Robust Control Framework for Lateral Dynamics of Autonomous Vehicle Using Barrier Lyapunov Function

This article presents the robust lateral control of an autonomous vehicle in the presence of unknown lateral tire forces, road curvature angle, and parametric uncertainties. The sliding mode control (SMC) with barrier Lyapunov function is implemented to guarantee the system robustness while maintaining the outputs of the system in realistic bounds. Following the model reduction approach, the slow and fast dynamics of the system are separately controlled using the proposed control technique. The efficacy of the proposed control technique is examined by comparing the simulation results with conventional sliding mode control in two-time scales.


I. INTRODUCTION
The development of autonomous vehicles has remained an interesting field of research for academia, the car manufacturing industry, and other companies for the last three decades [1]. The autonomy of the vehicles is beneficial to society in several ways including passenger comfort, reduction in road accidents, and optimal fuel consumption. With the advancements in the development of autonomous vehicles, there are several challenges associated with it, including vehicle and passenger safety, fuel efficiency, and efficient maneuvering in different environments on different terrains [2]. The efficient maneuvering of the autonomous vehicle is of paramount importance, therefore significant research is being carried out to address the problems including vehicle stabilization in a cluttered environment, path tracking in presence of uncertainties and disturbances, localization, navigation, obstacle avoidance, and steering control while catering road bank angle and slippage etc. [3]- [7].
Path tracking control is a key concern for the autonomous vehicle during motion. The path tracking control enables the autonomous vehicle to track the desired trajectory by adjusting the lateral and longitudinal motion of the vehicle.
The associate editor coordinating the review of this manuscript and approving it for publication was Luigi Biagiotti .
In the car-like autonomous vehicle, the longitudinal dynamics become dominant when moving along the straight path while moving on the curved path, the lateral dynamics become more significant [8]. Autonomous vehicles require longitudinal and lateral control to achieve the desired path tracking [9]. The longitudinal control is to cater to the challenges of desired acceleration and operations of accelerating and braking of the vehicle [10]. On the other hand, lateral control deals with the challenges of safety, lane-keeping, lane changing, and achieving the desired lateral position [11], [12]. This article focuses on the robust control design for the lateral dynamics of the autonomous vehicle moving on the curve path.
Lateral control for the autonomous vehicle has remained a challenging task for researchers due to the complicated lateral dynamics. Several studies are found in the literature to design the different control topologies for the autonomous vehicle lateral dynamics including LQR [13], PID [14], feedback linearization [15] backstepping [16], [17], gain scheduling [18], sliding mode control (SMC) [19], [20], and fuzzy logic [21], [22]. The steering control for the autonomous vehicle using SMC and the backstepping controller is proposed in [19]. In [23] the sliding mode variable structure is adopted to control the lateral dynamics of the vehicle. The SMC in conjunction with gain scheduling and disturbance observer is presented in [24] for the vehicle path tracking.
Generally, the autonomous car-like vehicles are modeled with the 3 degrees of freedom (DOF) bicycle model. Moreover, with the assumption that the longitudinal dynamics are smooth and entirely controlled by the longitudinal controller, the system model is reduced to 2-DOF i.e. yaw and lateral position. This modifies the autonomous vehicle model into a single-input multi-output (SIMO) system. The SIMO model of the autonomous vehicle has two outputs, yaw angle, and lateral position. Intuitively, the lateral position is a slow state while the yaw angle is a fast state. Hence, the autonomous vehicle lateral dynamics can be reduced to slow and fast dynamics. These fast and slow systems are usually controlled by devising a two-time scale-based controller approach. Mostly, SIMO systems are controlled using energy-based controllers [25], intelligent controllers [26], [27], hierarchical controllers [28]- [31], and multi-time scale based controllers [32], [33]. This article uses the reducedorder model based on the two-timescale approach to design the control for lateral dynamics of the autonomous vehicle.
Another important prospect of the autonomous vehicle is its safety and stability when prone to an uncertain environment. Due to saturation and nonlinear tire-terrain dynamics, the vehicle can maneuver in unsafe operating range. One of the solutions to this problem lies in restricting the system states in certain bound. These constraints are usually fulfilled by using a control barrier function [34], [35]. In [34], the barrier function in conjunction with SMC is used to constraint the output of the quadrotor with modeling uncertainties and disturbances. In [35], a hierarchical controller-based approach is used for lateral control for the vehicle. The authors have used the barrier function to constraint the yaw rate and sideslip angle for the multi-input multi-output (MIMO) model and linear tire dynamics. Although several other studies have employed barrier function [36], [37], no significant research exists regarding barrier function with SMC for the autonomous vehicles' lateral dynamics in the presence of nonlinear tire-forces, curvature angle, and parametric uncertainties.
In this article, the control of the lateral dynamics of the carlike autonomous vehicle is considered, while moving on the curved path. The vehicle is modeled using the 2-DOF bicycle model by incorporating the impact of the lateral tire forces, parametric uncertainties, and road curvature. This modifies the autonomous vehicle model into a SIMO system with yaw and lateral position as its outputs and steering angle as the input. The control design becomes challenging in the presence of nonlinear tire forces and road curvature in the vehicle model. Following the approach used in [38]- [40], the system model is reduced into slow and fast subsystem with lateral position having slow dynamics while the yaw having fast dynamics. This allows designing the two separate controllers by inducing an auxiliary control input in slow dynamics which reduces the complexity of the control design. To keep the states in lateral position and yaw angle in realistic bounds in presence of the aforementioned disturbances, the approach of [34] is modified for both subsystems.
Furthermore, the finite-time convergence advantage of the control approach helps in choosing the controller gains systematically. Finally, the comparison of the proposed control technique with conventional SMC is carried out using numerical simulation, which proves the efficacy of the proposed technique.
The overall contribution of this article is stated as follows: i) The proposed control guarantees the tracking of lateral dynamics in presence of modeling uncertainties, lateral tire forces due to slip and road banking. ii) The yaw angle and lateral position are bounded by certain realistic upper and lower bounds. iii) different from existing literature, the proposed methodology provides a novel control scheme for SIMO perturbed systems under state constraints.
The remaining article is structured in the following way: the necessary lemmas and assumptions for control design are defined in Section II. The lateral dynamics of the autonomous vehicle are modeled in Section III. In Section IV, the conventional SMC and SMC with barrier function is developed and implemented for the reduced-order system model for controlling the lateral dynamics of the autonomous vehicle. In Section V MATLAB/Simulink based simulation result are presented and discussed. Section VI concludes the paper.

