Collision-avoidance steering control for autonomous vehicles using neural network-based adaptive integral terminal sliding mode

Abstract

Due to the complex driving conditions confronted by an autonomous vehicle, it is significant for the vehicle to possess a robust control system to achieve effective collision-avoidance performance. This paper proposes a neural network-based adaptive integral terminal sliding mode (NNAITSM) control scheme for the collision-avoidance steering control of an autonomous vehicle. In order to describe the vehicle’s lateral dynamics and path tracking characteristics, a two-degrees-of-freedom (2DOF) dynamic model and a kinematic model are adopted. Then, an NNAITSM controller is designed, where a radial basis function neural network (RBFNN) scheme is utilized to online approximate the optimal upper bound of lumped system uncertainties such that prior knowledge about the uncertainties is not required. The stability of the control system is proved via Lyapunov, and the selection guideline of control parameters is provided. Last, Matlab-Carsim co-simulations are executed to test the performance of the designed controller under different road conditions and vehicle velocities. Simulation results show that compared with conventional sliding mode (CSM) and nonsingular terminal sliding mode (NTSM) control, the proposed NNAITSM control scheme owns evident superiority in not only higher tracking precision but also stronger robustness against various road surfaces and vehicle velocities.

Keywords

Autonomous vehicle neural networks sliding mode control vehicle dynamics and control

1 Introduction

With the development of automobile industry and automation technology, autonomous vehicle has become a research focus, which involves multi-disciplinary theories and methodologies [1 –3]. In order to reduce the burden of mankind, autonomous vehicles are designed to possess numerous functions. In [4], a distributed algorithm based on dual decomposition is developed to solve the coordinated parking problem, which optimizes the parking performance of an autonomous vehicle to a great extent. Adaptive cruise control (ACC) and cooperative adaptive cruise control (CACC) systems are effective strategies implemented on autonomous vehicles to reduce traffic jams [5, 6]. For instance, a variable time headway spacing strategy is proposed for CACC, which is able to enhance the stability of the traffic flow, improve the road capacity and alleviate the traffic congestion [7]. Path planning is another important issue for autonomous vehicles [8, 9]. In [10], based on the analysis of the merits and shortages of conventional path planning schemes including artificial potential fields and optimal control, Rasekhipour et al. propose a model predictive path-planning controller such that the system can distinguish obstacles distinctly while planning an optimal path via vehicle dynamics.

As a key function of an autonomous vehicle, path following can enable the vehicle to track a given reference path accurately while maintaining lateral and longitudinal stability [11, 12]. Because of the complex road-tyre dynamics during vehicle lateral movement, the nonlinear characteristics of tyres cannot be ignored, which increases the difficulty of control design. To eliminate path tracking errors and ensure vehicle stability, a number of control schemes are proposed, such as H_∞ control [13], model predictive control [14], neural network control [15] and fuzzy logic control [16]. Specifically, a backstepping sliding mode controller is designed for the emergency steering control of an autonomous vehicle such that the vehicle can avoid collisions in time and maintain its stability [17]. In [18], the original path-tracking problem with parameter variations is formulated as a T-S fuzzy system with norm-bounded uncertainties, and a fuzzy observer-based output feedback control scheme is proposed to achieve strong robustness against parametric uncertainties.

Owning the distinguished ability of approximating arbitrary linear or nonlinear functions in adjustable accuracy under various conditions, neural networks have been a hot spot attracting considerable attentions from researchers and engineers. Thanks to its remarkable benefits, neural network strategy has been widely utilized in numerous applications such as dealing with pathology problems [19], malware forecasting [20] and intelligent control [21 –23]. Adaptive control is another popular control scheme utilized to estimate time-varying system information under a wide range of perturbations and parametric uncertainties [24, 25]. For instance, a nested adaptive scheme is proposed for the control of vehicle steer-by-wire systems in [26]. The nested adaptation law is designed to adaptively estimate the self-aligning torques acting on vehicle front wheels during the steering procedure such that a corresponding feedforward control input can be designed to compensate for the disturbing effect caused by self-aligning torques.

Sliding mode control is a specific nonlinear control strategy, whose working principle is to force the system trajectory to ‘slide’ along a predetermined sliding surface [27 –29]. Due to its remarkable advantages including quick response, high precision and strong robustness, sliding mode control is widely utilized in many applications such as [30, 31]. However, due to the existence of a discontinuous term in the conventional sliding mode (CSM) control input, chattering phenomenon is normally generated as well. To overcome this shortage, a terminal sliding mode (TSM) control scheme is proposed [32, 33]. Furthermore, a nonsingular terminal sliding mode (NTSM) control strategy is presented to address the singularity problem existing in the TSM control [34, 35]. Recently, a so-called integral terminal sliding mode (ITSM) control scheme is introduced, which can achieve fast convergence rate and chattering-alleviation effect [36, 37]. The sliding mode, adaptive and neural network strategies possess unique advantages, respectively. However, the integration of them is still rare, especially in the applications of autonomous vehicles. Therefore, it motivates us to combine the benefits of neural network, adaptive and sliding mode schemes in the collision-avoidance steering control of autonomous vehicles.

