Intelligent Vehicle Path Tracking Control Based on Improved MPC and Hybrid PID

In this study, to improve the accuracy of path tracking in intelligent vehicles, we propose an intelligent vehicle path-tracking control method based on improved model predictive control (MPC) combined with hybrid proportional-integral-derivative (PID) control theory. In the lateral control, a constraint on the side deflection of the front wheel is added based on traditional MPC and a relaxation factor is introduced to improve the stability of vehicle control for the driving stability. In longitudinal control, a hybrid PID controller is designed for different road conditions to improve the accuracy of control of vehicle speed. We present the results of a co-simulation using Carim and MATLAB/Simulink and a test with a sample vehicle, which show that the proposed path tracking controller can greatly improve the path tracking accuracy and stability of an intelligent vehicle. The model-based prediction, rolling optimization solution, feedback control, and the addition of a constraint on the side deflection of the front wheel as well as a relaxation factor can ensure the lateral driving stability of an intelligent vehicle. The proposed approach achieved a lateral error of less than 1%, and the yaw angle was controlled within 4°. The longitudinal speed control based on hybrid PID controller can improve the response speed of the system and meet the real-time requirements of vehicle driving.

have been developed to perform path tracking control in 23 automated systems designed to drive intelligent vehicles, the 24 most important of which include preview control (a form of  ory, which improved tracking accuracy overall. However, its 30 The associate editor coordinating the review of this manuscript and approving it for publication was Jerry Chun-Wei Lin . performance was inconsistent under different road conditions 31 and driving speeds. Tang et al. [8] used kinematic MPC to 32 deal with road curvature disturbance, along with yaw rate PID 33 feedback control to eliminate uncertainty and modeling errors 34 and a vehicle sideslip angle compensator to correct motion 35 model prediction. The robustness of this method to time delay 36 was average. Shuo et al. [9] proposed the MPC-Fuzzy control 37 strategy, which reduced the amount of computation required 38 along with the steering error between the tracking accuracy 39 and controller, but failed to make corresponding improve-40 ments in longitudinal tracking. Lin et al. [10] proposed a 41 control algorithm combining MPC and fuzzy PID control, 42 which was designed to solve the problem of rapid vehicle 43 response owing to complex path tracking and stability control 44 models, but did not consider all relevant aspects of vehicle 45 dynamics. In terms of the effect of an MPC objective function 46 on performance. A controller based on a combination of MPC 47 vehicle's path tracking accuracy, the process from data input 92 to data processing and acquisition of PID values based on an 93 adaptive PID control algorithm was time-consuming. More-94 over, intelligent vehicles adhere to rigorous requirements to 95 ensure real-time performance; therefore, the performance of 96 this method was not ideal for real vehicle tests. Xie et al. [20] 97 proposed the use of yaw moment information for tracking 98 bias compensation and coordinate vehicle stability control, 99 and used MPC to calculate the steering angle of the front 100 wheels and to control the vehicle along a reference path 101 through automatic steering. However, the torque distribution 102 in this study was simplified, and the tires of the vehicle were 103 not fully utilized to distribute the yaw moment. Goli and 104 Eskandarian [21] optimized tracking stability and tracking 105 time in the process of path tracking and used a multi-objective 106 optimization method to set the parameter value. Although the 107 complexity of the process of manually adjusting the param-108 eters was reduced, the performance of this method in path 109 tracking was poor under complex road conditions.

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To solve the above problems, in this study, we propose 111 an intelligent path tracking control method based on an 112 improved MPC and hybrid PID. The main contributions of 113 this study are as follows. 114 1) The objective function of the vehicle model predictive 115 control is set based on a model of vehicle kinematics and 116 dynamics, and the prediction model is improved in real time 117 using the feedback correction characteristics of MPC. Then, 118 the established objective function is solved according to the 119 improved model to obtain the optimal value for the angle of 120 the front wheels.

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2) In the lateral control, a constraint is added on the sideslip 122 angle of the front wheels, and a relaxation factor is introduced 123 to ensure the stability of the lateral control. In the longitudinal 124 control, a hybrid PID controller is designed for different road 125 conditions to ensure the accuracy, stability, and real-time 126 operation of the longitudinal control of an intelligent vehicle. 127 3) Finally, a co-simulation and real-vehicle experiment 128 were carried out using CarSim and MATLAB/Simulink soft-129 ware. The research results show that precise control of intel-130 ligent vehicle path tracking can be achieved under different 131 road conditions and speeds. The lateral error was less than 132 1%, and the yaw angle was controlled within −4 • to 2 • . 133 Moreover, the computation time of the program was shorter, 134 and it exhibited better real-time performance in path tracking. 135 Overall, the improved MPC and hybrid PID methods reduced 136 the calculation time of the program and improved its real-time 137 path-tracing performance compared to prior methods.