II. PRELIMINARIES
The nonlinear second-order system is given bẏ (1) where the systems' states are x 1 , x 2 ∈ R, input u ∈ R, output y ∈ R and f and g are the smooth functions and uncertainties are given by D (x, t) . The initial conditions are 10 x 20 ]. The output y (t) should track the desired output y d (t) while remaining in a certain bound i.e.
|y (t)| ≤ c ∀t ≥ 0, where c is a positive constant. Definition 1 [34]: For a systemẋ = f (x), a barrier Lyapunov function V (x) is a scalar function, which is continuously differentiable such that x ∈ D and V (x) → ∞ as x approaches to ∂D, and satisfies V (x) ≤ a, ∀t ≥ 0 for some positive constant a.
Assumption 1: For any c > 0, there exist positive con- and V 2 are positive definite and continuously differentiable functions in their respective domain as V 1 (x 1 ) → ∞ as Here βs and β are class K ∞ functions. Let V 1 and V 2 are to the set Lemma 2 [42], [43]: For any positive constant a, b, the following inequality holds ∀x such that |x| < a: where 0 < ≤ 1, and ∀x such that |x| < a.

III. AUTONOMOUS VEHICLE MODELLING
In this section, the lateral dynamical modeling of an autonomous 4-wheeled vehicle with front wheels steering is presented. The simplified bicycle model is used to derive the vehicle model [12]. Since on the curved path, the autonomous vehicle motion is considerably affected by the tire-terrain interaction and banking forces, therefore the effect of these forces is incorporated into the model.