In this paper, a neural network-based adaptive integral terminal sliding mode (NNAITSM) controller is designed for the lateral movement control of an autonomous vehicle to obtain effective collision-avoidance performance. In the NNAITSM control, an radial basis function neural network (RBFNN) system is introduced to estimate the optimal upper bound of lumped system uncertainties, an adaptive law with the function of updating the sliding variable parameters is utilized to optimize the control performance, and an ITSM control element is designed to guarantee the system stability and convergence property. The contributions of this paper are mainly reflected in the following 3 aspects:

(1) The consideration of the lumped system uncertainties in the vehicle dynamic model, and the introducing of an RBFNN system to estimate the optimal upper bound of lumped system uncertainties such that the gain of the discontinuous term in the sliding mode control will neither be chosen with an overlarge value to cause extra chattering nor be chosen with an insufficiently large value to lose the system’s stability.

(2) The combination of neural networks, adaptive method and sliding mode control in the application of vehicle lateral motion control such that the merits of these 3 strategies can be integrated and the collision-avoidance steering control performance of autonomous vehicles can be comprehensively improved.

(3) The consideration and arrangement of different road-tyre adhesion coefficients and vehicle velocities in the Matlab/Simulink-Carsim simulation, which is able to demonstrate strong robustness of the proposed NNAITSM control scheme in dealing with various road conditions and vehicle velocities compared with the benchmark controllers.

The rest part of this paper is organized as follows. In Section 2, a two-degree-of-freedom (2DOF) dynamic model and a vehicle kinematic model are combined to describe the autonomous vehicle’s collision-avoidance steering behavior. In Section 3, the architecture of the RBFNN system is illustrated, the design of the NNAITSM controller is described, and the stability of the control system is proved via Lyapunov. In addition, a CSM controller and an NTSM controller are designed for comparison. In Section 4, Matlab/Simulink-Carsim co-simulation results are shown and analyzed in detail with the help of bar charts. Finally, Section 5 concludes this paper.

2 Plant modeling

A plant model consisting of a vehicle dynamic model and a vehicle kinematic model is identified in this section.

2.1 Vehicle dynamics model

A model capable of describing the autonomous vehicle’s essential lateral dynamics precisely is of great importance for the collision-avoidance control design. To achieve this goal, a typicl 2DOF dynamic model is adopted here [15], [17]:

$\begin{matrix} {\begin{matrix} \dot{β} = \frac{F_{fl} + F_{fr} + F_{rl} + F_{rr}}{{mv}_{x}} - γ \\ \dot{γ} = \frac{l_{1} (F_{fl} + F_{fr}) - l_{2} (F_{rl} + F_{rr})}{I_{z}} \end{matrix} \end{matrix}$ (1) where β and γ represent the sideslip angle and the yaw rate of the vehicle body, respectively; F_fl and F_fr denote the lateral forces exerted on the left and right tyres of the front axle, respectively; F_rl and F_rr denote the lateral forces acting on the left and right tyres of the rear axle, respectively; m is the mass of the vehicle; v_x represents the longitudinal velocity of the vehicle, l₁ and l₂ are the lengths from the center of gravity (CG) of the vehicle to the front axle and the rear axle, respectively; I_z denotes the yaw moment of inertia of the vehicle body.

2.2 Vehicle kinematics model

Fig. 1 shows the schematic of the vehicle’s trajectory-tracking characteristics. It is the best approach that both the lateral error e and the heading error Δ_φ = φ - φ_r can be eliminated in the path-following procedure. Nevertheless, with single control input, i.e., the steering angle δ_f, only one of the errors can get reduced. Therefore, a projected error e_m is introduced here to integrate the lateral error e and the heading error Δ_φ [17]. Then, a simplified vehicle kinematics model based on the projected error e_m is presented as follows [15]:

$\begin{matrix} {\begin{matrix} e_{m} = e + x_{m} \sin (Δ_{φ}) \\ \dot{e} = v_{x} \sin (Δ_{φ}) + v_{y} \cos (Δ_{φ}) \\ \dot{d} = v_{x} \cos (Δ_{φ}) - v_{y} \sin (Δ_{φ}) \\ Δ_{φ} = φ - φ_{r} \end{matrix} \end{matrix}$ (2) where x_m denotes the constant projected distance; φ and φ_r are the vehicle heading and the heading of the reference path, respectively; v_y is the lateral velocity at CG; d represents the distance along the reference path.

Fig.1

Architecture of the control system.

Under the normal assumption of a small Δ_φ, differentiating e_m and Δ_φ in (2) yields

$\begin{matrix} {\begin{matrix} e_{m} = e + x_{m} Δ_{φ} \\ {\dot{e}}_{m} = \dot{e} + x_{m} {\dot{Δ}}_{φ} \\ \dot{e} = v_{x} Δ_{φ} + v_{y} \\ {\dot{Δ}}_{φ} = \dot{φ} - {\dot{φ}}_{r} = γ - ρ \dot{d} \end{matrix} \end{matrix}$ (3) where ρ denotes the path curvature. According to (3), we can get

$\begin{matrix} {\begin{matrix} {\ddot{Δ}}_{φ} = \dot{γ} - \dot{ρ} \dot{d} - ρ \ddot{d} \\ \ddot{e} = {\dot{v}}_{x} Δ_{φ} + v_{x} {\dot{Δ}}_{φ} + {\dot{v}}_{y} \end{matrix} \end{matrix}$ (4) and

