# Adaptive Terminal Sliding Mode Trajectory Tracking Control for Autonomous Vehicles Considering Completely Unknown Parameters and Unknown Perturbation Conditions

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- A non-singular terminal sliding model control protocol is proposed to track the ideal trajectory and guarantee the preview error converging to zero in a finite time. The chattering issue encountered by the conventional sliding mode controller is effectively addressed by the proposed method.
- (2)
- The proposed controller is integrated with the adaptive algorithm, eliminating the need for prior knowledge of vehicle parameters and perturbation bounds. This ensures a more flexible and robust system capable of dynamically adjusting to varying conditions.
- (3)
- To illustrate the effectiveness of the proposed method, the CarSim–Matlab joint simulations and real-world experimental studies are conducted. The proposed method is compared with the conventional controllers and verified under various driving conditions.

## 2. Modeling and Problem Description

#### 2.1. Modeling of Vehicle Three-Degrees-of-Freedom Dynamics

_{fax}= F

_{1x}+ F

_{2x}, F

_{rax}= F

_{3x}+ F

_{4x}, and F

_{fay}= F

_{1y}+ F

_{2y}.

_{ray}= F

_{3y}+ F

_{4y}.

_{fby}= F

_{2y}− F

_{4y}.

_{i}is the input torque of wheel I, and ΔT

_{i}is the torque loss. R is the wheel radius.

_{i}is the cornering stiffness of wheel i, and β

_{i}is the sideslip angle of wheel i.

_{f}, c

_{r}represent the constant part of the actual cornering stiffness, Δc

_{i}is the uncertain and nonlinear part, respectively, and there exists a positive constant c

_{s}to make $\left|\Delta {c}_{i}\right|\le {c}_{s}/2$.

#### 2.2. Problem Formulation

_{x}$\gg $ v

_{y}. Therefore, the lateral velocity and lateral acceleration can be simplified as follows:

_{a}= T

_{1}+ T

_{2}+ T

_{3}+ T

_{4}. Substituting (4), (8), and (9) into (1)–(3) yields the equation of the velocity. The sideslip angle and the angular velocity of the transverse pendulum are given as follows:

_{v}, d

_{β}, and d

_{γ}are the unmodeled and perturbed terms.

_{f}, which can effectively reduce the complexity of control. By approximating the ideal trajectory through the position and the heading angle of the intelligent vehicle, the complex trajectory tracking control is simplified into the distance error and direction error, and then the tracking control of the designed path is realized.

_{0}and then switches to another straight lane. A and B are the entrance and exit points of the transition path, respectively, and d

_{s}is the distance between them.

_{v}and the direction of the tangent to the desired path ψ

_{s}, where ψ

_{v}is expressed as follows:

_{s}, which is the tangential velocity at the point in the relationship between v

_{s}and the actual velocity of the vehicle, v, is given as follows:

## 3. Trajectory Tracking Controller Design

#### 3.1. Controller with the Known Vehicle Parameters

_{x}, the sideslip angle β, the longitudinal acceleration a

_{x}, and the yaw rate of the vehicle γ can be measured by the inertial sensors. Therefore, e, ψ, v

_{x}, β, a

_{x}, and γ are known state quantities in this study.

_{f}, c

_{r}, l

_{f}, l

_{r}, and I

_{z}are known in the vehicle dynamics model, while the disturbances are upper-bounded and in a known state. The second-order nonlinear system with preview error is established as follows:

**Lemma**

**1**

**.**For a nonlinear system, the system is finite-time stable if V(x) is assumed to be a smooth positive definite function ($U\subset {R}^{n}$) and f(x) is a continuous function with ${V}^{\prime}(x)f(x)\le -c{(V(x))}^{\alpha}$, where c > 0 and 0 < α < 1. The convergence time T

_{r}is as follows:

**Theorem**

**1.**

_{m}+ η

_{d}+ |S| and (d

_{m}≥ |d

_{σ}|, η

_{d}> 0). Then, the path following error can be finite-time converged to zero. The convergence time is estimated by the following:

**Proof.**

_{1}(t) and x

_{2}(t) as x

_{1}and x

_{2}, respectively. □

_{r}is

_{2}is as follows:

_{1}can converge to zero in finite time, so it can be obtained that x

_{2}also converges to zero in finite time. By solving (27), the time for the system to converge to the origin along the sliding mode surface S = 0 is as follows:

_{r}+ t

_{s}) from any initial state. This proof is completed.

