2020 International Conference on Emerging Trends in Information Technology and Engineering (ic-ETITE) 978-1-7281-4142-8/20/$31.00 ©2020 IEEE 10.1109/ic-ETITE47903.2020.291 2020 International Conference on Emerging Trends in Information Technology and Engineering (ic-ETITE) Control of Autonomous Vehicle for Lateral Dynamics using Sliding Mode and Input-to State Stability Methods Aman Parkash Department of Electrical Engineering, National Institute of Technology, Kurukshetra, India-136119 amanparkash@gmail.com Abstract— This paper considers the stabilization problem of a surface autonomous vehicle and develops a control method for efficient lane keeping. The proposed control law is composed of two components; first is Lyapunov stability based sliding mode control (SMC) designed for the vehicle dynamics for stable and robust operation, the other one is Input-to-StateStability (ISS) based control designed for the lane keeping performance. It is shown that the proposed controller for nonlinear dynamics is easier due to its simple derivation. The closed-loop system performance and robustness analysis has been presented through a simulation exercise. The results show that the stability is maintained with negligible lateral deviation even in bending motion. Keywords— sliding mode control (SMC), Input-to-StateStability (ISS) control, nonlinear dynamics etc. I. INTRODUCTION Over past two decades, autonomous driving has attracted the attention of multidisciplinary research. Research on autonomous vehicle has been proved significantly successful in avoiding accidents, improving safety, contributing to the optimization of traffic flow, reducing of CO2 emissions, enhancing the mobility of elderly people and unconfident drivers [15]. To achieve the autonomous driving a pre-defined path (it may be circular, tortuous and spiral), the lateral and longitudinal problem have to be studied. However, the dynamics of longitudinal and lateral dynamics are coupled; it is supposed that the curvature of path is small. Some papers [1]-[4] have discussed both of the dynamics lateral and longitudinal. A model of the autonomous vehicle has been discussed in [9]. The controllability test for this nonlinear dynamics has been derived via lie bracket method [7],[8],[12] . Many control techniques have been presented for solving the lateral control problem of autonomous vehicle. The Papers [13], [14], [18] have proposed the H2 and H based control laws; while [5], [6] have proposed the control law based on PD-P control. LQR and MPC based control laws have also been reported by some papers [2], [13], [19]. These control laws have complex mathematical calculation and take much computational time for solving the dynamics of the autonomous vehicle. This paper considers the lateral dynamics of an autonomous vehicle [9]. The aim of this work is