# Adaptive Robust Path Tracking Control for Autonomous Vehicles Considering Multi-Dimensional System Uncertainty

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{∞}robust steering torque compensation controller was designed to reduce the lateral offset [17]. Though H

_{∞}robust control guarantees the system robustness under a certain parameter perturbation range, the control performance is conservative. Adaptive control is considered as another suitable candidate. Akermi et al. proposed a path-tracking architecture with the combination of sliding mode control, fuzzy logic, and perturbations observer. The SMC gain is automatically adjusted by fuzzy organ [18]. Ao et al. developed the super twisting sliding model control algorithm based on Lyapunov theory and applied back-stepping technology. The system robustness is enhanced and the chattering phenomenon is attenuated [19]. Sun et al. proposed the adaptive non-singular fast terminal sliding mode (NFTSM) control for yaw stability control of a bus. Meanwhile, the robust least-squares allocation method is adopted for braking force distribution of each tire, which significantly improves the vehicle lateral stability under special driving conditions [20]. Most research focuses on the variation in the vehicle tire-cornering stiffness under different working conditions. Except for the vehicle state parameters, the system uncertainty of the vehicle chassis subsystem also has an important impact on the control performance.

- (1)
- The NFTSM controller is adopted for the vehicle lateral tracking control considering the vehicle dynamic model uncertainty and parameter disturbance, which has a faster convergence rate and transient response than a linear sliding model controller. In addition, the RBFNN is introduced to estimate and compensate the nonlinear uncertainty terms in real time, which enhances the control performance;
- (2)
- The steering system dynamics are established and the model reference adaptive control (MRAC) is utilized for the steering torque control to overcome the model uncertainty caused by the dissipation of production and system degradation

## 2. Vehicle Dynamic Modeling

_{y}and orientation error e

_{ψ}should be eliminated. The relationship of e

_{y}and e

_{ψ}can be expressed as Equation (1) based on vehicle kinematic model:

_{d}is the desired heading angle. With the small angle assumption and simplified linear tire model, the vehicle dynamic model is described as:

_{f}, l

_{r}represent the distance from the center of gravity to the front and rear axle, respectively. r is the vehicle yaw rate. F

_{yf}, F

_{yr}denote the lateral force of the front and rear axle, respectively, which can be calculated by the cornering stiffness C

_{f}, C

_{r}and steering wheel angle δ.

## 3. Path-Tracking Control Algorithm

#### 3.1. Control Law Design

_{y}| ≤ 1, Equation (6) can be approximated by ignoring the high order terms as:

_{y}| ≤ 1, the ${\dot{e}}_{y}$ can be approximated by:

_{eq}is equivalent control term, δ

_{sw}is switching control term, and ς is system uncertainty. The equivalent control term is designed as:

_{1}, λ

_{2}, λ

_{3}are positive constants.

#### 3.2. RBFNN-Based System Uncertainty Estimator

^{*}and f, which are expressed as:

^{*}and bias ε

^{*}, respectively, n is number of nodes in the hidden layer of network, and $\sigma ={\left[{\sigma}_{1},{\sigma}_{2},\dots ,{\sigma}_{n}\right]}^{T}$ is Gaussian radial basis function, expressed as:

_{i}and b

_{i}are network parameters of the ith radial basis function. In this paper, the steering wheel angle δ and state vectors ${e}_{y},{\dot{e}}_{y},{e}_{\psi},{\dot{e}}_{\psi}$ are chosen as the network input.

^{*}and weight of the neural network is bounded, the update law of weight and error compensation term can be designed as:

_{w}, η

_{w}, Γ

_{ε}, η

_{ε}are positive constants and $\tau ={\dot{e}}^{\alpha -1}>0$.

#### 3.3. System Performance Analysis

#### 3.3.1. System Stability

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

^{*}, ε

^{*}satisfies ‖w

^{*}‖ ≤ w

_{max}and |ε

^{*}| ≤ ε

_{max}. Based on the F-norm properties [25], we obtain:

#### 3.3.2. Finite Time Convergence

_{σ}, K

_{ε}are positive constants. Based on Equations (29) and (30), we can obtain the following inequality by setting $\Vert s\Vert ={\mathsf{\Delta}}_{3}\ge \mathrm{max}\left({\mathsf{\Delta}}_{1},{\mathsf{\Delta}}_{2}\right)$ and substituting Equations (27) and (28) into (26):

_{s}satisfies:

_{3}. Meanwhile, it can be perceived that the tracking performance is better with smaller Δ

_{3}, which can be acquired by tuning parameters q and β.

## 4. MRAC for Active Steering System

_{f}is the eventual front steering angle. J, b, K represent the nominal moment of inertia, damping coefficient, and angular stiffness of the steering system, respectively. T

_{δ}is the output torque of the steering motor.

_{p}that the closed–loop system is bounded to and the state vectors x

_{p}that track the reference signals. Here the second-order low-pass filter is adopted to describe the reference model:

_{r}and ξ

_{r}are the cut-off frequency and damping coefficient of the filter.