A. LATERAL DYNAMIC MODEL
One of the major challenges for an autonomous vehicle is to keep its lane while moving on the curved path. Unlike the straight path, the effect of lateral forces becomes significant. The 4-wheeled autonomous vehicle motion on a curved path is shown in Fig. 1. The XY-plane represents the reference inertial frame. The center of gravity of the vehicle is denoted by CG, the steering angle is given by δ, while ν x and ν y are the longitudinal and lateral speed, respectively. The yaw angle is denoted by ϕ. Assumption 2: Forces on the rear and front wheels of vehicle satisfy F i = F i , where i ∈ {f , r} where F f , F f , F r and F r are the forces on front-left, front-right, rear-left, and rear-right wheels, respectively. Remark 1: Following Assumption 2, the combined force on the rear tires F r and the front tires F f become twice of the individual wheel forces such that F i = 2F i = 2F i , i ∈ {f , r}. Therefore, the autonomous vehicle illustrated in Fig. 1 can be modeled by the rear wheeled drive bicycle model while neglecting the roll and pitch motion.
Assumption 3: We assume that an autonomous vehicle is moving with constant longitudinal speed.
Remark 2: By Assumption 3, the conventional 3-DOF planar model is reduced to the 2-DOF model i.e., lateral position y, and yaw angle ϕ.
Under Assumption 2 and Assumption 3, the lateral dynamic model of the autonomous vehicle is given bÿ where F fy , F ry are lateral front and rear tire forces, respectively and F b = mg sin (θ b ) is the road banking force where θ b is the road banking angle. Mass of vehicle and gravitational acceleration is denoted by m and g, respectively. J is the yaw moment of inertia, d f and d r are the distances of the front and rear tire from the vehicle's center of gravity.

B. TIRE FORCES
For the autonomous vehicle, the forces acting on tires are of paramount importance. The tire-terrain interaction forces are generally modeled as the nonlinear function of longitudinal slip angle, lateral slip angle, and camber angle, etc. In this article, the Dugoff's tire model is used to determine the lateral tire forces. It is an empirical model, which provides relations for the longitudinal and lateral forces as function of the slip angle and slip ratio. Thus, Dugoff model is a simpler model that accounts for the coupling between the side and longitudinal forces as compared to other linear and nonlinear empirical tire models [44]. The lateral tire forces using this model are given as follows.
F fy =F fy +F fy (10) where the front and rear nominal forces are represented as F fy andF ry while forces due to sideslip are denoted byF fy andF ry . The front and rear nominal forces are calculated bȳ F ry = 2C rᾱr (13) where C f ,ᾱ f are nominal cornering stiffness and approximate slip angles of the front tire, and C r andᾱ r are nominal cornering stiffness and approximate slip angles of the rear tire respectively, and can be expressed as: Forces due to sideslip are calculated as follows: (17) where the S x is the longitudinal slip ratio. Front and rear tire sideslip angles are given by α f and α r whileC f andC r are cornering stiffness coefficients. The α f and α r are given by The function f (λ) and λ i s are defined by whereC The maximum road-tire friction coefficient is denoted by µ max , the coefficients of longitudinal and lateral tire stiffness areC x andC y [45].

C. INPUT AFFINE MODEL
By plugging in the expressions for forces in (8) and (9), the resulting input affine autonomous vehicle model is given bÿ We define the yaw and lateral motion in terms of error dynamics as:ė 1 =ẏ + v x (ϕ − ϕ d ) and e 2 = ϕ − ϕ d . The resulting model of the autonomous vehicle in the error dynamics is given by: Furthermore, we define 1 = e 1 ; 2 =ė 1 ; 3 = e 2 ; 4 =ė 2 , then the final system takes the form: where

D. MODEL REDUCTION
We define the system parameters using additive uncertainty form as follows: where nominal parameters areκ i andγ m , the perturbation terms are represented as κ i and γ m . Now, the system model with nominal parameters is given by ˙1 = 2 2 =κ 1 2 +κ 2 3 +κ 3 4 +γ 1 δ + 1 ( , δ, 1 , ω 1 ) Here, the lateral and yaw perturbations are given by 1 and 2 , respectively while corresponding overall perturbation in κ i and γ m is denoted by 1 and 2 .
First, we assume that the control input δ is designed such that the dynamics of (31) are much faster than the dynamics of (30) so within a short period, 3 reaches its quasi-steady-state¯3, while 4 and δ reach to zero. Then the slow dynamics are approximated by ˙1 = 2 2 =κ 1 2 +κ 2¯3 + 2 ( , δ, 1 , ω 1 ) where¯3 is a quasi-steady-state and acts as the auxiliary input for the slow dynamics. Assumption 4: The perturbation terms 1 and 2 are uniformly bounded as | 1 | ≤ ρ 1 and | 2 | ≤ ρ 2 .