to obtain tracking of vehicle using stable and robust control. The Akhilesh Swarup Department of Electrical Engineering, National Institute of Technology, Kurukshetra, India-136119 a.swarup@ieee.org proposed control is a combination of Sliding Mode Control (SMC) and Input to State Stability (ISS) control. The simulation study using proposed control demonstrates stable and very close tracking. Further, robustness analysis of vehicle tracking performance has been investigated. This paper is organized as follows. Section II describes the system dynamics of autonomous vehicle, problem formulation. Some mathematical results which are essential for designing the controller are presented in section III. Design of the controller is presented in section IV. Results and simulation outputs are described in section VI. Conclusions are discussed in section VII. NOMENCLATURE m mass of the vehicle [kg] ߜ steering angle [rad] ݒ௫ longitudinal velocity of the vehicle [m/s] ݒ௬ lateral velocity of the vehicle [m/s] ߚ ratio of the lateral velocity and the longitudinal velocity ߰ heading error [rad] ߰ሶ yaw rate [rad] ݕ lateral deviation of the car [m] ߩ path curvature [m-1] ܥ cornering stiffness of front tyre [N/rad] ܥ cornering stiffness of rear tyre [N/rad] ܮ distances of the front tyre to the mass centre [m] ܮ distances of the front tyre to the mass centre [m] ܶ driver’s preview time II. DESCRIPTION OF SYSTEM DYNAMICS AND PROBLEM FORMULATION A. Description of System Dynamics Assumption 1: Assuming that the longitudinal speed, ݒ௫ is a constant and positive. The model of the autonomous vehicle has been discussed in [9]. The Lateral dynamics is described as ʹܥ ʹܥ ܮ ߚሶ ൌ ߜ െ ߰ሶ െ ିଵ ൬ߚ ߰ሶ൰ ݉ݒ௫ ݉ݒ௫ ݒ௫ ʹܥ ܮ ିଵ െ ൬ߚ െ ߰ሶ൰ǡ ݉ݒ௫ ݒ௫ 978-1-7281-4141-1/$31.00 ©2020 IEEE 978-1-7281-4142-8/$31.00 ©2020 IEEE Authorized licensed use limited to: University of Otago. Downloaded on May 31,2020 at 10:28:22 UTC from IEEE Xplore. Restrictions apply. 1 2020 International Conference on Emerging Trends in Information Technology and Engineering (ic-ETITE) ߰ሷ ൌ ʹܥ ܮ ʹܥ ܮ ܮ ߜെ ିଵ ൬ߚ ߰ሶ൰ ܫ௭ ܫ௭ ݒ௫ ʹܥ ܮ ܮ ିଵ െ ൬ߚ െ ߰ሶ൰ǡሺͳሻ ܫ௭ ݒ௫ in centre lane of the road with road curvature ߩሺݐሻ=0 to achieve zero equilibrium. 3) After designing the controller for the system (1) and (2) separately, and road curvature ߩሺݐሻ for all ݐ Ͳ, we will design a feedback controller, ߜ which is composite of two controller i.e. sliding mode controller, ߜୗେ and ISS property based controller, ߜ୍ୗୗ for achieving any desired equilibrium. The control effort also will be bounded all t>0 . Control Objectives: a) The control objectives of proposed controller are lane keeping with minimum deviation. b) Designing of simple controller. c) Robustness to external disturbances and internal parameter variations. d) Maintaining of the stability. (a) (b) Fig.1. Side-slip angles (a) Front tyre