**4. Results**

In this section, we show the effect of SMVSC and the fuzzy logical controller, respectively. The simulating parameters of SMVSC are shown as follows: (*xr*, *yr*, *<sup>θ</sup>r*) = (0, 0, <sup>0</sup>), *ωr* = 0, *vr* = 0, (*xc*, *yc*, *<sup>θ</sup>c*) = (2, 2, <sup>0</sup>), *ωc* = 0, and *vc* = 0. The results are shown in Figures 11–15.

**Figure 12.** Ordinate *Y* time response diagram.

**Figure 13.** Course *θ* time response diagram.

**Figure 14.** Speed *V* time response diagram.

**Figure 15.** *ϕ* time response diagram.

As we can see from Figures 11–13, the ideal position and pose (*xr*, *yr*, *<sup>θ</sup>r*) and the initial position and pose (*xc*, *yc*, *<sup>θ</sup>c*) are changed from the initial error of (−2, −2, 0) to (0, 0, <sup>0</sup>), by the control laws *vc* and *ϕ<sup>c</sup>*, which are shown in Figures 14 and 15. In other words, according to control laws *vc* and *ϕ<sup>c</sup>*, the vehicle is driven from the initial position (*xc*, *yc*, *<sup>θ</sup>c*) and pose to the ideal position and pose (*xr*, *yr*, *<sup>θ</sup>r*). It is noteworthy that the convergence time of (*xe*, *ye*, *<sup>θ</sup>e*) in ten seconds. In Figures 14 and 15, the curvilinear trend is smooth, so that it can be implemented in reality [26].

The result of the fuzzy logical controller is given in Figure 16, which is the trajectory of the vehicle's center point of the rear axle.

**Figure 16.** The trajectory of the fuzzy control parking to a different initial position.

As shown in Figure 16, the simulating parameters are shown as follows: *w* = 2.5, *h* = 5.3, speed *v* = −1 m/s, the initial position and pose of magenta trajectory is (*<sup>x</sup>*, *y*, *θ*) = (7, 12, <sup>0</sup>), the initial position and pose of green trajectory is (*<sup>x</sup>*, *y*, *θ*) = (7, 6.5, <sup>0</sup>), and both of the trajectories are outside the area of the effective parking position, so they park unsuccessfully. The initial position and pose of the red trajectory is (*<sup>x</sup>*, *y*, *θ*) = (7, 9, <sup>0</sup>), which is inside the red box, which allows for parking successfully. According to the experiments, if the initial position is within the area of the red box, shown in Figure 16, the fuzzy logical controller described in Section 3 can park the vehicle successfully.

As discussed in the Section 1 (Introduction), this paper aims to park the vehicle from an initial position, which is out of the black box, by combining SMVSC with the fuzzy logical control. Figure 17 shows the effect of this method.

**Figure 17.** Trajectory of two different control methods.

The magenta trajectory is controlled by the hybrid method. The red trajectory is controlled by the fuzzy logical controller, which is the same as the hybrid method. The simulating parameters of Figure 17 are shown as follows: the ideal position and pose (*xr*, *yr*, *<sup>θ</sup>r*) are the same as that of Figure 16 (*<sup>x</sup>*, *y*, *θ*) = (7, 9, <sup>0</sup>). The initial position and pose are (*xc*, *yc*, *<sup>θ</sup>c*) = (20, 12, <sup>0</sup>), which is out of black box. The result of the simulating shows that the hybrid method can park the vehicle into the target slot successfully, but the fuzzy logical control cannot park successfully (the left side of the vehicle is not in the slot). The result shows that the hybrid method can expand the range of the effective parking position.
