Adaptive Second Order Recursive Terminal Sliding Mode Control for a Four-Wheel Independent Steer-by-Wire System

An adaptive second order recursive terminal sliding mode (SO-RTSM) is developed for the steering angle control of a four-wheel independent steer-by-wire (4WI-SbW) system. A recursive sliding mode structure is presented based on which the reaching phase is eliminated. Moreover, the finite time convergence of the tracking error is guaranteed compared to common integral sliding mode control (ISMC). Furthermore, to suppress chattering, instead of a first-order sliding mode used in a conventional ARTSM controller, a second order nonsingular terminal sliding mode is adopted such that reaching control input is obtained in an integral form. The advantage of the proposed controller is that the chattering can be effectively reduced without decreasing the control accuracy. Experiments are carried out on a real 4WI-SbW vehicle to demonstrate the fast convergence rate and small chattering of the proposed controller.


I. INTRODUCTION
Recently, steer-by-wire (SbW) system has been investigated in many works because of its advantages of an active steering operation. For example, adaptive control is adopted in [1], [2] based on the parameter identification of the SbW system. In [3], adaptive disturbance rejection control (ADRC) is discussed for the control of an SbW system at different ground friction levels. However, although the steering shaft is cancelled compared a traditional steering system, the kinematic relationship between two steered wheels is still fixed by rack and pinion steering mechanism in an SbW system. To achieve a more flexible steering control, four-wheel independent steer-by-wire (4WI-SbW) system has drawn many attentions.
The 4WI-SbW technology can be potentially applied in wheeled vehicles to realize a more intelligent and complex motion. In 2015, NASA adopted a four-wheel independent steering system in a modular robotic vehicle (MRV) developed for a space application. In 2019, the company of AEV Robotics introduced a lightweight modular vehicle The associate editor coordinating the review of this manuscript and approving it for publication was Mark Kok Yew Ng . system (MVS), where an electric four-wheel independent steering system is installed. In 2020, Jaguar Land Rover(JLR) launched a four-wheel individually-steered self-driving car ''Project Vector'' for an urban traffic environment. Compared to conventional steering system or SbW system, the mechanical connection between tire and steering wheel or between two tires are both cancelled where each tire is directly-steered by an individual motor [4]. Instead of a ''hard'' mechanical connection, flexible communication bus (such as CAN, FlexRay, etc.) is used for a ''soft'' connection. Since each tire is connected to an independent steering mechanism, it gets rid of the shortcomings and limitations of the traditional rack and pinion steering system and potentially improves the driving flexibility and safety [5], [6]. Explicitly, the 4WI-SbW system can provide the vehicle a smaller turning radius and a more flexible driving mode. At the same time, there exists a huge research prospects for the active safety control of a 4WI-SbW system installed vehicle. However, the vehicle is affected by many uncertainties at a real driving condition, and each steering tire is subject to a complex external disturbance. In the process of steering motion, if there exists a large wheel steering angle control error or a serious lag, tire sideslip and drag may occur, that will further aggravate tire wear, and the vehicle will not be able to drive normally in serious cases [7], [8]. Because of the uncertainty of the system parameters and the nonlinear characteristics of the tire, the controller design for a 4WI-SbW system is often a difficult work [9], [10]. Therefore, to obtain a desired steering angle control performance, robust controller with higher response speed and control accuracy is required.
Due to the advantages of fast convergence rate and strong robustness against system uncertainties, sliding mode control (SMC) has been widely used in many motion control systems such as feed drive system [11], mechanical transmission [12], automotive safety [13], etc. However, during the reaching phase, the system is sensitive to system uncertainties and noises and the sliding mode invariance can not be guaranteed any more [14]. Integral sliding mode technique (ISMC) provides a method to eliminate the reaching phase and as a result the global convergence time is also reduced [15], [16]. However, the conventional ISMC can only achieve asymptotic convenience since a linear sliding function is often used [17], [18]. In [19], motivated by ISMC, a recursive sliding mode structure is proposed based on which the finite time convergence is achieved but the singular problem may occur. In [20], nonsingular adaptive recursive terminal sliding mode (ARTSM) is first developed. However, the control input still contains a switching operation that may cause system chattering. As the main drawback of SMC, severe chattering may damage the mechanical structure or even make the system unstable. There exist several ways in the literatures to alleviate this phenomenon, e.g., boundary layer technique [21], disturbance compensation [22], filter technique [23], etc. The widely used boundary layer technique can lead to the loss of the invariance that defacts the control accuracy [21]. Alternatively, based on high order sliding mode (HOSM), the control input is obtained in an integration of the discontinuous switching term. The advantage of HOSM control is that the chattering is effectively suppressed while the sliding mode invariance maintains [24].
In this paper, activated by HOSM, an adaptive second order recursive terminal sliding mode (SO-RTSM) control method is proposed for the 4WI-SbW system. First, a dynamic model of the 4WI-SbW system with parametric uncertainty is presented. Second, based on the concept of ISMC, the controller is constructed based on a recursive integral terminal sliding mode. Moreover, a second order terminal sliding mode is combined in the recursive sliding mode to obtain a continuous reaching control input. Finally, to verify the superior advantages of fast convergence rate and small chattering of the proposed controller, experiments are conducted on a real 4WI-SbW vehicle compared to a conventional ARTSM controller. The main contribution of this paper is listed in the following: (a) A recursive sliding mode structure is developed such that the reaching phase is eliminated and moreover, the finite time convergence of the tracking error to zero is ensured compared to ISMC.
(b) A second order nonsingular terminal sliding mode is introduced in the RTSM, based on which the reaching control input is achieved in an integral form of the signum function instead of a conventional switching term as that in ARTSM, which can effectively suppress chattering. This paper is organized as follows. Section II presents the structure and the dynamic model of the 4WI-SbW system. In Section III, the SO-RTSM controller is discussed in detail where the stability analysis and parameter selection guideline are also provided. To demonstrate the advantages of proposed controller, section IV presents the experiment results on a real 4WI-SbW vehicle. Finally, conclusions and future work are given in Section V.

