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Abstract

To solve the problem of understeer and oversteer for autonomous vehicle under high-speed emergency obstacle avoidance conditions, considering the effect of steering angular frequency and vehicle speed on yaw rate for four-wheel steering vehicles in the frequency domain, a feed-forward controller for four-wheel steering autonomous vehicles that tracks the desired yaw rate is proposed. Furthermore, the steering sensitivity coefficient of the vehicle is compensated linearly with the change in the steering angular frequency and vehicle speed. In addition, to minimize the tracking errors caused by vehicle nonlinearity and external disturbances, an active disturbance rejection control feedback controller that tracks the desired lateral displacement and desired yaw angle is designed. Finally, CarSim® obstacle avoidance simulation results show that an autonomous vehicle with the four-wheel steering path tracking controller consisting of feed-forward control and feedback control could not only improve the tire lateral forces but also reduce tail flicking (oversteer) and pushing ahead (understeer) under high-speed emergency obstacle avoidance conditions.

Keywords

Active disturbance rejection control autonomous vehicle four-wheel steering high-speed emergency obstacle avoidance trajectory tracking

Introduction

When the distance between an autonomous vehicle and an obstacle is less than the minimum safe distance, steering obstacle avoidance will be implemented as the longitudinal brake is not sufficient to guarantee effective obstacle avoidance.¹ Steering obstacle avoidance is similar to the “elk test.” High-speed emergency obstacle avoidance for autonomous vehicles involves two main aspects: how to achieve effective obstacle avoidance and how to improve handling stability and prevent secondary accidents.^2,3

As early as 1989, Nelson⁴ proposed a quintic polynomial to describe an ideal obstacle avoidance trajectory that provided continuous curvature. Subsequently, many researchers used the trajectory to study emergency obstacle avoidance. Lee⁵ studied the improvement of stability and rollover resistance in high-g lateral maneuvers. Soudbakhsh and Eskandarian⁶ proposed an automated trajectory following system based on a linear quadratic method. Hassanzadeh et al.⁷ used a polynomial trajectory to calculate feed-forward control inputs and used a feedback controller based on a proportional derivative (PD) controller to compensate the yaw angle error caused by nonlinearity and uncertainty in autonomous collision avoidance control. Soudbakhsh et al.⁸ proposed a neighboring optimal controller for a collision avoidance system, and the entire obstacle avoidance time was controlled to be within 1.6 s. Shim et al.⁹ proposed a collision-free reference trajectory and a model predictive control controller that tracks the reference trajectory. Wang et al.¹⁰ presented a collision avoidance system consisting of a path planner and a robust tracking controller.

An active disturbance rejection control (ADRC) is used to minimize the tracking errors caused by vehicle nonlinearity and external disturbances. The ADRC can observe and compensate the external disturbances and model uncertainties without high mathematical model accuracy. The ADRC was first proposed by Professor Han Jingqing of the Chinese Academy of Sciences and further developed and improved by Professor Gao Zhiqiang of Cleveland State University. Sang et al.¹¹ proposed an active front steering control method that includes a PD feed-forward controller and an ADRC feedback controller. Wu et al.¹² used an active steering controller based on ADRC to follow the ideal yaw rate under high-speed emergency conditions.

The traditional four-wheel steering (4WS) vehicles use a proportional control strategy based on zero sideslip angle to become light and sensitive at low speed and stable and heavy at high speed.^13,14 Nevertheless, owing to the neglect of tracking yaw rate, these vehicles experience understeer at high speed and oversteer at low speed.^15,16 To solve the problem of understeer and oversteer for 4WS vehicles, Citroën Company (Paris, France) proposed the rear-wheel follow-up technology. Through the tilt steering principle of the drag arm suspension and the self-deflecting rubber bushing, the vehicles with peugeot steering system (PSS) technology could have an oversteer tendency at the beginning of cornering or emergency obstacle avoidance and an understeer tendency at the end of cornering or emergency obstacle avoidance. The representative vehicle type of the technology was DC7160AXC16V, but this technology was a passive control.¹⁷ Fukui et al.¹⁸ proposed a rear-to-front steer angle ratio based on the steering wheel operative speed: When the operative speed of the steering wheel was low, the rear wheels of 4WS vehicles were steered in the same direction as the front ones; in contrast, when the operative speed of the steering wheel was high, the rear wheels of 4WS vehicles were steered in the opposite direction to the front ones. In this article, the operative speed of the steering wheel is reflected by the steering angular frequency (ω).

The rest of this article is organized as follows. The second section describes the establishment of a 4WS vehicle model based on two-degree-of-freedom (2-DOF). In the third section, polynomial obstacle avoidance trajectories based on a ground inertial coordinate system are planned, and the trajectories are transformed into the initial inputs of an autonomous vehicle based on a vehicle coordinate system. In the fourth section, the obstacle avoidance trajectories are transformed into the initial inputs of an autonomous vehicle. In the fifth section, considering the effect of the steering angular frequency and vehicle speed (u_c ) on the yaw rate, a feed-forward controller based on linear compensation is used to track the desired yaw rate, thus preventing the yaw rate reduction and the steering lag caused by the increase in ω and u_c . In the sixth section, an ADRC feedback controller that tracks the desired lateral displacement and desired yaw angle is used to minimize the tracking errors caused by vehicle nonlinearity and external disturbances. The obstacle avoidance simulations are implemented in the seventh section. Finally, conclusions of this study are presented.

