Figure 5.Schematic diagram of MPC
MPC algorithm is widely used for automatic parking in low-speed conditions [41]. For different parking accuracy problems of different parking scenarios, researchers have made different solutions based on MPC. For example, Qiu et al. designed an MPC-based autonomous parallel parking trajectory tracking algorithm for solving the problem of parallel parking in a narrow space, and the simulation results showed that the method could make the vehicle park safely, quickly, and accurately in the parking space [42].Wang designed two MPC- and PP-based vertical parking trajectory tracking controllers with automatic reverse parking charging as the application scenario, and combined with PI-based The simulation results proved the feasibility and stability of the controller [43]. Song et al. proposed a tracking control method combining MPC and Iterative learning control (ILC), and the simulation results proved that the method has higher tracking accuracy than open-loop control, linear quadratic regulator (LQR) and traditional MPC The simulation results proved that this method has higher tracking accuracy than open-loop control, linear quadratic regulator (LQR) and traditional MPC algorithm [44].
The MPC algorithm also has many applications in low-speed steering conditions [45]. For example, in the literature [46-50], the MPC algorithm was improved to improve the path tracking accuracy and driving stability of driverless cars under right-angle turns, continuous curves and arc curves, and MPC-based integrated control algorithm was proposed and the tracking accuracy and driving stability were verified. There are also studies combining MPC with other algorithms, for example, Shi et al. proposed a path tracking algorithm based on MPC and PID, added front wheel side bias constraints on the basis of traditional MPC and introduced relaxation factors, and designed hybrid PID controllers for different road conditions to improve the accuracy of vehicle speed control, and simulation results proved that the algorithm greatly improved the stability and tracking accuracy of vehicle control [ 51].
MPC algorithm also has many applications in low-speed complex driving conditions [52]. Considering that the longitudinal speed, road curvature, and adhesion coefficient changes under low-speed complex driving conditions have a large impact on the tracking accuracy and stability of unmanned vehicles, the literature [53-55] proposed an optimized MPC controller to improve the tracking performance and stability by considering the influence of road constraints, but the study had the problem of being unable to track the reference path with large curvature changes. Therefore, the literature [56,57] proposed the use of Nonlinear Model Predictive Control (NMPC) control method, and the simulation results showed that the tracking performance and stability of the NMPC controller were substantially improved under low-speed large curvature conditions. In addition, Mohammad Rokonuzzaman et al. proposed an MPC controller that can be applied to vehicle models of different complexity, which can avoid more complex models in the tracking process, improve computational efficiency, and track better at low speeds compared with the traditional MPC controller [58]; Jin et al. proposed a path tracking algorithm for driverless cars considering driver characteristics. The drivers were classified into normal, conservative and aggressive types, and the path tracking controller was designed according to the MPC algorithm, and the experimental results showed that the speed tracking error of the controller did not exceed 2% and the lateral tracking error was less than 0.13m [59].
For MPC algorithm, with the increase of constraint dimension, the computational volume of its optimal solution will increase, and the difficulty of its optimal solution will further increase with the increase of vehicle speed, so the real-time of computation is a major bottleneck of MPC in path tracking control technology at present. In addition, the traditional MPC has limited ability to handle system uncertainty, and it is often difficult to achieve the established tracking target task when the system model is perturbed.
3.Research status of high-speed working condition path tracking algorithm
Compared with the low-speed conditions where only the path tracking accuracy needs to be considered, the high-speed path tracking control needs to ensure not only the accuracy of path tracking but also the driving stability of the vehicle due to the vehicle dynamics characteristics, and the complex dynamics model brings certain challenges to the research in this area and has become a hot spot for scholars’ research in recent years [60]. At present, the path tracking algorithms with more applications in high-speed conditions are LQR, MPC, ADRC, H∞, SMC, RL, etc. [61].
3.1 LQR algorithm
Linear Quadratic Regulator (LQR) is a linear system given in a state space model, and in vehicle path tracking, the feedforward and feedback control system of LQR is used to achieve closed-loop optimal control through feedback of the state [62,63].The advantage of LQR algorithm is that, by effectively combining with steering feedforward It can solve the steady-state tracking error when the vehicle is driving in a curve, and its steady-state error tends to zero when driving in a curve at medium speed, thus greatly improving the tracking performance. Therefore, the LQR algorithm is very suitable for highway and city driving scenarios with smooth paths, and has good vehicle speed control performance, which is also used in engineering.
