Abstract
Path tracking is a crucial function for achieving unmanned driving. This paper addresses the challenge of low tracking accuracy and poor stability in driverless trucks caused by uncertain model parameters and steady-state errors during path tracking. A linear quadratic regulator (LQR) controller optimization by an improved genetic algorithm has been designed. Firstly, the paper formulates the dynamic model of a two-degree-of-freedom vehicle as well as the model for tracking error. Subsequently, path tracking control is achieved through the utilization of feedforward control and LQR feedback control algorithms. Secondly, the weight coefficient of the LQR controller is enhanced through the utilization of an improved GA in order to boost the precision of path tracking. Ultimately, the devised controller undergoes simulation and validation in the TruckSim-Matlab/Simulink platform across diverse operational circumstances. The findings from the simulation demonstrate that the controller, optimized through improvements in the genetic algorithm, exhibits excellent tracking accuracy and stability.
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1 Introduction
In the past few years, the persistent expansion of global trade volume has resulted in a shortage of truck drivers in both domestic and foreign logistics and transportation industries. This shortage is evident in nations such as the United States, the United Kingdom, Germany, China, Japan, India, and Brazil [1]. The deficit of truck drivers in the United States has intensified during the recent years, despite the truck transportation industry's efforts to meet the growing demand. Consequently, the utilization of driverless truck technology has surfaced as a substantial remedy to tackle the shortage of truck drivers in numerous countries. As a fundamental and critical component of autonomous driving, the main aim of path tracking control is to make the actual route of the vehicle as close as possible to the pre-planned route, minimizing lateral and heading errors. As a result, improving the precision and firmness of the trajectory tracking regulator becomes imperative and urgent.
The development of unmanned driving path tracking control began early, and currently the main methods for tracking control include Proportional–Integral–Differential (PID) control [2], Sliding Mode control [3], Fuzzy logic control [4], Model Predictive control (MPC) [5], and LQR control [6]. Among them, LQR control is a feedback control method designed based on optimization principles [7, 8]. The primary principle is to establish a linearized model for the path tracking error of the vehicle system and solve the optimal feedback control parameters through quadratic optimization within the cyclic control cycle [9]. The LQR regulator is widely utilized in unmanned vehicles to accurately follow paths due to its exact performance, easy-to-follow algorithm, suitability for non-linear systems, and ability to take into account dynamic limitations. Currently, numerous scholars have conducted research on this topic and proposed various solutions. Hu et al. [10] introduced a fuzzy LQR with preview PID angle compensation that exhibits high tracking accuracy while maintaining stability across varying vehicle speeds. Xu et al. [11] developed a multi-point preview lateral controller based on the monorail dynamics model, implementing a LQR approach. They utilized the system's disturbance term as a state vector to construct an augmented LQR problem, resulting in enhanced path tracking accuracy. Wang et al. [12] presented an enhanced lateral path control strategy for vehicle navigation. The foundation of this strategy is constructed using an LQR algorithm while integrating a fuzzy control approach to actively adapt the weighting coefficients of LQR in real-time. This, in turn, enhances the precision of tracking, stability of steering, and effectiveness of computation. Although the aforementioned enhancements to the LQR have yielded favorable outcomes, there exists potential for additional advances in both control accuracy and stability.
This paper proposes the utilization of an improved genetic algorithm to optimize a LQR controller, aiming to enhance the tracking precision and stability of the controller. Firstly, a classic LQR controller was designed using the Two-Degree-of-Freedom vehicle dynamics lateral tracking error model. In order to eradicate the persistent deviation of the model's equilibrium state, feedforward control compensation was implemented based on this model. Secondly, this paper analyzes the effect of weight coefficients of the LQR controller on the vehicle's tracking capability. In order to achieve a balance between control stability and tracking accuracy, an enhanced genetic algorithm is applied for optimize the LQR weighting factors. Finally, the study concludes by conducting simulation experiments on both the pre-optimized and post-optimized LQR controllers. The results consistently demonstrate the superior performance of the GA-LQR controller devised in this research.