Linear 4WS vehicle model

To facilitate implementation and simplicity, a linear 4WS vehicle model based on 2-DOF is used to design a trajectory tracking controller.^19,20 Regardless of rolling, the state-space equation of the vehicle mode can be written as follows

\{\begin{cases} \dot{X} = A X + B U \\ Y = C X + D U \end{cases}

where $X = [\begin{matrix} β & r \end{matrix}]$ is the state vector, $U = [\begin{matrix} δ_{f} \\ δ_{r} \end{matrix}]$ is the input vector, and Y is the output vector. A, B, C, and D are the parameters of the vehicle mode. These parameters are described as follows

\begin{array}{l} A = [\begin{matrix} - \frac{C_{a f} + C_{a r}}{m u_{c}} & \frac{b C_{a r} - a C_{a f}}{m u_{c}^{2}} - 1 \\ \frac{b C_{a r} - a C_{a f}}{I_{z}} & - \frac{a^{2} C_{a f} + b^{2} C_{a r}}{I_{z} u_{c}} \end{matrix}] \\ B = [\begin{matrix} \frac{C_{a f}}{m u_{c}} & \frac{C_{a r}}{m u_{c}} \\ \frac{a C_{a f}}{I_{z}} & \frac{- b C_{a r}}{I_{z}} \end{matrix}] \\ C = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] \\ D = [0] \end{array}

where m is the vehicle mass, u_c is the longitudinal speed based on the vehicle coordinate system, a and b are the longitudinal distances from the vehicle center of gravity to the front and rear axles, respectively, I_z is the yaw moment of inertia, β is the sideslip angle, r is the yaw rate, and $δ_{f}$ and $δ_{r}$ are the front- and rear-wheel steering angles, respectively.

A vehicle model (D-class sedan) in CarSim® was used in the following obstacle avoidance simulations. The main parameters of the vehicle model are presented in Table 1.

Table 1.

Parameters of a D-class sedan.

Model parameter	Notation	Unit	Value
Yaw moment of inertia	I_z	kg·m²	4192
Vehicle mass	m	kg	1530
Sprung mass of vehicle	m_s	kg	1370
Distances from vehicle CG to front axle	a	m	1.11
Distances from vehicle CG to rear axle	b	m	1.66622
Distances from CG to ground	h_a	m	0.54
Vehicle wheelbase	l	m	2.77622
Initial transmission ratio of $δ_{sw}$ to $δ_{f}$	i ₀		23
Cornering stiffness of front wheel	$C_{a f}$	N rad⁻¹	195874
Cornering stiffness of rear wheel	$C_{a r}$	N rad⁻¹	140574

CG: center of gravity

Planning of obstacle avoidance trajectories

Two coordinate systems of autonomous vehicles are shown in Figure 1(a), where OXY is a ground inertial coordinate system and oxy is a vehicle coordinate system, Ω represents the yaw angle based on the ground inertial coordinate system, the initial coordinate of obstacle avoidance is (0,0), the final coordinate of obstacle avoidance is $(2 x_{e}, 0)$ , y_e is the total width of the obstacle, and the coordinate of the desired trajectory is $(x_{d}, y_{d}, Ω_{d})$ . A one-way four-lane road condition diagram for obstacle avoidance is shown in Figure 1(b), where the width of each lane is 3.5 m; the width of the autonomous vehicle is 2 m; and the road adhesion coefficient is set to 1. The above road conditions satisfy the SAE standards J3087_201710.²¹

Figure 1.

Obstacle avoidance diagrams: (a) two coordinate systems of autonomous vehicles and (b) a one-way four-lane road condition for obstacle avoidance.

Polynomial trajectory

A quintic polynomial can be used to describe a desired obstacle avoidance trajectory, which is not only simple and flexible but also accurate.²² The functions of y_d on x_a and x_b for the desired obstacle avoidance trajectory can be expressed as follows

\{\begin{cases} y_{d} (x_{a}) = y_{e} \{10 {(x_{a} / x_{e})}^{3} - 15 {(x_{a} / x_{e})}^{4} + 6 {(x_{a} / x_{e})}^{5}\} \\ 0 \leq x_{a} \leq x_{e} \\ y_{d} (x_{b}) = y_{e} \{10 {((2 x_{e} - x_{b}) / x_{e})}^{3} - 15 {((2 x_{e} - x_{b}) / x_{e})}^{4} + 6 {((2 x_{e} - x_{b}) / x_{e})}^{5}\} \\ x_{e} \leq x_{b} \leq 2 x_{e} \end{cases}

Cosine trajectory

A cosine function can also be used to describe a desired obstacle avoidance trajectory.²³ The function of y_d on x for the desired obstacle avoidance trajectory can be expressed as follows

\{\begin{cases} y_{d} (x) = \frac{y_{e}}{2} \{1 - cos (\frac{π}{x_{e}} x)\} \\ 0 \leq x \leq 2 x_{e} \end{cases}

Conversion effects of two different trajectories

According to the road condition in Figure 1(a), when $x_{e} = 37.5 m$ , $y_{e} = 6 m$ , and $u_{c} = 90 km h^{- 1}$ , the conversion effects of two different obstacle avoidance trajectories are shown in Figure 2.