IV. CONTROLLER DESIGN
In this section control law is devised for a reduced-order system of (31)- (32). Two separate control inputs are designed with (31)-(32) acting as the fast and slow systems, respectively. The fast control input δ tracks the auxiliary control input¯3 which eventually result in the stabilization of slow dynamics. In the proceeding text, first conventional SMC is designed for (31)-(32), then SMC with barrier function is designed.

A. CONVENTIONAL SMC DESIGN
To design the conventional SMC, we consider the following sliding surface for slow and fast dynamics, respectively.
The following Theorem proposes a controller based on the sliding mode control technique. Theorem 1: For a nonlinear SIMO system (31)-(32) with the sliding manifolds (33)-(34), considering Assumption 4, for control input: and with an auxiliary control¯3: where K i > ρ i for i = 1, 2 and p 2 p 1 , which guarantees the asymptotic stability of the system and the tracking of the desired lateral position and yaw angle is achieved. The initial conditions are defined to be 0 = 10 , 20 , 30 , 40 .

B. SMC WITH BARRIER FUNCTION
Theorem 2: Under Assumption 1 for the system (31)-(32) and the sliding manifolds (33)- (34), the control input sign (s 2 ) +κ 5 3 with an auxiliary control¯3 defined bȳ with K i > ρ i ∀i = 1, 2, p, r ∈ R and p, r > 0, then the origin of (31)-(32) is stable. Moreover, the output of the system is guaranteed to track the reference while maintaining the constraints 1 < and 3 < h.
Proof: We design an auxiliary control¯3 based on the SMC with barrier function. For sliding surface (33) and Lyapunov function candidate and the time derivative of V 1 iṡ By employing auxiliary control input (38) Now to design a controller for slow dynamics, consider the sliding surface (34), with Lyapunov function candidate and the time derivative of V 1 iṡ s 2 = p 2 4 − p 2˙3 +κ 4 2 +κ 5 3 +κ 6 4 +γ 2 δ + 2 ( , δ, 2 , ω 2 ) −¨3 s 2 = −p 2˙3 +κ 4 2 +κ 5 3 + 4 (p 2 +κ 6 ) +γ 2 δ sgn (s 2 ) +κ 5 3 + 4 (p 2 +κ 6 ) By employing control input (37) Corollary 1: Theorem 2 not only guarantees the output trajectory tracking but also enables to stabilize the slow and fast dynamics in the finite time. Moreover, if the initial conditions are defined to be 0 = 10 , 20 , 30 , 40 and with a = √ 2 p, then the upper bound on the settling time function T slow : R 2 → R + is given as: while the convergence of fast states is guaranteed in By using the fact that V = 0; ∀t ≥ T , the solution of the above equation and is given by; where V 0 is the initial value of the Lyapunov function and is given by; Since¯3 is an auxiliary input that is zero initially and 30 = The solution of the above equation is 1 = * 10 e −p 1 t . Therefore, for the decaying signal after 4τ , only 2% of * 10 will be available. The time required for 1 to reach its equilibrium point (98% settling time criteria) is calculated by 0.02 * 10 = e −p 1 t * Hence, the total settling time for the slow states to converge to the origin is Following a similar methodology, the convergence of fast states is guaranteed in Remark 3: The T slow and T fast facilitate the selection of controller gains that result in the finite-time stability of slow and fast dynamics such that T fast T slow and hence model reduction is validated.