2 Vehicle Path Tracking Control Model
2.1 Vehicle Dynamic Model
The vehicle's lateral dynamics are influenced by a multitude of factors. The complexity of the vehicle model in controller design needs to be minimized for theoretical analysis, requiring certain assumptions to be made for driverless trucks:
-
(1)
Neglecting the influence of lateral and longitudinal aerodynamics;
-
(2)
Assuming that the vehicle solely moves in a two-dimensional plane without accounting for the effects of the suspension system;
-
(3)
By solely considering the lateral characteristics of the tire, it is understood that the lateral force acting on the tire exhibits a linear relationship with the lateral angle;
-
(4)
Neglecting the impact of the steering system and utilizing the front wheel angle as the direct input;
-
(5)
Neglecting the transfer of load between the front and rear axles.
Considering the aforementioned assumptions, driverless trucks exhibit only two degrees of freedom: motion in the horizontal direction along the y-axis and movement in the direction of orientation around the z-axis. Here is the corresponding linear model depicting the two degrees of freedom, as depicted in Fig. 1, where O-x-y-z is the coordinate system of the vehicle.
Assuming a uniform speed for the vehicle and considering a small front wheel angle \(\delta\) (less than 5°), expressed as \(\cos \delta \approx 1\), the vehicle dynamics equation can be represented as:
where the moment of inertia about the z-axis is denoted as \(I_{z}\); the vehicle mass is represented by \(m\); the vehicle’s heading rate is given by \(\dot{\varphi }\); the lateral forces of the front and rear wheels are indicated as \(F_{yf}\) and \(F_{yr}\); the values representing the distances from the vehicle's center of mass to the front and rear axles are identified as \(a\) and \(b\); the lateral stiffness of the front and rear wheels is represented as \(C_{f}\) and \(C_{r}\); the longitudinal speed of the vehicle is denoted by \(v_{x}\); and finally, the lateral velocity of the vehicle is given by \(v_{y}\).
Simplify the above equation and define \(v_{y} = \dot{y}\), thus obtaining the matrix form of the vehicle dynamics equation:
2.2 Lateral Error Model for Vehicle Path Tracking
By applying coordinate transformation to the existing model depicting the vehicle's dynamics with two degrees of freedom, we can obtain a lateral error model specifically designed for path tracking, illustrated in Fig. 2.
Among them, the lateral error \(e_{d}\) is defined as the shortest distance between the vehicle centroid and the reference path projection point. The heading angle error \(e_{\varphi } = \varphi - \theta_{r}\) may be described as the difference between the true heading angle \(\theta\) of the vehicle and the specified reference heading angle \(\theta_{r}\). In real-life control applications, it becomes essential to develop a controller capable of eradicating these two errors instantaneously. This guarantees real-time accurate tracking of the intended path by the vehicle. The first derivatives of the lateral error \(\dot{e}_{d}\) and heading angle error \(\dot{e}_{\varphi }\) are computed by differentiating the lateral error \(e_{d}\) and heading angle error \(e_{\varphi }\) with respect to time.
By combining Eq. (1), we can obtain the second derivatives of both the lateral error \(\ddot{e}_{d}\) and the heading angle error \(\ddot{e}_{\varphi }\).
By further modifying the above mentioned formula, the derivation of state space equations capturing lateral and heading errors in unmanned vehicle steering is achievable. The obtained equations are as follows:
The state space equation for the tracking error of vehicle dynamics can be derived as follows:
3 Design of Path Tracking Controller
The algorithm flowchart of the path tracking controller is depicted in Fig. 3. By combining the given path planning information with real-time vehicle status and pose information, the path tracking error can be calculated. To address the hysteresis in algorithm control, a prediction module is introduced following the real-time vehicle status information. This enables the algorithm to predict road conditions, thereby enhancing control accuracy.