II. PLANT MODELING
In a four-wheel independent steer-by-wire (4WI-SbW) system, each tire is directly steered by a steering arm connected to an individual steering motor. As shown in Fig. 1, the plant model of each wheel can be described by where δ is the steering angle of the wheel. J , c and k are the equivalent rotational inertia, damping and stiffness, respectively. M w denotes the unknown uncertain ground torque. τ is the active steering torque impacted on the wheel given by where F is the driving force of the steering motor and r is the radius of the steering arm. M f is the friction torque given by where T v is the viscous friction coefficient, T c is the viscous friction level and sgn(·) denotes the standard signum function. M z is the tire alignment torque given by [25] where F z is the vertical tire force, D is the horizontal distance from the tire center to the kingpin and γ represents the kingpin inclination angle. VOLUME 8, 2020 Substituting (2)-(4) to (1), the plant model can be reformed as δ and the relationship of sin(δ) = δ+ δ has been used with δ a small uncertain number. In this paper, parameter uncertainties are assumed to satisfy where The symbol · 0 represents the nominal value of each parameter and · denotes the corresponding uncertain value. The control objective is to design a robust controller to achieve accurate and fast steering angle tracking performance of the 4WI-SbW system in the presence of system uncertainties. Define the tracking error where δ d is the steering angle command supposed to be thrice differentiable. Combining (11) with (5), the error dynamics is given by By substituting (8)-(10) into (12), the error dynamics can be rewritten as where with d the reformatted uncertainty to the system (13). From (15), one can find that d is linearly affectly by δ andδ. Then, the derivative of d can reasonably be assumed to be bounded by where a i (i = 0, 1, 2, 3) is a positive number, which exists but are unknown.

III. CONTROL DESIGN
In this section, for the 4WI-SbW system to track the steering angle command fast and accurately under model uncertainties and disturbances, a robust controller based on an adaptive second order recursive terminal sliding mode (SO-RTSM) is presented. To achieve this goal, a recursive terminal sliding mode structure is constructed based on which the tracking error converges to the origin in finite while the reaching phase is eliminated. Moreover, an adaptive law is also introduced to compensate for the uncertainties without knowing the upper bound of the uncertainty a prior. Finally, selection of the control parameters is discussed.

