## 5 Path Following Control Design
### 3 DOFALOS Guidance Law
#### 5.1.1 3 DOF Adaptive Look-Ahead Distance
The adjustment strategy at different speeds is shown in Figure 9. Note that, if the path is a straight line or has no curvature, \(\Delta^{\prime}_{\text{min}\,/\,\text{max}}=\Delta_{\text{min}\,/\,\text{max}}\):  
\[\Delta=\Delta^{\prime}_{\text{min}}+\left(\Delta^{\prime}_{\text{max}}-\Delta ^{\prime}_{\text{min}}\right)e^{-\frac{y_{0}^{2}}{u}}. \tag{22}\]  
Figure 8: Convergence performance under of different speed. (**a**) high forward speed; (**b**) low forward speed. Figure 7: Comparison of different \(\Delta\). 3 DOFALOS is established as follows:  
\[ \begin{split}\Delta\left(y_{e},c,u\right)&=\underbrace{ \left[\Delta_{\text{min}}-\Delta_{\text{min}}\operatorname*{sat}\left(c/c_{ \text{max}}\right)\right]}_{\Delta_{\text{min}}^{\prime}}\\ &+\left\{\underbrace{\left[\Delta_{\text{max}}-\Delta_{\text{max }}\operatorname*{sat}\left(c/c_{\text{max}}\right)\right]}_{\Delta_{\text{max }}^{\prime}}-\underbrace{\left[\Delta_{\text{min}}-\Delta_{\text{min}} \operatorname*{sat}\left(c/c_{\text{max}}\right)\right]}_{\Delta_{\text{min}}^ {\prime}}\right\}e^{-\frac{y_{e}^{2}}{\pi}}\\ &=\left\{\left(\Delta_{\text{max}}-\Delta_{\text{min}}\right) \left[1-\frac{1}{2\operatorname*{sat}\left(c/c_{\text{max}}\right)}\right] \right\}e^{-\frac{y_{e}^{2}}{\pi}}+\left[\Delta_{\text{min}}-\frac{\Delta_{ \text{min}}}{2}\operatorname*{sat}\left(\frac{c}{c_{\text{max}}}\right) \right],\end{split}  \tag{23}\]  
where \(c_{max}\
eq 0\). Just two inputs, \(\Delta_{\text{min}}\) and \(\Delta_{\text{max}}\), are required to determine the range of look-ahead distance. Equation (23) is suitable for any curve tracking. If \(c_{max}=0\), i.e., \(c\equiv 0\), 3 DOFALOS can be simplified to 2 DOFALOS, as shown in Equation (24):  
\[\Delta\left(y_{e},u\right)=\Delta_{\text{min}}+\left(\Delta_{\text{max}}- \Delta_{\text{min}}\right)e^{-\frac{y_{e}^{2}}{u}}. \tag{24}\]  
In summary, 3 DOFALOS considers three degrees of freedom, \(y_{e}\), \(c\) and \(u\). Firstly, on the basis of ALOS only considering \(y_{e}\), this paper optimizes the range of \(\Delta\) by using \(c\). If the reference path is curved, the improved ALOS makes \(\Delta\) selected in a smaller range \(\left(\Delta_{\text{min}}^{\prime},\Delta_{\text{max}}^{\prime}\right)\). Then, additional factor \(u\) is considered to optimize \(\Delta\) to employ suitable adjustment strength. At high speed, the adjustment should be weakened. At low speed, the adjustment should be enhanced. Table 2 summarizes the comparison of ALOS and improved ALOS 3 DOFALOS in this paper. Figure 10 depicts the intensity of look-ahead distance under three degrees of freedom. The change of hue from cyanine to pink corresponds to the change of the intensity of look-ahead, i.e., from small to large. The evolution of look-ahead distance with the other two degrees of freedom under the interpolation of the constant curvature (\(c=0.5\) m\({}^{-1}\)), the constant cross-track error(\(y_{e}=0\),\(10\) m) and \begin{table}
\begin{tabular}{c c c c c} \hline
**Algorithm** & **Improved Parameters** & **Input** & **Equation** & **Source** \\ \hline ALOS & \(\Delta_{\text{min}}\),\(\Delta_{\text{max}}\) & \(y_{e}\) & (20) & Fossen [35] \\ Improved ALOS & \(\Delta_{\text{min}}^{\prime}\Delta_{\text{max}}^{\prime}\) & \(y_{e},c\) & (21) & This paper \\ Improved ALOS(3 DOFALOS) & \(\Delta\) & \(y_{e}\),\(c\),\(u\) & (22)–(23) & This paper \\ \hline \end{tabular}
\end{table} 
Table 2: ALOS algorithm evolution. Figure 9: Relationship between look-ahead, cross-track error and speed. the constant forward speed (\(u\) = 0.5,1,1.5 m/s) are given, respectively. Significantly, when \(y_{e}>\) 5 m, the look-ahead distance is generally small. The curvature and the forward speed have little effect on it. The main function of the look-ahead is convergence. When \(y_{e}<\) 5 m, the look-ahead is greatly affected by curvature and speed. The main function of the look-ahead is guidance. When \(c<\) 0.3 m\({}^{-1}\), the look-ahead distance is generally large. When \(c>\) 0.3 m\({}^{-1}\), the look-ahead distance is generally small.