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1. Introduction
Trajectory tracking control of autonomous ground vehicles (AGVs) plays an essential role in domestic and industrial use for increasing safety, accuracy, and efficiency demands, such as self-guiding uncrewed driving vehicles, platform supplying, and minesweeping [1]. Critical to the trajectory tracking problem of an AGV is its capability for accurate and reliable control to follow the desired trajectory by the planner [2]. The high dynamic nonlinearity, uncertainties and disturbances, and the mechanical limitations of the vehicle itself make the AGV trajectory tracking control challenging.
Various research results exist in the literature. Some existing works investigate the trajectory tracking problem of AGVs utilizing the kinematic model [3, 4]. However, the driving condition may be limited without reckoning the dynamics model. For this reason, the trajectory tracking control problem of a class of autonomous vehicles is investigated with parametric uncertainties, external disturbances, and overactuated features by introducing a novel adaptive hierarchical control framework with the linear matrix inequality technique [5]. Then, an augmented state variable and a nonlinear observer are trained to design a torque overlay-base robust steering wheel angle control with the backstepping method [6]. A robust H∞ dynamic output-feedback controller is designed to control the vehicle motion without using the lateral velocity information [7]. Besides, an accurate and efficient clothoid approximation approach is presented using Bezier curves [8]. A model predictive control law incorporating neural-dynamic optimization is introduced [9].
Compared with the above approaches, the finite-time control method has faster convergence speed, better disturbance rejection property, and better robustness against uncertainties [10]. Further, a fixed-time control method can guarantee global exponential convergence within a designated time independent of the initial conditions [11]. Thus, a fixed-time control scheme of AGVs might be of great significance. Apart from that, the actuator dead-zones with uncertainty issues have seldom been investigated in the trajectory tracking control design. Due to the mechanical limitations of the vehicle itself, dead zones in yaw actuators should be considered. Dead zones are commonly observed in servo valves or DC servo motors of AGVs but are not discussed in the above papers [12, 13]. When the control input falls in the dead zone, the actuator will have no output signal, resulting in steady-state errors and degraded performance in the system. How to address the actuator dead-zone issue in the fixed-time trajectory tracking control problem of AGVs still requires extensive research.
The trajectory tracking control law should achieve disturbance rejection to enhance accuracy and reliability. The LQR technique investigates a hierarchical adaptive trajectory tracking control for autonomous vehicles [14]. A compensation method combining NTSM and ARDC is presented [4]. A kinematic MPC handling the disruptions on road curvature and a PID feedback control of yaw rate rejecting uncertainties and modeling errors are designed to follow the reference trajectory [15]. The intelligent control method of interactive control paradigm-based robust lateral stability controller design for autonomous automobile trajectory tracking with uncertain disturbance is proposed [16]. Above all, the EKF, LQR, and MPC are usually simplified with limited available model parameters. However, time-varying and nonlinear terms always exist, which cannot be regarded as white Gaussian noise. The intelligent control method generally depends on experience and enough off-line training. Although the ARDC method may effectively solve the uncertain systems and be easy to implement, observer error convergence’s settling time is inaccurate. The DOBC method is a good option for rejecting disturbances in such a case. Mainly, a fixed-time observer can estimate the perturbance within a given settling time [17]. However, since the existence of dead zones, it is difficult to obtain the complete control signal. Thus, existing fixed/finite-time observers cannot directly be applied in the tracking control problem.
Motivated by the above observations, this paper is aimed at giving a fixed-time observer-based control method, rendering the AGVs with yaw actuator dead zones to follow the reference trajectory. To this end, time-varying yaw guidance and corresponding state transformation are introduced to convert trajectory tracking control into a simple yaw angle tracking problem. A fixed-time observer-based control scheme solving the actuator dead-zones is constructed for the transformed system, whose dwell times are independent of the initial conditions. The main contributions of this paper are threefold:
(1) A novel time-varying yaw guidance law is proposed based on the line of sight method, which can overcome the undesired terms involved in the error dynamics
(2) Uncertainties, external disturbances, and dead zones are simultaneously considered in the tracking control problem, making the designed controller robust to more extensive and challenging. Note that there is seldom any previous work addressing the control issues in one controller
(3) Based on fixed-time control theory, a feedback compensation control scheme based on the fixed-time observer is proposed for the tracking problem of AGVs. Compared with the method suggested in the reference [17], the dwell time can be chosen independently of initial conditions
This paper is organized as follows. Section System Modeling and the Objective explains the modeling of autonomous ground vehicles and the objective. Section Main Results contains the time-varying yaw guidance law, state transformations, trajectory tracking control framework, and convergence analysis. Numerical simulations and discussions are presented in Section Simulation.