_{p}asymptotically tracks x

_{m}for any bounded reference signal δ [26].

_{x}, γ

_{δ}are the parameters learning rate, then the time derivative of Lyapunov function V

_{3}is:

## 5. Simulation Results and Discussion

#### 5.1. Double-Lane Change (DLC) Maneuver

_{∞}robust control. Here, the tracking performance is compared under two conditions. One is when the control system is healthy (without any parameter perturbation or external disturbance) and the other one has tire-cornering stiffness variation, as stated above. The simulation results are shown in Figure 6.

_{∞}controller is regulated by solving linear matrix inequalities considering the potential system uncertainty, which obtains a wide range of robustness by sacrificing certain control accuracy. The tracking error is largest when system is healthy, which reaches 0.31 m, although the tracking performance of the proposed NFTSM control is inferior to MPC. Its maximum lateral position error is 0.18 m, only about 7% worse than MPC. However, when parameter perturbation occurs, MPC no longer generates the optimal steering command. The tracking performance becomes the worst among these three controllers, and has the largest tracking error. In contrast, the H

_{∞}controller and NFTSM controller have better robustness against the tire-cornering stiffness variation. The proposed NFTSM controller, in particular, modifies the control law in real time by estimating system uncertainty online, which significantly reduces the tracking error compared with the fixed control gain of the H

_{∞}controller. Hence, compared with MPC and the H

_{∞}controller, the proposed control algorithm has better adaptability and a more balanced tracking performance in complex working conditions.

#### 5.2. Slalom-like Maneuver

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Gray, A.; Ali, M.; Gao, Y.; Hedrick, J.K.; Borrelli, F. A unified approach to threat assessment and control for automotive active safety. IEEE Trans. Intell. Transp. Syst.
**2013**, 14, 1490–1499. [Google Scholar] [CrossRef] - Kim, H.; Kim, D.; Shu, I.; Yi, K. Time-varying parameter adaptive vehicle speed control. IEEE Trans. Veh. Technol.
**2015**, 65, 581–588. [Google Scholar] [CrossRef] - Lian, Y.; Zhao, Y.; Hu, L.; Tian, Y. Longitudinal collision avoidance control of electric vehicles based on a new safety distance model and constrained-regenerative-braking-strength-continuity braking force distribution strategy. IEEE Trans. Veh. Technol.
**2015**, 65, 4079–4094. [Google Scholar] [CrossRef] - Li, S.; Li, K.; Rajamani, R.; Wang, J. Model predictive multi-objective vehicular adaptive cruise control. IEEE Trans. Control Syst. Technol.
**2010**, 19, 556–566. [Google Scholar] [CrossRef] - Amer, N.H.; Hairi, Z. Modelling and Control Strategies in Path Tracking Control for Autonomous Ground Vehicles: A Review of State of the Art and Challenges. J. Intell. Robot. Syst.
**2017**, 86, 225–254. [Google Scholar] [CrossRef] - Kapania, N.R.; Gerdes, J.C. Design of a feedback-feedforward steering controller for accurate path tracking and stability at the limits of handling. Veh. Syst. Dyn.
**2015**, 53, 1687–1704. [Google Scholar] [CrossRef] [Green Version] - Salehpour, S.; Pourasad, Y.; Taheri, S.H. Vehicle path tracking by integrated chassis control. J. Central South Univ.
**2015**, 22, 1378–1388. [Google Scholar] [CrossRef] - Xu, S.; Peng, H. Design, analysis, and experiments of preview path tracking control for autonomous vehicles. IEEE Trans. Intell. Transp. Syst.
**2019**, 21, 48–58. [Google Scholar] [CrossRef] - Ji, J.; Khajepour, A.; Melek, W.W.; Huang, Y. Path planning and tracking for vehicle collision avoidance based on model predictive control with multiconstraints. IEEE Trans. Veh. Technol.
**2016**, 66, 952–964. [Google Scholar] [CrossRef] - Chen, Y.; Hu, C.; Wang, J. Human-Centered Trajectory Tracking Control for Autonomous Vehicles with Driver Cut-In Behavior Prediction. IEEE Trans. Veh. Technol.
**2019**, 68, 8461–8471. [Google Scholar] [CrossRef] - Guo, H.; Liu, J.; Cao, D.; Chen, H.; Yu, R.; Lv, C. Dual-envelop-oriented moving horizon path tracking control for fully automated vehicles. Mechatronics
**2018**, 50, 422–433. [Google Scholar] [CrossRef] [Green Version] - Funke, J.; Brown, M.; Erlien, S.M.; Gerdes, J.C. Collision avoidance and stabilization for autonomous vehicles in emergency scenarios. IEEE Trans. Control. Syst. Technol.
**2016**, 25, 1204–1216. [Google Scholar] [CrossRef] - Brown, M.; Funke, J.; Erlien, S.; Gerdes, J.C. Safe driving envelopes for path tracking in autonomous vehicles. Control Eng. Pr.
**2017**, 61, 307–316. [Google Scholar] [CrossRef] - Jing, H.; Hu, C.; Yan, F.; Chadli, M.; Wang, R.; Chen, N. Robust H∞ output-feedback control for path following of autonomous ground vehicles. Mech. Syst. Sig. Pro.
**2016**, 70–71, 414–427. [Google Scholar] - Guo, J.; Luo, Y.; Li, K. Robust gain-scheduling automatic steering control of unmanned ground vehicles under velocity-varying motion. Veh. Syst. Dyn.
**2018**, 57, 595–616. [Google Scholar] [CrossRef] - Guo, J.; Wang, J.; Hu, P.; Li, L. Robust guaranteed-cost path-following control for autonomous vehicles on unstructured roads. Proc. Inst. Mech. Eng. Part D J. Automob. Eng.
**2017**, 232, 896–908. [Google Scholar] [CrossRef] - Chen, W.; Zhao, L.; Wang, H.; Huang, Y. Parallel Distributed Compensation /H∞ Control of Lane-keeping System Based on the Takagi-Sugeno Fuzzy Model. Chin. J. Mech. Eng.
**2020**, 33, 61. [Google Scholar] [CrossRef] - Akermi, K.; Chouraqui, S.; Boudaa, B. Novel SMC control design for path following of autonomous vehicles with uncertainties and mismatched disturbances. Int. J. Dyn. Control
**2018**, 8, 254–268. [Google Scholar] [CrossRef] - Ao, D.; Huang, W.; Wong, P.K.; Li, J. Robust backstepping super-twisting sliding mode control for autonomous vehicle path following. IEEE Access
**2021**, 9, 123165–123177. [Google Scholar] [CrossRef] - Sun, X.; Wang, Y.; Cai, Y.; Wong, P.K.; Chen, L. An adaptive nonsingular fast terminal sliding mode control for yaw stability control of bus based on STI tire model. Chin. J. Mech. Eng.
**2021**, 34, 79. [Google Scholar] [CrossRef] - Norouzi, A.; Masoumi, M.; Barari, A.; Sani, S.F. Lateral control of an autonomous vehicle using integrated backstepping and sliding mode controller. Pro. Inst. Mech. Eng. Part K J. Mult. Dyn.
**2019**, 233, 141–151. [Google Scholar] [CrossRef] - Yang, L.; Yue, M.; Ma, T. Path Following Predictive Control for Autonomous Vehicles Subject to Uncertain Tire-ground Adhesion and Varied Road Curvature. Int. J. Control Autom. Syst.
**2019**, 17, 193–202. [Google Scholar] [CrossRef] - Aiguo, W.; Liu, H.; Dong, N. Nonsingular Fast Terminal Sliding Mode Control of Robotic Manipulators Based on Neural Networks. Trans. Chin. Soc. Agri. Mach.
**2018**, 44, 395–404. [Google Scholar] - Pham, C.V.; Wang, Y.N. Robust Adaptive Trajectory Tracking Sliding mode control based on Neural networks for Cleaning and Detecting Robot Manipulators. J. Intell. Robot. Syst.
**2015**, 79, 101–114. [Google Scholar] [CrossRef] - Yong, F.; Yu, X.; Man, Z. Non-singular terminal sliding mode control of rigid manipulators. Automatica
**2002**, 38, 2159–2167. [Google Scholar] - Ioannou, P.A.; Sun, J. Robust Adaptive Control; Prentice-Hall, Inc.: Hoboken, NJ, USA, 1995. [Google Scholar]