$\begin{matrix} \begin{matrix} {\ddot{e}}_{m} = & \ddot{e} + x_{m} {\ddot{Δ}}_{φ} \\ = & {\dot{v}}_{y} + {\dot{v}}_{x} Δ_{φ} + v_{x} (γ - ρ \dot{d}) + x_{m} (\dot{γ} - \dot{ρ} \dot{d} - ρ \ddot{d}) . \end{matrix} \end{matrix}$ (5) By substituting 1) into (5), we further get

$\begin{matrix} \begin{matrix} {\ddot{e}}_{m} = & {\dot{v}}_{y} + {\dot{v}}_{x} Δ_{φ} + v_{x} (γ - ρ \dot{d}) \\ + x_{m} [- \dot{ρ} \dot{d} - ρ \ddot{d} + \frac{l_{1} (F_{fl} + F_{fr}) - l_{2} (F_{rl} + F_{rr})}{I_{z}}] . \end{matrix} \end{matrix}$ (6)

Restricted by economic and technical conditions, it is usually difficult to measure the tyre lateral forces directly. Therefore, an effective estimation technique is desired such that the lateral forces acting on the tyres of the vehicle can be accurately estimated. For this aim, we adopt the following estimation method [17]:

$\begin{matrix} {\hat{F}}_{L} & = [\begin{matrix} {\hat{F}}_{fl} & {\hat{F}}_{fr} \\ {\hat{F}}_{rl} & {\hat{F}}_{rr} \end{matrix}] = [\begin{matrix} μ {\hat{C}}_{FL} α_{f} & μ {\hat{C}}_{FR} α_{f} \\ μ {\hat{C}}_{RL} α_{r} & μ {\hat{C}}_{RR} α_{r} \end{matrix}] \end{matrix}$ (7) where C_FR and C_FL represent the cornering stiffnesses of the right and left tyres in the front axle, respectively; C_RR and C_RL denote the cornering stiffnesses of the right and left tyres in the rear axle, respectively; α_f and α_r are the slip angles of the front-axle tyres and the rear-axle tyres, respectively; μ denotes the tyre-road friction coefficient; and $\hat{*}$ denotes the estimation of ∗. In addition, by a small-angle approximation, we can simplify the tyre slip angles as

$\begin{matrix} \begin{matrix} {\begin{matrix} α_{f} = \arctan [\frac{v \sin (β) + l_{1} γ}{v \cos (β)}] - δ_{f} \approx β + \frac{l_{1} γ}{v_{x}} - δ_{f} \\ α_{r} = \arctan [\frac{v \sin (β) - l_{2} γ}{v \cos (β)}] \approx β - \frac{l_{2} γ}{v_{x}} \end{matrix} \end{matrix} \end{matrix}$ (8) where δ_f is the steering angle of the front wheels.

To better present the tyre cornering characteristics, the so-called Pacejka nonlinear cornering stiffnesses are adopted and shown as follows [17, 38]:

$\begin{matrix} {\begin{matrix} {\hat{C}}_{FL} = C_{F 0} \sin [2 \arctan (\frac{{\hat{F}}_{Gfl}}{G_{f 0}})] \\ {\hat{C}}_{FR} = C_{F 0} \sin [2 \arctan (\frac{{\hat{F}}_{Gfr}}{G_{f 0}})] \\ {\hat{C}}_{RL} = C_{R 0} \sin [2 \arctan (\frac{{\hat{F}}_{Grl}}{G_{r 0}})] \\ {\hat{C}}_{RR} = C_{R 0} \sin [2 \arctan (\frac{{\hat{F}}_{Grr}}{G_{r 0}})] \end{matrix} \end{matrix}$ (9) where C_F0 and C_R0 are the nominal cornering stiffnesses of the front and rear axles, respectively; G_f0 and G_r0 are the load factors of the front and rear axles, respectively.

Combining the longitudinal and lateral accelerations, the vertical load acting on each wheel can be estimated by [39]

$\begin{matrix} {\begin{matrix} {\hat{F}}_{Gfl} = (g \frac{l_{2}}{2} - {\dot{v}}_{x} \frac{h}{2} - {\dot{v}}_{y} \frac{l_{2} h}{t} + {\dot{v}}_{x} {\dot{v}}_{y} \frac{h^{2}}{gt}) \frac{m}{L} \\ {\hat{F}}_{Gfr} = (g \frac{l_{2}}{2} - {\dot{v}}_{x} \frac{h}{2} + {\dot{v}}_{y} \frac{l_{2} h}{t} - {\dot{v}}_{x} {\dot{v}}_{y} \frac{h^{2}}{gt}) \frac{m}{L} \\ {\hat{F}}_{Grl} = (g \frac{l_{1}}{2} + {\dot{v}}_{x} \frac{h}{2} - {\dot{v}}_{y} \frac{l_{1} h}{t} - {\dot{v}}_{x} {\dot{v}}_{y} \frac{h^{2}}{gt}) \frac{m}{L} \\ {\hat{F}}_{Grr} = (g \frac{l_{1}}{2} + {\dot{v}}_{x} \frac{h}{2} + {\dot{v}}_{y} \frac{l_{1} h}{t} + {\dot{v}}_{x} {\dot{v}}_{y} \frac{h^{2}}{gt}) \frac{m}{L} \end{matrix} \end{matrix}$ (10) where t is the track width, h is the height of CG, and L is the wheelbase.