side-slip angle (b) rear type side-slip angle [9]. Where is ratio of lateral velocity and the longitudinal velocity of the vehicle. ሶ is the yaw rate (rad. /sec) and is the input of the vehicle i.e. steering angle. Condition of the road is not plain everywhere. There may be gradients on the road. The vehicle dynamics become very complex if we consider the path with gradients. So formulating assumption below. Assumption 2: Assuming that the path surface is plain and has no gradients on the path. For achieving the target in case of lane keeping, the relationship is obtained between vehicle and the reference trajectory i.e. centre lane of the plain. Graphical definition of variables are given in Fig.2.The dynamics [9] of the lane keeping can be described by ݕሶ ൌ ݒ௫ ߚ ܶ ݒ௫ ߰ሶ ݒ௫ ߰ , (2) ߰ሶ ൌ ߰ሶെݒ௫ ߩ where ߩ denotes the curvature of the lane keeping trajectory. B. Problem Formulation The objective is to develop a feedback controller for stabilizing the vehicle itself and keeping the centre lane of the road with the condition of road curvature, ߩሺݐሻ for all t>0. The scheme of designing the feedback controller, ߜ will have following steps 1) To design a sliding mode controller, ߜୗେ for vehicle dynamics (1) with bounded control effort for all t>0. This controller will stabilize the yaw rate, ߰ሶ and ߚ to achieve zero equilibrium. 2) To design a ISS property based controller, ߜ୍ୗୗ for lane keeping dynamics (2). This controller will keep the vehicle Fig.2. Graphical definition [9] of variables ݕ and ߰ in lane keeping cases [9]. III. PRELIMINARIES This section provides the results required for designing the Lyapunov based sliding mode controller (SMC). A result has been formulated in the form of a theorem explained below, will be used to develop controller in section IV. Theorem 1: Consider the dynamics for two dimensional system described by ߦଵሶ ൌ ݍଵଵ ߦଵ ߣଵ ሺߦଶ ሻ, (3) ߦଶሶ ൌ ݂ଵ ሺߦଵ ǡ ߦଶ ሻ ܾݑ Where ߦଵ and ߦଶ are the state of the system and nonlinear mappings are ߣଵ ǣԸ ՜ Ը and ݂ଵ ǣԸଶ ՜ Ը and ݑis the control input. Assume that ܾ ്0 , ݍଵଵ ൏ Ͳ and The function ߣଵ ሺߦଶ ሻ is continuous at ߦଶ ൌ Ͳ. Then there exists a state-feedback control ݑሺߦଵ ǡ ߦଶ ሻ such that the zero equilibrium of the closed-loop system is globally exponentially stable . The controller is given by ݑሺߦଵ ǡ ߦଶ ሻ ൌ െ ͳ ሾ݂ ሺߦ ǡ ߦ ሻ െ ܭௗ ߶ሺݏሻሿሺͶሻ ܾሺߦଶ ߦଵ ߣሻ ଶ ଵ ଶ 2 Authorized licensed use limited to: University of Otago. Downloaded on May 31,2020 at 10:28:22 UTC from IEEE Xplore. Restrictions apply. 2020 International Conference on Emerging Trends in Information Technology and Engineering (ic-ETITE) ͳ ۓǡ݂݅ ݏ ξʹǡ ଶ ξʹ ۖ ට ۖ ͳ െ ൫ξʹ െ ݏ൯ ǡ݂݅ ʹ ൏ ݏ ξʹǡ ۖ ξʹ ξʹ ߶ሺݏሻ ൌ ݏ ǡ ݏǡ݂݅ െ ۔ ʹ ʹ ۖ ଶ ξʹ ۖ ටͳ െ ൫ξʹ ݏ൯ ǡ݂݅ െ ξʹ ݏ൏ െ ǡ ʹ ۖ ەെͳǡ݂݅ ݏ൏ െξʹǤ (5) Proof: Consider the Sliding surface [17] is selected as follows ିଵ ߲ ݏൌ ൬ ߣ൰ ߦଵ ߲ݐ Where ߣ Ͳ and r is the order of the system. For ʹௗ order system, ݎൌ ʹ ݏൌ ߦଶ ߣߦଵ Now consider the Lyapunov candidate ଵ ܪൌ ݏଶ and its derivative ଶ ሶ ܪൌ ݏݏሶ ൌ ሺߦ ߣߦ ሶ ሻ൫ߦሶ ߣߦሶ ൯ ଶ ଵ ଶ ଵ ൌ ݂ଵ ሺߦଵ ǡ ߦଶ ሻሺߦଶ ߦଵ ߣሻ ߣሺߦଶ ߦଵ ߣሻ൫ݍଵଵ ߦଵ ߣଵ ሺߦଶ ሻ൯ + ܾݑሺߦଶ ߦଵ ߣሻ Substituting eq. (9)into above eq. we get ܪሶ ൌ െܭௗ ߶ሺݏሻ, by A2 ,ߣ Ͳandܭௗ Ͳǡ we conclude that ܪሶ ൏ Ͳ For all ሺߦଵ ǡ ߦଶ ሻ ് ሺͲǡͲሻ. Furthermore, ܪሶ ൌ Ͳ ߦଵ ൌ ߦଶ ൌ Ͳ. The ISS property based controller is developed following the Proposition-1 and Remark-6 of the paper [10]. Formulating the controller with the help of following proposition. Lemma 1: consider the two dimensional system which has two subsystems can be described as ȯሶ ൌ ݂క ሺȯሻ ݄క ሺȯሻୡ (6) Where ȯ ൌ ሾȯଵ ǡ ȯଶ ሿǡ ݂క ǣԸଶ ՜ Ըଶ and ݄క ǣԸଶ ՜ Ըଶ . There exists a controllerୡ such that the dynamics of the overall system which is described in (6) is zero equilibrium of the system. ଶ ୧ ୡ ൌ െ Ԗ୧ ߶ ൬ ȯ୧ ൰ ሺሻ Ԗ୧ ୧ୀଵ Where the function ߶ሺǤ ሻ is defines in (5) There exists כ୧ Ͳ and Ԗכ୧ Ͳ , such that for any כ୧ ԖሺͲǡ כ୧ ሻ and Ԗכ୧ ԖሺͲǡ Ԗכ୧ ሻ the overall closed-loop dynamics of the system is stable. IV. PROPOSED CONTROL DESIGN FOR AUTONOMOUS VEHICLE In this section, the proposed controller is designed to achieve zero equilibrium of the system. The overall interconnected system for designing the controller is given in Fig.3. The controller viz. sliding mode and ISS property based controller is designed as Consider the overall system (1)-(2) and define the variables ݔଵ and ݔଶ as ݔଵ ൌ ߚ ௩ೣ ߰ሶǡݔଶ ൌ ߚ െ ߰ሶ ௩ೣ (8) And the system (1)-(2) can be described with the new variable as ݔଵሶ ൌ ݃ଵଵ ݔଵ ݃ଵଶ ݔଶ ݂ଵ ሺݔଶ ሻ ܾଵ ߜሚǡ (9) ݔଶሶ ൌ ݃ଶଵ ݔଵ ݃ଶଶ ݔଶ ݂ଶ ሺݔଶ ሻ ܾଶ ߜሚ , ܮ ܮ ݔଵ ݔቇ ݒ௫ ߰ ǡ ܮ ܮ ܮ ܮ ଶ ݒ௫ ݒ௫ ݔଵ െ ݔቇǡ ܶ ݒ௫ ቆ ܮ ܮ ܮ ܮ ଶ ݒ௫ ݒ௫ ሶ ߰ሶ ൌ ݔെ ݔെߩ ݒǡሺͳͲሻ ܮ ܮ ଵ ܮ ܮ ଶ ௫ ݕሶ ൌ ݒ௫ ቆ Where ߜሚ ൌ ሺߜ െ ିଵ ݔଵ ሻ, is the auxiliary signal, ݒ௫ ݃ଵଵ ൌ ݃ଶଵ ൌ െ݃ଵଶ ൌ െ݃ଶଶ ൌ െ ܮ ܮ ʹܥ ܮଶ ʹܥ ʹܥ ʹܥ ܮ ܮ ܾଵ ൌ ቆ ቇǡ ܾଶ ൌ ൬ െ ൰ǡ ܫ௭ ݒ௫ ݉ݒ௫ ݉ݒ௫ ܫ௭ ݒ௫ ʹܥ ܮ ܮ ʹܥ െ ൰ ିଵ ݔଶ ǡ ܫ௭ ݒ௫ ݉ݒ௫ ʹܥ ܮଶ ʹܥ ቇ ିଵ ݔଶ ǡ ݂ଶ ሺݔଶ ሻ ൌ ቆ ܫ௭ ݒ௫ ݉ݒ௫ ݂ଵ ሺݔଶ ሻ ൌ ൬ By utilising the theorem-2 of paper [9] the Eq. (9) can be rewritten as ݔଵሶ ൌ ݍଵଵ ݔଵ ߣଵ ሺݔଶ ሻ, ݔଶሶ ൌ ݂ଵ ሺݔଵ ǡ ݔଶ ሻ ܾଶ ߜሚሺͳͳሻ By utilising the controller from Eq. (4) of theorem-1and the controller from Eq. (7) of lemma-1 there exists a feedback controller which is combining of sliding mode based control and ISS property based control such that the closed-loop dynamics converges to zero and this dynamics will be globally exponentially stable . The controller for system (10)-(11) is designed as ߜሚ ൌ ߜୗେ ߜ୍ୗୗ ሺͳʹሻ after back substitution of auxiliary signal ߜሚ we get the control signal, ߜ for overall system dynamics which is defined in Fig. 3. ߜ ൌ ߜୗେ ߜ୍ୗୗ ିଵ ݔଵ Where ߜୗେ ൌ െ (13) ͳ ሾ݂ ሺݔ ǡ ݔሻ െ ܭௗ ߶ሺݏሻሿǡ ܾଶ ሺݔଶ ݔଵ ߣሻ ଶ ଵ ଶ Fig.3. Block diagram of the vehicle dynamics control. 3 Authorized licensed use limited to: University of Otago. Downloaded on May 31,2020 at 10:28:22 UTC from IEEE Xplore. Restrictions apply. 