Figure 2.

Conversion effects of two different obstacle avoidance trajectories: (a) lateral displacementm (b) yaw angle, (c) yaw rate, and (d) curvature.

We hope that the planned obstacle avoidance trajectory is as smooth as possible and the curvatures at the beginning and end are as close to as possible. Figure 2(d) shows that the curvatures of the polynomial trajectory at the beginning and end of obstacle avoidance are zero. Hence, in this study, the polynomial trajectory is used in the following obstacle avoidance simulations.

Initial inputs of autonomous vehicle

Considering the derivative of the function $y_{d} (x)$ with respect to x, the curve length and the critical collision time for the obstacle avoidance trajectory are expressed as

\{\begin{cases} \int_{0}^{x_{e}} \sqrt{1 + {(y_{d}^{'} (x))}^{2}} d x = u_{c} t_{e} \\ 0 \leq x \leq x_{e} \end{cases}

where t_e is the critical collision time for the obstacle avoidance trajectory.

Transforming the function $y_{d}^{'} (x)$ into the inverse tangent trigonometric function, the function of $Ω_{d}$ on x is expressed as follows

\{\begin{cases} Ω_{d} (x) = arctan (y_{d}^{'} (x)) \\ 0 \leq x \leq 2 x_{e} \end{cases}

Considering the derivative of the function of $Ω_{d}$ with respect to t, the function of the desired yaw rate r_d with respect to t is formulated as

\{\begin{cases} r_{d} (t) = Ω_{d}^{'} (t) \\ 0 \leq t \leq 2 t_{e} \end{cases}

Combined with the desired yaw rate and road adhesion coefficient conditions, the desired steering wheel angle $δ_{sw}$ with respect to t under steady-state condition is expressed as follows^24,25

\{\begin{cases} r_{d d} (t) = r_{d} (t), \begin{matrix} |r_{d} (t)| \leq g μ / u_{c} \end{matrix} \\ r_{d d} (t) = g μ / u_{c}, \begin{matrix} |r_{d} (t)| > g μ / u_{c} \end{matrix} \\ δ_{sw} (t) = r_{d d} (t) / G_{r d} \\ 0 \leq t \leq 2 t_{e} \end{cases}

where μ is the road adhesion coefficient, $G_{r d}$ is the desired steering sensitivity coefficient (desired yaw rate/desired steering wheel angle) and $G_{r d} = \frac{u_{c}}{i_{0} l + i_{0} l K u_{c}^{2}}$ ²⁶, and K is the stability factor and $K = \frac{m (b C_{a r} - a C_{a f})}{l^{2} C_{a f} C_{a r}}$ . The conversion results under four different obstacle avoidance trajectories are shown in Figure 3.

Figure 3.

Conversion results under four different obstacle avoidance trajectories: (a) lateral displacement, (b) yaw angle, (c) yaw rate, and (d) steering wheel angle.

As shown in Figure 3(d), the desired steering wheel angle produced by the desired obstacle avoidance trajectory is similar to a sinusoidal waveform. We can use the steering angular frequency (ω) to reflect the operative speed of the steering wheel. By comparing the black and blue lines, we can observe that y_e only affects the amplitude of the desired steering wheel angle. By comparing the black, green, and blue lines, we can observe that x_e and u_c affect the ω and amplitude of the desired steering wheel angle.

For a front-wheel steering (FWS) autonomous vehicle, its initial inputs are formulated as follows

\{\begin{cases} \frac{δ_{f d 1} (t)}{δ_{sw} (t)} = \frac{1}{i_{0}} \\ \frac{δ_{r d 1} (t)}{δ_{sw} (t)} = 0 \\ 0 \leq t \leq 2 t_{e} \end{cases}

where $δ_{f d 1}$ and $δ_{r d 1}$ represent the desired front- and rear-wheel steering angles of the FWS autonomous vehicle, respectively.

For a 4WS autonomous 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 Its transfer functions from the steering wheel angle to the front- and rear-wheel steering angles are expressed as follows

\{\begin{cases} \frac{δ_{f d 2} (s)}{δ_{sw} (s)} = \frac{I_{z} s + m b u_{c} + \frac{l a C_{a f}}{u_{c}}}{l C_{a f}} \cdot \frac{G_{r d}}{1 + T_{r} s} \\ \frac{δ_{r d 2} (s)}{δ_{sw} (s)} = - \frac{I_{z} s - m a u_{c} + \frac{l b C_{a r}}{u_{c}}}{l C_{a r}} \cdot \frac{G_{r d}}{1 + T_{r} s} \end{cases}

where s is the Laplace operator, $δ_{f d 2}$ and $δ_{r d 2}$ represent the front- and rear-wheel initial inputs of the 4WS autonomous vehicle, respectively, and $T_{r} = \frac{1}{\sqrt{\frac{l^{2} C_{a f} C_{a r} (1 + K u_{c}^{2})}{u_{c}^{2} m I_{z}}}}$ .

