## 3 LOS Guidance Algorithms
### Adaptive Compensation of the Sideslip Angle  
**Assumption 5**.: _The heading controller can follow the reference heading angle perfectly, so \(\psi=\psi_{d}\). Let \(\dot{\beta}\) denote the adaptive estimate of \(\beta\), and \(\dot{\beta}=\beta-\dot{\beta}\) is the parameter estimation error._  
The ALOS can be represented as (18) and (19).  
\[\psi_{d}=\gamma_{p}-acr\tan(\frac{1}{\Delta}y_{e}+\dot{\beta}) \tag{18}\]  
\[\dot{\beta}=\kappa\frac{Uy_{e}}{\sqrt{\Delta^{2}+\left(y_{e}+\Delta\dot{ \beta}\right)^{2}}} \tag{19}\]  
For (15), the velocity \(U_{p}\) can be seen as used to stabilize \(x_{e}\).  
\[U_{p}=-U\frac{\Delta}{\sqrt{\Delta^{2}+\left(y_{e}+\Delta\dot{\beta}\right)^{2 }}}+\gamma x_{e} \tag{20}\]  
and:  
\[\dot{\theta}=\frac{U_{p}}{\sqrt{{x^{\prime}}_{p}\left(\theta\right)^{2}+{y^{ \prime}}_{p}\left(\theta\right)^{2}}} \tag{21}\]  
Figure 2: LOS guidance geometry for curved paths.  
where \(0<\Delta_{\min}<\Delta<\Delta_{\max}\) is the user-specified lookahead distance. Applying (17) with \(0<U_{\min}\leq U\leq U_{\max}\) renders the origin \((x_{e},y_{e},\tilde{\beta})=(0,0,0)\) uniformly globally asymptotically stable (UGAS).  
\[\sin(\arctan(-\frac{y_{e}+\Delta\hat{\beta}}{\Delta}))=-\frac{y_{e}+\Delta\hat {\beta}}{\sqrt{\Delta^{2}+\left(y_{e}+\Delta\hat{\beta}\right)^{2}}} \tag{22}\]  
\[\cos(\arctan(-\frac{y_{e}+\Delta\hat{\beta}}{\Delta}))=-\frac{\Delta}{\sqrt{ \Delta^{2}+\left(y_{e}+\Delta\hat{\beta}\right)^{2}}} \tag{23}\]  
Then:  
\[\dot{y}_{e}=-U\frac{y_{e}-\Delta\hat{\beta}}{\sqrt{\Delta^{2}+\left(y_{e}+ \Delta\hat{\beta}\right)^{2}}}-\dot{\gamma}_{p}x_{e} \tag{24}\]  
Similarly,  
\[\dot{x}_{e}=\dot{\gamma}_{p}y_{e}-U\frac{\Delta}{\sqrt{\Delta^{2}+\left(y_{e} +\Delta\hat{\beta}\right)^{2}}}-U_{p} \tag{25}\]  
Design the Lyapunov function \(V_{1}=\frac{1}{2}x_{e}^{2}+\frac{1}{2}y_{e}^{2}+\frac{1}{2}\kappa\hat{\beta}^ {2}\) with \(\kappa>0\). From (24) and (25), we can get that:  
\[\dot{V}_{1}= x_{e}\dot{x}_{e}+y_{e}\dot{y}_{e}+\kappa\hat{\beta}\hat{\beta}\] \[= -\gamma x_{e}^{2}-\frac{Uy_{e}^{2}}{\sqrt{\Delta^{2}+\left(y_{e} +\Delta\hat{\beta}\right)^{2}}}+\hat{\beta}(\frac{U\Delta y_{e}}{\sqrt{\Delta ^{2}+\left(y_{e}+\Delta\hat{\beta}\right)^{2}}}+\frac{1}{\kappa}\hat{\beta}) \tag{26}\]  
Since \(\hat{\beta}=-\hat{\beta}\), we can substitute (19) into (26). This gives:  
\[\dot{V}_{1}<-\gamma x_{e}^{2}-\frac{Uy_{e}^{2}}{\sqrt{\Delta^{2}+\left(y_{e} +\Delta\hat{\beta}\right)^{2}}}<0 \tag{27}\]  
The speed of the USV is chosen as:  
\[U=l\sqrt{\Delta^{2}+\left(y_{e}+\Delta\hat{\beta}\right)^{2}} \tag{28}\]  
where \(l>0\); we obtain:  
\[\dot{V}_{1}\leq-\gamma x_{e}^{2}-ly_{e}^{2}<0 \tag{29}\]  
**Theorem 1**.: _The origin \((x_{e},y_{e},\hat{\beta})=(0,0,0)\) is uniformly globally exponentially stable (UGES) if the adaptive law of \(\theta\) is (21), \(\psi_{d}\) is set as (18) and U is equal to (28)._  
Proof of Theorem 1.: The first Lyapunov function \(V_{1}\) is set as positive definite and radially unbounded, while its derivative is quadratically negative definite when asserting (18), (21) and (28). Therefore, by the stability theory of [PERSON], the origin \((x_{e},y_{e},\hat{\beta})=(0,0,0)\) is UGES. This is actually a theoretical conclusion. In actual engineering, due to the interference of various factors and the restriction of actual system, the result of this idea is difficult to achieve.