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