By combining (6), (7) and (8), we can get the following equations

$\begin{matrix} {\begin{matrix} {\ddot{e}}_{m} = w_{1} + w_{2} + w_{3} δ_{f} + d_{l} \\ w_{1} = {\dot{v}}_{y} + {\dot{v}}_{x} Δ_{φ} + v_{x} (γ - ρ \dot{d}) \\ w_{2} = x_{m} [\frac{l_{1} μ (v_{x} β + l_{1} γ) ({\hat{C}}_{FL} + {\hat{C}}_{FR})}{v_{x} I_{z}} - \frac{l_{2} μ α_{r} ({\hat{C}}_{RL} + {\hat{C}}_{RR})}{I_{z}} - \dot{ρ} \dot{d} - ρ \ddot{d}] \\ w_{3} = - x_{m} \frac{l_{1} μ ({\hat{C}}_{FL} + {\hat{C}}_{FR})}{I_{z}} . \end{matrix} \end{matrix}$ (11) In (11), the expressions of ${\hat{C}}_{FL}$ , ${\hat{C}}_{FR}$ , ${\hat{C}}_{RL}$ and ${\hat{C}}_{RR}$ can be obtained by (9) and (10); d_l is used to denote the lumped system uncertainties, which satisfies

$\begin{matrix} | d_{l} | ⩽ {\bar{d}}_{l} \end{matrix}$ (12) where ${\bar{d}}_{l} > 0$ is the upper bound of the lumped system uncertainties d_l.

According to the aforementioned analysis, we can finally express the plant model with the lumped system uncertainties shown as follows:

\begin{matrix} {\begin{matrix} {\dot{x}}_{1} = x_{2} \\ {\dot{x}}_{2} = w_{1} + w_{2} + w_{3} u + d_{l} \end{matrix} \end{matrix}

(13) where x₁ = e_m; u = δ_f is the control input of the closed-loop system, i.e., the front wheel steering angle. In addition, a high-fidelity vehicle model is

Table 1

Parameters in model

Symbol	Description	Value
l ₁	Distance from front axle to CG	1.04 m
l ₂	Distance from rear axle to CG	1.56 m
t	Track width	1.48 m
h	Height of CG	0.54 m
x _m	Projected distance	8 m
m	Vehicle mass	1300 kg
I _z	Yaw inertia	1343 kg · m²
C _F0	Front axle cornering stiffness	56500 N/rad
C _R0	Rear axle cornering stiffness	66500 N/rad
G _f0	Load factor of front axle	5700 N
G _r0	Load factor of rear axle	6200 N

Fig.2

Architecture of the control system.

established by Carsim, and the values of the parameters existing in the model are given in Table 1.

3 Control design

Our objective is to design a robust controller for the lateral motion control of the autonomous vehicle such that the vehicle can steer to avoid collisions in time and effectively. To achieve this goal, an NNAITSM controller is designed in this section, where an RBFNN approximator is adopted to learn the optimal upper bound of lumped system uncertainties in real time. Specifically, Fig. 2 shows the architecture of the closed-loop control system.

3.1 NNAITSM control design

On the basis of the plant model described in (13), the reference command x_r and the tracking error e of the control system are defined as [15]:

$\begin{matrix} {\begin{matrix} x_{r} = 0 \\ e = x_{1} - x_{r} = x_{1} \\ \dot{e} = {\dot{x}}_{1} - {\dot{x}}_{r} = x_{2} . \end{matrix} \end{matrix}$ (14) Then, a sliding variable s is defined as follows [37, 40]:

$\begin{matrix} {\begin{matrix} s = \dot{e} + λ_{1} (t) e + λ_{2} (t) e_{a} \\ {\dot{e}}_{a} = e^{\frac{q}{p}} (t) \end{matrix} \end{matrix}$ (15) where the initial condition of e_a is expressed as $e_{a} (0) = - [\dot{e} (0) + λ_{1} (0) e (0)] / λ_{2} (0)$ with λ₁ (0) and λ₂ (0) being the initial conditions of λ₁ (t) and λ₂ (t), respectively; p and q are two positive odd integers satisfying p > q > 0; λ₁ and λ₂ are two control parameters that are updated online in real time by the following adaptive law [37]:

$\begin{matrix} {\begin{matrix} {\dot{λ}}_{1} = - ζ_{1} se \\ {\dot{λ}}_{2} = - ζ_{2} s \int_{0}^{t} e^{\frac{q}{p}} d τ \end{matrix} \end{matrix}$ (16) in which ζ₁, ζ₂ > 0 are two adaptation gains tuned to determine the adaptation rates of the control parameters.

In order to online learn the optimal upper bound of lumped system uncertainties, the following RBFNN system is adopted [41, 42]:

$\begin{matrix} k_{1} = {\hat{σ}}^{T} Φ (X) \end{matrix}$ (17) where k₁ is the estimation of the optimal upper bound of lumped system uncertainties; X = [x₁ x₂] ^T is the input of the RBFNN system, where x₁ and x₂ are given in (13); $\hat{σ}$ is the weight vector of the RBFNN system; vector Φ (X) ∈ R^N is a Gaussion function with the following definition:

$\begin{matrix} Φ_{i} (X) = \exp (- \frac{| | X - g_{i} | |^{2}}{{z_{i}}^{2}}) \\ i = 1, 2, . . ., N - 1, N . \end{matrix}$ (18) In (18), g_i and z_i are the predetermined center and width of the N neurons, respectively; the estimated weights are adjusted by the following algorithm:

$\begin{matrix} \dot{\hat{σ}} = η Φ (X) | s | \end{matrix}$ (19) where η > 0 is the adjusting gain to be designed.

