Fig 1.
Desired path for path-following control is plotted in this figure. This path comprises of four parts: a straight part, two curves, and a clothoid part. The straight part is used to prove the fulfilment of the proposed controller in this paper by setting the starting point, which is deviated from the desired path. By using the two curves and the clothoid part, the ability of the controller to perform path-following control will be demonstrated.
Table 1.
Parameters of vehicle model for path-following control.
Fig 2.
(A) Corresponding lateral tire force, Fy, as a function of the slip angle of the tire, α with different vertical tire loads. (B) the initial slope of the function (red line) when the vertical tire load is 3187.16 N, considering the vehicle mass, i.e., 967 N/deg or 55405 N/rad, which are both values of tire cornering stiffness Cr and Cf, respectively.
Fig 3.
An example of critical regions and the associated cost function are illustrated in this figure. In this figure, , where x(t), u2(t), and ysp(t) are the state vector, second input shown in Eq (2), and set point of the output within the prediction horizon Ny, respectively. The proposed controller predicts the second input, which can be obtained from the desired path, along Ny.
Fig 4.
Explicit MPC controller structure.
Structure of an explicit MPC controller for path-following control constructed using a vehicle model from CarSim. Based on the values of the parameter vector θ, the block “critical regions” selects a critical region CRi; then, the block “MPC feedback law” calculates the control action by applying the first MPC feedback law to the selected critical region CRi.
Fig 5.
Determination of weighting factors for the state.
(A) and (B) The effects of q1 and q2, respectively, which are weighting factors of the state, as given in Eq (13), where e1(t) indicates the lateral position error of the vehicle with respect to the desired path. It is demonstrated that for achieving path following control with error variables, the weighting factor of the position error must be large, whereas the weighting factor of the position error derivative must be small. (C) The lateral acceleration of the sprung mass with different values of q1. As ride comfort is typically evaluated according to the sprung mass of the vehicle, this figure shows a very large q1 deteriorates ride comfort.
Fig 6.
Simulation results with different ranges of prediction horizon.
This figure shows the simulation results when the range of the prediction horizon Ny is varied while the input horizon Nu is fixed at 3. As Ny increases, the input dynamics, i.e., the steering wheel angle, changes in advance; this consequently reduces the lateral position error because a longer Ny improves the prediction ability of the controller. However, we found that an extremely long Ny leads to an increase in the steering wheel angular velocity, which deteriorates ride comfort.
Fig 7.
Comparison of optimization controllers.
This figure shows the dynamics of the states of the LQR controllers, MPC controller, and explicit MPC controller. It is proved that LQR1 cannot fulfil the constraints as set Eq (12) and that the MPC controller consumes more time than the explicit MPC controller in the first 100 simulation runs (0.51 s in the case of the explicit MPC controller and 35.71 s in the case of the MPC controller). Moreover, LQR2 is designed to limit the maximum values of the state dynamics in the constraints by adjusting the weighting matrices; nevertheless, a high steering wheel angular velocity, which reduces ride comfort, persist.
Fig 8.
Paths of controllers and driver model.
This figure shows the paths of the LQR controller, explicit MPC controller, and driver model. It can be observed that in the case of the LQR controller, the deviation from the center line of the desired path is larger than that in the case of the explicit MPC controller, whereas path-following control performed using the explicit MPC controller is similar to that performed using the driver model in CarSim. The details of the error variables are shown in Fig 9.
Fig 9.
Simulation results for optimization controllers and the driver model.
Simulation results for the path-following controllers obtained using the LQR and explicit MPC methods as well as those obtained using the driver model are shown in this figure. Both controllers use the same weighting matrices to solve the optimization problem. From the results of the error variables, in particular, from the result of the lateral position error, the superiority of the explicit MPC controller over the LQR controller is demonstrated.
Fig 10.
Lateral position errors at different vehicle speeds.
This figure shows the lateral position error that occurred when the vehicle speed was 18 m/s, 20 m/s, 22 m/s, and 24 m/s. The position error increases as the vehicle speed increases.