Safety research on stabilization of autonomous vehicles based on improved-LQR control Open Access

Autonomous driving has become the solution to the traffic safety and intelligent network connectivity vehicles in recent years. Among the related functionalities (e.g., environment perception, path planning, motion prediction, collision avoidance, and path tracking) in entirely automated vehicles,¹ path tracking is the most important technology that calculates the steering angles to drive the car on the route of the destination.

Path tracking has the main mission to derive the appropriate front wheel steering angle and achieve the predefined goal, which can ensure that the lateral displacement error tends to zero as soon as possible.² To achieve this aspect, a lot of control approaches have been proposed in recent years, such as model predictive control (MPC), optimal preview control, and linear quadratic regulator (LQR). For various control goals and research mission, the choice of the control approaches will also be distinctive. In another study,³ a gain scheduled H_∞ yaw stability controller was designed to guarantee both parameter disturbance robustness and track accuracy adjustability, taking vehicle mass and moment of inertial as the scheduled variables. A novel path tracking controller for a 4-wheel-steering (4WS) electric vehicle is proposed according to the linear parameter varying system model.⁴ In combination with the feedback system, the tracking control is adapted to various longitudinal speeds and coefficients of road friction and minimizes the error caused by the disturbance. The implicit MPC method, for example, mainly avoids crash circumstances by designing the shifting horizon trajectory tracking system using the classical control method.⁵ The shifting sample time and prediction horizon are considered in the control strategy that aims to tackle modeling error more effectively. The mentioned dual-envelop-oriented shifting horizon that applies to path tracking finally increases path tracking precision effectively. The control needs could be satisfied by the target function, which has been through a receding horizon combined with the entire control structure and MPC theory.⁶ The results for distinguishing road maneuvering on slippery surfaces display the beneficial aspects of the control methods developed on the 8 degrees of freedom (8DOF) electric vehicle (EV) model offline simulation platform. The controller corrects the deviation problem between the actual movement and the intent of the driver. On top of a linear error system, a discrete-time preview controller has been designed to augment the perturbations within the finite preview window.⁷ The strategies are involved with MPC and particles swarm-optimization (PSO) methods.⁸ The MPC module is utilized to deliver virtual inputs in the upper processor and then assign the actual inputs in the bottom control system using sequential quadratic programming (SQP). When considering the case of calculation times, the PSO controller is more efficient and stable than the MPC-SQP controller, whereas the latter controller proved the superior performance than before on the path tracking aspect. The vehicle speed preview control system is designed to improve safety and comfort in high-speed cornering situations with the aid of high-definition road curvature information by reducing speed before the vehicle corners as a human driver.⁹ Taking the preview essence of the MPC into account, the method is used in combination with a new module of the vehicles’ kinematical and road parameters, which include the road curvature, yaw rate, lateral speed control acceleration, and performance index. A new control regime was introduced by the combination of MPC and PFT (prevision-following theory) to track autonomous vehicles, in attempt to optimize tracking accuracy and lateral stability.¹⁰ For the design of MPC and PFT controllers and linear time varying (LTV)-MPC, the non-linear model of vehicle dynamics was continuously linearized. The advantage of this study is that it is possible to extend the effective reference path into an unalterable horizon. There are many control approaches combined with the envelope method to continue the path tracking analysis.^11–14 The collision avoidance is the main applied situation of the MPC control strategy.¹⁵ The path tracking control strategy also matters the stability research,^16–19 and the MPC and Lyapunov methods are involved. MPC controllers are designed for over-actuated vehicles to maintain the yaw stability.²⁰ The bio-inspired LQR approach is used in off-road vehicle path tracking, and it builds the dynamic model and realizes the accurate path track.²¹ The integrated path planning and tracking controller aims to generate optimal control input, guaranteeing the minimum collision risk by unifying MPC.²² The adaptive-prediction-horizon model prediction control-based path tracking controller for a four-wheel independent control electric vehicle, unlike traditional model prediction control with fixed prediction horizon, for its adjustment considered the vehicle dynamic stability by introducing exponentially decreasing weight,²³ and the vehicle dynamic features are also considered when implementing MPC-based path tracking for autonomous vehicles.^24–26 Research on autonomous vehicle path tracking control considered the roll stability,²⁷ and the simulation results verify the design purpose. Active front-wheel steer (AFS)/direct yaw-moment control (DYC) integrated control helps build the envelope control framework for autonomous ground vehicle (AGV), and it applies the feedforward-feedback controller to calculate the desired steering angle to track the desired path.²⁸ There are different control approaches, and different kinds of vehicles are considered in the path tracking area;^29–33 this has been a valuable engineering problem.