3.1 LQR Controller Design
The essence of the LQR controller is to seek the optimal control law for control variables and state variables while operating within linear constraints. Even when the following objective functions reach their minimum values.
where the control variable of the system is denoted as \({\mathbf{U}}\); the system's state parameter is denoted as \({\mathbf{X}}\); the weighted coefficient \({\mathbf{Q}}\) representing the error represents the extent of significance attached to the corresponding control target; the control weighting matrix is denoted as \({\mathbf{R}}\).
The feedback control law of LQR controller is:
In the formula: \({\mathbf{K}} = (R + {\mathbf{B}}^{{\mathbf{T}}} {\mathbf{PB}})^{ - 1} {\mathbf{B}}^{{\mathbf{T}}} {\mathbf{PA}}\), where the control gain coefficient is denoted as \({\mathbf{K}}\); \({\mathbf{P}}\) is a constant matrix, consisting of a positive definite solution to the Riccati equation as follows.
Substituting Eq. (12) into Eq. (9) yields:
By introducing a comprehensive deviation of lateral distance deviation and heading angle deviation \(e_{rr}\) as the system state variables, and controlling the front wheel angle \(\delta_{f}\), the LQR controller can acquire the ideal control law for the front wheel angle:
where \({\mathbf{K}} = [k_{1} ,k_{2} ,k_{3} ,k_{4} ]\) is the feedback gain matrix of the LQR controller; \({\mathbf{e}}_{{{\mathbf{rr}}}} = [e_{d} ,\dot{e}_{d} ,e_{\varphi } ,\dot{e}_{\varphi } ]^{T}\) is the comprehensive deviation amount.
3.2 Feedforward Control
Substituting Eq. (15) into Eq. (8) yields the following result:
Currently, irrespective of the magnitude of \({\mathbf{k}}\), it is impermissible for \({\dot{\mathbf{X}}}\) to be zero. Solely employing LQR feedback control will induce an enduring stagnant error. As a consequence, to eradicate the effect of the \({\mathbf{C}}\dot{\theta }_{r}\) parameter, the feedforward control variable \(\delta_{ff}\) is implemented. The system's state variable, with the inclusion of feedforward control, can be expressed as:
In the event that there is zero steady-state error, When \({\dot{\mathbf{X}}} = 0\), the equation for the state variable of the system is as follows:
According to Eq. (18), it is inferred that attaining the optimal control effect requires determining a suitable value \(\delta_{ff}\) that ensures the system exhibits zero steady-state error. The system's equation for steady-state error is determined by utilizing Eq. (7) and is given by the following expression:
From Eq. (19), it is evident that:
In order to attain sideways deviation \(e_{d} = 0\), the feedforward control amount \(\delta_{ff}\) would be:
where \(\rho = \frac{{\dot{\theta }_{r} }}{{v_{x} }}\) is the path curvature.
3.3 LQR Controller Optimized Based on GA
Genetic algorithm [13, 14] is a series of search algorithms inspired by natural evolution theory. The algorithm's exceptional convergence, efficient computational time, and robustness make it a popular choice for parameter optimization problems. And there are not many mathematical requirements for the optimization problem being solved. Owing to its characteristics, the search process does not rely on the intrinsic properties of the problem and can accommodate any type of objective function and constraint, regardless of whether they are linear or nonlinear, discrete or continuous in nature. In the design of LQR controllers, choosing appropriate Q and R values is crucial for achieving effective tracking with LQR controllers. The specific process of optimizing matrices Q and R using genetic algorithm is shown in Fig. 4.
Firstly, encode the parameters, including the fundamental parameters of the vehicle along with the coefficients of the LQR state space equation. Secondly, a population is generated, and each individual population value is assigned sequentially to the randomly generated weight coefficients of LQR, within a feasible range. Reuse the generated weight coefficients Q and R to calculate the optimal feedback gain matrix K of LQR. Finally, the Simulink model is then executed to compute the pertinent parameters, and based on these parameters, the fitness function is evaluated to determine the fitness value of each particle. Once the termination condition is satisfied, the optimal value will be displayed.