For example, Yin et al. used the LQR controller to obtain the desired longitudinal acceleration, and predicted the driving risk based on the collision time and following time distance on the basis of collision avoidance, and the simulation results showed that the algorithm has high stability [64]. Although the conventional LQR controller can make the system stable, the disturbance of the disturbance term makes the steady-state error of the system not 0. For this reason, some scholars have introduced a road curvature feedforward link to the conventional LQR controller to eliminate the lateral displacement steady-state error. For example, Meng et al. designed an LQR control strategy based on foresight information [65], literature [66,67] proposed LQR controller with feedforward control, Liu et al. designed a multimodal controller based on the combination of MPC and LQR, by judging the path tracking curvature, the controller selects the corresponding control mode, and the simulation results show that the introduction of feedforward control can eliminate the system The simulation results show that the introduction of feedforward control can eliminate the steady-state error of the system and obtain good control performance [68]. However, because there is no pre-sight distance at the control point at the center of mass, the feedforward term can only react passively, which will produce more obvious overshoot. For example, the literature [69] designed an LQR controller with pre-scanning PID corner compensation and verified that the controller has high tracking accuracy under high-speed conditions by real vehicle tests.
For the lateral motion control problem, Ma et al. disassembled the dynamics model of the vehicle into a lateral error dynamics model, used a fuzzy control method to adjust the weight coefficients of the LQR in real time according to the vehicle state, and designed an update mechanism based on the cosine similarity, and the simulation results showed that the tracking accuracy and computational efficiency of the algorithm were greatly improved at 25 m/s vehicle speed [70].Gu et al. added a lateral error integral, constructed an energy function, and introduced a genetic algorithm to achieve optimization of the LQR controller, and the simulation results showed that the algorithm can effectively reduce the overshoot of the system and improve the convergence speed and stability of the system compared with the traditional LQR algorithm [71].
The LQR algorithm is usually based on a linear vehicle dynamics model, and when the vehicle motion does not satisfy the assumptions that the dynamics model steers to small angles or linearizes the tire dynamics, the tracking performance of the LQR algorithm is significantly reduced, which leads to control failure. Therefore, the combination of LQR and feedforward control cannot solve all tracking control problems, so the LQR algorithm is often combined with other path tracking algorithms in engineering for layered control to complement each other’s strengths and weaknesses.
3.2 MPC algorithm
MPC algorithm is not only applicable to low-speed operating conditions, but also applicable to high-speed operating conditions. For the design of MPC controller in this condition, scholars often divide into hierarchical control and centralized control.
Hierarchical control refers to the use of two or more controllers to control different targets independently, and this control method is relatively simple to implement and can be designed in a coordinated manner based on transverse and longitudinal path tracking control algorithms [72]. For example, in the literature [73], MPC controllers were used to handle perturbations in pavement curvature, PID feedback control to suppress instability and modeling errors; MohammadRokonuzzaman et al. proposed an MPC algorithm designed with a neural network-based vehicle learning dynamic model [74]; Yao et al. proposed an MPC path with longitudinal speed compensation in the predictive time domain tracking controller and longitudinal speed compensation strategy [75]; Cui et al. designed a traceless Kalman filter and proposed a multi-constraint model prediction controller (MMPC) [76]; Wael Farag et al. proposed a framework for a path tracker (SDC) for self-driving cars based on the NMPC approach [77]. After simulation verification, all of the above decentralized control methods improve the path tracking accuracy and vehicle driving stability under high-speed conditions.
In hierarchical control, lateral control is usually implemented by an MPC controller, and longitudinal control is implemented by another independent controller. However, hierarchical control is often coordinated only for longitudinal control, ignoring the coupling between the two. When encountering kinematic coupling enhancement with large changes in road curvature or sharp changes in vehicle speed or kinetic coupling enhancement under extreme operating conditions such as high vehicle speed and low road adhesion coefficient, the controller may suffer from large overshoot, leading to the problem of tracking failure.