2. System Modeling and the Objective
2.1. System Modeling
The configuration of the AGV (i.e., the position and orientation) investigated in this paper can be described by two independent coordinates or two degrees of freedom, i.e., surge and yaw. For such AGVs, an inertial frame is used, as depicted in Figure 1. The states
[figure omitted; refer to PDF]
The states
where
In addition, due to the mechanical limitations of the vehicle itself, dead zones are commonly observed in servo valves or DC servo motors of AGVs. Hence, by combining (1)-(2) with the actuator dead-zone characteristic, the nonlinear model of the AGV considered in this paper is described as follows:
where
2.2. The Objective
Define the reference trajectory as follows:
Define the error states
This paper is aimed at establishing a control methodology that can stabilize the trajectory tracking error system in (6) while ensuring that all closed-loop signals are bounded.
Remark 1.
The trajectory tracking problem is transformed into the yaw angle tracking control problem, and it is efficient to reduce the complexity of controller design by dimension reduction. Due to the dead-zone nonlinearities of the input
3. Main Results
3.1. Guidance Law Design
Define the coordinate transformation as follows:
The dynamics of
Lemma 2.
Stabilization of the system
Proof.
The Jacobian matrix between
The above lemma indicates that the trajectory tracking problem is to design desired yaw angle
Remark 3.
In literature [4], the
In this paper, a novel idea of designing
Lemma 4.
The state
Proof.
To stabilize state
The solution
then the following conditions are discussed:
(1) If
(2) If
(3) If
In summary, the parameter
The longitudinal velocity
Thus, to make the vehicle perfectly follows the reference trajectory, we only need to design the velocity
Lemma 5.
If the longitudinal velocity
Proof.
Substituting (15) into (8), the dynamics of
States
In the after parts, we assume that longitudinal velocity has been closed-loop controlled and focuses on maintaining the yaw angle φ of the AGV on the desired value.
3.2. Improved Fixed-Time Disturbance Observer
In this subsection, a fixed-time disturbance observer considering inputs dead-zones is proposed. In [4], the ARDC method is utilized to estimate the uncertainties and disturbances; however, the estimation error converges to zero asymptotically. The fixed-time control method has a faster convergence speed, and the settling time is dependent on the initial conditions. This issue prohibits its applicability in practical vehicle systems.
Define the desired yaw angle of AGV as
Assumption 6.
The unknown term
Then, the yaw angle tracking problem can be transformed to stabilize the system (17). To facilitate the subsequent analysis process, we introduce the definition of fixed-time control from [19–21].
Definition 7.
Consider a system as
Following the above definition, we define
where
Due to the existence of dead zones, the complete control signal cannot be obtained. So the traditional fixed or finite time observer cannot be directly applied. This section introduces the compensation term
Lemma 8.
Consider system (17) under Assumption 6, the observer (19) can estimate the term
Proof.
Define the estimation errors
Since
The proof is completed.☐
3.3. Controller Design
The trajectory tracking control strategy is designed based on the fixed-time observer in this subsection. Consider the above disturbance observer (19), the system (17) with dead-zone characteristics can be described as following second-order nonlinear form:
For the second-order nonlinear system (24), the control law in this subsection is designed as follows:
Lemma 9.
The control law (25) can ensure that the system (24) converges to zero in fixed time with the settling time
Proof.
Since
According to results in [22], the system (29) converges to zero before the fixed time
In the above theorem, the state transformation is introduced:
Moreover, we have
According to Lemma 9, the proposed control law can stabilize the second-order nonlinear system (17) with the actuators’ dead zones in the fixed-time. Finally, the trajectory tracking control theorem for AGVs with dead zones is given.
Theorem 10.
Consider the AGV dynamics in (1)-(3) with the dead zones (4), combining with the fixed-time state observer (19) and the control law (25). Under Assumption 6, the error signals
Proof.
According to Lemmas 8 and 9, the fixed-time state observer (19) and the control law (25) can stabilize the transformed system in the fixed time. With Lemmas 2–5, the trajectory tracking control can be proved.☐
4. Simulation
To verify the effectiveness of the presented method, we use CarSim and MATLAB platforms to carry out the trajectory tracking problem. We used a general AGV model derived by [4]. In this simulation, the proposed guidance laws (9), fixed-time state observer (19), and the control law (25) are applied. Let the AGV have the following initial conditions:
Table 1 shows all the variables involved in the AGV model. The parameters of actuator dead-zones and control parameters are chosen as follows:
Table 1
The model parameters of the AGV.