**Figure 5.**The simulation results for scenario 1: (

**a**) Global vehicle trajectory; (

**b**)Steering wheel angle; (

**c**) Lateral position error; (

**d**) Orientation error.

**Figure 6.**The comparison for three controllers: (

**a**) Global vehicle trajectory with healthy system; (

**b**) Lateral position error with healthy system; (

**c**) Global vehicle trajectory with parameter perturbation; (

**d**) Lateral position error with parameter perturbation.

**Figure 7.**The simulation results for scenario 2: (

**a**) Global vehicle trajectory; (

**b**) Steering wheel angle; (

**c**) Lateral position error; (

**d**) Orientation error.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chen, M.; Ren, Y.; Ou, M.
Adaptive Robust Path Tracking Control for Autonomous Vehicles Considering Multi-Dimensional System Uncertainty. *World Electr. Veh. J.* **2023**, *14*, 11.
https://doi.org/10.3390/wevj14010011

**AMA Style**

Chen M, Ren Y, Ou M.
Adaptive Robust Path Tracking Control for Autonomous Vehicles Considering Multi-Dimensional System Uncertainty. *World Electric Vehicle Journal*. 2023; 14(1):11.
https://doi.org/10.3390/wevj14010011

**Chicago/Turabian Style**

Chen, Mengyuan, Yue Ren, and Minghui Ou.
2023. "Adaptive Robust Path Tracking Control for Autonomous Vehicles Considering Multi-Dimensional System Uncertainty" *World Electric Vehicle Journal* 14, no. 1: 11.
https://doi.org/10.3390/wevj14010011