2020 International Conference on Emerging Trends in Information Technology and Engineering (ic-ETITE) ଵ ଶ ߜ୍ୗୗ ൌ Ԗଵ ߶ ൬ ൰ െ Ԗଶ ߶ ൬ ߰ ൰ǡ Ԗଵ Ԗଶ Where the function ߶ሺǤ ሻ is defines in (5) The proposed controller (based on SMC and ISS property) is described as ߜሚ ൌ ߜୗେሺୣሻ ߜ୍ୗୗሺୣሻ ሺͳͷሻ after back substitution of auxiliary signal ߜሚ and using the ߜ ൌ ߜ െ ߜ we get the control signal, ߜ ݏൌ ݔଶ ߣݔଵ ǡ ݂ଶ ሺݔଵ ǡ ݔଶ ሻ ൌ ݂ଵ ሺݔଵ ǡ ݔଶ ሻሺݔଶ ݔଵ ߣሻ ߣሺݔଶ ݔଵ ߣሻ൫ݍଵଵ ݔଵ ߣଵ ሺݔଶ ሻ൯ , ݔଵ ൌ ܾଵ ݔଶ െ ܾଶ ݔଵ , ݍଵଵ ൌ ݃ଵଵ െ ߣଵ ሺݔଶ ሻ ൌ ݔଶ ቂܾଵ ݃ଶଶ െ ܾଶ ݃ଵଶ ܾଵ భ ሺܾଵ ݃ଶଵ െ ܾଶ ݃ଵଵ ሻቃ, మ ݂ଵ ሺݔଵ ǡ ݔଶ ሻ ൌ ݃ଶଵ భ ௫మ ି௫భ మ ఒమ ሺ௫మ ሻ ௫మ భ మ ݃ଶଵ , െ ܾଶ ఒభ ሺ௫మ ሻ ௫మ ߜ ൌ ߜ ߜୗେሺୣሻ ߜ୍ୗୗሺୣሻ ିଵ ݔଵ ሺͳሻ Where ߜୗେሺୣሻ ൌ െ Note: After back substitution of states ݔଵ and ݔଶ we get the signal ߚ and ߰ሶ respectively for overall system dynamics which is defined in Fig. 3. Values 1625 1.12 ܥ 195940 ܮ ܫ௭ ܥ ܶ ݔଵ ൌ ܾଵ ݔଶ െ ܾଶ ݔଵ , ݍଵଵ ൌ ݃ଵଵ െ భ మ ఒమ ሺ௫మ ሻ ௫మ ݃ଶଵ , െ ܾଶ ሺܾଵ ݃ଶଵ െ ܾଶ ݃ଵଵ ሻቃ, and݂ଵ ሺݔଵ ǡ ݔଶ ሻ ൌ ݃ଶଵ 1.48 భ మ భ ௫మ ି௫భ మ ఒభ ሺ௫మ ሻ ௫మ ݃ଶଶ ݔଶ ݂ଶ ሺݔଶ ሻ ଵ Ͳ , ଶ Ͳǡ Ԗଵ Ͳ , Ԗଶ Ͳ and ߣ Ͳ are positive constants. 1500 170390 2 V. STATE FEEDBACK DESIGN FOR TRACKING ACHIEVING ANY EQUILIBRIUM POINT In previous section we have designed proposed controller achieving the zero equilibrium of the system but this section provides the design for overall dynamics of the system (1)(2) with positive and feasible road curvature ߩሺݐሻ. Define the target value ݔଵ , ݔଶ as ݔଵ ൌ ߚ ߰ሶ ǡݔଶ ൌ ߚ െ ߰ሶ and ௩ೣ ݂ଶ ሺݔଵ ǡ ݔଶ ሻ ൌ ݂ଵ ሺݔଵ ǡ ݔଶ ሻሺݔଶ ݔଵ ߣሻ ߣሺݔଶ ݔଵ ߣሻ൫ݍଵଵ ݔଵ ߣଵ ሺݔଶ ሻ൯ , ߣଵ ሺݔଶ ሻ ൌ ݔଶ ቂܾଵ ݃ଶଶ െ ܾଶ ݃ଵଶ ܾଵ TABLE I. VEHICLE PARAMETERS Symbol m ܮ ሾ݂ଶ ሺݔଵ ǡ ݔଶ ሻ െ ܭௗ ߶ ሺୣ ሻሿ, ଵ ଶ ୣ ൰ െ Ԗଶ ߶ ൬ ߰ ൰ǡ Ԗଵ Ԗଶ Where the function ߶ ሺǤ ሻ is defines in (5) ሶ ǡ ୣ ൌ ݔଶ ߣݔଵ ߜ୍ୗୗሺୣሻ ൌ Ԗଵ ߶ ൬ ݃ଶଶ ݔଶ ݂ଶ ሺݔଶ ሻ , ଵ మ ሺ௫మ ା௫భ ఒሻ ௩ೣ the error signal defined as ݕ ൌ ݕ െ ݕ , ߰ ൌ ߰ െ ߰ , ݔଵ ൌ ݔଵ െ ݔଵ , ݔଶ ൌ ݔଶ െ ݔଶ and ߜ ൌ ߜ െ ߜ By utilizing the system (9)-(10), the error dynamics of overall system is described by ܮ ܮ ݔଵ ݔቇ ݒ௫ ߰ ǡ ܮ ܮ ܮ ܮ ଶ ݒ௫ ݒ௫ ݔଵ െ ݔቇǡ ܶ ݒ௫ ቆ ܮ ܮ ܮ ܮ ଶ ݒ௫ ሶ ݒ௫ ሶ ൌ ߰ ݔെ ݔǡ ܮ ܮ ଵ ܮ ܮ ଶ ݔሶଵ ൌ ݃ଵଵ ݔଵ ݃ଵଶ ݔଵ ݂ଵ ሺݔଶ ሻ െ ݂ଵ ሺݔଶ ሻ ܾଵ ߜሚ ݔሶ ଶ ൌ ݃ଶଵ ݔଵ ݃ଶଶ ݔଶ ݂ଶ ሺݔଶ ሻ െ ݂ଶ ሺݔଶ ሻ ܾଶ ߜሚ , ݕሶ ൌ ݒ௫ ቆ Where ߜሚ ൌ ሺߜ െ ିଵ ݔଵ ሻ, is the auxiliary signal. For tracking error converges to zero equilibrium ሺ ݕ ሺݐሻ െ ݕ ሺݐሻሻ ൌ ሺ ߰ ሺݐሻ െ ߰ ሺݐሻሻ ൌ Ͳǡ ௧՜ஶ ௧՜ஶ ሺ ߚሺݐሻ െ ߚ ሺݐሻሻ ൌ ሺ ߰ሶሺݐሻ െ ߰ሶ ሺݐሻሻ ൌ Ͳ, ௧՜ஶ ሺ ߜ െ ߜ ሻ ൌ Ͳand ௧՜ஶ ௧՜ஶ the bounded input for such that ȁߜሺݐሻȁ ൏ for all Ͳ. VI. RESULTS AND DISCUSSIONS This section discusses the comparative studies for showing the robustness of control law against road curvature, vehicle mass and longitudinal velocity respectively. The results are simulated in MATLAB/SIMULINK. The dynamics of the system are considered as explained in (1)-(2) and the nominal parameters of the vehicles are considered as given in Table1. Note that we take the values for road