## 4 Results and Discussions
### Simulation Results  
The reference path is generated for validation and evaluation according to the real GPS data. The target tracking speed is set to 60 km/h, and the constraints of yaw rate and the comfortable acceleration are set to 10 deg/s and 0.2 g, respectively. In the path tracking control with the target tracking speed set to 60 km/h, the reference trajectory results of the PID controller and the MPC controller are shown in Figure 6a,b, respectively. The lateral and heading errors are shown in Figure 7a,b, respectively. The simulation results show that, at a tracking speed of 60 km/h, both PID control and MPC control meet the requirements of lateral tracking accuracy. The two controllers' lateral and heading errors are within 15 cm and 6 deg, respectively. Thus, compared with the lateral tracking of PID Control, the MPC exhibits better performance. The simulation result of tracking the target speed is shown in Figure 6b. Since a single PID controller can only achieve tracking control of a single target, the PID controller only carries closed-loop control based on the lateral position error feedback without interfering with the vehicle's Longitudinal control. The longitudinal speed of the vehicle always maintains 60 km/h. The tracking control based on MPC controller involves multiple constraint processing and multiple target tracking. In constraint processing, the target speed as a soft constraint can deviate from a particular value at the expense of slack penalty terms to ensure better vehicle stability during path tracking. As shown in Figure 6b, at 28 s, when the vehicle is travelling close to a large curvature curve, if the vehicle is turned at a speed of 60 km/h, the stability and driving safety of the vehicle cannot be guaranteed. In the design of the MPC controller, the yaw rate and lateral acceleration are unbreakable hard constraints on vehicle dynamics. When optimising the objective function in the feasible region, the optimal objective function will be solved at the expense of speed tracking accuracy. When the curvature decreases (after the turn is completed), the longitudinal target speed of the vehicle will be tracked again under the premise of ensuring the accuracy of lateral tracking. Therefore, the tracking control based on MPC is a multi-target coordinated control of the vehicle. Figure 6: Path tracking results of two controllers, PID\(-\)based and MPC-based: (**a**) The path tracking of two controllers; (**b**)The tracking velocity based on MPC. Figure 7 The lateral errors and heading angle errors comparison of two controllers, PID and MPC: (**a**) The lateral errors of two controllers; (**b**) The heading angle errors of two controllers. The comparison of the longitudinal and lateral acceleration during the path tracking of the autonomous vehicle is shown in Figure 8. As shown in Figure 8a, the longitudinal acceleration of the vehicle is always 0 in PID control, because only a single lateral tracking target is controlled in the path tracking. The lateral acceleration of PID control is shown by the black line in Figure 8b. The red line shows the longitudinal acceleration control based on the MPC in Figure 8a. To ensure that the vehicle can track stably, comfortably and safely, the path tracking of an autonomous vehicle under the MPC needs to combine the curvature of the reference trajectory and the vehicle stability constraints to control the longitudinal speed of the vehicle accordingly. Compared with the PID-based control, the average lateral acceleration of MPC drops from \(-0.0606\) to \(-0.0376\), the maximum acceleration fell from \(0.4549\) to \(0.22\) and the standard deviation decrease from \(0.1194\) to \(0.0682\). Figure 8: _Cont._  
Figure 7: The lateral errors and heading angle errors comparison of two controllers, PID and MPC: (**a**) The lateral errors of two controllers; (**b**) The heading angle errors of two controllers.