#### 3.2. Controller with the Unknown Vehicle Parameters

_{δ}, F

_{γ}, F

_{β}, b, and d

_{m}in Equation (19) are usually not available in real time, and an adaptive estimation of them is required. The proof is completed.

**Lemma**

**2**

**.**For a nonlinear system, if there exists a continuous function V(x) and real numbers λ

_{1}> 0, λ

_{2}> 0, 0 < γ < 1, and 0 < ε < ∞ such that

**Theorem**

**2.**

_{1}, η

_{2}, η

_{3}, η

_{11}, η

_{22}, and η

_{33}are all positive constants, then the sliding mold surface S can converge to within the domain of the origin in finite time (t

_{c}+ t

_{m}).

**Proof.**

_{3}:

_{3}and bringing in the controller (34) yields

_{1}, ζ

_{2}, and C are denoted by the following:

_{3}converges in finite time to the following field:

_{c}is the set value and its value is as follows:

_{c}within the definable interval ε (ε > 0).

_{1}| > ε, the Lyapunov function is established, and the differential function of V

_{4}is as follows:

_{1}| can converge to within the interval ε in finite time which means that the error can converge to within the interval ε in finite time. The time to converge to within the interval is given by the calculation of

_{c}+ t

_{m}) into the definable interval ε. This completes the proof.