The working of the presented RBFNN approximator is based on the following assumptions:

Assumption 1: Given a positive constant Δ₁ that is sufficiently small, the optimal weights of the RBFNN system, namely, σ^∗ satisfies

$\begin{matrix} | {σ^{*}}^{T} Φ (X) - {\bar{d}}_{l} | < Δ_{1} \end{matrix}$ (20) where ${\bar{d}}_{l}$ is the upper bound of lumped system uncertainties.

Assumption 2: The difference between the upper bound and the lumped system uncertainties satisfies the following inequality:

$\begin{matrix} {\bar{d}}_{l} - | d_{l} (t) | > Δ_{2} > Δ_{1} . \end{matrix}$ (21)

For the convenience of control design, the neural network approximator’s working conditions are summarised as follows:

$\begin{matrix} {\begin{matrix} | {σ^{*}}^{T} Φ (X) - {\bar{d}}_{l} | < Δ_{1} \\ {\bar{d}}_{l} - | d_{l} (t) | > Δ_{2} > Δ_{1} \\ k_{1} = {\hat{σ}}^{T} Φ (X) \\ \dot{\hat{σ}} = η Φ (X) | s | . \end{matrix} \end{matrix}$ (22)

Theorem 1. For our system described in (13), the tracking error e can realize an asymptotic convergence under the following control law

$\begin{matrix} \begin{matrix} u = - \frac{1}{w_{3}} [λ_{1} \dot{e} + λ_{2} e^{\frac{q}{p}} + w_{1} + w_{2} + k_{1} sign (s) + k_{2} s] \end{matrix} \end{matrix}$ (23)

where k₁ is the estimation of the optimal lumped system uncertainties given by (17); k₂ > 0 is to be designed; λ₁ and λ₂ are control parameters updated by the adaptation law given by (16).

Proof. A Lyapunov function is chosen as

$\begin{matrix} V = \frac{1}{2} s^{2} + \frac{1}{2 η} {\tilde{σ}}^{T} \tilde{σ} \end{matrix}$ (24) where s is the sliding variable defined in (15) and $\tilde{σ}$ is the difference between the optimal weights and estimates weights in the RBFNN system, i.e., $\tilde{σ} = σ^{*} - \hat{σ}$ . Then, evaluating the first-order derivative of V along the trajectories of the closed-loop system (13) yields

$\begin{matrix} \dot{V} = & s \dot{s} + \frac{1}{η} {\tilde{σ}}^{T} (\dot{σ^{*}} - \dot{\hat{σ}}) \\ = & s (\ddot{e} + {\dot{λ}}_{1} e + λ_{1} \dot{e} + {\dot{λ}}_{2} e_{a} + λ_{2} {\dot{e}}_{a}) - \frac{1}{η} {\tilde{σ}}^{T} \dot{\hat{σ}} \\ = & s ({\ddot{x}}_{1} + {\dot{λ}}_{1} e + λ_{1} \dot{e} + {\dot{λ}}_{2} e_{a} + λ_{2} {\dot{e}}_{a}) - \frac{1}{η} {\tilde{σ}}^{T} \dot{\hat{σ}} \\ = & s (w_{1} + w_{2} + w_{3} u + d_{l} + {\dot{λ}}_{1} e + λ_{1} \dot{e} + {\dot{λ}}_{2} e_{a} \\ + λ_{2} {\dot{e}}_{a}) - \frac{1}{η} {\tilde{σ}}^{T} \dot{\hat{σ}} . \end{matrix}$ (25) Substituting the control input u given by (23) into (25) yields

$\begin{matrix} \dot{V} & = & s {w_{1} + w_{2} + d_{l} + {\dot{λ}}_{1} e + λ_{1} \dot{e} + {\dot{λ}}_{2} e_{a} + λ_{2} {\dot{e}}_{a} \\ - [λ_{1} \dot{e} + λ_{2} e^{\frac{q}{p}} + w_{1} + w_{2} + k_{1} sign (s) + k_{2} s]} \\ - \frac{1}{η} {\tilde{σ}}^{T} \dot{\hat{σ}} . \end{matrix}$ (26)

Rearranging (26) and combining it with the adaptation law in (16), we get

$\begin{matrix} \dot{V} = & s [d_{l} + {\dot{λ}}_{1} e + {\dot{λ}}_{2} e_{a} - k_{1} sign (s) - k_{2} s] - \frac{1}{η} {\tilde{σ}}^{T} \dot{\hat{σ}} \\ = & s [d_{l} - ζ_{1} {se}^{2} - ζ_{2} s (\int_{0}^{t} e^{\frac{q}{p}} d_{τ})^{2} - k_{1} sign (s) - k_{2} s] - \frac{1}{η} {\tilde{σ}}^{T} \dot{\hat{σ}} \\ = & d_{l} s - k_{1} | s | - \frac{1}{η} {\tilde{σ}}^{T} \dot{\hat{σ}} - ζ_{1} s^{2} e^{2} - ζ_{2} s^{2} (\int_{0}^{t} e^{\frac{q}{p}} d_{τ})^{2} - k_{2} s^{2} \\ \leq & d_{l} s - k_{1} | s | - \frac{1}{η} {\tilde{σ}}^{T} \dot{\hat{σ}} . \end{matrix}$ (27) Finally, we substitute the neural network conditions as expressed in (22) into (27) and obtain