Feed-forward control for 4WS

The yaw rate response of the 4WS autonomous vehicle mentioned in the previous section is a first-order lag system. Its yaw rate frequency-domain response diagrams under different ω and u_c conditions are shown in Figure 4, where the input is the steering wheel angle and the output is the yaw rate.

Figure 4.

Actual yaw rate frequency-domain response diagrams under different u_c conditions: (a) sys1 represents $u_{c} = 50 km h^{- 1}$ , (b) sys2 represents $u_{c} = 90 km h^{- 1}$ , and (c) sys3 represents $u_{c} = 130 km h^{- 1}$ .

Figure 4 shows that the amplitude of the actual yaw rate decreases with the increase in ω, and its response lag at high speed becomes more serious. It can be observed that ω and u_c have a considerable influence on the yaw rate. In the process of following the desired yaw rate r_d in equation (7), the actual yaw rate for the 4WS autonomous vehicle will be lower than the desired yaw rate owing to the yaw rate frequency-domain characteristics of the 4WS vehicle in the case of large ω and high u_c , thus increasing the understeer for the 4WS vehicle. For details, refer the work of Liu et al. ²⁸

We propose a linear compensation method for the first-order yaw rate lag system to prevent the yaw rate reduction and steering lag caused by the increase in ω and u_c . This method is applied to the trajectory tracking of the autonomous vehicle as a feed-forward control. The accurate curve and error curve for the first-order yaw rate lag system under different ω conditions are shown in Figure 5, where the vertical axis (dB) represents the decibels of yaw rate response; the horizontal axis represents ω; the constraint range of ω is $0.1 / T_{r} (rad s^{- 1}) \leq ω \leq 10 / T_{r} (rad s^{- 1})$ .

Figure 5.

First-order yaw rate frequency-domain response: (a) accurate curve and (b) error curve

To make the autonomous vehicle follow r_d accurately in the case of large ω and high u_c , combined with the $G_{r d}$ in equation (7) and Figure 5, the actual steering sensitivity coefficient G_r (yaw rate/steering wheel angle) is compensated linearly with the change in ω and u_c . The linear compensation method is described as follows.

First, if the steering angular frequency is $ω = 0$ , then the yaw rate in this angular frequency state is a steady-state value, and the decibels of the yaw rate are expressed as follows

L (0) = 20 lg G_{r d} dB

Second, if the range of the steering angular frequency $(ω = 2 π f = 2 π \frac{1}{t_{e}})$ is $0.1 / T_{r} (rad s^{- 1}) \leq ω \leq 1 / T_{r} (rad s^{- 1})$ , then the error occurring in this angular frequency state is expressed as follows

Δ L (2 π \frac{1}{t_{e}}) = 20 lg (\sqrt{1 + T_{r}^{2} {(2 π \frac{1}{t_{e}})}^{2}}) dB

and the decibels of the yaw rate in this state are expressed as follows

L (2 π \frac{1}{t_{e}}) = L (0) - Δ L (2 π \frac{1}{t_{e}}) dB

If the range of the steering angular frequency $(ω = 2 π f = 2 π \frac{1}{t_{e}})$ is $1 / T_{r} (rad s^{- 1}) \leq ω \leq 10 / T_{r} (rad s^{- 1})$ , then the error occurring in this state is expressed in equation (11), and the decibels of the yaw rate in this state are expressed as follows

L (2 π \frac{1}{t_{e}}) = L_{0} (2 π \frac{1}{t_{e}}) - Δ L (2 π \frac{1}{t_{e}}) dB

where

L_{0} (2 π \frac{1}{t_{e}}) = L (0) + 20 (lg (1 / T_{r}) - lg (2 π \frac{1}{t_{e}}))

Third, when $0.1 / T_{r} (rad s^{- 1}) \leq ω \leq 10 / T_{r} (rad s^{- 1})$ , combined with equations (12) and (13), the G_r value after compensation is expressed as follows

G_{r} = 10^{\frac{L {(0)}^{2}}{20 L (2 π \frac{1}{t_{e}})}}

Finally, substituting the G_r value into equation (9), the front- and rear-wheel feed-forward inputs of the 4WS autonomous vehicle are expressed as follows

\{\begin{cases} {\dot{x}}_{1} = - \frac{1}{T_{r}} + u \\ y_{a} = c_{1} x + d_{1} u \\ c_{1} = (- \frac{1}{T_{r}} + \frac{m b u_{c}^{2} + l a C_{a f}}{I_{z} u_{c}}) \frac{G_{r} I_{z}}{l C_{a f} T_{r}} \\ d_{1} = \frac{G_{r} I_{z}}{l C_{a f} T_{r}} \\ {\dot{x}}_{2} = - \frac{1}{T_{r}} + u \\ y_{b} = c_{2} x + d_{2} u \\ c_{2} = - (- \frac{1}{T_{r}} + \frac{- m a u_{c}^{2} + l b C_{a r}}{I_{z} u_{c}}) \frac{G_{r} I_{z}}{l C_{a r} T_{r}} \\ d_{2} = - \frac{G_{r} I_{z}}{l C_{a r} T_{r}} \\ 0 \leq t \leq 2 t_{e} \end{cases}

where $y_{a} = δ_{f 2}$ , which represents the front-wheel feed-forward input of the 4WS autonomous vehicle; $y_{b} = δ_{r 2}$ , which represents the rear-wheel feed-forward input of the 4WS autonomous vehicle.

