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Thanks to their road safety potential, automated vehicles are rapidly becoming a reality. In the last decade, automated driving has been the focus of intensive automotive engineering research, with the support of industry and governmental organisations. In automated driving systems, the path tracking layer defines the actuator commands to follow th...
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... of the vehicle w.r.t. the reference path are provided in Section 3.1, which is followed by: i) the physics-based vehicle model formulations, presented according to their increasing complexity level in Sections 3.2-3.4; ii) the tyre modelling approaches, in Section 3.5; and iii) the discussion on neural network prediction models (Section 3.6). Fig. 2 shows the schematic of a generic single track model, also known as bicycle model, commonly adopted for PT applications, which includes indication of the reference path. With a few exceptions ( Hang et al., 2021;Wurts et al., 2021), a common assumption is to neglect the rear wheel steering, δ r . In these conditions, the longitudinal ...
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... kinematic single track (ST) model assumes zero slip angles on the front and rear tyres, and neglects the inertial effects. Based on Fig. 2, the longitudinal and lateral positions of the centre of gravity, X and Y, and the yaw angle ψ can be used as states. The time derivatives of the three states can be expressed as ( Kong et al., 2015;Law et al., 2018;L. Tang et al., ...
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... respect to the vehicle model in Fig. 2, the lateral control objectives can be typically achieved by defining the MPC cost function in terms of: i) lateral position error, e y , at the vehicle centre of gravity, w.r. t. the closest point on the reference path; and ii) heading angle error, e ψ , between the body frame orientation and the tangent to the reference path, ...
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... path; and ii) heading angle error, e ψ , between the body frame orientation and the tangent to the reference path, evaluated at the centre of gravity. Both e y and e ψ are defined w.r.t. a curvilinear reference system. Specific cost function formulations can include the evaluation of these errors at a look-ahead distance, see e y,p and e ψ,p in Fig. 2 ...
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... and icy pavements, and compares the trajectory tracking performance of the proposed NNMPC with that of an MPC using a bicycle model as prediction model. Differently from the physics-based MPC, the NNMPC does not need the estimation of the tyre-road friction level. prediction, corresponding to the controller prediction horizon, during constant ( Fig. 22(a)) and increasing ( Fig. 22(b)) steering angle operation. The dotted vertical line represents the condition when the online NN has sufficient data, and starts adapting its weights and biases. The adaptive NN requires a certain number of data and learning iterations to provide better performance than the physics-based dynamic model and ...
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... the trajectory tracking performance of the proposed NNMPC with that of an MPC using a bicycle model as prediction model. Differently from the physics-based MPC, the NNMPC does not need the estimation of the tyre-road friction level. prediction, corresponding to the controller prediction horizon, during constant ( Fig. 22(a)) and increasing ( Fig. 22(b)) steering angle operation. The dotted vertical line represents the condition when the online NN has sufficient data, and starts adapting its weights and biases. The adaptive NN requires a certain number of data and learning iterations to provide better performance than the physics-based dynamic model and the offline-trained NN. To ...
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... nonlinear (NL) dynamic vehicle model until the online-trained NN provides better performance. The online NN training process can be conducted on a separate processing core, and therefore it is not part of the real-time MPC optimisation. The proposed strategy is tested on a generic trajectory, and compared with the benchmarking LMPC and NMPC. Fig. 23 shows the prediction error comparison of the NL and NN models, calculated as the difference between the predicted and measured states during the tracking task, while Fig. 24 reports an extract of the trajectories generated by the PT ...
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... and therefore it is not part of the real-time MPC optimisation. The proposed strategy is tested on a generic trajectory, and compared with the benchmarking LMPC and NMPC. Fig. 23 shows the prediction error comparison of the NL and NN models, calculated as the difference between the predicted and measured states during the tracking task, while Fig. 24 reports an extract of the trajectories generated by the PT ...
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... only actuating the steering system, in an emergency manoeuvre with an obstacle popping up on a straight road with μ = 0.5, from an initial speed of 60 km/h, which pushes the vehicle to limit handling conditions. The differential braking actuation provides quick yaw response, which prevents the vehicle from violating the road boundaries, see Fig. 25. A similar solution, limited to the front steering angle actuation, had already been proposed by Funke et al. (2016), including experimental tests on an automated vehicle, which highlight swift and effective reactions to sudden emergency scenarios, achieved by relaxing the sideslip angle ...
