is one of the most important functions of autonomous vehicles. First, small turning radius circular bends considering bend quadrant and travel direction restrictions are planned by polar coordinate equations. Second, an estimator of a vehicle state parameter and road adhesion coefficient based on an extended Kalman filter is designed. To improve the convenience and accuracy of the estimator, the combined slip theory, trigonometric function group fitting, and cubic spline interpolation are used to estimate the longitudinal and lateral forces of the tire model (215/55 R17). Third, to minimize the lateral displacement and yaw angle tracking errors of a four-wheel steering (4WS) vehicle, the front-wheel steering angle of the 4WS vehicle is corrected by a model predictive control (MPC) feed-back controller. Finally, CarSim® simulation results show that the 4WS autonomous vehicle based on the MPC feed-back controller can not only significantly improve the curved path tracking performance but also effectively reduce the probability of drifting or rushing out of the runway at high speeds and on low-adhesion roads.

1. Introduction

Path tracking control enables autonomous vehicles to travel accurately, smoothly, and safely along a predetermined path and to remain stable under all driving conditions [1]. As early as 1980, MacAdam [2] proposed an optimal preview path tracking control method, which could minimize the tracking error in channel-changing simulation. Currently, this control method has been applied to the path tracking model in CarSim® software [3]. In addition, the model predictive control (MPC) method can easily handle the nonholonomic constraints of a vehicle system and solve the trade-off problem between path tracking and driving safety [4]. Due to these advantages, many researchers have studied the MPC path tracking method. Borrelli et al. [5] presented a nonlinear MPC approach for active front-wheel steering control. Katriniok and Abel [6] used a predictive controller based on a linear time-varying model (LTV-MPC) to minimize tracking error and response lag. Kim et al. [7] used an MPC path tracking control method based on quadratic programming (QP) optimization to improve the path tracking performance of autonomous vehicles. Schwarting et al. [8] used a receding horizon planner based on nonlinear model predictive control (NMPC) to ensure safety collision avoidance. However, the above control methods ignore the influence of high speeds and low-adhesion roads on curved path tracking.

Road adhesion coefficient plays an important role in vehicle electronic control systems. Accurate road adhesion coefficient estimation is critical to improve the path tracking performance of autonomous vehicles. In recent years, to estimate accurate road adhesion coefficient, scholars have proposed various estimation methods. Hahn et al. [9] reported a road friction coefficient estimation algorithm based on a differential global positioning system (DGPS). Wenzel et al. [10] proposed an estimation method based on a double extended Kalman filter (DEKF). Enisz et al. [11] used an extended Kalman filter (EKF) method to estimate the instantaneous and maximum values of road friction coefficient. Zhao et al. [12] designed an EKF method based on a braking dynamics model considering the load transfer of front and rear axles.

For steering on a wet and slippery road, when the front-wheel loses most of the lateral force (grip), the vehicle (front-drive) will push ahead or rush out of the runway (understeer); when the rear wheel loses most of the lateral force (grip), the vehicle (rear-drive) will drift out or have a tail flicking (oversteer) [13]. To reduce the understeer and oversteer in path tracking, Hang and Chen [14] used a path tracking integrated control based on four-wheel steering (4WS) and direct yaw-moment control to improve the path tracking performance of vehicles on low-adhesion roads. It is generally known that the handling stability and driving safety of 4WS vehicles are much higher than those of front-wheel steering (FWS) vehicles [15, 16]. As early as 1986, Sano et al. [17] proposed a speed function and a steering angle function for 4WS vehicles. In a small steering wheel angle range or at high speed, the rear wheels of the 4WS vehicle were steered in the same direction as the front ones. In a large steering wheel angle range or at low speed, the rear wheels of the 4WS vehicle were steered in the opposite direction to the front ones. This greatly improves the handling stability and driving safety of 4WS vehicles. In [18, 19], by setting the zero-sideslip angle of 4WS vehicle, the stability at high speed and the sensitivity at low speed were significantly improved. In a recent study by Liu et al. [20], a feed-forward controller that can adjust the steering sensitivity coefficient of the 4WS vehicle based on zero-sideslip angle according to the steering angular frequency and vehicle speed is proposed, but this controller is not suitable for low-adhesion curved roads.