Feedback control for 4WS

In the actual trajectory tracking process of an autonomous vehicle, uncertainties such as tire nonlinearity, load transfer, road adhesion, and crosswind disturbance seriously affect the trajectory tracking performance of the autonomous vehicle. To solve this problem, an ADRC feedback controller is used to track the desired yaw angle and desired lateral displacement and minimize the tracking errors between the actual output values and the desired values. The ADRC controller has good control effect in dealing with nonlinearity, decoupling, and time-varying problems. The ADRC controller consists of a tracking differentiator, an extended state observer (ESO), and a state error compensation law.²⁹ The state equation of the second-order linear control system for ADRC is expressed as

\{\begin{cases} {\dot{x}}_{1} = x_{2} \\ {\dot{x}}_{2} = f_{1} (β, r, t) + w_{1} (t) + b_{1} U_{f} + b_{1} u_{1} \\ {\dot{x}}_{3} = x_{4} \\ {\dot{x}}_{4} = f_{2} (β, r, t) + w_{2} (t) + b_{2} U_{f} + b_{2} u_{2} \\ y_{1} = x_{1} \\ y_{2} = x_{3} \\ f_{1} (β, r, t) = \frac{b C_{a r} - a C_{a f}}{m u_{c}} r - \frac{C_{a f} + C_{a r}}{m} β \\ f_{2} (β, r, t) = \frac{b C_{a r} - a C_{a f}}{I_{z}} β - \frac{a^{2} C_{a f} + b^{2} C_{a r}}{I_{z} u_{c}} r \\ b_{1} = \frac{C_{a f}}{m} \\ b_{2} = \frac{a C_{a f}}{I_{z}} \end{cases}

where y ₁ and y ₂ are the outputs of the control system, that is $y_{1} = y$ and $y_{2} = Ω$ ; u ₁ and u ₂ are the feedback controls of the control system; and w ₁ and w ₂ are the internal and external disturbances of the control system, respectively. Setting $y_{1}^{*} = y_{d}$ and $y_{2}^{*} = Ω_{d}$ , the algorithms of the ADRC control system are expressed as follows³⁰

\{\begin{cases} f h = f h a n (v_{11} - y_{1}^{*}, v_{12}, r_{0}, h) \\ {\dot{v}}_{11} = v_{12} \\ {\dot{v}}_{12} = f h \\ e = z_{11} - y_{1} \\ f e = f a l (e, 0.5, h) \\ f e_{1} = f a l (e, 0.25, h) \\ {\dot{z}}_{11} = z_{12} - β_{01} e \\ {\dot{z}}_{12} = z_{13} - β_{02} f e + b_{1} u_{1} \\ {\dot{z}}_{13} = - β_{03} f e_{1} \\ e_{1} = v_{11} - z_{11} \\ e_{2} = v_{12} - z_{12} \\ u_{0} = - f h a n (e_{1}, c e_{2}, r_{1}, h_{1}) \\ u_{1} = (u_{0} - z_{13}) / b_{1} \end{cases}

\{\begin{cases} f h = f h a n (v_{21} - y_{2}^{*}, v_{22}, r_{0}, h) \\ {\dot{v}}_{21} = v_{22} \\ {\dot{v}}_{22} = f h \\ e = z_{21} - y_{2} \\ f e = f a l (e, 0.5, h) \\ f e_{1} = f a l (e, 0.25, h) \\ {\dot{z}}_{21} = z_{22} - β_{01} e \\ {\dot{z}}_{22} = z_{23} - β_{02} f e + b_{2} u_{2} \\ {\dot{z}}_{23} = - β_{03} f e_{1} \\ e_{1} = v_{21} - z_{21} \\ e_{2} = v_{22} - z_{22} \\ u_{0} = - f h a n (e_{1}, c e_{2}, r_{1}, h_{1}) \\ u_{2} = (u_{0} - z_{23}) / b_{2} \end{cases}

where h and h ₁ are the integral steps, and $h = 0.01$ , $h_{1} = 0.3$ ; r ₀ and r ₁ are the tracking speed factors, and $r_{0} = 30$ , $r_{1} = 5$ ; $c = 1$ ; $β_{01}$ , $β_{02}$ , and $β_{03}$ are the ESO parameters, and $β_{01} = 100$ , $β_{02} = 300$ , $β_{03} = 1000$ ; $sign ()$ is a symbolic function; $f a l ()$ is a nonlinear function, as shown in equation (19); and $f l a n ()$ is a tracking differential function, as shown in equation (20)