curvature, mass of the vehicle and longitudinal velocity are 1625 kg, 0.02 m-1 and 10 m/s respectively as a reference. A. Robustness to Road Curvature Variations (14) To perform the test for demonstrating the robustness we take curvature of the road is constant i.e. for nominal system ߩ ൌ ͲǤͲʹ, and for variation in parameter , ߩ ൌ ͲǤͲʹ Ͳʹ טΨ . Constant road curvature represented the uniform circular motion. Note that trajectories start from straight path and the suddenly change into uniform circular path. The simulation results for nominal value and values with variation of ͲʹטΨ are displayed in Fig.4. 4 Authorized licensed use limited to: University of Otago. Downloaded on May 31,2020 at 10:28:22 UTC from IEEE Xplore. Restrictions apply. 2020 International Conference on Emerging Trends in Information Technology and Engineering (ic-ETITE) (a) (a) (b) (b) (c) (c) (d) Fig. 4. Robustness against road curvature variations for trajectories (a) lateral deviation, (b) heading error, (c) beeta and (d) yaw rate. In Fig. 4.we simulate the results for lateral deviation, heading error, beeta and yaw rate. In this figure solid and blue line shows the results for nominal values; and other two lines i.e. black-dashed line and thin-red line are represented for variation of െʹͲΨ and ʹͲΨ respectively. From fig. 4.our controller perform well against robustness despite of variation in road curvature; we can see that just change steady-state values with small variations. B. Robustness to vehicle mass The mass of the vehicle is not always same it depends on how many persons in the vehicle and fuel. The results for nominal mass of the vehicle and mass with variation of ͲʹטΨ are displayed in fig.5. (d) Fig. 5.Robustness against vehicle mass for trajectories (a) lateral deviation, (b) heading error, (c) beeta and (d) yaw rate. Fig.5. shows that the controller for closed-loop system performs well against robustness with the variations by ͲʹטΨ . C. Robustness to longitudinal velocity The results for lateral deviation, heading error, beeta and yaw rate are displayed with the nominal values and variation of longitudinal velocity by ͲʹטΨ . 5 Authorized licensed use limited to: University of Otago. Downloaded on May 31,2020 at 10:28:22 UTC from IEEE Xplore. Restrictions apply. 2020 International Conference on Emerging Trends in Information Technology and Engineering (ic-ETITE) (a) (a) (b) (c) (b) (d) Fig. 6. Robustness against longitudinal velocity for trajectories (a) lateral deviation, (b) heading error, (c) beeta and (d) yaw rate In this paper our main focused on the lane keeping with minimum deviation. From the simulation results from fig.4 to Fig. 6 the ݕ , lateral deviation is almost near to zero with less peak and our controller perform well against variation in parameters by ͲʹטΨ . D. Stable Cut off Point for Robustness The