The centralized control structure, from the perspective of the system as a whole, can more fully consider the coupling between the horizontal and vertical motion control during the control process. For high-speed large curvature or low road adhesion coefficient working conditions, some scholars have conducted in-depth research. For example, Huang et al. proposed an MPC controller based on the vehicle error state-space equation [78]; Tian et al. proposed an adaptive path tracking strategy based on MPC algorithm to coordinate active front wheel steering and direct transverse sway moment, and used recursive least squares with forgetting factor to identify the lateral deflection stiffness of the rear tire, and based on the MPC control framework, used the side slip angle, transverse sway angular velocity and zero moment methods to construct optimization constraints [79]; literature [9] proposed an MPC controller based on line-walk time variation; Fan et al. designed an improved MPC control method from a three-degree-of-freedom vehicle dynamics model by analyzing the vehicle transverse sway stability and adding envelope constraints to the model [80]; Tian et al. proposed a strategy for forced switching MPC path tracking control and coordinated the active front wheel steering with the external transverse sway moment [81]; Liu et al. proposed a path tracking control method adaptive to lateral adhesion [82]. Simulation results show that the above MPC algorithm-based centralized controllers have good tracking accuracy and driving stability under high-speed large curvature or low road adhesion coefficient conditions.
For dynamic obstacle avoidance problem, RezaHajiloo et al. proposed an integrated controller based on MPC algorithm using differential braking to improve the lateral flexibility and responsiveness of the vehicle, and the simulation results showed that the obstacle avoidance capability, path tracking capability and driving stability of the controller were better [83].Shi et al. realized the tracking of fifth-order polynomial obstacle avoidance path by driverless car based on MPC algorithm Sun et al. proposed an optimized MPC path tracking steering controller based on a linear model to reduce the lateral path tracking error at high speed, and the simulation verified the tracking performance of the controller at high speed[9] . . Overall, most of the current research on dynamic obstacle avoidance problems at high speeds focuses on simple scenarios with only static obstacles, and adding obstacle trajectory prediction to the research on dynamic obstacle avoidance is a future research development direction.
For the acceleration overtaking conditions of self-driving cars, Zhang et al. proposed a desired transverse swing angular velocity calculation method considering path curvature, lane change time, and longitudinal vehicle speed, and established a multi-objective MPC topologizable coordination control strategy for self-driving car trajectory tracking, and the simulation results showed that the proposed trajectory tracking control strategy not only can accurately track the planned path, but also has high lateral stability [85].
The centralized control structure is from the perspective of the system as a whole, which can more fully consider the coupling between the horizontal and vertical motion control in the process of control, but the increase in the system model dimension, complexity, computation and application cost brings difficulties to the controller design and application compared to the hierarchical control.
3.3 ADRC algorithm
Other algorithms applied to the path tracking control of driverless cars under high-speed conditions are the Active Disturbance RejectiongControl (ADRC), which attributes all uncertainties acting on the controlled object to unknown perturbations and estimates and compensates them by designing extended state observations [86].
For example, Kang et al. proposed an improved selfadverse control (IADRC) method, including an improved extended state observer and an LQR-based error compensator, and designed a vehicle path tracking controller based on IADRC considering lateral stability, and the simulation results showed that the controller has better path tracking effect and robustness against disturbances [87].Yun et al. proposed a new method for high-speed Wang et al. proposed an ANN-based ADRC method for high-speed automatic emergency vehicle avoidance technology, and the simulation results proved that the method has better path tracking accuracy and robustness at different vehicle speeds, and the tracking error is smaller at 60 km/h lateral wind interference at 100 km/h vehicle speed [88].Wang et al. proposed an integrated feedforward-feedback and ADRC compensation lateral control algorithm and achieved better tracking effect and stability [89]. And Yang et al. verified that the tracking performance of nonlinear ADRC is slightly worse, but it has strong robustness [90].
At present, the research of ADRC-based path tracking is still in the stage of simulation or simple experiments, and compared with linear ADRC, nonlinear ADRC stability proof and control parameter optimization still need to be studied in depth.
3.4 Robust control
Robust control is a class of research methods for uncertain systems, which is based on parametric theory and structural singular values, and provides a more complete theoretical system for model error and uncertainty description. h∞ control takes external disturbances into account, and it is able to find a control gain K that makes the system stability and error convergence optimal [91].