The rest of this paper is organized as follows. First of all, small turning radius circular bends (single circle center) considering different restrictions on bend quadrant and travel direction are planned by polar coordinate equations in Section 2. Then, in Section 3, an estimator based on the extended Kalman filter for the vehicle state parameter and road adhesion coefficient is designed. To improve the convenience and accuracy of the estimator, we use the combined slip theory, trigonometric function group fitting, and cubic spline interpolation to estimate the longitudinal and lateral forces of the tire model (215/55 R17). Next, in Section 4, a 4WS vehicle based on a MPC feed-back controller is used to improve the curved path tracking performance at high speeds and on low-adhesion roads, and the lateral displacement and yaw angle tracking errors of the 4WS vehicle in path tracking are minimized. Furthermore, CarSim® simulation effects for the optimal preview feed-back and MPC feed-back controllers under different curved paths, vehicle speeds, and road adhesion coefficients are compared in Section 5. Finally, Section 6 presents the conclusions of the work.

2. Circular Bend and Vehicle Model

2.1. Circular Bend

It is necessary for autonomous vehicles to plan a predetermined curved path in curved path tracking [21–25]. The predetermined path (target path) is usually obtained by high-precision vehicle positioning system, combining the global positioning system (GPS) with the inertial navigation system (INS) [26]. In the absence of electronic map (E-map) data, to facilitate calculations and simulations on small turning radius circular bends (single circle center), the circular bends used as target paths can be planned by polar coordinate equations.

A polar coordinate equation can express a circular bend simply and accurately, and the arc length of the circular bend is defined as follows:where is the maximum radian of the circular bend and is the turning radius (m).

To plan all forms of circular bends based on the ground inertial coordinate system, we introduce constraints on bend quadrant and travel direction. The equations of the radian with respect to for the circular bends under different restrictions are described as follows.(1)Travel clockwise:(2)Travel anticlockwise:where is the driving time; is the radian value corresponding to time ; represents the longitudinal speed based on vehicle coordinate system; is the first quadrant; is the second quadrant; is the third quadrant; and is the fourth quadrant. Then, the coordinate equations for the circular bends are expressed as follows:where is the circle center coordinate; is the desired lateral displacement based on the ground inertial coordinate system; and is the desired longitudinal displacement based on the ground inertial coordinate system.

After the derivative of the function of with respect to , the function of the desired yaw angle with respect to is expressed as follows:where represents the desired yaw angle based on the ground inertial coordinate system.

After the derivative of the function of with respect to , the function of the desired yaw rate with respect to is formulated as follows:

Taking the restriction of traveling clockwise as an example, Figure 1 shows the right-angle circular bends under four different quadrants, where is the corner angle and (deg).

Taking the restrictions of traveling anticlockwise and the second quadrant () as an example, Figure 2 shows the four circular bend diagrams with acute, right, obtuse, and U-shaped angles that meet the restrictions.

(a)

(b)

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Finally, the target paths can be converted into the desired front and rear wheel steering angles and used as initial inputs of autonomous vehicles. For a 4WS vehicle, the yaw rate response is generally set to a first-order lag response and the sideslip angle response is generally set to 0 [27–31]. Its initial inputs can be formulated as follows:where is a Laplace operator; ; is the desired steering sensitivity coefficient and ; is the stability factor and ; and are initial inputs of front and rear wheel steering angles for 4WS vehicles, respectively; and .

2.2. Vehicle Model

A vehicle model (D-class Sedan) of CarSim® is used in the following simulations. The vehicle model is front-drive (Generic 200 KW powertrain with an automatic transmission). A tire model (215/55 R17) used in the vehicle model is a standard lookup table tire model. The main parameters for the vehicle model are shown in Table 1.

3. Estimator Based on EKF

3.1. Vehicle State Parameter Estimation

Ignoring the effects of the suspension system, road roughness, and air resistance, a three-degree-of-freedom (3DOF) vehicle dynamics model considering lateral, longitudinal, and yaw motions is used to estimate the longitudinal speed, sideslip angle, and yaw rate of vehicles [32]. The state space equation of the vehicle dynamics model is shown as follows:

The measurement equation of the vehicle model is shown as follows:where is the front-wheel steering angle, r is the yaw rate based on the vehicle coordinate system, is the sideslip angle based on the vehicle coordinate system, is the longitudinal acceleration based on the vehicle coordinate system, is the lateral acceleration based on the vehicle coordinate system, the input vector is , the state vector is , and the measurement vector is . After the partial differential of equations (8) and (9) with respect to , the Jacobian matrices and H in Figure 3 are calculated as follows:

Then, the flow chart of EKF algorithm is shown in Figure 3, where ; is the sampling time; is the covariance matrix of process noise; is the covariance matrix of measurement noise; is the state space equation; is the state estimation vector at time ; is the state error covariance matrix at time ; is the state estimation vector at time ; and is the state error covariance matrix at time . Given the initial values and and and can be updated cyclically. The above estimation process is continually looped until the number of set steps.