V. SIMULATION RESULTS
This section presents the numerical simulation results of the controller designed for lateral control of the autonomous vehicle. The conventional SMC and SMC with barrier function are compared to examine the performance of the proposed technique. The results are obtained for tracking a realistic smooth trajectory given as ϕ d = −sech 2 (2t − 17) + sech 2 (2t − 42) and The parameters for the autonomous vehicle are given in Table-1. The vehicle is assumed to be moving with a constant longitudinal speed v x = 20ms −1 while initial conditions for the lateral position, lateral speed, yaw angle, and yaw rate are set to 0m, 0ms 1 , 0 rad and 0 rad/s, respectively. The slip ratio S x is assumed to be between 0.1 to 0.4. The road banking angle, which is an unknown switching disturbance, is presumed to be 0.087rad. The lateral tire forces are modelled using the Dugoff tire model. It is observed that the upper bound on 1 and 2 were ρ 1 = 0.6 and ρ 2 = 6.7, respectively. The vehicle parameters are perturbed 12% of their nominal values. The simulations are performed for a fixed step size of 2ms.
The absolute bound on the lateral position c is defined by c = 20.75m while yaw angle is bounded by h c = 1.0524rad. Since maximum absolute values of y d and ϕ d are 20m and 1rad, therefore by Assumption 2, the b = The settling time functions (39) and (40) are taken into consideration to evaluate the controller gains. By setting T fast = 1.10s and T slow = 7.86s, we obtained the controller gains p 1 = 0.5, p 2 = 3, K 1 = 0.5, K 2 = 2, p = 1, r = 1. For simulations of both conventional SMC and SMC with barrier function, the same controller gains are utilized. It is stressed that T fast and T slow only help in choosing gains. The cumulative controller may settle in different time and hence we cannot observe these times in simulation results.
The comparison graphs of the lateral position of the autonomous vehicle for tracking y d is given in Figure 2. Both conventional SMC and SMC with barrier function track the reference It can be observed that the tracking performance of conventional SMC deteriorates when the reference lateral trajectory changes. The comparison of lateral position error is given in Figure 3. It is evident that the conventional SMC defies the desired error bound c . On the other hand, SMC   with barrier function ensures that lateral position remains in desired bound during trajectory tracking. Furthermore, it is evident from Figure. 2,3, that SMC with barrier function exhibits better lateral position tracking as compared to conventional SMC.
In Figure 4, the yaw angle tracking of the autonomous vehicle for conventional SMC and SMC with barrier function is shown. Both controllers exhibit their robustness against disturbances and parametric uncertainties while tracking the desired yaw reference. Moreover, the results of Figure 5 depict that SMC with barrier function outperforms the conventional SMC. The conventional SMC not only displays relatively higher error but also violates the yaw error constraint h c . Conversely, SMC with barrier function satisfies the desired output constraints and hence demonstrates improved performance as compared to conventional SMC.  The control input for both controllers is compared in Figure 6. From the results, it can be deduced that SMC with barrier function results in less chattering as compared to conventional SMC. Since, in both controllers the resulting control effort is proportional to the magnitude of the disturbances; therefore, the centre of magnitude of the control effort is non-zero.

VI. CONCLUSION
In this paper, a robust control technique is proposed for the autonomous vehicle lateral dynamics in the presence of lateral tire forces, road curvature, and parametric uncertainties. The autonomous vehicle model is reduced to slow and fast dynamics and SMC with barrier function is employed to track the system's desired outputs while restricting the output in certain bounds. From the numerical simulations, it is evident that SMC with barrier function achieved better tracking performance as compared to conventional SMC. RAJA AMER AZIM received the B.E. degree in mechanical engineering from the University of Engineering and Technology, Taxila, Pakistan, in 1993, the M.S. degree in mechanical engineering from the University of New South Wales, Australia, in 2001, and the Ph.D. degree in mechanical engineering, in 2015. He joined the Faculty of Department of Mechanical Engineering with the College of EME, National University of Sciences and Technology, Islamabad, Pakistan. With over two decades of experience in mechanical product development in research and development and academic organizations. He is currently serving as an Assistant Professor with the Department of Mechanical Engineering, College of EME, National University of Sciences and Technology. He has various reputed journal and conference publications to his name. He is working on several funded research projects. His research interests include vehicle dynamics and control systems, mechanical system design, and autonomous vehicle systems.
HAMEED ULLAH received the B.Sc. degree in electrical engineering from UET, Peshawar, Pakistan, in 2015, and the M.S. degree in electrical engineering from the National University of Sciences and Technology, Islamabad, Pakistan, in 2019. His research interests include the flexible link robotic manipulators, robust state estimation, high gain observer, linear and nonlinear control systems, robotics, modeling, UAVs, sliding mode control, and robust control for underactuated mechanical systems.