Considering the fact that studies on improving the path planning algorithm of the path tracking controller are few, the main research objective of this paper lies in two aspects: (1) the path tracking controller depends on classical LQR theory, and furthermore, the improved path planning algorithm is applied; and (2) under different working conditions, comparing the improved-LQR and traditional MPC on a closed test road proves the effectiveness of the improved-LQR controller; and (3) To achieve the safety research of autonomous vehicles, the safe driving envelope is applied for analyzing the simulation results.

The remainder of this paper is organized as follows: Sec. II describes the dynamic vehicle model for the LQR controller. In Sec. III, the improved-LQR controller is designed. Section IV presents findings that rely on simulations and analysis. Finally, Sec. V summarizes the conclusion and further research direction.

In this section, a planar vehicle model is used to describe the vehicle dynamics, as shown in Fig. 1; only vehicle lateral movement and yaw motion are considered. Assuming that the longitudinal speed of the vehicle is constant, in the same direction of the coordinate system, the left and right wheels are regarded as a single wheel. Moreover, the front wheel steering angle is equal to the rear wheel steering angle. Thus, the simplification of the bicycle model has been done.

In Fig. 1, the vehicle kinematic relation is obtained as follows:

where x₀ and y₀ mean the longitudinal and lateral directions of the coordinate system, γ is the yaw rate, β is the slip angle, and φ and v_x are the yaw angle and longitudinal velocity, respectively. Assuming that the sideslip angle is small, the relative relation could be expressed as sin(φ + β) = φ + β and cos(φ + β) = 1. Equations (1)–(3) are transformed into

The lateral dynamics of the 2-DOF bicycle model can be interpreted as Eqs. (7) and (8), in agreement with Newton’s second law,

where v is the lateral velocity of the vehicle; I_z is the moment of the inertia about the z-axes; l_f and l_r mean the distances to the front and rear axles, respectively, from the center of gravity of the vehicle; and F_yf and F_yr are the lateral tire forces of the front and rear axles, respectively. Moreover, the nonlinear tire characteristics should be considered to improve the performance of the controller. In addition, without the impacts of the braking/driving force, the lateral tire force adopting a magic formula (MF) tire model is

where C_y1 represents the shape factor, β_y1 is the stiffness factor, D_y1 is the peak value, and E_y1 is the curvature value. To achieve the optimal performance of the target controller, the MF tire model expressed in Eq. (9) can be applied to make clear the front and rear lateral tire forces as a nonlinear form,

where δ_f means the front wheel steering angle, β is the sideslip angle of the vehicle body, and $Cαf*$ and $Cαr*$ are cornering stiffness variations of front and rear axles, respectively; these two parameters can be derived as

where C_αf and C_αr are the linear cornering stiffnesses of front and rear axles, respectively, and α_f and α_r are the front and rear tire slip angles, respectively. Furthermore, the approximation formula could be derived from the geometric relationship

Then, the above vehicle dynamic model can be simplified as

and setting the desired yaw rate

the deviation of the yaw angle can be determined as

Deriving the change rate of lateral deviation, we obtain

where the change rate of the lateral deviation is

The state variable could be selected as $X=e1ė1e2ė2$ and the front wheel steering angle as the control value. After the selection of the state variables, the matrix is

where the state space formula of the vehicle system is

where u is the control value.

In this part, we offer a path controller applied with the LQR theory in order to ensure that tracking is accurate, and the vehicle movement is optimally deviated. The main intention of the path tracking controller design is the balancing of high precision tracking performance and safe lateral stabilization on safety test road.

In modern control theory, the linear system expressed by the state-space formula is the main research target of the LQR (Linear Quadratic Regulator), designing the quadratic function related to input or the state of the controlled objective as the target function.

A continuous linear system can be described as

Meanwhile, Eq. (20) is also the state formula of the path tracking deviation. Based on this principle, we can further analyze the response characteristics of the path tracking deviation under the action of front steering angle input. The desired response characteristics require the tracking deviation to be fast and stably converge to be zero within an equilibrium range, and the control input value of the front steering angle should be as small as possible. Then, this has become a multi-objective optimal control problem. The optimization target function could be expressed as a weighted sum of the cumulative tracking deviation of the tracking process and the cumulative control input,

where Q is semi-positive definite state weighting matrix and R is the positive definite weighted matrix and usually taken as a diagonal matrix. Larger matrix elements mean that the tracking deviation should converge to zero as soon as possible, and the control input could be limited in a small value period. The cost function of quadratic type is defined in Eq. (23), and the following control law could acquire the minimum cost function:

where

where matrix P is calculated using the Riccati algebraic equation of continuous time, and the Riccati formula is as follows:

However, continuous systems need to be discretized in real control simulations. Now, we list Eq. (22) as $ẋt=Ax(t)+Bu(t)$⁠, multiplying both sides of the equation by e^−Ax,

after the shifting changes of Eq. (27), we obtain

and integrate both sides of the above equation simultaneously,

Assuming that the input is constant at each time step and defining T the time step, $xk=xKT$⁠, we obtain following transformation:

and using the permutation integral method and noting u(τ) = u(k), we can obtain

After the complete discretization, the optimal control for front wheel steering as the control objective can be defined as

where A_d and B_d are the discretized values and P is the solution to the Riccati equation, which is given as follows:

Combining Eq. (32) and state feedback principle shown in Fig. 2, the closed-loop optimal control can be achieved through state feedback,

Then, the optimal state feedback front wheel angle could be calculated using the benefit of the state feedback regulator,

Considering the path curvature and the understeer characteristics of the vehicle, it is also necessary to add feedforward control δ_ff so as to eliminate the steady state error. The total control input of the front steering angle is

The improved path planning optimization algorithm, which is selected as the enhancement of the controller, will be discussed in this part, and the overall framework of the improved-LQR controller is shown in Fig. 3.

The whole control algorithm is primarily based on the LQR control with path algorithm optimization on the transfer path error. The criteria of vehicle driving safety and vehicle dynamic constraints are also considered. The path information is obtained using the optimization layer to transfer the location error parameter, which is sequentially transferred to the feedforward control module and the optimal control output after the error information is processed by the LQR. The vehicle steering angle is the control input, and then, a prediction module of feedback control is added so that the framework of the control algorithm is completed. Now, the path optimization algorithm will be introduced as follows:

where x_r and y_r are the matching planning points of the reference trajectory as the input of the controller, θ_r is the angle between the direction of normal and tangential, and k_r is the path projection point. Traversing $xr,yr$ in Eq. (37) and then finding the closet planning point $x,y$ and naming it as d_min, we obtain

The above two equations are the tangent vector and the normal vector of the matching point, respectively. The distance error vector is

Due to Eqs. (37)–(39), the various types of path error points for the path planning can be obtained as follows:

where e_d and e_s are the distance errors of the tangent direction and normal direction, the matching point angle in the traveling direction is

the vector of e_d is

where e_φ is the yaw angle error of the vehicle, $ṡ$ is a direction value, and finally, the error value will be the output mentioned in the above equations.

In this section, a simulation study is proposed to evaluate the improved-LQR path tracking control behavior. The simulations were carried using Carsim/Simulink involved with various driving scenarios on the desired closed test road, and the designed road is shown in Fig. 4. The closed-loop test road is designed according to the test track standard GB/T12678-90. The road consisted of two straight sections of 50 and 150 m, two continuous curves of 50 m radius, an outer arc of 100 m diameter, and two curves of 50 m radius. The main purpose is to test the driving stability of the vehicle and the performance of the proposed controller under various conditions.

The experimental conditions include the speed scenario and road adhesion coefficient scenario, and the two scenarios are verified at low medium and high speed and low medium and high degree road friction coefficients.

The vehicle parameters are shown in Table I, and the vehicle model is a D-class car. Furthermore, the simulation step size is 0.025 s, and the matrices and variables used for the improved-LQR are

In order to value the simulation, an evaluation of the control performance in terms of the path planning optimization algorithm has been done, and three types of speed conditions and road coefficient conditions are designed with reference to a previous study.² Test A examined the performance of the improved-LQR at different velocities compared to the MPC control, and Test B examined the performance of the improved-LQR under different friction coefficient conditions.

In this subsection, the improved-LQR is tested at diverse vehicle velocities on the specific trial road (road adhesion u = 0.8). At certain steady speeds (60, 70, and 90 km/h), the autonomous vehicle drives, while other control parameters remain the same. By comparing tracking accuracy and the corresponding vehicle parameters of the proposed LQR and MPC on the test road, we verify whether the control performance of the improved-LQR in vehicle handling stability is reliable and whether the errors converge.

Figure 5 illustrates simulation outcomes of the improved-LQR of the 60 km/h case; the path tracking error performance of the two controllers at this velocity is illustrated in Fig. 8. The improved-LQR tracking error is 0.102 m, and the MPC tracking error is 0.441 m. The yaw rate has shown a kind of convergence under the improved-LQR, while the improved-LQR has also better performance in the steering angle than MPC.

The simulation results of the improved-LQR for the scenario of 70 km/h are shown in Fig. 6(a), while the tracking error is 0.215 m for the improved-LQR and 0.495 m for MPC.

Figure 6(b) shows a steady fluctuation, and the peak value of the LQR is 2.13 deg/s less than that of MPC. Figure 6(c) shows the change in the sideslip angle, although the fluctuation is gentle and the improved-LQR still realizes the objective of optimization. Figure 6(d) shows that the peak value of the steering angle is 5.22° under MPC control and 4.56° under improved-LQR control.

