## 5 Simulink Results and Discussion
### Analysis of FLOS Simulation Results
#### 5.1.1 FLOS Navigation Algorithms in Case 1  
In Case 1, a straight-line desired trajectory is designed. Additionally, a lateral wind disturbance of \(v\) = 0.5 m/s is applied at 15 s. The results can be found in Figures 6-8.  
In Figure 6, point \(i\) (\(i\) = 1,2... 6) is the navigation point. The start point is (2 m, 1 m) and the heading angle is 0 rad. At 15 s, the lateral velocity \(v\) = 0.5 m/s is applied to the CDR at position (21.5 m, 21.5 m). In this case, the traditional LOS has a position tracking error that cannot be eliminated as the red line shows. The ELOS, ALOS, FLOS can adjust the yaw angle to eliminate the position error. The heading angle and sideslip angle are shown in Figure 7.  
Figure 6: Four LOS methods are used to track the desired path. (**a**) CDR tracks eight navigation points. (**b**) Tracking the desired path with the lateral velocity \(v\) = 0.5 m/s. At 15 s, the lateral velocity \(v\) = 0.5 m/s is applied to the CDR, which caused side-slip angle.  
There is no lateral velocity until at 15 s, so the desired heading angle should be 0.785 rad. The FLOS, ELOS, and LOS are all capable of planning the desired heading angle. At 15 s, the lateral velocity \(v\) = 0.5 m/s and the longitudinal speed \(u\) = 2 m/s, the desired heading angle should be 1.031 rad, a side-slip angle of 0.245 rad appears. ALOS convergence speed is the slowest. The reason is the adaptive rate of ALOS only considers the position error \(y_{e}\). ELOS considers the position error \(y_{e}\) and \(x_{e}\), the heading angle error is also introduced into ESO, so it has faster observation speed than ALOS. However, due to the large position error between the current position of the robot with the target position at the initial time, there is a large error in the side-slip angle observed by ELOS at the initial time. FLOS takes the error between \(y_{e}\) with the observed value of \(y_{e}\) as the input state of the side-slip angle observer, so it avoids the large observation error at the initial time. The exponential term is introduced into FLOS algorithm to further improve the planning speed. The position error of \(y_{e}\) as shown in Figure 8.  
Without lateral velocity, all four methods can ensure that the position error \(y_{e}\) converges to 0. At 15 s, the lateral velocity appears. Since the traditional LOS algorithm does not consider the lateral velocity, there is a static state error of about 0.5 m that cannot be eliminated. The maximum \(y_{e}\) error of FLOS is about 0.12 m, smaller than that of ELOS (about 0.18 m) and ALOS (about 0.4 m).  
To present more clearly the advantages of the FLOS algorithm we have proposed in Case 1, the quantitative comparison is shown in Table 2.  
Figure 8: Position error \(y_{e}\). The traditional LOS algorithm does not consider the lateral velocity. Therefore, there is a static state error of about 0.5 m that cannot be eliminated.  
Figure 7: The desired heading angle is planned by the four LOS algorithms. (**a**) Desired heading angle. The desired heading is 0.785 rad when there is no lateral velocity. (**b**) Side-slip angle. The side-slip angle is 0.245 rad when there is lateral velocity \(v\) = 0.5 m/s.  
It should be notes that the continuous parameterized reference trajectory is discretized. Therefore, the time constraint of the desired trajectory is abandoned. The robot should reach the \(k+1\)th navigation point along a straight line from the \(k\)th point. The intersection of the robot's current position and the vertical line of the desired path is taken as the position of the virtual robot, so \(x_{e}\) is always 0. We only discuss the convergence of \(y_{e}\).