For example, Tian et al. proposed a robust control strategy based on MPC and H∞ and verified that the controller can improve path tracking accuracy and ensure vehicle lateral and lateral stability under extreme conditions of high speed and large curvature, while showing superiority in suppressing parameter uncertainty, modeling errors and external disturbances [92].Feng et al. designed a static feedback controller based on H∞ observer. Simulation results under different operating conditions showed that the proposed controller was effective regardless of road condition variations, vehicle longitudinal speed variations, and external disturbances [93]. To address the fact that model uncertainty and noise are two factors that degrade the path tracking accuracy and system stability in autonomous vehicle systems, Li et al. proposed an adaptive robust controller to solve this problem [94].
Because the robust control theory-based unmanned vehicle path tracking controller has a large amount of operations and relies more on a high-performance hardware processor, the current study only verified the effectiveness of the controller under simulation conditions, and there is a lack of in-depth research in practical engineering applications.
3.5 Sliding mode control
Sliding mode control (SMC) is a relatively simple and superior control performance of the control method, is essentially a nonlinear control, its nonlinear performance is the control of the discontinuity, and the system ”structure” is not fixed, can be in the dynamic process according to the current state of the system It has the advantages of sliding mode designability, few adjustment parameters, fast response time, and insensitivity to disturbances [95-98].
For example, Cao et al. proposed a model-based optimal trajectory tracking architecture, in which the SMC controller with a fuzzy adaptive preview time policy can effectively track the target path and avoid large lateral accelerations [99]. Chen et al. used hierarchical control and designed the upper layer controller using SMC control method to reduce the heading deviation and lateral deviation during path tracking while ensuring the driving stability of the vehicle [101]. Some papers also combine SMC and particle swarm optimization or replace the symbolic function with a continuous function to improve the commonly used exponential convergence law to improve the control performance [102].
The above methods only address the tracking robustness from the perspectives of modeling error, parameter uncertainty and external disturbances, without specifically analyzing the specific action laws of various uncertainties, and more in-depth experimental research on real vehicles is needed for practical applications in engineering.
3.6 Reinforcement learning
ReinforcementLearning (RL), a class of adaptive optimal control algorithms, has received increasing attention in solving complex control problems [103].
To solve the problem of feature representation and online learning capability in learning control of uncertain dynamic systems, a multicore online RL method for path tracking control was proposed in the literature [24], where a multicore feature learning framework was designed based on pairwise heuristic planning, and simulations under S-curve and urban road conditions verified that the controller has better tracking accuracy and stability performance than LQR controller and PP controller.Ma et al. combined RL and PID were combined to propose a self-seeking optimal path tracking control based on the interactive learning mechanism of the RL framework to achieve online optimization of the PID control parameters, and the simulation and real vehicle tests proved that the control method has better tracking performance in high-speed conditions (maximum speed above 100 km/h) [104].
A challenge for RL-based controllers is the need to design accuracy reward functions and utilize them efficiently to avoid falling into local optima. Moreover, despite the fact that simulation can provide a large amount of data to RL, the gap between engineering applications and simulation is still the main reason that prevents its diffusion in practical engineering.
3.7 Other algorithms
In addition to the above-mentioned path-tracking algorithms with more applications in high-speed conditions, new control algorithms continue to emerge as the research progresses. For example, Huang et al. of Hunan University studied an adaptive enhanced path tracking system (AEPTS) [60], and Sun proposed a new algorithm using a hierarchical control structure and local linearization of the nonlinear path tracking model [105]. Others study model-based adaptive Q-matrix linear quadratic Gaussian (LQG) control [106], some use pure tracking algorithms with rolling time-domain strategies, etc. [107]. These algorithms have been validated by simulations or real vehicle tests to verify the performance of path tracking under high-speed conditions.
4. Analysis of the characteristics of the path tracking algorithm
In order to facilitate the selection of path tracking control algorithms under different working conditions, the characteristics of different control algorithms are listed and compared in this paper.
The PID, PP, MPC and Stanley algorithms, which are more frequently used in the path tracking control algorithm under low speed conditions, and their advantages and disadvantages are shown in Table 1.
Table 1.Path tracking algorithm at low speed conditions