Symbol | Quantity | Value |
Mass of the vehicle | 1723 kg | |
Vehicle yaw moment of inertial | 4175 kg·m | |
The distances from vehicle C.G. to the front axis center | 1.232 m | |
The distances from vehicle C.G. to the rear axis center | 1.468 m | |
Vehicle nominal front axle cornering stiffness | N/rad | |
Vehicle nominal rear axle cornering stiffness | N/rad |
4.1. Disturbance Observer Compensation
To illustrate the advantage of the proposed disturbance observer, the trajectories of our improved disturbance observer and conventional fixed-time disturbance observer are shown in Figures 2 and 3.
[figure omitted; refer to PDF]
In this comparison, we aim at the compensation result of the dead zone of the actuator in the improved fixed time disturbance observer (Improved-FTDO) and compare it with the conventional fixed time disturbance observer (FTDO) [17]. The results show that after considering the dead zone of the actuator, the improved FTDO method proposed is better than the conventional FTDO method because the compensation problem is considered in this paper.
4.2. Actuator Compensation
The simulation results are shown in Figures 4–7. To evaluate the effectiveness of our method, we consider the convergence comparisons between the control methods of ADCC and NTSM-ADRC in [4, 22], respectively. The same driving conditions are carried out, whose road adhesion coefficient is 0.8. We now look at the differences in the performance of yaw angle changing
[figure omitted; refer to PDF]
In Table 1, only the nominal values of the parameters are given, and there are still many parametric uncertainties. For example, (1) the tire deformation; (2) the equivalent lateral stiffness
Moreover, the IFDO-ADCC can dynamically estimate and compensate the system for total disturbances in fixed time. The simulation results are shown in Figure 5. We can see that the IFDO-ADCC has faster estimation error convergence and good disturbance estimation performance, which reflects the excellent robustness to the uncertainties. The trajectory tracking performances of AGVs by the controllers of IFDO-ADCC, ADCC, and NTSM-ARDC are given in Figure 6. We can see that three controllers can track the reference trajectory well. Figure 7 shows the corresponding tracking error of the vehicle under different controllers. It can be seen that the trajectory tracking performance of the proposed IFDO-ADCC method is superior to ADCC and NTSM-ARDC.
4.3. Overall Situation
In Figure 8, the CarSim and MATLAB simulation results of the trajectory tracking based on the proposed guidance laws (9), fixed-time state observer (19), and the control law (25) are given. It shows that the proposed algorithms can be applied in actual scenarios.
[figure omitted; refer to PDF]
Compared with the ADCC method and NTSM-ARDC proposed in [4], the case of input dead zones is considered in the trajectory tracking performance in our paper. Although the control input falls in the dead-zone at a specific time when the actuator has no output signal, the performance in trajectory tracking errors is guaranteed, proving the effectiveness of the method proposed in our paper.
5. Conclusions
In this paper, based on a global diffeomorphism transformation, a fixed-time observer control method is proposed for the trajectory tracking problem of the AGV system. Trajectory tracking of AGVs subject to actuator saturation is still an open problem. Energy consumption optimal control of AGVs is another interesting problem to be studied in the future. Additionally, some nonlinear system control methods can also be applied for AGVs in the future. For example, an adaptive super-twisting sliding mode control technique is developed for forn-link rigid robotic manipulators [23]. An adaptive nonsingular integral-type second-order terminal sliding mode tracking control for nonlinear systems with uncertainties is proposed [24].
Acknowledgments
The supports of this paper are the Project Supported by Scientific Research Fund of Zhejiang Provincial Education Department under Grant Y202146005 and the General Projects of Science and Technology under Grant 2021XJKJ04.
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Abstract
This paper investigates the trajectory tracking problem of autonomous ground vehicles (AGVs). The dynamics considered feature external disturbances, model uncertainties, and actuator dead zones. First, a novel time-varying yaw guidance law is proposed based on the line of sight method. By a state transformation, the AGV is proved to realize trajectory tracking control under the premise of eliminating guidance deviation. Second, a fixed time dead zone compensation control method is introduced to ensure the yaw angle tracking of the presented guidance. Furthermore, an improved fixed-time disturbance observer is proposed to compensate for the influence of the actuator dead zone on disturbance observation. Finally, the trajectory tracking control strategy is designed, and simulation comparison shows the effectiveness of the compensate method. The CarSim–MATLAB cosimulation shows that the proposed control strategy effectively makes the AGV follow the reference trajectory.
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