## 6 Simulation and Experiment
### Experiment
#### 6.2.1 Path Following for a Lawnmower
It is apparent that the convergence effect is better under the _APLOS_ algorithm. At the apex, _LOS_ with constant look-ahead distance has a serious overshoot, which is similar to the first experiment. Due to the increased speed, _APLOS_ with a time-varying look-ahead distance also has a slight overshoot compared with the first experiment. However, the tracking performance of _APLOS_ is still better than _LOS_. Furthermore, _APLOS_ based AUV sails along the desired path with small deviations considering drift angle. Figure 18b illustrates the comparison of cross-track error under two methods. _APLOS_ is denoted by the red line and _LOS_ is denoted by the blue line. The comparison further illustrates the cross-track error of _APLOS_ is mostly smaller than _LOS_. Therefore, the adaptive method and proportional guidance method significantly improve the path tracking performance and greatly reduce the cross-track error. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithms** & **MAE(m)** & **RMSE(m)** \\ \hline LOS & 2.6088 & 3.4903 \\ APLOS & 1.7838 & 2.7833 \\ Improvement & 31.62\% & 20.25\% \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Performance metric of lawnmower path following (\(u\) = 1.5 m/s). Figure 18c presents the time evolution of look-ahead distance. _LOS_ (the blue line) employs a constant look-ahead 15 m. _APLOS_ (the red line) employs varied look-ahead distance according to the cross-track error and the speed. Figure 18d gives the time history of the forward speed (the pink line), the sway speed (the black line) and the drift angle (the black line). It is obvious that the drift angle of AUV is time-varying. Combining Figure 18d with Figure 18a, _APLOS_ can resist the effects of drift angles. However, because of drift angle, the resultant motion of _LOS_ based AUV is not coincident with the desired course so that it deviates from the desired path. It is necessary to consider the drift angle. _LOS_ and _APLOS_ both have good heading control performance, as shown in Figure 18e. The current heading denoted by the blue line fits the desired heading denoted by the red line well. Therefore, the final tracking performance mainly depends on the guidance law. Table 8 summarizes the related indicators. The MAE and RMSE of _APLOS_ are 1.7838 m and 2.7833 m, reduced by 32% and 20%, respectively. Hence, compared with conventional _LOS_, _APLOS_ has improved a lot under ocean currents. Figure 18: Lawnmower path following (\(u=1.5\) m/s). (**a**) comparison of trajectory; (**b**) comparison of cross-track error; (**c**) comparison of look-ahead distance; (**d**) comparison of velocity and drift angle; (**e**) comparison of heading effect. Table 9 provides the fluctuations of MAE and RMSE of two methods after increasing speed. It is clear that the fluctuation of _APLOS_ is smaller than _LOS_. Hence, under the adaptive and proportional method, the path following performance of AUV is more stable for different speeds of AUV. The two experiments illustrate that _APLOS_ offers the great tracking performance. In the convergence stage, _APLOS_ automatically adjusts the look-ahead distance by time-varying cross-track error and speed. AUV converges to the path faster and more stably. In the guidance stage, _APLOS_ reflects the role of a proportional method considering drift angle, which fits the reference path better compared with _LOS_.