**Remark**

**1.**

## 4. Simulation Analysis

#### 4.1. Simulation Results of Three Sliding Mode Controllers

#### 4.2. Robustness of Non-Singular Terminal Sliding Mode with Parameter Adaptation

#### 4.2.1. Robustness to Vehicle Speed

#### 4.2.2. Robustness to Road Adhesion Coefficients

## 5. Experimental Verification of Trajectory Tracking Controller

#### Algorithm Experimental Validation Results

^{−2}rad. The error between the actual trajectory and the ideal trajectory does not exceed 4 cm. Both errors fluctuate within a small range. The smaller error indicates that the controller performs very well. Figure 8d shows the output value of the front wheel steering during trajectory tracking, and from the figure, the corner change during the control process is smooth with no corner mutation. In summary, the NTSM controller can ensure that the unmanned vehicle can track the reference path with high accuracy, and the whole control process is relatively smooth, which meets the design requirements.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Xavier, N.; Bandyopadhyay, B. Practical Sliding Mode Using State Depended Intermittent Control. IEEE Trans. Circuits Syst. II: Express Briefs
**2021**, 68, 341–345. [Google Scholar] [CrossRef] - Liang, Z.; Zhao, J.; Dong, Z.; Wang, Y.; Ding, Z. Torque Vectoring and Rear-Wheel-Steering Control for Vehicle’s Uncertain Slips on Soft and Slope Terrain Using Sliding Mode Algorithm. IEEE Trans. Veh. Technol.
**2020**, 69, 3805–3815. [Google Scholar] [CrossRef] - Nguyen, N.P.; Oh, H.; Moon, J. Continuous Nonsingular Terminal Sliding-Mode Control With Integral-Type Sliding Surface for Disturbed Systems: Application to Attitude Control for Quadrotor UAVs Under External Disturbances. IEEE Trans. Aerosp. Electron. Syst.
**2022**, 58, 5635–5660. [Google Scholar] [CrossRef] - Guo, S.; Orosz, G.; Molnar, T.G. Connected Cruise and Traffic Control for Pairs of Connected Automated Vehicles. IEEE Trans. Intell. Transp. Syst.
**2023**, 24, 12648–12658. [Google Scholar] [CrossRef] - Chen, X.; Xu, B.; Qin, X.; Bian, Y.; Hu, M.; Sun, N. Non-Signalized Intersection Network Management With Connected and Automated Vehicles. IEEE Access
**2020**, 8, 122065–122077. [Google Scholar] [CrossRef] - Qin, Z.; Chen, L.; Fan, J.; Xu, B.; Hu, M.; Chen, X. An Improved Real-Time Slip Model Identification Method for Autonomous Tracked Vehicles Using Forward Trajectory Prediction Compensation. IEEE Trans. Instrum. Meas.
**2021**, 70, 7501012. [Google Scholar] [CrossRef] - Subroto, R.K.; Wang, C.Z.; Lian, K.L. Four-Wheel Independent Drive Electric Vehicle Stability Control Using Novel Adaptive Sliding Mode Control. IEEE Trans. Ind. Appl.
**2020**, 56, 5995–6006. [Google Scholar] [CrossRef] - Wen, S.; Guo, G. Distributed Trajectory Optimization and Sliding Mode Control of Heterogenous Vehicular Platoons. IEEE Trans. Intell. Transp. Syst.
**2022**, 23, 7096–7111. [Google Scholar] [CrossRef] - Ge, X.; Han, Q.-L.; Wu, Q.; Zhang, X.-M. Resilient and Safe Platooning Control of Connected Automated Vehicles Against Intermittent Denial-of-Service Attacks. IEEE/CAA J. Autom. Sin.
**2023**, 10, 1234–1251. [Google Scholar] [CrossRef] - Wang, B.; Su, R. A Distributed Platoon Control Framework for Connected Automated Vehicles in an Urban Traffic Network. IEEE Trans. Control Netw. Syst.
**2022**, 9, 1717–1730. [Google Scholar] [CrossRef] - Scheffe, P.; Henneken, T.M.; Kloock, M.; Alrifaee, B. Sequential Convex Programming Methods for Real-Time Optimal Trajectory Planning in Autonomous Vehicle Racing. IEEE Trans. Intell. Veh.
**2023**, 8, 661–672. [Google Scholar] [CrossRef] - Ju, Z.; Zhang, H.; Li, X.; Chen, X.; Han, J.; Yang, M. A Survey on Attack Detection and Resilience for Connected and Automated Vehicles: From Vehicle Dynamics and Control Perspective. IEEE Trans. Intell. Veh.
**2022**, 7, 815–837. [Google Scholar] [CrossRef] - Tang, L.; Yan, F.; Zou, B.; Wang, K.; Lv, C. An Improved Kinematic Model Predictive Control for High-Speed Path Tracking of Autonomous Vehicles. IEEE Access
**2020**, 8, 51400–51413. [Google Scholar] [CrossRef] - Wang, H.; Liu, B. Path Planning and Path Tracking for Collision Avoidance of Autonomous Ground Vehicles. IEEE Syst. J.
**2022**, 16, 3658–3667. [Google Scholar] [CrossRef] - Hu, C.; Chen, Y.; Wang, J. Fuzzy Observer-Based Transitional Path-Tracking Control for Autonomous Vehicles. IEEE Trans. Intell. Transp. Syst.
**2021**, 22, 3078–3088. [Google Scholar] [CrossRef] - Guan, Y.; Ren, Y.; Li, S.E.; Sun, Q.; Luo, L.; Li, K. Centralized Cooperation for Connected and Automated Vehicles at Intersections by Proximal Policy Optimization. in IEEE Trans. Veh. Technol.