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The establishment of a vehicle kinematics model and a 140 dynamics model is the basis for the analysis and research 141 of the intelligent vehicle control system and the design of 142 the controller [22]. A vehicle kinematics model is established 143 based on the position of an intelligent vehicle in space and the 144 current driving speed and other geometric variable changes 145 over time. In contrast, the vehicle dynamics model describes 146 the state and the physical laws affecting the motion of 147 intelligent vehicles from the mechanical perspective, includ-148 ing a physical model and model composed of differential 149 equations.  In this work, we used a 3-DOF kinematic model of the entire 158 vehicle, as shown in Fig.1.

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In Fig. 1, (X f , Y f ) and (X r , Y r ) are the center coordinates The following expression may be obtained according to the 171 kinematic constraints of intelligent vehicles.

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Ẋ r sin ϕ −Ẏ r cos ϕ = 0 By combining equation (1) and equation (2), we obtain According to the geometric relationship between the front 176 and rear wheels of an intelligent vehicle, By combining the above equations, the kinematic model of 179 the intelligent vehicle can be obtained as follows. The balance equations of an intelligent vehicle along the x, y 195 and z axis can be obtained according to Newton's second law, 196 as given below.
In equation (6) above, m is the mass of the intelligent 199 vehicle, a and b are the distance between the center of mass 200 of the intelligent vehicle and the front and rear axles, respec-201 tively, and I z is the moment of inertia of the intelligent vehicle 202 around the z axis.

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The expressions of the longitudinal and lateral forces of the 204 intelligent vehicle in the x and y axis directions are given as 205 follows.
In equation (7) above, F lf and F lr denote the longi-208 tudinal force respectively received by the front and rear 209 wheels, whereas F cf and F cr are similarly the transverse force 210 VOLUME 10, 2022 received by the front and rear wheels, F xf and F xr are the force along the X axis received by the front and rear wheels, F yf and 212 F yr are the force along the Y axis received by the front and 213 rear wheels.

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Assuming that the lateral acceleration of an intelligent 215 vehicle during driving is a y ≤ 0.4g and the tire force has 216 a linear relationship with the sideslip angle, the longitudinal 217 and lateral forces on the front and rear wheels of the intelli-218 gent vehicle are as follows.
In equation (8) Combining the above equations, the nonlinear dynamic 231 model of an intelligent vehicle can be obtained as follows. The state variable of the reference system at any time is 251 related to its state variable and control variable at a given time. 252 where ξ dyn is the state variable, and u dyn is the control vari-254 able.

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According to Taylor's formula (12), high-order terms other 256 than the first order can be omitted from equation (11) to 257 obtain the linear time-varying equation (13).
To achieve faster real-time control of the entire system, 264 (13) must be discretized as follows. where: Suppose that N p and N c are the prediction time domain 271 and control time domain in the model predictive control, 272 respectively, and the controller can predict the state variables 273 of the system as follows.
To allow an intelligent vehicle to track the path planned by the 280 upper layer quickly and accurately, an objective function must 281 be established. Because intelligent vehicles may encounter a 282 sudden change in the control variable when driving, a relax-283 ation factor must be added to the objective function. To this 284 end, the objective function is established as follows.

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To accelerate the solving speed of the solver, the above 300 optimization problem is converted into a quadratic program-301 ming problem as follows.