Based on the optimization design method described above, the Table 1 displays the optimal values of the final LQR weight matrix.
4 Simulation Verification
In order to confirm the efficiency of the LQR controller, optimized using the proposed improved genetic algorithm, a joint simulation experiment was conducted using TruckSim and Matlab/Simulink. The primary parameters of the vehicle are shown in Table 2.
In this article, the double lane change scenario is employed as the benchmark path, depicting the standard driving conduct of cars navigating through barriers. The automobile is put to the test on a road with a friction coefficient of 0.85, experiencing two different velocities: 30 and 60 km/h. Simulations are conducted for the LQR and GA-optimized LQR controllers at both speeds. Figure 5 shows the tracking effect of vehicle lateral displacement. The diagram demonstrates that the car exhibits satisfactory performance when following the designated route at both velocities. Nevertheless, the LQR controller designed in this study, using GA-optimized weight coefficients, exhibits superior tracking performance and enhanced stability.
In Fig. 6, the tracking accuracy of the LQR controller and the GA-LQR controller in managing lateral errors is showcased in the context of double line shifting situations, precisely at velocities of 30 and 60 km/h. According to the results findings, the LQR controller has the ability to consistently restrict the maximum tracking discrepancy to 0.2848 m at different velocities. The lateral error of the GA-LQR controller designed in this article is considerably lower compared to that of the LQR controller. At a velocity of 30 km/h, the GA-LQR controller can effectively limit the maximum lateral error to 0.0704. This value, however, is approximately 49.7% greater than that of traditional LQR controllers. At a velocity of 60 km per hour, the GA-LQR controller attains a peak lateral deviation of 0.1429. This represents an enhancement of approximately 49.8% compared to the effectiveness of the traditional LQR controller.
This implies that the LQR controller designed in this article can more efficiently minimize errors and enhance the precision of vehicle tracking during path tracking.
Figure 7 illustrates the impact of dual line shifting conditions on the yaw error tracking for both the LQR and the GA-LQR controller at speeds of 30 and 60 km/h. The simulation results demonstrate that the yaw error remains within 0.1145 rad across various vehicle speeds, and the observed variation is insignificant. These findings indicate the vehicle's tracking process maintains a relatively stable performance. The GA-LQR controller successfully limits the maximum yaw error to 0.0984 at a speed of 30 km/h, and 0.1114 at a speed of 60 km/h. These values demonstrate an improvement over the performance of traditional LQR controllers.
Figure 8 depicts the performance of the front wheel angle tracking for both the LQR controller and the GA-optimized LQR controller under dual lane shifting conditions, at speeds of 30 and 60 km/h. The simulation results reveal a gradual increase in the front wheel angle as the vehicle speed rises, without any step changes. And the front wheel angle is effectively controlled within a range of −10° to 10° at various vehicle speeds, which strongly suggests a stable tracking process. Overall, the GA-LQR controller demonstrates satisfactory levels of tracking accuracy and stability.
5 Conclusions
This study addresses the issue of low tracking accuracy and poor stability in unmanned trucks during path tracking under traditional LQR controllers. To combat these problems, the inclusion of feedforward control methods is proposed to compensate for the front wheel angle and mitigate steady-state errors resulting from the simplification of the vehicle dynamics model. The weight coefficients are optimized and adjusted using a genetic algorithm to achieve higher tracking accuracy while simultaneously addressing the vehicle's driving stability. The outcomes of the simulation demonstrate that the GA-LQR controller, developed in this article, exhibits commendable tracking accuracy and stability when subjected to the double lane reference path. This research achievement offers innovative insights for designing and optimizing path tracking controllers for driverless trucks. In future research, it is necessary to conduct further validation of the proposed control strategy using an actual vehicle.