3.2. Tire Longitudinal and Lateral Forces Estimation

Professor Pacejka put forward a “magic formula” tire model in 1987, which estimates the tire longitudinal and lateral forces by constructing the trigonometric function group. The tire longitudinal and lateral force estimation equations under a single working condition are shown as follows:where ; is the slip ratio; and are the stiffness factors of the tire longitudinal and lateral forces, respectively; and are the curve shape factors of the tire longitudinal and lateral forces, respectively; and are the curve peak factors of the tire longitudinal and lateral forces, respectively; and and are the curvature factors at the curve peak of the tire longitudinal and lateral forces, respectively. These factors are represented by equations containing several different coefficients [33].

Although the “magic formula” has a high fitting accuracy for the tire longitudinal and lateral forces, these coefficients in the above factors are difficult to fit [34, 35]. In order to estimate the longitudinal and lateral forces of the tire model (215/55 R17) conveniently and accurately, we use the combined slip theory, trigonometric function group fitting, and cubic spline interpolation to estimate longitudinal and lateral forces of the tire model.

Firstly, combined with the longitudinal and lateral force tabular data of the tire model (the tabular data are measured on a laboratory or on-road tester, and they are all measured under the condition that the road adhesion coefficient is 1), tabular data under 10 different tire loads are selected, which are represented by the purple, pink, green, cyan, and blue lines in Figure 4. Using the trigonometric function group to fit these tabular data, the fitting results are represented by the five types of black lines in Figure 4. These factors after fitting are shown as follows (RMSE is the root mean square error):(1)Tire load 1725: is 9.342; is 2.753; is 1891.4; is 1.123; and RMSE is 0.0096(2)Tire load 3500: is 9.909; is 2.694; is 3698.8; is 1.114; and RMSE is 0.0089(3)Tire load 6100: is 10.17; is 2.626; is 5382.1; is 1.109; and RMSE is 0.0077(4)Tire load 6950: is 9.943; is 2.573; is 6971.6; is 1.112; and RMSE is 0.0066(5)Tire load 9005: is 9.029; is 2.565; is 8564.5; is 1.126; and RMSE is 0.0055(6)Tire load 2105: is 11.37; is 1.528; is 2729.5; is −0.5744; and RMSE is 0.0016(7)Tire load 3995: is 11.37; is 1.528; is 5459.1; is −0.5744; and RMSE is 0.0016.(8)Tire load 6120: is 11.37; is 1.528; is 8188.6; is −0.5744; and RMSE is 0.0016(9)Tire load 7900: is 11.37; is 1.528; is 10918; is −0.5744; and RMSE is 0.0016(10)Tire load 10100: is 11.37; is 1.528; is 13648; is −0.5745; and RMSE is 0.0016

(a)

(b)

Secondly, the cubic spline interpolation is used to estimate these factors under other tire loads. Taking the estimation under a certain tire load as an example, the circular calling format of the cubic spline interpolation is shown as follows:where ; z is a certain tire load; ; y_z is the interpolation result; and is the function about cubic spline interpolation. Similarly, the remaining factors can be estimated. Then, the longitudinal force estimation equation (under pure braking/driving conditions) and lateral force estimation equation (under pure steering conditions) under any tire load can be obtained.

Finally, using the combined slip theory [36, 37], the tire longitudinal and lateral forces under joint conditions can be described as follows:where ; ; ; ; is the tire load; ; ; , , , and are the left front, left rear, right front, and right rear wheel longitudinal forces under joint conditions, respectively; and , , , and are the left front, left rear, right front, and right rear wheel lateral forces under joint conditions, respectively.

3.3. Road Adhesion Coefficient Estimation

A 3DOF vehicle dynamics model involving tire lateral and longitudinal forces under joint conditions is used to estimate the road adhesion coefficient [38]. The state space equation of the vehicle dynamics model is shown as follows:where the input vector ; the state vector , , , , and are the left front, right front, left rear, and right rear wheel adhesion coefficients, respectively; and the measurement vector . After the partial differential of equation (14) with respect to , the Jacobian matrices and in Figure 3 are calculated as follows:

The flow chart of the road adhesion coefficient estimation algorithm is similar to that of the vehicle state parameter estimation, as shown in Figure 3.

3.4. Verification of the Estimator

The overall block diagram of the estimator for vehicle state parameter and road adhesion coefficient is shown in Figure 5.