\{\begin{cases} f a l (e, 0.5, h) = {\begin{array}{l} | e |^{0.5} sign (e), & | e | > h \\ \frac{e}{h^{0.5}}, & | e | \leq h \end{array} \\ f a l (e, 0.25, h) = \{\begin{array}{l} | e |^{0.25} sign (e), & | e | > h \\ \frac{e}{h^{0.25}}, & | e | \leq h \end{array} \end{cases}

\{\begin{cases} d = r_{0} h^{2}, a_{0} = h c x_{2}, y = x_{1} + a_{0} \\ a_{1} = \sqrt{d (d + 8 | y |)} \\ a_{2} = a_{0} + sign (y) (a_{1} - d) / 2 \\ s_{y} = (sign (y + d) - sign (y - d)) / 2 \\ a = (a_{0} + y - a_{2}) s_{y} + a_{2} \\ s_{a} = (sign (a + d) - sign (a - d)) / 2 \\ f h a n (x_{1}, c x_{2}, r_{0}, h) = - r_{0} (a / d - sign (a)) s_{a} - r_{0} sign (a) \end{cases}

The general process of the ADRC is described as follows. First, the desired target is scheduled to transition using the tracking differentiator. The purpose is to increase the adjustable ranges of the output parameters and provide effective error signals. The tracking targets values $(v_{i j})$ are expressed as $v_{11} \to y_{d}$ , $v_{12} \to {\dot{y}}_{d}$ , $v_{21} \to Ω_{d}$ , and $v_{22} \to {\dot{Ω}}_{d}$ .

Second, the state parameters and disturbances of the system are estimated using the ESO. The estimated target values $(z_{i j})$ are expressed as $z_{11} \to y$ , $z_{12} \to \dot{y}$ , $z_{13} \to f_{1} (β, r, t) + w_{1} (t) + b_{1} U_{f}$ , $z_{21} \to Ω$ , $z_{22} \to \dot{Ω}$ , and $z_{23} \to f_{2} (β, r, t) + w_{2} (t) + b_{2} U_{f}$ .

Third, the state errors of the lateral displacement and yaw angle are compensated dynamically by the state error compensation laws in equations (17) and (18) in real time.

Finally, the flowchart of the control algorithm of the ADRC control system is shown in Figure 6. When $x_{e} = 37.5 m$ , $y_{e} = 6 m$ , $u_{c} = 90 km h^{- 1}$ , $y_{d} (x_{a})$ shown in equation (2), and $y_{d} (x_{b}) = 6 m$ , the ADRC control effects are shown in Figure 7.

Figure 6.

Flowchart of the control algorithm of the ADRC control system.

Figure 7.

ADRC control effects: (a) lateral displacement, (b) yaw angle, (c) lateral velocity, (d) yaw rate, (e) disturbance values, and (f) compensated values.

Obstacle avoidance simulations with CarSim

CarSim enables simulated vehicles to perform complex behaviors such as drifting and slipping under extreme conditions. Figure 8 shows animation skid marks based on large lateral slip angles generated as the vehicle slides sideways in an extreme maneuver, where the sizes of the yellow arrows indicate the magnitudes of tire forces in the lateral and vertical directions.

Figure 8.

Tire force arrows and skid marks in an extreme maneuver.

Three different path tracking controllers

CarSim has its own path tracking controller based on an optimal preview control, named controller 1. It is located in “Steering by the Closed-loop Driver Model” of CarSim, as shown in Figure 9, where the driver preview time is 0.5 s, the driver time lag is 0 s, the low-speed dynamic limit is10 km h⁻¹, the maximum steering wheel angle is 720°, the maximum steering wheel angle rate is 1200° s⁻¹, and the lateral offset of the target path is Y_d . The detailed control process is described in the literatures.^31,32

Figure 9.

Optimal preview control model in CarSim®.

We expect that the controller proposed in this article is more effective than controller 1 in tracking the obstacle avoidance trajectory. Three different trajectory tracking controllers for autonomous vehicles in the following simulations are described as follows:

The only optimal preview feedback controller for an FWS autonomous vehicle is defined as controller 1, which is an FWS path tracking controller.

The only ADRC feedback control for a 4WS autonomous vehicle is defined as controller 2, which is a 4WS path tracking controller.

The feed-forward control and ADRC feedback control for a 4WS autonomous vehicle is defined as controller 3, which is also a 4WS path tracking controller.

The red, blue, and green vehicles in the following simulations represent controller 1, controller 2, and controller 3, respectively. Overall block diagrams of controllers 2 and 3 are shown in Figure 10. Considering the tire adhesion limits of 4WS vehicles, the restrictions of the simulation road and vehicle driving in the following simulations are shown as follows

\{\begin{matrix} 0.1 / T_{r} (rad s^{- 1}) \leq ω = 2 π \frac{1}{t_{e}} \leq 10 / T_{r} (rad s^{- 1}) \\ - 0.7 (rad s^{- 1}) \leq r_{d max} \leq 0.7 (rad s^{- 1}) \\ R_{min} = \frac{18 t_{e}^{2} u_{c}^{2}}{60 \sqrt{3} y_{e}} \geq 40 (m) \end{matrix}

where $r_{d max}$ is the maximum desired yaw rate of the vehicle; $R_{min}$ is the minimum radius of the obstacle avoidance trajectory.