variations of three parameters over steady-state lateral deviation has been studied. This study provides and insights affect the robustness of vehicle dynamic. (c) Fig.7 The variations parameters over steady-state lateral deviation for (a) road curvature (b) mass (c) longitudinal velocity. Fig. 7(a) depicts the steady state lateral deviation with respect to road curvature (ߩ) and shows that steady-state value sharply decreases after the ߩ ൌ ͲǤͲͶ͵ͷ. Therefore for a stable operation the safe value of road curvature (ߩ) may be choose Ͳ to ͲǤͲͶ͵ͷ. 6 Authorized licensed use limited to: University of Otago. Downloaded on May 31,2020 at 10:28:22 UTC from IEEE Xplore. Restrictions apply. 2020 International Conference on Emerging Trends in Information Technology and Engineering (ic-ETITE) Fig. 7(b) depicts the steady state lateral deviation with respect to mass of the vehicle and shows that steady-state value decreases to zero at mass1625 kg and then increases within the range from 1400 kg to 4200 kg. Therefore for a stable operation the safe value of mass with two percent tolerance may be choose ͳͶͲͲkg toʹͷͲͲ kg. Fig. 7(c) depicts the steady state lateral deviation with respect to longitudinal velocity (ݒ௫ ) and shows that steadystate values very near to zero at 10 m/s and 12.8 m/s respectively but decreases before 10 m/s and after 12.8 m/s. Therefore for a stable operation the safe value of may be choose ͻǤͺm/s to ͳʹǤͺ m/s. VII. CONCLUSIONS A nonlinear control scheme is developed for surface autonomous vehicle. The design method consists of sliding mode control (SMC) and Input-to-State-Stability (ISS) based control. The design of ISS possesses to have the property of boundedness. Convergence and stability of autonomous vehicle are ensured with the proposed controller. Robustness to external disturbances and internal parameter variations has been investigated. The performance of vehicle dynamics has been presented through simulation study in various cases. The effect on lateral deviation due to variations in longitudinal velocity, path curvature and vehicle mass has been analysed. This analysis provides the range of parameters for acceptable performance and hence will be useful selecting these parameters for lane following motion. The steady state performance of vehicle dynamics can be improved further by use of optimized controller gain values. REFERENCES [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [1] [2] Alcalá, Eugenio, et al. "Comparison of two non-linear model-based control strategies for autonomous vehicles." 2016 24th Mediterranean Conference on Control and Automation (MED). IEEE, 2016. Attia, Rachid, Rodolfo Orjuela, and Michel Basset. "Coupled longitudinal and lateral control strategy improving lateral stability for autonomous vehicle." 