A. CONSTRUCTION OF THE SO-RTSM CONTROLLER
To construct the controller, the following nonsingular terminal sliding function σ is first introduced as follows: where the parameters λ 1 , λ 2 are selected such that the polynomial, which corresponds to the system (17), is Hurwitz. Besides, γ 1 , γ 2 are selected to satisfy where sgn(x) is the standard signum function. Note that for a > 0, ∀x ∈ R the function sig(x) a is smooth and monotonically increasing and always returns a real number. It has been proven that when σ = 0 in (17), the tracking error e converges to zero in a finite time [27] (Referred to as t σ ). Next, we propose a recursive integral terminal sliding function s as follows: where σ is with (17), and σ I is designed in the following form:σ where the parameter κ > 0. In addition, to eliminate the reaching phase the initial integral value of σ I in (20) is set as By substituting (22) into (20), it can be verified that the initial sliding variable s(0) = 0 which implies that the control system is enforced to start on the sliding surface at the initial time such that the reaching time is eliminated [19]. Since the initial states of the 4WI-SbW system are available in practice, σ I (0) can be calculated by When s = 0 holds in (20), the sliding variable σ converges to zero in a finite time t s given by The derivation of (24) is addressed in Appendix A.
Finally, we give the form of the SO-RTSM controller based on the recursive integral terminal sliding function. The controller will be constructed in the following form: with τ 0 , τ 1 the equivalent control input and the reaching control input to be designed, respectively. By using (13), (17) and (20) and lettingṡ = 0, d = 0 in (13), we can obtain the following equivalent control input Further, an integral form control input is introduced where the control parametersâ 0 ,â 1 ,â 2 andâ 3 are updated by the following adaptive laẇ with η i > 0 andâ i (0) ≥ 0, (i = 0, 1, 2, 3). Remark 1: An alternative dynamics of σ I is utilized in (21) in addition to the power function form used in [20]. Although a signum function is adopted in (21), the control input (26) is continuous. Explicitly, the integral term σ I is obtained due to the use of the second order sliding function (17).
Remark 2: In [20], a first-order nonsingular terminal sliding mode is utilized. However, serious chattering may occur due to the signum function in the reaching control input. In this paper, a second order sliding function is used such that an integral term is obtained in the reaching control input (27) instead of a conventional switching operation. Then, the total control input (25) is theoretically chattering-free, which will benefit control of the 4WI-SbW system.

B. STABILITY ANALYSIS
The result for the proposed SO-RTSM controller is summarized in the following theorem and stability analysis is provided.
Proof of Lemma 1 is in Appendix B. Theorem 1: Consider the 4WI-SbW system in (5) with the parameter uncertainties (7)- (10). Then, under the SO-RTSM controller in (25), the tracking error e converges to zero in a finite time.
Proof : First, from (17), (20) and (25), the derivative of the sliding function s in (20) becomeṡ Next, choose the Lyapunov function as where µ i > 0,ã i =â i −a i (i = 0, 1, 2, 3). Solving the derivative of (33) along the system trajectories and substituting (32) and (27) to it yieldṡ µ iãiȧi −(a 0 + a 1 |δ| + a 2 δ + a 3 δ )|s| +(a 0 + a 1 |δ| + a 2 δ + a 3 δ )|s| = −(a 0 + a 1 |δ| + a 2 δ + a 3 δ − Note thatâ i ≤ a i (i = 0, 1, 2, 3) from Lemma 1, (34) becomeṡ Define the following symbols where i = 0, 1, 2, 3. Then, if σ s , σ i > 0, (34) can be rewritten aṡ where is a positive constant satisfying ≤ ϒ, It is obvious that σ s > 0, and for any η i (i = 0, 1, 2, 3), , there exist positive constants µ i , a i such that σ i > 0, ϒ ≥ . Therefore, the inequality (36) satisfies the finite time stability criterion in Appendix C. More specific, V will converge from any initial condition V (0) to zero in a finite time t V given by This implies that the sliding variable s and the estimation error a i will converge to zero in a finite time as well. Besides, when s = 0, σ and e will successively converge to zero in the finite time of t s and t σ . Therefore, the tracking error e will converge from any initial condition to zero in the finite time of t c = t V + t s + t σ . This completes the proof. Remark 3: Based on the RTSM in (17) and (20), the system state passes along the sliding surfaces s = 0 and σ = 0 successively and as a result the tracking error converges to zero. As given in (22), s(0) = 0 can be assured by selecting an initial value of the integral element such that the reaching phase is eliminated. Moreover, different from the conventional ISMC [28], finite time convergence is achieved based on the proposed RTSM. Besides, when s(0) = 0, the value of V (0) is also reduced and as a result the time for V to reach the origin, i.e., t V is reduced from (38).
Remark 4: In practical applications, the sliding variable s is often chattering around zero and may result in conservatively large estimation of a i as described in (28)- (31). To solve this problem, dead zone technique can be used for a practical implementation [29], [30]. The adaptive law is then given by where i = 0, 1, 2, 3, > 0 is the threshold. It is clear from (39) that when s is within the threshold,â i will retain its present value. One can verify that when |s| < , (36) still holds, i.e., the finite-time stability is still guaranteed.
This completes the proof.