**2020**, 69, 12597–12608. [Google Scholar] [CrossRef] - Bai, W.; Xu, B.; Liu, H.; Qin, Y.; Xiang, C. Robust Longitudinal Distributed Model Predictive Control of Connected and Automated Vehicles With Coupled Safety Constraints. IEEE Trans. Veh. Technol.
**2023**, 72, 2960–2973. [Google Scholar] [CrossRef] - Xu, H.; Xiao, W.; Cassandras, C.G.; Zhang, Y.; Li, L. A General Framework for Decentralized Safe Optimal Control of Connected and Automated Vehicles in Multi-Lane Signal-Free Intersections. IEEE Trans. Intell. Transp. Syst.
**2022**, 23, 17382–17396. [Google Scholar] [CrossRef] - Liu, Y.; Zhou, B.; Wang, X.; Li, L.; Cheng, S.; Chen, Z.; Li, G.; Zhang, L. Dynamic Lane-Changing Trajectory Planning for Autonomous Vehicles Based on Discrete Global Trajectory. IEEE Trans. Intell. Transp. Syst.
**2022**, 23, 8513–8527. [Google Scholar] [CrossRef] - Mohamed, M.; Rabik, S.; Vasanth, T. Muthuramalingam. Implementation of LQR based SOD control in diode laser beam machining on leather specimens, Optics & Laser Technology. Automatica
**2024**, 170, 110328. [Google Scholar] - Li, Q.; Ding, B. Design of Backstepping Sliding Mode Control for a Polishing Robot Pneumatic System Based on the Extended State Observer. Machines
**2023**, 11, 904. [Google Scholar] [CrossRef] - Ye, Y.; Wang, Y.; Wang, L.; Wang, X. A modified predictive PID controller for dynamic positioning of vessels with autoregressive model. Ocean Eng.
**2023**, 284, 115176. [Google Scholar] [CrossRef] - Fu, Q.; Wu, J.; Yu, C.; Feng, T.; Zhang, N.; Zhang, J. Linear Quadratic Optimal Control with the Finite State for Suspension System. Machines
**2023**, 11, 127. [Google Scholar] [CrossRef] - Gao, Z.; Wu, Z.; Hao, W.; Long, K.; Byon, Y.-J.; Long, K. Optimal Trajectory Planning of Connected and Automated Vehicles at On-Ramp Merging Area. IEEE Trans. Intell. Transp. Syst.
**2022**, 23, 12675–12687. [Google Scholar] [CrossRef] - Chen, N.; van Arem, B.; Alkim, T.; Wang, M. A Hierarchical Model-Based Optimization Control Approach for Cooperative Merging by Connected Automated Vehicles. IEEE Trans. Intell. Transp. Syst.
**2021**, 22, 7712–7725. [Google Scholar] [CrossRef] - Kim, S.; Lee, J.; Han, K.; Choi, S.B. Vehicle Path Tracking Control Using Pure Pursuit With MPC-Based Look-Ahead Distance Optimization. IEEE Trans. Veh. Technol.
**2024**, 73, 53–66. [Google Scholar] [CrossRef] - Liang, Z.; Zhao, J.; Liu, B.; Wang, Y.; Ding, Z. Velocity-Based Path Following Control for Autonomous Vehicles to Avoid Exceeding Road Friction Limits Using Sliding Mode Method. IEEE Trans. Intell. Transp. Syst.
**2020**, 23, 1947–1958. [Google Scholar] [CrossRef] - Wu, Y.; Wang, L.; Zhang, J.; Li, F. Path Following Control of Autonomous Ground Vehicle Based on Nonsingular Terminal Sliding Mode and Active Disturbance Rejection Control. IEEE Trans. Veh. Technol.
**2019**, 68, 6379–6390. [Google Scholar] [CrossRef] - Labbadi, M.; Djemai, M.; Boubaker, S. A novel non-singular terminal sliding mode control combined with integral sliding surface for perturbed quadrotor. Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng.
**2022**, 236, 999–1009. [Google Scholar] [CrossRef] - Lei, Q.; Zhang, W. Adaptive non-singular integral terminal sliding mode tracking control for autonomous underwater vehicles. IET Control Theory Appl.
**2017**, 11, 1293–1306. [Google Scholar] - Shen, H.; Pan, Y.J.; Ahmad, U.; He, B. Pose Synchronization of Multiple Networked Manipulators Using Nonsingular Terminal Sliding Mode Control. IEEE Trans. Syst. Man Cybern. Syst.
**2021**, 51, 12. [Google Scholar] [CrossRef] - Guo, J.; Luo, Y.; Li, K. An Adaptive Hierarchical Trajectory Following Control Approach of Autonomous Four-Wheel Independent Drive Electric Vehicles. IEEE Trans. Intell. Transp. Syst.
**2018**, 19, 2482–2492. [Google Scholar] [CrossRef] - Liang, Z.; Wang, Z.; Zhao, J.; Wong, P.K.; Yang, Z.; Ding, Z. Fixed-Time Prescribed Performance Path-Following Control for Autonomous Vehicle With Complete Unknown Parameters. IEEE Trans. Ind. Electron.
**2023**, 70, 8426–8436. [Google Scholar] [CrossRef] - Liang, Z.; Shen, M.; Li, Z.; Yang, J. Model-Free Output Feedback Path Following Control for Autonomous Vehicle With Prescribed Performance Independent of Initial Conditions. IEEE/ASME Trans. Mechatron.
**2023**, 10, 1–12. [Google Scholar] [CrossRef] - Ghasemi, M.; Nersesov, S.G. Finite-time coordination in multiagent systems using sliding mode control approach. Automatica
**2014**, 50, 1209–1216. [Google Scholar] [CrossRef] - Yu, J.; Shi, P.; Zhao, L. Finite-time command filtered backstepping control for a class of nonlinear systems. Automatica
**2018**, 92, 173–180. [Google Scholar] [CrossRef]