Figure 10.

Overall block diagrams: (a) controller 2 and (b) controller 3.

Obstacle avoidance simulation ( $x_{e} = 37.5 m$ , $y_{e} = 6 m$ , and $u_{c} = 60 km h^{- 1}$ )

The road simulation conditions are set as follows. The initial coordinate of the autonomous vehicle is (0,0), the initial coordinate for obstacle avoidance is (262.5,0), the coordinate of the obstacle is (300,0), and the arc length of the obstacle avoidance trajectory is 76.35 m in equation (4): $x_{e} = 37.5 m$ , $y_{e} = 6 m$ , and $u_{c} = 60 km h^{- 1}$ . The steering sensitivity coefficient for the green vehicle is $G_{r} \approx 0.26$ in equation (14), and that of the red and blue vehicles is $G_{r d} \approx 0.25$ in equation (7). The animation effects, lateral displacements, and yaw angles for the three different vehicles in the simulation are shown in Figure 11, where $t_{e} \approx 2.29 s$ , $r_{d max} \approx 0.40 rad s^{- 1}$ , $R_{min} \approx 42.07 m$ , $ω \approx 2.74 rad s^{- 1}$ , and the critical collision time $X = \frac{300 - x_{e}}{u_{c}} + t_{e} \approx 18.04 s$ . The steering angles and lateral forces for the three different vehicles in the simulation are shown in Figure 12.

Figure 11.

Simulation results ( $x_{e} = 37.5 m$ , $y_{e} = 6 m$ , and $u_{c} = 60 km h^{- 1}$ ): (a) animation effect, (b) lateral displacement, and (c) yaw angle.

Figure 12.

Steering angles and lateral forces ( $x_{e} = 37.5 m$ , $y_{e} = 6 m$ , and $u_{c} = 60 km h^{- 1}$ ): (a) and (b) red vehicle, (c) and (d) blue vehicle, and (e) and (f) green vehicle.

As shown in Figures 11 and 12, the critical collision coordinates of the red, blue, and green vehicles were (18.04,6.015), (18.04,5.934), and (18.04,6.004), respectively. Even if the red and green vehicles have different front- and rear-wheel steering angles, they all achieve effective obstacle avoidance in this case, without sideslip and collision with obstacles.

Obstacle avoidance simulation ( $x_{e} = 37.5 m$ , $y_{e} = 6 m$ , and $u_{c} = 100 km h^{- 1}$ )

The road simulation conditions are the same as those in the “Obstacle avoidance simulation ( $x_{e} = 37.5 m$ , $y_{e} = 6 m$ , and $u_{c} = 60 km h^{- 1}$ )” section. The obstacle simulation values are $x_{e} = 37.5 m$ , $y_{e} = 6 m$ , and $u_{c} = 100 km h^{- 1}$ . The steering sensitivity coefficient for the green vehicle is $G_{r} \approx 0.47$ in equation (14) and that of the red and blue vehicles is $G_{r d} \approx 0.40$ in equation (7). The animation effects, lateral displacements, and yaw angles for the three different vehicles in the simulation are shown in Figure 13, where $t_{e} \approx 1.37 s$ , $r_{d max} \approx 0.66 rad s^{- 1}$ , $R_{min} \approx 42.07 m$ , $ω \approx 4.57 rad s^{- 1}$ , and $X \approx 10.83 s$ . The steering angles and lateral forces for the three different vehicles in the simulation are shown in Figure 14.

Figure 13.

Simulation results ( $x_{e} = 37.5 m$ , $y_{e} = 6 m$ , and $u_{c} = 100 km h^{- 1}$ ): (a) animation effect, (b) lateral displacement, and (c) yaw angle.

Figure 14.

Steering angles and lateral forces ( $x_{e} = 37.5 m$ , $y_{e} = 6 m$ , and $u_{c} = 100 km h^{- 1}$ ): (a) and (b) red vehicle, (c) and (d) blue vehicle, and (e) and (f) green vehicle.

When u_c increased to 100 km h⁻¹, compared to the “Obstacle avoidance simulation ( $x_{e} = 37.5 m$ , $y_{e} = 6 m$ , and $u_{c} = 60 km h^{- 1}$ )” section, $r_{d max}$ increased to 0.66 rad s⁻¹ and ω increased to 4.57 rad s⁻¹. Figures 13 and 14 show the simulation results for the three different vehicles. The critical collision coordinates of the red vehicle were (10.83, 7.232), and it lost control and rushed out of the runway after obstacle avoidance owing to severe tail flicking (oversteer). The critical collision coordinates of the blue vehicle were (10.83, 5.245), and it had apparent pushing ahead (understeer) and collision with obstacles, but it did not lose control or have a tail flicking because of the high stability of the 4WS path tracking controller. The critical collision coordinates of the green vehicle were (10.83, 6.008), and it had a slight overshoot (the maximum deviation was approximately 2 m) compared with the blue vehicle, but because the G_r was compensated linearly with the change in ω and u_c , it did not collide with the obstacles.