A joint adhesion road is used in the verification. The features of the joint adhesion road are detailed as follows: a road of a total length of about 321 m, with the first 28 m a high- section (a road adhesion coefficient of 0.85), and the rest a low- section (a road adhesion coefficient of 0.5). In addition, a two-way single-lane road is used in the estimation, and it is a right-angled circular bend (the corner angle is 90 deg), as shown in Figure 6(a), where the width of each lane is 6 m, the blue dashed line is the target path, and the turning radius is 37.5 m. The estimated road adhesion coefficient and vehicle state parameters at are shown in Figures 6(b)–6(e), where a uniformly distributed random signal (the minimum value is 0 and the maximum value is 0.02) is added to the actual outputs of CarSim® vehicle considering uncertain disturbances. The parameter settings of vehicle state parameter estimation are shown as follows: ; ; ; ; and can be updated cyclically in the form of matrix. The parameter settings of road adhesion coefficient estimation are shown as follows: ; ; ; ; and can be updated cyclically in the form of matrix. Figure 6 shows that even in the case of a small turning radius and low-adhesion road, the estimator can quickly and accurately estimate the vehicle state parameters and road adhesion coefficient.

(a)

(b)

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(e)

4. MPC Feed-Back Controller

MPC generally includes predictive model, rolling optimization, and feed-back correction. It mainly compensates for system errors through feed-back correction and compensates for the effects caused by time variance and interference through rolling optimization [39, 40]. Since the effects of the tire nonlinearity, tire load transfer, road low-adhesion, and crosswind, the actual trajectories often deviate from the desired trajectories in path tracking [20]. To minimize these deviations, a MPC feed-back controller is used to follow the desired lateral displacement and the desired yaw angle and minimize the tracking errors between the desired values and the actual values. First, a nonlinear dynamic model considering the transformation between the vehicle coordinate system and the ground inertial coordinate system is described as follows:where is the rear-wheel steering angle; is the lateral speed based on the vehicle coordinate system; is the lateral speed based on the ground inertial coordinate system; is the longitudinal speed based on the ground inertial coordinate system; and is the yaw angle based on the ground inertial coordinate system. Then, it is transformed intowhere

To apply equation (17) to the MPC system, it is discretized as follows:where is the sampling time. It is further converted intowhere the state vector ; the input vector ; the output vector ; and ; and and are the dimensions of the state vector and output vector, respectively.

Next is a rolling optimization process, we can convert the MPC problem into a quadratic programming problem [41]. Its mathematical model is expressed as follows:where is the input increment; is the quadratic matrix of quadratic programming; is the one item vector of quadratic programming; is the coefficient matrix in linear inequality constraint; is the right-end vector in linear inequality constraint; ; ; and is the relaxation factor. Its optimization objective function is set towhere and are weighting matrices; is the weight coefficient; is the prediction time domain; and is the control time domain.

To avoid a sudden change in the input vector, the problem of constrained optimization for each step is shown as follows [42, 43]:

Calling the quadratic programming function (quadprog), which is expressed as follows:where and .

After solving each control cycle, a series of input increments in the control time domain are obtained:

Finally, there is a process of feed-back correction. The first element of the control sequence acts on the MPC system:where represents the feed-back control (front-wheel steering angle). The above control process is continually looped until path tracking is completed.

5. Curved Path Tracking Simulations

5.1. Two-Path Tracking Controllers

Two-path tracking controllers for autonomous vehicles in the following simulations are shown as follows:(1)The optimal preview feed-back controller for an FWS autonomous vehicle is defined as controller 1, which is an FWS path tracking controller. CarSim® has its own optimal preview path tracking control model located in “Steering by the Closed-loop Driver Model.” The red vehicle in the following simulations represents controller 1.(2)The MPC feed-back controller for a 4WS autonomous vehicle is defined as controller 2, which is a 4WS path tracking controller. The blue vehicle in the following simulations represents controller 2. The block diagram of controller 2 is shown in Figure 7, where and .

5.2. Circular Bend Simulations at Low Speeds and on High-Adhesion Roads

For the circular bend simulation, the coordinates of the target path can be updated continuously by polar coordinate equations depending on , u_c, , and variables.

The target path of this simulation is a right-angled circular bend (travel anticlockwise and second quadrant), which is shown in Figure 2(b) (the corner angle is 90 deg). A one-way four-lane road is applied to the simulation, where the width of each lane is 3.5 m, the turning radius is 37.5 m, the initial coordinates of the autonomous vehicle are , the ending coordinates of the autonomous vehicle are , the circle center coordinates are , and the road adhesion coefficient of the dry asphalt road is set to 0.85. The road conditions meet the SAE standards J3087_201710 [44]. The path tracking animation effects and steering angles for the red and blue vehicles at and are shown in Figure 8. The longitudinal and lateral displacements, lateral accelerations, yaw angles, and yaw rates for the red and blue vehicles at and are shown in Figure 9. The detailed constraints of the MPC are shown as follows: the input increment constraints, ; the input vector constraints, ; the output vector constraints: ; ; ; ; ; ; ; and . The detailed optimal preview parameter settings are shown as follows: the driver preview time is 0.5 s; the driver time lag is 0 s; the low-speed dynamic limit is 10 km/h; the maximum steering wheel angle is 720 deg; and the maximum steering wheel angle rate is 1200 deg/s.