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The tire sideslip angle plays an important role in the char-328 acteristics of vehicle tires. The sideslip angle is the basis for 329 studying vehicle handling and stability. Thus, it is necessary 330 to calculate the constraint of the tire sideslip angle of the 331 front and rear wheels according to the state quantity. This 332 relationship is given as follows.
To ensure that the tire lateral force changes linearly, the 336 maximum tire sideslip angle is limited to 5 • on a road with 337 large adhesion coefficient and 2 • on a road with small adhe-338 sion coefficient.
where the output u(t) is the linear combination of the e(t) 351 proportion, integral, and differential, and K P , T i and T d are the 352 proportional coefficient, integral time constant, and derivative 353 time constant, respectively. 354 Fig. 4 shows the PID control logic diagram. The difference 355 between the actual speed and the reference speed of the driv-356 ing wheel is used as the deviation input of the PID controller. 357 Reset is defined as a state in which the torque is at zero or in 358 the braking state. At the same time, the PID debug enable port 359 is set. The Vehicle Spy 3 software product was used to adjust 360 the PID value online through the CAN network to obtain the 361 best PID value under various working conditions by the debug 362 enable port.      Table 1, the parameters of the vehicle dynam-394 ics model are listed in Table 2. 395 To highlight the performance of intelligent vehicle control 396 based on the improved MPC and hybrid PID controllers, 397 MPC controllers and the improved MPC and hybrid PID 398 controllers were used to perform path tracking tests. The 399 co-simulation diagram of the improved MPC, hybrid PID 400 controller, and CarSim is shown in Fig. 6. 401 Fig. 7 is a simulation diagram of traditional 402 MPC-controlled path tracing control. It may be observed 403 from Fig. 7 that when the speed was 18 km/h, the intelligent 404 vehicle achieved good path tracking performance, but when 405 the speed increased to 36 km/h, the path tracking exhibited 406 a significant deviation, which increased with the travel dis-407 tance. According to the path tracking curve in Fig. 7, when the 408 speed was 36 km/h, the intelligent vehicle did not turn until 409 37 m, which obviously lagged behind that of the reference 410 path, leading to a deviation in path tracking. According 411 to Figs. 8-9, under conditions of high vehicle speed, the 412 intelligent vehicle exhibited poor lateral controllability, and 413 even sideslip occurred in some cases.

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As may be observed from Figs. 10 to 12, when the intelli-415 gent vehicle speed reached 54 km/h, controlling the steering 416 of the intelligent vehicle became difficult owing to the high 417 speed, and the vehicle sideslip angle, lateral speed, and lateral 418 acceleration increased significantly, leading to a large devia-419 tion in path tracking. This result shows that traditional MPC 420 controllers exhibit poor control performance and stability 421 under high-speed working conditions. In summary, the path 422 94138 VOLUME 10, 2022     of the hybrid PID, the accuracy of the path tracking was very 433 high, with an error below 1%. When the speed was 54 km/h, 434 the path tracing effect was slightly worse than at low speed. 435 However, with the extension of time or the increase of driving 436 distance, the tracing accuracy can also reach the accuracy at 437 low speed.

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As shown in Figs. 14-16, when the vehicle speed was 439 18 km/h, the intelligent vehicle exhibits a high lateral 440 dynamic stability threshold owing to the low vehicle speed. 441 At this point, the steering limit was high, which can ensure 442 the flexibility of the intelligent vehicle and enhance its lat-443 eral control ability. As the vehicle speed increased, the slip 444 angle, lateral speed, and lateral acceleration increased more 445 obviously; however, according to the tracing trajectory in 446 Fig. 14, it may be observed that the vehicle was still within 447 the controllable range.         speed and the target vehicle speed; thus, the intelligent vehicle 500 gradually approaches and coincides with the reference path. 501  In the working conditions of low torque and excessively fast 522 rotation speed, where the changes in torque acceleration and 523 speed are abrupt, a large PID value controller is used. The 524 targeted setting of the PID controller can ensure the stability 525 of the longitudinal control, real-time vehicle tracking, and 526 stability. Simultaneously, CarSim and MATLAB/Simulink 527 were used for co-simulation to verify the effectiveness of the 528 algorithm.

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To address the path tracking problem in the automatic driving 531 of intelligent vehicles, in this study, we have proposed a 532 control method based on an improved MPC and hybrid PID. 533 The effectiveness of the algorithm was verified through sim-534 ulations and real vehicle tests. We have drawn the following 535 conclusions based on the results.

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(1) Model-based prediction, rolling optimization solution, 537 feedback control, and the addition of front wheel sideslip 538 angle constraints and relaxation factors can improve the sta-539 bility of intelligent vehicles in lateral driving; the lateral error 540 was less than 1%.

541
(2) The proposed hybrid PID controller can realize longi-542 tudinal speed control of an intelligent vehicle, improve the 543 response speed of the system, and improve the real-time 544 performance of path tracking. When the vehicle speed is 545 greater than 18 km/h and less than 54 km/h, the yaw angle is 546 deliberately controlled within −4 • to 2 • , which can improve 547 the driving safety of intelligent vehicles.

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(3) The proposed controller based on the improved MPC 549 and hybrid PID can ensure that an intelligent vehicle tracks 550 the target path quickly and stably under medium and low 551 speeds and various complex working conditions, and exhib-552 ited higher path tracking accuracy than prior methods.