Obstacle avoidance simulation ( $x_{e} = 50 m$ , $y_{e} = 6 m$ , and $u_{c} = 100 km h^{- 1}$ )

The road simulation conditions are set as follows. The initial coordinate of the autonomous vehicle is (0,0), the initial coordinate for obstacle avoidance is (250,0), the coordinate of the obstacle is (300,0), and the arc length of the obstacle avoidance trajectory is 101.02 m in equation (4): $x_{e} = 50 m$ , $y_{e} = 6 m$ , and $u_{c} = 100 km h^{- 1}$ . The steering sensitivity coefficient for the green vehicle is $G_{r} \approx 0.44$ in equation (14), and that of the red and blue vehicles is $G_{r d} \approx 0.40$ in equation (7). The animation effects, lateral displacements, and yaw angles for the three different vehicles in the simulation are shown in Figure 15, where $t_{e} \approx 1.82 s$ , $r_{d max} \approx 0.38 rad s^{- 1}$ , $R_{min} \approx 73.65 m$ , $ω \approx 3.46 rad s^{- 1}$ , and $X \approx 10.82 s$ . The steering angles and lateral forces for the three different vehicles in the simulation are shown in Figure 16. To better observe the animation, the height of the camera in the simulation increases.

Figure 15.

Simulation results ( $x_{e} = 50 m$ , $y_{e} = 6 m$ , and $u_{c} = 100 km h^{- 1}$ ): (a) animation effect, (b) lateral displacement, and (c) yaw angle.

Figure 16.

Steering angles and lateral forces ( $x_{e} = 50 m$ , $y_{e} = 6 m$ , and $u_{c} = 100 km h^{- 1}$ ): (a) and (b) red vehicle, (c) and (d) blue vehicle, and (e) and (f) green vehicle.

When x_e increased to 50 m, compared to the “Obstacle avoidance simulation ( $x_{e} = 37.5 m$ , $y_{e} = 6 m$ , and $u_{c} = 100 km h^{- 1}$ )” section, ω decreased to 3.46 rad s⁻¹, $r_{d max}$ decreased to 0.38 rad s⁻¹, and $R_{min}$ increased to 73.65 m. Figure 15 shows the simulation results for the three different vehicles. The critical collision coordinates of the red vehicle were (10.82, 6.593); although oversteer had reduced, the FWS vehicle was still out of control. The critical collision coordinates of the blue vehicle were (10.82, 5.643), and understeer was weakened by the decrease in ω compared to the “Obstacle avoidance simulation ( $x_{e} = 37.5 m$ , $y_{e} = 6 m$ , and $u_{c} = 100 km h^{- 1}$ )” section. The critical collision coordinates of the green vehicle were (10.82, 6.009), and its overshoot was effectively reduced by the decrease in ω and the compensating G_r . Figure 16 indicates that controller 3 can produce greater tire lateral forces than the other two controllers, thus improving the safety and stability of the vehicle under high-speed emergency obstacle avoidance conditions.

Conclusions

The results of this study can be summarized as follows:

First, a 4WS path tracking controller can produce greater tire lateral forces than an FWS path tracking controller. Second, in terms of safety and stability under high-speed emergency obstacle avoidance conditions, the ADRC feedback controller combined with the feed-forward controller is superior to the ADRC feedback controller only. Third, for the real obstacle avoidance path tracking of 4WS autonomous vehicles, the obstacle avoidance paths can be continuously planned and updated depending on x_e , u_c , and y_e , and the inputs of the autonomous vehicles can be adjusted by the 4WS path tracking controller in real time. Finally, in future work, the path tracking controller based on coordinated control of steering and braking for 4WS autonomous vehicles on variable-adhesion-coefficient curved roads should be further studied.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by National Natural Science Foundation of China [Grant No. 51775268] and Funding of Jiangsu Innovation Program for Graduate Education [Grant No. KYLX16_0328].

ORCID iD

Runqiao Liu

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Emergency obstacle avoidance trajectory tracking control based on active disturbance rejection for autonomous vehicles

Abstract

Keywords

Introduction

Linear 4WS vehicle model

Planning of obstacle avoidance trajectories

Polynomial trajectory

Cosine trajectory

Conversion effects of two different trajectories

Initial inputs of autonomous vehicle

Feed-forward control for 4WS

Feedback control for 4WS

Obstacle avoidance simulations with CarSim

Three different path tracking controllers

Obstacle avoidance simulation ( x e = 37.5 m , y e = 6 m , and u c = 60 km h − 1 )

Obstacle avoidance simulation ( x e = 37.5 m , y e = 6 m , and u c = 100 km h − 1 )

Obstacle avoidance simulation ( x e = 50 m , y e = 6 m , and u c = 100 km h − 1 )

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

Obstacle avoidance simulation ( $x_{e} = 37.5 m$ , $y_{e} = 6 m$ , and $u_{c} = 60 km h^{- 1}$ )

Obstacle avoidance simulation ( $x_{e} = 37.5 m$ , $y_{e} = 6 m$ , and $u_{c} = 100 km h^{- 1}$ )

Obstacle avoidance simulation ( $x_{e} = 50 m$ , $y_{e} = 6 m$ , and $u_{c} = 100 km h^{- 1}$ )