Fig 1.
Reference road path for DLC maneuver.
This path is a built-in path in CarSim, having 200 m of longitudinal distance, and the vehicle is supposed to change the lane twice within this distance. The orange line represents the trajectory of the vehicle. Performance is evaluated by checking whether the vehicle collides with the traffic cones. During driving, the longitudinal velocity of the vehicle is assumed to be constant.
Fig 2.
Assuming that the longitudinal velocity is constant, the two degrees of freedom are represented by the lateral displacement, yltr(t), and the yaw angle δyaw(t). The values of the parameters can be found in Table 1.
Table 1.
Parameters of vehicle model.
Fig 3.
Lateral displacement of the vehicle with different prediction horizons.
This figure shows the tracking ability of the proposed controller with prediction horizons of 20, 40, and 60. When the prediction horizon is 20, a relatively large error in the lateral displacement appears as the prediction ability of the controller is degraded. On the other hand, when the prediction ability increases overwhelmingly a prediction horizon of 60 in this case, an extremely early steering is observed. Therefore, the prediction horizon is set to 40 for the controller.
Fig 4.
The generated critical regions of the eMPC controller can be seen in this figure, where ; 113 critical regions were formed based on the constraint set, of which 16 regions are feasible for obtaining an optimal control input when only z1 and z2 vary and the other parameters are fixed. Each region owns its unique sequence of optimal vectors, and the controller explicitly chooses a region based on the parameter values.
Fig 5.
Frameworks of eMPC for LTI systems (A) and LTV systems (B). In terms of parameter variation, there is no alternative way in Fig 5A to adjust the variation because the critical regions cannot be changed with respect to variation. In contrast, in Fig 5B, compensating for the state vector with an error compensator, enables the controller to be robust against such parameter variation. The main advantage of this approach is that, by simply adding a compensator, where only matrix multiplication is taken in its process, no modification of the critical regions is necessary.
Fig 6.
Simulation result of eMPC controller for the LTI system at a longitudinal velocity of 60 km/h.
This figure shows the fulfillment of the constraint set (14) of the eMPC controller. The vertical range of each window is identical with the constraints of each variable and input.
Fig 7.
Lateral displacement of eMPC controllers for LTI and LTV systems with driver model at 100 km/h longitudinal velocity.
The longitudinal velocity of the vehicle varies during the DLC maneuver despite the fact that this value is intended to be constant. An improved performance of the eMPC controller for the LTV system can be found in the lateral displacement error. After the longitudinal velocity starts varying, the error decreases compared to that of the eMPC controller for the LTI system.
Fig 8.
CarSim simulation of proposed controller and driver model.
This figure shows the lateral displacement of the proposed controller and the driver model in CarSim. At a high longitudinal velocity (100 km/h here), the performance of the driver model is deteriorated and nearly causes a collision with a traffic cone. In contrast, the proposed controller maintains a reasonable distance away from the traffic cones.
Fig 9.
States of the vehicle model during DLC maneuver.
In the implementation of the eMPC controller for the LTV system, the states are compensated regarding the parameter variation to improve the robustness of the controller. The proposed controller not only improves the tracking ability of the controller, but also enhances ride comfort as the lateral velocity is more restricted compared to the eMPC controller for the LTI system.