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... experimentally implemented on the 1/10-scale BARC vehicle platform. The algorithm, controlling the steering angle and longitudinal acceleration, is initialised by performing two laps with PT at constant speed, while the following learning iterations use the data from the last two laps. The controller is assessed on the oval-and L-shaped tracks in Fig. 28, which are covered in counter-clockwise direction. The first row of subplots refers to the trajectories used for initialisation and those achieved at laps 7 and 15, while the second row refers to the conditions to which the controller converges. The learning process induces the vehicle to progressively deviate from the original ...
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... and those achieved at laps 7 and 15, while the second row refers to the conditions to which the controller converges. The learning process induces the vehicle to progressively deviate from the original trajectory, by adopting a more aggressive behaviour that 'cuts' the corners and increases vehicle speed, with lateral acceleration peaks of ~1g (Fig. 29). The effectiveness of the learning strategy is evident from the lap times in Fig. 30, showing a progressive improvement until reaching a steady-state condition after ~30 ...
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... MPC in Kabzan et al. (2019) is implemented on the AMZ vehicle used in 2018 Formula Student Driverless event. The experimental test is carried out on the track in Fig. 31, where the top speed is limited to 15 m/s for safety reasons. Also in this case, two laps are used for algorithm initialisation. The speed profile and acceleration comparison in Fig. 32 highlights that the learning-based controller enables more aggressive driving than a nominal MPC, with maximum lateral accelerations that increase from 1.3 g to 2 g, since the learning-based controller is able to make use of the increased tyre grip due to the aerodynamic package. After five laps, the learning approach guarantees a lap ...
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Citations
... Since 2014, the EU has mandated vehicle stability controllers (VSCs), enhancing active safety during emergency manoeuvres, through the limitation of the yaw rate error, sideslip angle, and longitudinal tyre slip [1][2]. While these systems are effective in supporting the average human driver, they may be overly conservative for highly automated vehicles (AVs) [3]. In parallel, powertrain electrification and active chassis control systems offer new AV control opportunities [4]. ...
Path tracking (PT) controllers capable of replicating race driving techniques, such as drifting beyond the limits of handling, have the potential of enhancing active safety in critical conditions. This paper presents a nonlinear model predictive control (NMPC) approach that integrates multiple actuation methods, namely four-wheel-steering, longitudinal tyre force distribution, and direct yaw moment control, to execute drifting when this is beneficial for PT in emergency scenarios. Simulation results of challenging manoeuvres, based on an experimentally validated vehicle model, highlight the substantial PT performance improvements brought by: i) vehicle operation outside the envelope enforced by the current generation of stability controllers; and ii) the integrated control of multiple actuators.
... The choice of vehicle models for the path-following problem is diverse [16]. Considering vehicle control near handling limits during racing maneuvers, the selected model must capture the coupling of longitudinal and lateral motions, as well as the nonlinear behavior of tire forces. ...
This article presents a hierarchical control framework for autonomous vehicle trajectory planning and tracking, addressing the challenge of accurately following high-speed, at-limit maneuvers. The proposed time-optimal trajectory planning and tracking (TOTPT) framework utilizes a hierarchical control structure, with an offline trajectory optimization (TRO) module and an online nonlinear model predictive control (NMPC) module. The TRO layer generates minimum-lap-time trajectories using a direct collocation method, which optimizes the vehicle’s path, velocity, and control inputs to achieve the fastest possible lap time, while respecting the vehicle dynamics and track constraints. The NMPC layer is responsible for precisely tracking the reference trajectories generated by the TRO in real time. The NMPC also incorporates a preview algorithm that utilizes the predicted future travel distance to estimate the optimal reference speed and curvature for the next time step, thereby improving the overall tracking performance. Simulation results on the Catalunya circuit demonstrated the framework’s capability to accurately follow the time-optimal raceline at an average speed of 116 km/h, with a maximum lateral error of 0.32 m. The NMPC module uses an acados solver with a real-time iteration (RTI) scheme, to achieve a millisecond-level computation time, making it possible to implement it in real time in autonomous vehicles.
