## 5 Path Following Control Design
### 3 DOFALOS Guidance Law
#### 5.1.1 3 DOF Adaptive Look-Ahead Distance  
The key of the LOS guidance law is to introduce the suitable look-ahead distance, which essentially determines the convergence performance of path following. The traditional method employed a constant look-ahead. However, the cross-track error continues to decrease when AUV is converging to the path. If look-ahead is not selected properly, constant look-ahead distance easily leads to some problems, such as oscillation or failure to converge. [PERSON] and [PERSON] presented an adaptive look-ahead method Equation (20) considering the cross-track error in convergence stage, which improved the convergence performance [35]. On this basis, we hope to consider two other factors to make the look-ahead distance more adaptive:  
\[\Delta=\Delta_{\text{min}}+\left(\Delta_{\text{max}}-\Delta_{\text{min}}\right) e^{-y_{e}^{2}}, \tag{20}\]  
where \(\Delta\in\left(\Delta_{\text{min}},\Delta_{\text{max}}\right)\). \(\Delta_{\text{min}}\) and \(\Delta_{\text{max}}\) denote the predetermined minimum and maximum values of the look-ahead distance, respectively, which are mainly decided by the sensitivity of AUV itself. **Assumption 1**. : _The path curvature._  
Suppose that the reference path is a curve path with different curvature. Compared with constant look-ahead distance, as shown in Figure 6a, smaller look-ahead distance is more suitable in large curvature, and larger look-ahead distance is more fit in small curvature, as shown in Figure 6b. The saturated function \(sat(\cdot)\) is introduced to reduce the \(\Delta_{\text{min}}\) and \(\Delta_{\text{max}}\), aiming at \(\Delta\) in smaller scope. The descending function is constructed to describe the relationship between look-ahead distance and curvature. The results are described in Equation (21). \[\Delta^{\prime}_{\text{min/max}}=\Delta_{\text{min/max}}-\Delta_{\text{min/max }}\,\text{sat}\left(\frac{c}{c_{\text{max}}}\right) \tag{21}\]  
Figure 5: Convergence stage and guidance stage of path following. Figure 6: Variable curvature curve tracking. (**a**) Constant \(\Delta\); (**b**) Adaptive \(\Delta\). where \(\Delta^{\prime}_{\text{min}}\) and \(\Delta^{\prime}_{\text{max}}\) denote the redetermined minimum and maximum values of the look-ahead distance with respect to the path curvature, respectively. Consequently, \(\Delta\in\left(\Delta^{\prime}_{\text{min}},\Delta^{\prime}_{\text{max}}\right)\). Curve tracking under ALOS and improved ALOS method are illustrated in Figure 7. Obviously, the path following performance based on the improved ALOS considering the real-time curvature of the path is better, especially where the curvature of path is large. Hence, the improved ALOS can be able to adjust the range of the look-ahead distance in real time based on the time-varying curvature of the path. **Assumption 2**. : _The forward speed._  
Suppose that AUV is converging to the reference path. The look-ahead distance presents an increasing trend. Compared with ALOS mentioned above, a change norm is developed. If AUV sails at a high forward speed, the look-ahead distance should be quickly increased corresponding to the weak adjustment, as shown in Figure 8a. If AUV sails at a low forward speed, the look-ahead distance should be slowly increased corresponding to the strong adjustment, as shown in Figure 8b. Therefore, function \(L\left(u\right)\) is introduced to provide different correction strength. \(u\) expresses the current forward speed of AUV. The inverse proportional function (\(L\left(u\right)=1/u\)) is constructed to describe the relationship between look-ahead distance and the forward speed. The results are described in Equation (22). Suppose the adjustment range of the look-ahead distance is 10-15 m at a certain moment.