$\begin{matrix} \dot{V} \leq & d_{l} s - k_{1} | s | - \frac{1}{η} {\tilde{σ}}^{T} \dot{\hat{σ}} \\ = & d_{l} s - {\hat{σ}}^{T} Φ (X) | s | - (σ^{*} - \hat{σ})^{T} Φ (X) | s | \\ = & d_{l} s - {σ^{*}}^{T} Φ (X) | s | \\ = & d_{l} s - {\bar{d}}_{l} | s | + {\bar{d}}_{l} | s | - {σ^{*}}^{T} Φ (X) | s | \\ \leq & | d_{l} | | s | - {\bar{d}}_{l} | s | + ({\bar{d}}_{l} - {σ^{*}}^{T} Φ (X)) | s | \\ \leq & (| d_{l} | - {\bar{d}}_{l}) | s | + | {\bar{d}}_{l} - {σ^{*}}^{T} Φ (X) | | s | \\ \leq & - Δ_{2} | s | + Δ_{1} | s | \\ = & - (Δ_{2} - Δ_{1}) | s | \\ < & 0 \end{matrix}$ (28) when s ≠ 0.

Thus, the proof is completed.

Remark 1. From (15) we can see that through the setting of the initial value of e_a, the system can be forced to start on the sliding surface s = 0 [37]. Hence, the reaching phase can be eliminated implying a faster convergence rate and higher tracking accuracy. In addition, the employed adaptive law given by (16) is able to adaptively adjust the control gains in the sliding variable, which will further enhance the tracking precision of the control system.

Remark 2. From (24)–(28) it is theoretically verified that the estimated upper bound of lumped system uncertainties is able to converge to the optimal one in the sense of Lyapunov, which means that the designed NNAITSM controller owns the ability of leaning the optimal upper bound of lumped system uncertainties in the autonomous vehicles’ lateral motion control regardless of the variation of road conditions or vehicle velocities. This ability is novel and beneficial for our application since the gain of the switching term in sliding mode control will neither be set insufficiently large to loose the system’s stability nor be set overlarge to cause extra chattering in the control inputs. In summary, the RBFNN design in the control system brings a satisfactory balance between the tracking precision and control smoothness.}

3.2 Selection of control parameters

Up to now, a model to describe the autonomous vehicle’s trajectory-following behavior consisting of a dynamic model and a kinematic model has been fully presented. On the basis of the plant model, an NNAITSM controller has been designed and the stability of the control system has been verified via Lyapunov. In order to achieve satisfactory control performance, it is significant to select the values of control parameters carefully considering the tradeoff among system uncertainties, unmodeled dynamics and the chattering phenomenon. The parameters’ selection is based on theoretical analysis and simulation tests. Initially, the impact of increasing and reducing the control parameters’ values is theoretically analysed. Then, the setting of these parameters’ values is tested in simulation according to the theoretical instruction until appropriate values are found such that a satisfactory balance between the tracking accuracy and control smoothness can be achieved.

1) Selection of p and q:

According to [43] and [44], the control parameters p and q are suggested to be chosen as odd integers satisfying p > q > 0 to enhance the capability of handling parametric uncertainties and improve the control performance. Furthermore, when the value of $\frac{q}{p}$ is closer to 1, the nonlinear sliding variable s acts more like a linear PID-type sliding surface, which results in a slower convergence rate of the tracking errors. In our case, we find that the values of p = 7 and q = 5 are satisfactory.

2) Selection of k₂:

For our control system, a larger k₂ is able to reduce the chattering more but at the cost of amplifying the amplitude of control input signals. To tradeoff between these factors, we set k₂ = 200.

3) Selection of ζ₁ and ζ₂:

The adaptation gains ζ₁ and ζ₂ should be carefully selected such that the sliding variable parameters can be properly adjusted. Increasing the values of ζ₁ and ζ₂ can enhance the convergence rates and maintain smaller steady-state tracking errors in the path tracking. But overlarge ζ₁ and ζ₂ may cause the instability of the control system. Specifically, the steady-state tracking error is more sensitive to a larger ζ₁. It is found that the setting of ζ₁ = 16 and ζ₂ = 55 is appropriate.

4) Setting of RBFNN:

The parameter η determines the working performance of the RBFNN system to a large extent. According to the Lyapunov function as shown in (24), it can be seen that a larger η leads to a faster convergence rate of the estimated upper bound of lumped system uncertainties to the optimal one. However, an overlarge value will bring chaos to the system or even result in instability. Considering this tradeoff, we set η = 250. To strike a good balance between the approximation effect and computation load, the structure of the RBFNN system is selected with 6 nodes as shown in Fig. 3, and the corresponding centres, widths and initial values of the weights are set as follows to achieve a satisfactory performance:

$\begin{matrix} {\begin{matrix} g_{i} = [\begin{matrix} 1 & - 1 & 1 & - 1 & 1 & 1 \\ - 1 & 1 & - 1 & - 1 & 1 & 1 \end{matrix}] \\ {z_{i}}^{2} = 0.81 \\ \hat{σ} (0) = [0 0 0 0 0 0]^{T} . \end{matrix} \end{matrix}$ (29)

3.3 Controllers for comparison

To demonstrate the superiority of the presented NNAITSM control scheme, a CSM controller and an NTSM controller are also designed for comparison based on [27] and [34], respectively. For simplicity, the control design results are straightforwardly given.