... 2) NMPC: the nonlinear tire model -Magic Formula (MF), is employed to depict tire dynamics and later obtain the predictive vehicle states to follow the desired trajectory. The nonlinearities of tire forces have been proven effective for model accuracy enhancement but necessitate higher computational resources [38]. The constrained optimization problem is solved using CasADi/IPOPT toolbox [39]. ...
Koopman operator theory is a kind of data-driven modelling approach that accurately captures the nonlinearities of mechatronic systems such as vehicles against physics-based methods. However, the infinite-dimensional Koopman operator is impossible to implement in real-world applications. To approximate the infinite-dimensional Koopman operator through collection dataset rather than manual trial and error, we adopt deep neural networks (DNNs) to extract basis functions by offline training and map the nonlinearities of vehicle planar dynamics into a linear form in the lifted space. Besides, the effects of the dimensions of basis functions on the model accuracy are explored. Further, the extended state observer (ESO) is introduced to online estimate the total disturbance in the lifted space and compensate for the modelling errors and residuals of the learned deep Koopman operator (DK) while also improving its generalization. Then, the proposed model is applied to predict vehicle states within prediction horizons and later formulates the constrained finite-time optimization problem of model predictive control (MPC), i.e., ESO-DKMPC. In terms of the trajectory tracking of autonomous vehicles, the ESO-DKMPC generates the wheel steering angle to govern lateral motions based on the decoupling control structure. The various conditions under the double-lane change scenarios are built on the CarSim/Simulink co-simulation platform, and extensive comparisons are conducted with the linear MPC (LMPC) and nonlinear MPC (NMPC) informed by the physics-based model. The results indicate that the proposed ESO-DKMPC has better tracking performance and moderate efficacy both within linear and nonlinear regions.
... The objective of lateral controllers is to compute appropriate steering inputs to accomplish lateral maneuvers such as trajectory tracking, overtaking maneuvers, and obstacle avoidance [12]. Compared to longitudinal dynamics, lateral dynamics exhibit more complex nonlinear characteristics. ...
The development of intelligent transportation technology has provided a significant impetus for autonomous driving technology. Currently, autonomous vehicles based on Model Predictive Control (MPC) employ motion control strategies based on sampling time, which fail to fully utilize the spatial information of obstacles. To address this issue, this paper proposes a dual-layer MPC vehicle collision-free trajectory tracking control strategy that integrates spatial kinematics and vehicle dynamics. To fully utilize the spatial information of obstacles, we designed a vehicle model based on spatial kinematics, enabling the upper-layer MPC to plan collision avoidance trajectories based on distance sampling. To improve the accuracy and safety of trajectory tracking, we designed an 8-degree-of-freedom vehicle dynamic model. This allows the lower-layer MPC to consider lateral stability and roll stability during trajectory tracking. In collision avoidance trajectory tracking experiments using three scenarios, compared to two advanced time-based algorithms, the trajectories planned by the proposed algorithm in this paper exhibited predictability. The proposed algorithm can initiate collision avoidance at predetermined positions and can avoid collisions in predetermined directions, with all state variables within safe ranges. In terms of time efficiency, it also outperformed the comparative algorithms.
... Despite significant progress in autonomous driving technology, optimizing trajectory tracking control remains a challenging but crucial task [5]. Moreover, as the demand for autonomous vehicles continues to increase, it is crucial to refine trajectory tracking control algorithms to effectively address complex and various road scenarios [6,7], achieving a good balance between responsiveness, accuracy, and computational efficiency in dynamic driving environments and ensuring real-time adaptability. ...
... One main method to solve trajectory tracking control problems is model predictive control (MPC) [7,8]. MPC is a method for solving optimal control problems (OCPs) using the current state of the controlled system as the initial condition. ...
The prediction horizon is a key parameter in model predictive control (MPC), which is related to the effectiveness and stability of model predictive control. In vehicle control, the selection of a prediction horizon is influenced by factors such as speed, path curvature, and target point density. To accommodate varying conditions such as road curvature and vehicle speed, we proposed a control strategy using the proximal policy optimization (PPO) algorithm to adjust the prediction horizon, enabling MPC to achieve optimal performance, and called it PPO-MPC. We established a state space related to the path information and vehicle state, regarded the prediction horizon as actions, and designed a reward function to optimize the policy and value function. We conducted simulation verifications at various speeds and compared them with an MPC with fixed prediction horizons. The simulation demonstrates that the PPO-MPC proposed in this article exhibits strong adaptability and trajectory tracking capability.