References
Mittal N, Udayakumar PD, Raghuram G, Bajaj N (2018) The endemic issue of truck driver shortage—a comparative study between India and the United States. Res Transp Econ 71:76–84. https://doi.org/10.1016/j.retrec.2018.06.005
The AM, Thai SD, Dinh TD, Van CL, Banerjee S (2021) A combined backstepping and adaptive fuzzy PID approach for trajectory tracking of autonomous mobile robots. J Braz Soc Mech Sci Eng 43:156. https://doi.org/10.1007/s40430-020-02767-8
Sun Z, Zou J, He D, Zhu W (2022) Path-tracking control for autonomous vehicles using double-hidden-layer output feedback neural network fast nonsingular terminal sliding mode. Neural Comput Appl 34:5135–5150. https://doi.org/10.1007/s00521-021-06101-8
Hu C, Chen Y, Wang J (2021) Fuzzy observer-based transitional path-tracking control for autonomous vehicles. IEEE Trans Intell Transp Syst 22:3078–3088. https://doi.org/10.1109/TITS.2020.2979431
Tian J, Yang M (2023) Research on trajectory tracking and body attitude control of autonomous ground vehicle based on differential steering. PLoS ONE 18:e0273255. https://doi.org/10.1371/journal.pone.0273255
Tian J, Zeng Q, Wang P, Wang X (2021) Active steering control based on preview theory for articulated heavy vehicles. PLoS ONE 16:e0252098. https://doi.org/10.1371/journal.pone.0252098
Shauqee MN, Rajendran P, Suhadis NM (2021) An effective proportional-double derivative-linear quadratic regulator controller for quadcopter attitude and altitude control. Automatika 62:415–433. https://doi.org/10.1080/00051144.2021.1981527
Kumar M, Hildebrandt A-C, Strauss P, Kraus S, Stiller C, Zimmermann A (2020) Lateral trajectory stabilization of an articulated truck during reverse driving maneuvers. In: 2020 IEEE intelligent vehicles symposium (IV). IEEE, New York, pp 744–751
Qiu B, Wei L, Wang X, Li L, Zhou D, Wang Z (2023) Path tracking of autonomous vehicle based on adaptive preview trajectory planning with the consideration of vehicle stability. https://doi.org/10.1177/09544070221094112. Accessed 29 Nov 2023
Hu J, Zhong X, Chen R, Zhu L, Xu W, Zhang M (2022) Intelligent vehicle path tracking control based on fuzzy LQR. Autom Eng 44:17–25+43. https://doi.org/10.19562/j.chinasae.qcgc.2022.01.003
Xu S, Peng H (2020) Design, analysis, and experiments of preview path tracking control for autonomous vehicles. IEEE Trans Intell Transp Syst 21:48–58. https://doi.org/10.1109/TITS.2019.2892926
Wang Z, Sun K, Ma S, Sun L, Gao W, Dong Z (2022) Improved linear quadratic regulator lateral path tracking approach based on a real-time updated algorithm with fuzzy control and cosine similarity for autonomous vehicles. Electronics 11:3703. https://doi.org/10.3390/electronics11223703
Wang Y, Zhang Y, Zhang C, Zhou J, Hu D, Yi F, Fan Z, Zeng T (2023) Genetic algorithm-based fuzzy optimization of energy management strategy for fuel cell vehicles considering driving cycles recognition. Energy 263:126112. https://doi.org/10.1016/j.energy.2022.126112
Chen C, Ma R, Ma W (2023) GA-LQR for vehicle semi-active suspension with BiLSTM inverse model of magnetic rheological damper. Trans Can Soc Mech Eng. https://doi.org/10.1139/tcsme-2023-00271
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Song, W., Min, J., Zhang, T., Zhang, Y., Zhao, F. (2024). Research on Path Tracking Control of Driverless Trucks. In: Halgamuge, S.K., Zhang, H., Zhao, D., Bian, Y. (eds) The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering. ICACMVE 2023. Lecture Notes in Mechanical Engineering. Springer, Singapore. https://doi.org/10.1007/978-981-97-1876-4_49
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