Fig.3

Schematic of RBFNN with 6 nodes and 6 weights.

1) CSM control:

According to [27], the CSM control input is designed as

$\begin{matrix} \begin{matrix} {\begin{matrix} u_{csm} = - \frac{1}{w_{3}} [w_{1} + w_{2} + λ_{csm} \dot{e} + D sign (s_{csm})] \\ s_{csm} = \dot{e} + λ_{csm} e \end{matrix} \end{matrix} \end{matrix}$ (30)

where s_csm denotes a conventional sliding variable with λ_csm = 6 being the tracking bandwidth; w₁, w₂ and w₃ are given in (11), respectively.

2) NTSM control:

On the basis of [34], we design the NTSM control input u_ntsm as

$\begin{matrix} u_{ntsm} = & - \frac{1}{w_{3}} [w_{1} + w_{2} + \frac{1}{r λ_{ntsm}} | \dot{e} |^{2 - r} sign (\dot{e}) \\ + D sign (s_{ntsm})] . \end{matrix}$ (31) In (31), s_ntsm is an NTSM sliding variable with the expression of

$\begin{matrix} s_{ntsm} = e + λ_{ntsm} | \dot{e} |^{r} sign (\dot{e}) \end{matrix}$ (32) where λ_ntsm = 0.5 and r = 1.4.

4 Simulation results

In this section, Matlab/Simulink-Carsim co-simulations are carried out on the basis of a high-fidelity and full-vehicle model generated by Carsim. For the aim of verifying the superiority of the proposed control scheme, three cases are taken into consideration. Case 1 is with a low vehicle velocity of v = 60 km/h, and a low road adhesion coefficient μ = 0.4 representing a snowy road. Case 2 is with the same vehicle velocity as Case 1. However, the road adhesion coefficient is set as μ = 0.85 to mimic a dry asphalt road. Case 3 is still based on a dry asphalt road condition with μ = 0.85, but the vehicle velocity is set as v = 108 km/h. Hence, a comprehensive comparison in terms of different road conditions and vehicle velocities can be achieved for all three control strategies.

In this paper, we primarily focus on the control design for the path tracking procedure in the collision avoidance of the autonomous vehicle. Therefore, the path planning step is skipped here, and a trajectory model is straightforwardly provided as the reference command of the control system. The model is expressed as follows [15, 45]:

$\begin{matrix} {\begin{matrix} Y_{r} (x) = \frac{d_{n 1}}{2} [1 + \tanh (r_{1})] - \frac{d_{n 2}}{2} [1 + \tanh (r_{2})] \\ φ_{r} (x) = \arctan [d_{n 1} (\frac{1}{\cosh (r_{1})})^{2} (\frac{1.2}{d_{m 1}}) - d_{n 2} (\frac{1}{\cosh (r_{2})})^{2} (\frac{1.2}{d_{m 2}})] \end{matrix} \end{matrix}$ (33)

Fig.4

Lateral distances in Case 1.

Fig.5

Lateral accelerations in Case 1.

Fig.6

Sideslip angles in Case 1.

Fig.7

Yaw rates in Case 1.

Fig.8

Steering angles in Case 1.

where Y_r, φ_r and x are the reference lateral position, the reference yaw angle and the longitudinal position, respectively; r₁ = 0.096 (x - 60) -1.2, r₂ = 0.096 (x - 120) -1.2, d_m1 = 25, d_m2 = 25, d_n1 = 3.6, and d_n2 = 3.6.

4.1 Case 1: snowy road with low velocity

Normally, it is more possible for the autonomous vehicle to become unstable when steering on a low adhesion road, particularly in an emergent collision-avoidance situation. Thus, we choose a snowy road condition to test the performance of the NNAITSM control.

Figs. 4 and 5 show the lateral distances and the lateral accelerations of the vehicle, respectively. From Fig. 4 we can see that the vehicle’s lateral displacement is controlled to follow the reference command closely under all the three control schemes. However, as shown in Fig. 5, there is less chattering in the lateral acceleration under the NNAITSM control compared with other schemes, which indicates that a more stable and comfortable steering performance can be achieved by the NNAITSM controller.

The sideslip angles and yaw rates under three controllers are shown in Figs. 6 and 7, respectively. It can be seen that the vehicle’s double-lane-change maneuver leads to a relatively large sideslip angle. Thus, the tyres work in their nonlinear region and are saturated to a large extent. Though the yaw rates under three control schemes are with no big difference, the peak values of the sideslip angle under the NNAITSM control are still smaller than those of the other controllers, which means stronger stability of the vehicle.

Figs. 8 and 9 show the steering angles and tracking errors of the vehicle, respectively. It can be seen that the vehicle is controlled to follow the desired collision-avoidance path closely under all the three control schemes with the projected errors less than 0.1 m, which indicates that the collisions have been avoided successfully. It is evident that the NNAITSM control owns pretty high tracking precision compared with the other two control schemes due to the smallest peak errors whose values are less than 0.05 m.

Fig.9

Tracking errors in Case 1.