Symbolic | Unit | Description |
---|---|---|

m | kg | Vehicle mass |

l_{f} | m | Distance from center of gravity (CG) to front axle |

l_{r} | m | Distance from CG to rear axle |

δ_{f} | rad | Steering angle of front wheels |

d | m | Distance from CG to left/right wheel |

i | Wheel ID, and i = 1, 2, 3, 4 | |

I_{z} | kg·m^{2} | Yaw moment of inertia of vehicle |

γ | rad/s | Yaw rate of vehicle |

β | rad | Sideslip angle of vehicle |

β_{i} | rad | Sideslip angle of wheel i |

v | m/s | Total velocity of CG |

v_{i} | m/s | Speed of wheel i |

Parameter | Value | Parameter | Value |
---|---|---|---|

m | 1230 kg | I_{Z} | 1343 kg·m^{2} |

l_{f} | 1.04 m | c_{f} | 96,300 N/rad |

l_{r} | 1.56 m | c_{r} | 64,200 N/rad |

d | 1.48 m | g | 9.8 m/s^{2} |

R | 0.3 m | v_{d} | 50 km/h |

Controllers | Parameter |
---|---|

Sliding mold surface (22) | ξ = 0.4, p = 7, q = 5 |

Adaptive control laws, (40) | η_{1} = 0.4, η_{11} = 0.08, η _{2} = [0.5 1], η_{22} = [1 0.5], η _{3} = 5, η_{33} = 2 |

NTSM controllers, (39) | η_{d} = 5, L = 1.4, ksat = 8 |

Parameter | Value | Parameter | Value |
---|---|---|---|

m | 35.16 kg | I_{Z} | 2.188 kg·m^{2} |

l_{f} | 0.25 m | c_{f} | 1130 N/rad |

l_{r} | 0.25 m | c_{r} | 1130 N/rad |

d | 0.6 m | g | 9.8 m/s^{2} |

R | 0.128 m | v_{d} | 1.8 km/h |

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**MDPI and ACS Style**

Feng, C.; Shen, M.; Wang, Z.; Wu, H.; Liang, Z.; Liang, Z.
Adaptive Terminal Sliding Mode Trajectory Tracking Control for Autonomous Vehicles Considering Completely Unknown Parameters and Unknown Perturbation Conditions. *Machines* **2024**, *12*, 237.
https://doi.org/10.3390/machines12040237

**AMA Style**

Feng C, Shen M, Wang Z, Wu H, Liang Z, Liang Z.
Adaptive Terminal Sliding Mode Trajectory Tracking Control for Autonomous Vehicles Considering Completely Unknown Parameters and Unknown Perturbation Conditions. *Machines*. 2024; 12(4):237.
https://doi.org/10.3390/machines12040237

**Chicago/Turabian Style**

Feng, Chengyang, Mingyu Shen, Zhongnan Wang, Hao Wu, Zenghui Liang, and Zhongchao Liang.
2024. "Adaptive Terminal Sliding Mode Trajectory Tracking Control for Autonomous Vehicles Considering Completely Unknown Parameters and Unknown Perturbation Conditions" *Machines* 12, no. 4: 237.
https://doi.org/10.3390/machines12040237