(a)

(b)

(c)

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(b)

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(d)

It can be seen from Figures 8(b) and 8(c) that the red vehicle is an FWS vehicle and the blue vehicle is a 4WS vehicle. Even if the two vehicles have different front and rear wheel steering angles, Figures 9(a) and 9(c) show that they can still achieve effective path tracking without pushing ahead (understeer) or tail flicking (oversteer) at low speeds and on high-adhesion roads. It can be seen that there is little difference in the control effect under steady-state conditions in Figures 9(b) and 9(d).

5.3. Circular Bend Simulations at Low Speeds and on Low-Adhesion Roads

The target path of this simulation is the same as in Section 5.2. The road adhesion coefficient of the wet and slippery road is set to 0.5. The curved path tracking animation effects and steering angles for red and blue vehicles at and are shown in Figure 10. The longitudinal and lateral displacements, lateral accelerations, yaw angles, and yaw rates for red and blue vehicles at and are shown in Figure 11. The constraints of the MPC and the optimal preview parameter settings are the same as in Section 5.2.

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When drops to 0.5, Figure 10(a) shows that the red vehicle has obvious sideslip (the maximum deviation is about 4 m). Figure 10(a) shows that the blue vehicle has smaller sideslip (the maximum deviation is about 1 m), and its path tracking performance is significantly better than that of the red vehicle on the low-adhesion road. The reason for this phenomenon is that the rear wheels of the blue vehicle are steered in the opposite direction to the front ones when entering the corner, and its rear wheels are steered in the same direction as the front ones when driving out of the corner, as shown in Figures 10(b) and 10(c). This situation greatly improves the safety and stability of the 4WS vehicle in path tracking on a small turning radius and low-adhesion curved road [45]. Figure 11 shows that the yaw angle, yaw rate, and lateral acceleration values of the blue vehicle are closer to the desired values, and the vehicle maintains good path tracking performance even on low-adhesion roads with small turning radius.

5.4. True Variable Curvature Curved Road Simulations at High Speeds and on Low-Adhesion Roads

For the curved path tracking on real roads, the polar coordinate equation (4) are replaced by the coordinates of the electronic map (E-map), and the coordinates of the target path can be updated continuously by high-precision vehicle positioning system combining GPS with INS.

A true variable curvature curved road is shown as follows: the road adhesion coefficient of the wet and slippery road is set to 0.5; the minimum turning radius of the bend ① is ; the minimum turning radius of the bend ② is ; and the minimum turning radius of the bend ③ is . The coordinates of the true variable curvature curved road are shown in Figure 12(a). The cross slopes of the true variable curvature curved road are shown in Figure 12(b).

(a)

(b)

The constraints of the MPC are shown as follows: the output vector constraints is , and other constraints are the same as in Section 5.2. The optimal preview parameter settings are the same as in Section 5.2. The curved path tracking effects for red and blue vehicles at and are shown in Figure 13. Figure 13 shows that the red vehicle loses control and rushes out of the runway when passing the bend ③, but the blue vehicle maintains good path tracking performance even at high speeds and on low-adhesion roads.

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6. Conclusions

The results of this paper can be summarized as follows.

Since the combined slip theory, trigonometric function group fitting, and cubic spline interpolation are used, it is more convenient and accurate in estimating longitudinal and lateral forces and road adhesion coefficients. Compared to the FWS vehicle with an optimal preview path tracking controller, the 4WS vehicle with a MPC path tracking controller can more effectively improve safety and stability at high speeds and on low-adhesion roads. In the future, the path tracking test in the real 4WS autonomous vehicles needs to be implemented and the suspension and braking control systems for 4WS autonomous vehicles in curved path tracking need to be studied further.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare no conflicts of interest.

Authors’ Contributions

R.L. put forward the original concept, proposed the control strategy, and wrote the article. M.W., J.W., and N.S. gave their valuable suggestions on the research design. Furthermore, R.L., M.W., J.W., and N.S. analyzed and discussed the simulation results.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 51775268), Funding of Jiangsu Innovation Program for Graduate Education (Grant no. KYLX16_0328), and Zhejiang Provincial Public Welfare Technology Application Research Project of China (Grant no. 2017C35011).

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Copyright © 2020 Runqiao Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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