C. CONTROL PARAMETERS SELECTION
In a practical implementation, it has to compromise the impacts of control signal smoothness, energy consumption and measurement noises, etc. The control parameters selection guideline for the proposed SO-RTSM controller will be presented in the following and their values for the 4WI-SbW system will be selected. 1) Selections of λ i , γ i (i = 1, 2): Large values of λ i and γ i in (17) lead to a faster convergence rate of e to zero along the sliding surface σ = 0. However, large values of them also cause an increased control input signal as given in (26). In this study, we choose λ 1 = 100, λ 2 = 26, γ 1 = 18 35 and γ 2 = 36 53 . 2) Selections of λ, κ: As given in (24) and (26), a larger λ or κ in (20) indicates a smaller convergence time of σ after s = 0 but at the cost of an increased control input. Besides, a larger λ will also increase the amplitude of the integral element that implies a smaller steady-state tracking error. For the 4WI-SbW system, we choose λ = 20, κ = 50.
3) Selections of η i (i = 0, 1, 2, 3): The parameters η i can be selected sufficiently small to achieve a fast estimation of the control gain as in (28)- (31). However, it may cause overestimation or even lead to control input saturation of the steering motor. Based on the experimental performance, we choose η 0 = 0.5, η 1 = η 2 = η 3 = 2.

IV. EXPERIMENTAL RESULTS
To demonstrate the proposed SO-RTSM controller for the 4WI-SbW system, experiments are carried out on a real 4WI-SbW vehicle shown in Fig. 2. The vehicle is with double-wishbone independent suspension and each wheel is driven by an independent hub motor. The steering angle of each wheel is obtained by the real-time working stroke of the steering motor. A magnetic navigation sensor is installed at the bottom of the experimental vehicle for detecting the lateral position error to a planed magnetic route. The proposed control algorithm is implemented in MATLAB/Simulink and runs on a dSPACE-DS1005 real-time platform with a sampling period of 5 ms. The software of ControlDesk installed on a PC is used to collect the experimental data from the dSPACE platform. Fig. 3 shows the control structure in the experiments.
The reference steering angle is a sinusoidal sweep signal with a time-varying amplitude and frequency.

Case 1) Asphalt Ground Test:
A commonly used asphalt ground is first adopted. As we can see from Fig. 4, the tracking errors of both controllers increase as the frequency of the steering angle command grows and the peak error occurs when the steering direction changes. At either large amplitudes or high frequencies, the control accuracy of SO-RTSM is better than that of ARTSM. Specifically, the tracking error of ARTSM increases to 1.2 deg, which is much larger compared to that of SO-RTSM (0.4 deg). This is because the sliding error under the switching control input is increased due to the existence of the sampling period in a real discrete control system. As shown in Fig. 5, the control input of SO-RTSM responses faster than that of ARTSM. Moreover, due to the integral operation,   the amplitude of the control input chattering of SO-RTSM is effectively suppressed compared to that of ARTSM.