... Path planning and tracking control are fundamental and important components of autonomous vehicles (AVs). An investigation in reference [1] highlights that the research topic of path tracking has significantly grown in recent years. For secure and efficient driving, autonomous vehicles need to track precisely the reference trajectory generated by the path planning module. ...
... In the past years, to improve the path tracking control performance of AVs, many significant results have been reported with the applications of advanced linear and nonlinear control techniques, such as PID [2,3], LQR [4,5], MPC [6,7] and SMC [8,9]. Model predictive control (MPC) is a highly competitive solution with respect to the other possible control technologies [1]. An advantage is the better tracking performance during high speed and medium-to-high lateral acceleration conditions, compared to the kinematic or geometry-based path tracking methods, such as the pure pursuit [10] and Stanley [11] methods. ...
Despite its excellent performance in path tracking control, the model predictive control (MPC) is limited by computational complexity in practical applications. The neural network control (NNC) is another attractive solution by learning the historical driving data to approximate optimal control law, but a concern is that the NNC lacks security guarantees when encountering new scenarios that it has never been trained on. Inspired by the prediction process of MPC, the deviation sequence neural network control (DS-NNC) separates the vehicle dynamic model from the approximation process and rebuilds the input of the neural network (NN). Taking full use of the deviation sequence architecture and the real-time vehicle dynamic model, the DS-NNC is expected to enhance the adaptability and the training efficiency of NN. Finally, the effectiveness of the proposed controller is verified through simulations in Matlab/Simulink. The simulation results indicate that the proposed path tracking NN controller possesses adaptability and learning capabilities, enabling it to generate optimal control variables within a shorter computation time and handle variations in vehicle models and driving scenarios.
... Compared with general feedback control and optimal control algorithms, MPC algorithms are especially good at solving multi-objective problems under optimal control systems [9]. With the development of online optimization and computer hardware, model predictive control has become an area of increasing interest in vehicle active safety [10,11] and path tracking [12]. Considering the problem of poor tracking of an automated vehicle when it slows down to change lanes in a given vicinity, Li et al. [13] proposed a lane change detection algorithm using two controllers and a steering tracking algorithm based on the combination of an MPC controller and a PI controller. ...
For continuously changing road conditions and vehicle operating states, the exactitude of vehicle path tracking has not been secured by model predictive control based on linear lateral stiffness. An amended square root cubature Kalman filter method based on the minimization of a new covariance of interest is proposed to calculate the tire lateral deflection force in real time. The ratio of the estimated tire force to the linear tire force was used as a ratio to adjust the lateral deflection stiffness, and an adaptive model predictive controller was built based on the vehicle path-tracking error model to correct the tire lateral deflection stiffness. Finally, an analysis based on the joint CarSim and Simulink simulation platform shows that compared to a conventional model predictive control (MPC) controller, a trajectory-following controller built based on this method can effectively reduce the lateral distance error and heading error of an autonomous vehicle. Especially under low adhesion conditions, the conventional MPC controllers will demonstrate large instability during trajectory tracking due to the deviation of the linear tire force calculation results, whereas the adaptive model predictive control (AMPC) controllers can correct the side deflection stiffness by estimating the tire force and still achieve stable and effective tracking of the target trajectory. This suggests that the proposed algorithm can improve the effectiveness of trajectory tracking control for autonomous vehicles, which is an important reference value for the optimization of autonomous vehicle control systems.
... MPC has been widely used in traffic control due to its capability of solving multivariable optimization problems, systematically accounting for constraints on both state and control actions, and considering the anticipated future behavior of the system [23]. Recently, some researchers have conducted a review of model predictive path tracking (PT) control for automated road vehicles [24], considering the following MPC methods for PT control: linear MPC [25,26], linear time-varying MPC [27], linear parameter-varying MPC [28], nonlinear MPC [29], hybrid MPC [30], neural network MPC [31], robust MPC [32], and learning MPC [33]. Some researchers have proposed a learning-based model predictive control (LMPC) algorithm for a Formula Student (FS) autonomous vehicle to improve the dynamic model accuracy of the vehicle [34]. ...