2012 American Control Conference (ACC). IEEE, 2012. [19] Attia, Rachid, Rodolfo Orjuela, and Michel Basset. "Combined longitudinal and lateral control for automated vehicle guidance." Vehicle System Dynamics, Vol. 52, no. 2, pp. 261-279, 2014. Bae, Il, et al. "Integrated lateral and longitudinal control system for autonomous vehicles." 17th International IEEE Conference on Intelligent Transportation Systems (ITSC). IEEE, 2014. Cha, Young Chul, et al. "Controller design for lateral control of unmanned vehicle." 2011 11th International Conference on Control, Automation and Systems. IEEE, 2011. Cha, Young Chul, et al. "A lateral controller design for an unmanned vehicle." 2011 IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM). IEEE, 2011. A. Isidori ,Nonlinear control systems, Springr, New york, US, 1995. Jakubczyk B. (2002), Introduction to geometric nonlinear control; controllability and lie bracket , Mathematical control theory, ICTP publications. Jiang, Jingjing, and Alessandro Astolfi. "Lateral control of an autonomous vehicle." IEEE Transactions on Intelligent Vehicles, Vol. 3, no. 2, pp. 228-237, 2018. Kaliora, Georgia, and Alessandro Astolfi. "Nonlinear control of feedforward systems with bounded signals." IEEE Transactions on Automatic Control,Vol. 49, no. 11, pp. 1975-1990, 2004. Hassan K. Khalil ,Nonlinear systems, Third edition, Pearson inc., 2014. Mathieu, Johanna L., and J. Karl Hedrick. "Transformation of a mismatched nonlinear dynamic system into strict feedback form." Journal of Dynamic Systems, Measurement, and Control ,Vol. 133, pp. 041010(1-4),2011. Rafaila, Razvan C., and Gheorghe Livint. "H-infinity control of automatic vehicle steering." 2016 International Conference and Exposition on Electrical and Power Engineering (EPE). IEEE, 2016. Roselli, Federico, et al. "H control with look-ahead for lane keeping in autonomous vehicles." 2017 IEEE Conference on Control Technology and Applications (CCTA). IEEE, 2017. R. Rajamani ,Vehicle Dynamics and Control.Springer US, 2012. Rafaila, Razvan C., and Gheorghe Livint. "Nonlinear model predictive control of autonomous vehicle steering." 2015 19th International Conference on System Theory, Control and Computing (ICSTCC). IEEE, 2015. Slotine, Jean-Jacques E., and Weiping Li. Applied nonlinear control. Vol. 199. No. 1. Englewood Cliffs, NJ: Prentice hall, 1991. Sohnitz, Ina, and Klaus Schwarze. "Control of an autonomous vehicle: design and first practical results." Proceedings 199 IEEE/IEEJ/JSAI International Conference on Intelligent Transportation Systems (Cat. No. 99TH8383). IEEE, 1999. Rui, Wang, et al. "Research on bus roll stability control based on LQR." 2015 International Conference on Intelligent Transportation, Big Data and Smart City. IEEE, 2015. 7 Authorized licensed use limited to: University of Otago. Downloaded on May 31,2020 at 10:28:22 UTC from IEEE Xplore. Restrictions apply.