Case 2) Sandy Ground Test:
To further test the robustness of the proposed controller, a sandy ground is adopted and a load mass of 20 kg is putted on the vehicle. It can be seen from Fig. 6 that the tracking error under ARTSM significantly increases compared to Case 1). To the contrary, under SO-RTSM controller, the tracking error is still limited under a small value of 0.4 deg. Similar to Case 1), the control input of SO-RTSM shows much smaller chattering than that of ARTSM as shown in Fig. 5. This experiment verifies the stronger robustness of the proposed controller compared to ARTSM.

B. PATH FOLLOWING PERFORMANCE
A crab-steering motion is used to test the steering angle tracking performance of four wheels under a dynamic  uncertain environment. In a crab-steering motion, each wheel has the same steering angle such that the vehicle can complete a path following task without changing it heading angle. The crab-steering motion is potentially to realize a spacelimited driving and the 4WI-SbW vehicle can reach the target position more quickly than that with a conventional steering system. The magnetic navigation control method is adopted in the experiment, where the a magnetic route has been placed on the ground for the vehicle to track. The lateral position error between the center of vehicle and the magnetic route is measured by the magnetic navigation sensor and a traditional PID control method is used to generate the target steering angle for each wheel to track.
As given in Fig. 8, the stable error of the vehicle lateral position under SO-RTSM controller is about ±0.02 m and the maximum lateral error is less than 0.07 m. The reference steering angle signal from the PID controller is shown in Fig. 9 and the steering error of each wheel is  given in Fig. 10. It can be seen that for each wheel, the peak error is under 2.2 deg and converges into a small stable error bound of around 0.25 deg within a short time of 0.3 s, which implies the good performance of the proposed controller for the 4WI-SbW control. The control input variations of four wheels are as shown in Fig. 11. One can find that the peak current is less than 10.05 A and occurs at the moment when the target steering angle changes its orientation (Fig. 9). Fig. 12 gives the yaw rate and lateral acceleration of the vehicle. More specific, the maximum yaw rate is below 0.18 deg/s and the stable yaw rate is around ±0.1 deg/s that implies the vehicle heading angle changes small in the crab-steering motion. The dynamic crab-steering experiment demonstrate that the 4WI-SbW vehicle can accurately follow a desired path under the proposed SO-RTSM controller.

V. CONCLUSION
To improve the control performance of the 4WI-SbW system, an SO-RTSM controller is proposed in this paper. Lyapunov analysis proves that under the proposed controller, the steering angle error can converge to zero in finite time. Moreover, compared to conventional ARTSM, the proposed controller utilizes a second order nonsingular terminal sliding mode and as a result the reaching control input is obtained in a continuous form, which is useful to suppress the chattering of the control input signal. Besides, the proposed second order method maintains the sliding mode invariance during the sliding phase such that the control accuracy will not decrease when chattering is suppressed. In the future, we will do some work on the cooperative control strategy of four individual wheels and take the drive stability into account.

APPENDIX A
Once s = 0 holds in (20), it yields that σ = −λσ I , which implies that σ I has the same convergence time as σ .
Therefore, it can be concluded that for any σ (0) = 0, σ converges to zero in the finite time given by This completes the proof of (24).

APPENDIX B
Proof of Lemma 1: Supposing |s| = 0, from (28)-(31), a i (i = 0, 1, 2, 3) is increasing and there must exist a time instance t 1 such that a 0 +â 1 |δ| +â 2 δ +â 3 δ > 1 J 0 ḋ (46) From (32), from t = t 1 the adaptation gain is sufficiently large to make the sliding variable s decreasing. Simultaneously,â i (t) continues to increase until s = 0 is achieved in a finite time t. After that,â i (t) will retain its final valuê a i (t 1 + t). Due to its continuity property,â i (t 1 + t) is finite, i.e.,â i (t) is upper bounded. Therefore, there exists a positive number a i satisfyingâ i ≤ a i in (16). This completes the proof of Lemma 1.

APPENDIX C
Given the following first-order nonlinear differential inequalityV where β > 0, 0 < α < 1. V (x) represents a positive Lyapunov function with respect to the state x ∈ R. Then, for any given initial condition V (x(0)) = V (0), the function V (x) converges to the origin in the finite time given by The derivation refers to [31], [32] and references therein.