Figure 7 presents the simulation results of the improved-LQR for the scenario of 90 km/h, while the tracking error is 0.314 m for the improved-LQR and 0.605 m for MPC in Fig. 7(a). The peak value of the yaw rate of the improved-LQR is 21.6 deg/s, and the value of MPC is 24.5 deg/s in Fig. 7(b). Figure 7(c) shows a significant convergence change in the sideslip angle under the improved-LQR in contrast to MPC, and the peak values of both are almost close. Figure 7(d) shows that the steering angle decreases to 3.68° at high speed compared to MPC. Figure 8 presents the detailed error value of the path tracking.

According to the above analysis, there is a better performance of the proposed controller coming into our view. The vehicle response parameters show an upward fluctuation trend with increasing velocity at various velocities, both of which controllers maintain the stable tracking performance. The gained-over LQR has improvement in all aspects. Thus, the effectiveness of the proposed controller has been verified, and the proposed controller shows better performance than traditional MPC at three various velocities.

This subsection will manifest the validity of the improved-LQR controller under the influence of the road adhesion; the path following performance at the road adhesion values of 0.4, 0.6, and 0.9 is shown. We set the case that a vehicle drives at a speed of 65 km/h.

Figure 9 shows the simulation results of the improved-LQR for the scenario that road adhesion value u = 0.4; the path tracking error performance of the two controllers at this road adhesion value is illustrated in Fig. 12. The tracking error of the improved-LQR is 0.105 m, and the tracking error of another controller is 0.304 m. Figure 9(b) shows that the peak value of the yaw rate is 17.5 deg/s under MPC, and the value under the improved-LQR is 17.02 deg/s. Figure 9(c) shows the change in the sideslip angle under two controllers, and the change in the steering angle can be seen in Fig. 9(d).

Figures 10(a) and 12 show that the tracking error of the MPC is 0.386 m, while that of the improved-LQR is 0.194 m. Figure 10(b) shows the fluctuation of the yaw rate, and Figs. 10(c) and 10(d) shows the detailed value of the sideslip angle and steering angle, respectively. Figure 11(a) shows the comparison results of tracking error when u = 0.9. Figure 11(b) shows that the peak value of the yaw rate is 18.1 deg/s under MPC, and the value under the improved-LQR is 17.66 deg/s. Figures 11(c) and 11(d) present the comparing results of sideslip and steering angles. The detailed control performance value can be seen in Fig. 12.

From the above analysis on Test A and Test B, we demonstrate the efficiency and better control performance of the proposed controller compared to those of conventional MPC at different speeds and adherence levels. The improved-LQR has superiority aiming at any scenario in contrast to MPC. Now, we further analyze the stabilization of the proposed controller according to the safe driving envelope.

The safe driving envelope is not true that the vehicle will go out of control once the state leaves the envelope. The restricted area is actually steady-state assumptions according to empirical tests. It is still possible to explore a larger region, but it has been useful enough to guarantee that the vehicle drives stably.

Figure 13 shows the handling stabilization performance of the improved-LQR and the safety zone of the stable driving in the test scenarios mentioned above. The red envelope is the safety boundary, and details about the safe driving envelope can be found in Refs. 11 and 12. Figure 13(a) demonstrates the stabilization when the vehicle drives under the control of the proposed controller at three velocities, Fig. 13(b) shows the stabilization of the vehicle at different road adhesion values, and the evaluation indicator is within the safety boundary in each test scenario.

This paper has proposed an improved-LQR controller that applies a path planning algorithm to determine the optimal driving trajectory for autonomous vehicles. By using different calculating methods of path error points, the optimal driving trajectory will be the output from the improved-LQR. Moreover, the feedforward control and LQR are combined to enable the control of the vehicle steering angle, and then, the vehicle could be controlled to follow the desired trajectory. The simulation results demonstrated the superiority of the proposed improved-LQR control module to mediate conflicting objectives between the vehicle stability and better performance of the tracking control system, and in contrast to traditional MPC, several tests conducted at different speeds and road adhesion values were productive.

In future work, some further direction for the promotion of the improved-LQR controller will also be motivated, such as its application in real vehicle test circumstances.

This research was supported by the Natural Science Foundation of Zhejiang Province (Grant No. LGG18E080005), the Science and Technology Projects of Zhejiang Province (Provincial Key R&D Program: Grant No. 2020C01053), the National Natural Science Foundation of China (Grant Nos. 71874067, 71871078, and 51741810), and the Project of Zhejiang Education Department (Grant No. Y202044473).

The authors have no conflicts to disclose.

There was no ethical issue in this project.

H.L. investigated the study, performed the formal analysis and software testing, and wrote the original draft. P.L. conceptualized the study; performed the methodology; provided resources; wrote, reviewed, and edited the article; and acquired the funding. L.Y. supervised the study and administered the project. J.Z. performed the methodology and supervised the study. Q.L. conceptualized, investigated, and supervised the study; provided resources; and acquired the funding.

The data that support the findings of this study are available from the corresponding authors upon reasonable request.