4.2 Case 2: dry asphalt road with low velocity

Compared with a snowy road, it is much more usual for an autonomous vehicle to run on a dry asphalt road. Hence, whether the proposed NNAITSM controller can also work effectively in this scenario is of great importance. In this case, we set the road adhesion coefficient as μ = 0.85 to mimic the dry asphalt road condition, and the vehicle velocity as v = 60 km/h to exclude the impact of velocity compared with Case 1.

Figs. 10–13 show the lateral distances, lateral accelerations, sideslip angles and yaw rates under three control strategies, respectively. We can see that the autonomous vehicle can still maintain its stability. From Figs. 6 and 12, we can see that the amplitudes of the sideslip angles under three controllers in Case 2 all get reduced evidently in comparison with the corresponding ones in Case 1, which is due to the increasing of the road adhesion coefficient implying a larger friction between the tyres and road surface.

The steering angles and tracking errors are shown in Figs. 14 and 15. It can be seen that the NNAITSM controller takes action a bit earlier than the other controllers when the vehicle starts to steer. Thus, the peak errors under the NNAITSM control are forced to be 0.1 m and -0.1 m, which are much smaller than the corresponding ones under the other control schemes. Besides, the tracking error under the NNAITSM control is the first one to converge to zero or at least a very small region close to zero compared with the others.

Fig.10

Lateral distances in Case 2.

Fig.11

Lateral accelerations in Case 2.

Fig.12

Sideslip angles in Case 2.

Fig.13

Yaw rates in Case 2.

Fig.14

Steering angles in Case 2.

Fig.15

Tracking errors in Case 2.

Fig.16

Lateral distances in Case 3.

Fig.17

Lateral accelerations in Case 3.

Fig.18

Sideslip angles in Case 3.

Fig.19

Yaw rates in Case 3.

Fig.20

Steering angles in Case 3.

4.3 Case 3: dry asphalt road with high velocity

Sometimes, the autonomous vehicle will come across obstacles on a highway. In this scenario, the vehicle velocity is usually high and the collision-avoidance becomes more emergent, which is a challenge for the control system of the vehicle. In order to mimic the situation of a running vehicle on a high way, the road adhesion coefficient and the vehicle velocity are set as μ = 0.85 and v = 108 km/h.

Likewise, the lateral distances, lateral accelerations, sideslip angles and yaw rates in this case are sequentially shown in Figs. 16–19, respectively. It is evident that with the increasing of vehicle velocity, the errors between the actual lateral distances and the reference one all get increased compared with Cases 1 and 2. In addition, the amplitudes of the lateral accelerations, sideslip angles, and yaw rates under three controllers all get amplified. The reason lies in that with a much faster velocity, the steering behavior needs to be accomplished in a shorter time and the working conditions of the tyres approach closer to their limits.

From Fig. 20, we can see that with the RBFNN system to approximate the optimal upper bound of lumped system uncertainties and the effective adaptation law to tune the control parameters, the NNAITSM control input is with the smallest amplitude but without extra chattering among all controllers, especially compared with the CSM control. Moreover, the tracking precision of the NNAITSM control beats the rest ones as well in this case, which is demonstrated evidently by the smallest

Fig.21

Tracking errors in Case 3.

Fig.22

MAX errors and RMS errors.

peak error compared with the others as shown in Fig. 21.

4.4 Summary and comparison by bar charts

To show the superiority of the proposed control scheme more conveniently, bar charts are utilized with the indexes of max errors, RMS errors as shown in Fig. 22. It can be seen that both the max errors and RMS errors under the NNAITSM control are the smallest compared with the other two controllers, which proves the high tracking precision of the proposed control scheme. In addition, the high tracking precision of the NNAITSM control in all three cases also verifies its strong robustness against various road conditions and vehicle velocities. Therefore, the superiority of the presented control strategy has been comprehensively exhibited.

5 Conclusion

In this paper, an NNAITSM control scheme is proposed for the collision-avoidance steering control of an autonomous vehicle, where an RBFNN system is adopted to approximate the optimal upper bound information of lumped system uncertainties, an adaptive law is introduced to update the parameters in the sliding variable to achieve high tracking precision via Lyapunov, and an ITSM control component is used to guarantee a fast convergence rate and chattering-alleviation effect. The Matlab/Simulink-Carsim co-simulation results and bar charts evidently indicate that the proposed NNAITSM control scheme possesses not only higher tracking accuracy but also stronger robustness against various road conditions and vehicle velocities in comparison with a CSM controller and an NTSM controller.

In reality, the control of the steering angle must be carried out through an steer-by-wire (sbw) system installed in the vehicle. However, sometimes the steering motor in the sbw system might lose its efficacy. Besides, due to the impacts of road-tyre static friction and the backlash existing in the steering rack-and-pinion system, there exists time delay between the desired steering angle and the actual steering angle. These issues are the main limitations in actual implementation. Hence, our first future work is to investigate fault-tolerant control and time-delay system control to solve the possible limitations in actual applications. In addition, the factors of environment temperature, road roughness, wind impact, rolling resistance and sensor errors also affect the performance of the lateral motion control of autonomous vehicles. In the future, we will investigate the modeling of these factors, and propose novel and advanced control strategies with strong robustness to handle these impacts. Last but not the least, though Carsim is a professional software with high reputation in the area of vehicle dynamics simulation, there must exist modeling errors and measurement noises in reality. Therefore, our third future work is to construct a real autonomous vehicle as our experimental setup such that the designed control algorithms can be tested in a more realistic way.

6 Disclosure

The authors declare that they have no conflict of interest.

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