Traffic waves in traffic flow significantly impact road throughput and fuel consumption and may even lead to severe safety issues. Currently, in connected and autonomous environments, the jam-absorption driving (JAD) strategy shows good performance in dissipating traffic waves. However, the previous JAD strategy has mostly focused on wave dissipation without adequately assessing traffic efficiency and safety. To address this gap, an optimal control problem for JAD in mixed traffic is proposed to reduce traffic waves. The prediction model is developed using the car-following model within a model predictive control (MPC) framework. The Helly model is selected for the manual vehicle. This is because the Helly model is a linear model that describes the car-following phenomenon accurately without delay effect. In addition, the objective function of the prediction model considers both traffic safety and efficiency while satisfying mechanical and safety constraints. Simulation results indicate that the proposed methodology can effectively reduce traffic jams and improve traffic performance on a one-lane freeway. The optimal method is more applicable to complex traffic wave scenarios, providing a new perspective for reducing traffic jams on the freeway.
... For the last decade, PTC has been intensively explored [4][5][6]. Consequently, a huge deal of papers have been published to date in the field of PTC [4][5][6][7][8][9][10]. ...
... Three types of control input, named u 1 , u 2 and u 3, corresponding to input configurations IC#1, IC#2 and IC#3, are composed of three elements, δ f , δ r and ∆M z , as given in Equation (10) [31,32]. In Equation (10), B 2 {k} representing the k-th column of the matrix B 2 . ...
This paper presents a method to design a path tracking controller with a constraint on tire slip angles under low-friction road conditions. On a low-friction road surface, a lateral tire force is easily saturated and decreases as a tire slip angle increases by a large steering angle. Under this situation, a path tracking controller cannot achieve its maximum performance. To cope with this problem, it is necessary to limit tire slip angles to a value where the maximum lateral tire force is achieved. The most commonly used controllers for path tracking, linear quadratic regulator (LQR) and model predictive control (MPC), are adopted as a controller design methodology. The control inputs of LQR and MPC are front and rear steering angles and control yaw moment, which have been widely used for path tracking. The constraint derived from tire slip angles is imposed on the steering angles of LQR and MPC. To fully verify the performance of the path tracking controller with the constraint on tire slip angles, a simulation is conducted on vehicle simulation software. From the simulation results, it is shown that the path tracking controller with the constraint on tire slip angles presented in this paper is quite effective for path tracking on low-friction road surface.
... Liu et al. [13] systematically analysed the model complexity, optimal performance, computational cost, and application scenarios of these algorithms. Among them, MPC is also known as rolling timehorizon optimal control, whose main control idea is to build a predictive model and solve the optimal control strategy with an optimization algorithm [14]. Since MPC is well suited to handle multi-constraint objective problems such as vehicle dynamics constraints and actuator saturation constraints, it has gradually dominated research on automated driving motion control. ...
For complex and dynamic high‐speed driving scenarios, an adaptive model predictive control (MPC) controller is designed to ensure effective path tracking for automated vehicles. Firstly, in order to prevent model mismatch in the MPC controller, a tire cornering stiffness estimation algorithm is designed and a soft constraint on slip angle is added to further enhance the controller's precision in tracking trajectories and the vehicle's driving stability. Secondly, the improved particle swarm optimization (IPSO) method with dynamic weights and penalty functions is suggested to address the issue of insufficient accuracy in solving quadratic programming. Additionally, the standard particle swarm optimization (PSO) algorithm is used to seek the most suitable time horizon parameters offline to obtain the best time horizon data set under different vehicle speeds and adhesion coefficients, and then it is optimized online by an adaptive network‐based fuzzy inference system (ANFIS) to enhance the model predictive controller's adaptability in different operating conditions. Finally, simulation experiments are conducted under three different operating conditions: docked roads, split roads, and variable vehicle speeds. The results indicate that the designed adaptive MPC controller can accurately and stably track the reference trajectory in various scenarios.
