# Stabilization of Underactuated Surface Vessels: A Continuous Fractional Power Control Method
## 3 Main Results
### The Control Algorithm for USVs  
Based on the above discussions, we are now in a position to come back to the USV system in (1) and give the following controller,  
\[\tau_{u}=\eta_{1},\ \ \ \ \ \tau_{r}=\eta_{2}, \tag{40}\]  
where  
\[\eta_{1} =m_{11}\bigg{\{}d_{22}m_{11}^{-1}\Big{[}m_{11}d_{22}^{-1}\Big{(} \theta_{1}+\theta_{5}\Big{)}+m_{22}d_{22}^{-1}\theta_{4}\theta_{6}-\Big{(}d_{2 2}m_{33}-m_{22}d_{23}\Big{)}d_{22}^{-1}m_{22}^{-1}\theta_{3}\theta_{6}-\theta_ {2}\theta_{6}\] \[\quad-\Big{(}-\kappa_{1}sign(\theta_{2})|\theta_{2}|\phi_{1} \theta_{6}-\kappa_{2}\theta_{5}\Big{)}\Big{]}\bigg{\}}-m_{22}vr-0.5\Big{(}m_{23}+ m_{32}\Big{)}r^{2}+d_{11}u,\] \[\eta_{2} =m_{22}^{-1}\bigg{\{}\Big{(}m_{11}m_{33}-m_{22}m_{32}\Big{)} \Big{[}-\kappa_{3}\theta_{3}-\kappa_{4}\theta_{6}+\lambda|\theta_{2}|^{\phi_{ 2}}\sin t\Big{]}-\Big{(}m_{11}m_{22}-m_{22}^{2}\Big{)}uv\]\[-\left[m_{11}m_{32}-0.5m_{22}\Big{(}m_{23}+m_{32}\Big{)}\right]ur-\Big{(}m_{32}d_{23 }-m_{22}d_{32}\Big{)}r\bigg{\}},\]  
\(\kappa_{1}>0\), \(\kappa_{2}>0\), \(\kappa_{3}>0\), \(\kappa_{4}>0\) and \(\lambda\
eq 0\) are constants to be chosen, and  
\[ \begin{cases}\phi_{1}=0.5(1+\xi_{1})+0.5(1-\xi_{1})\cdot sign(\mid\phi_{2}\mid- 1),\\ \phi_{2}=0.5(1+\xi_{2})+0.5(1-\xi_{2})\cdot sign(\mid\phi_{2}\mid-1),\end{cases}  \tag{41}\]  
with \(\xi_{1}>0\) and \(\xi_{2}>0\) are such that \(\xi_{1}+\xi_{2}\leq 1\). We now have the following theorem, whose proof can be made along similar lines of Lemmas 1 and 2 and Theorem 1. For off-diagonal underactuated surface vessels (1) with the continuous controller in (40), the states \(x,y,\psi,u,v,r\) can be globally asymptotically converged to zero. To sum up with the main results, we give the following algorithm for implementation of the presented method and results in USVs. ```
Step 1. Choose parameters \(\kappa_{1}\), \(\kappa_{2}\), \(\kappa_{3}\), \(\kappa_{4}\), \(\lambda\), \(\xi_{1}\), \(\xi_{2}\) and initial values of \(x\), \(y\), \(\psi\), \(u\), \(v\), \(r\). Step 2. Let \(\epsilon=m_{23}m_{22}^{-1}\), \(\gamma_{1}=d_{22}m_{21}^{-1}\), \(\gamma_{2}=m_{11}m_{22}^{-1}\) and \(\gamma_{3}=d_{22}m_{23}m_{22}^{-2}-d_{23}m_{22}^{-1}\). Step 3. Make state transformation \(\theta_{1}\), \(\theta_{2}\), \(\theta_{3}\), \(\theta_{4}\), \(\theta_{5}\)\(\theta_{6}\) as \(\theta_{1}=\left[x+\epsilon(\cos\psi-1)\right]\cos\psi+\left(y+\epsilon\sin \psi\right)\sin\psi\), \(\theta_{2}=-\left[x+\epsilon\left(\cos\psi-1\right)\right]\sin\psi+\left(y+ \epsilon\sin\psi\right)\cos\psi\) \(-\gamma_{3}\gamma_{1}^{-1}\psi+\gamma_{1}^{-1}\big{(}v+\epsilon r\big{)}\), \(\theta_{3}=\psi\), \(\theta_{4}=v+er\), \(\theta_{5}=-\gamma_{2}\gamma_{1}^{-1}u-\left[x+\epsilon\left(\cos\psi-1 \right)\right]\cos\psi-\left(y+\epsilon\sin\psi\right)\sin\psi\), \(\theta_{6}=r\). Step 4. Calculate the parameters \(\phi_{1}\) and \(\phi_{2}\) \(\phi_{1}=0.5(1+\xi_{1})+0.5(1-\xi_{1})\cdot sign(\mid\phi_{2}\mid-1)\), \(\phi_{2}=0.5(1+\xi_{2})+0.5(1-\xi_{2})\cdot sign(\mid\phi_{2}\mid-1)\). Step 5. Compute virtual control inputs \(\omega_{1}\) and \(\omega_{2}\) \(\omega_{1}=-\kappa_{1}sign(\theta_{2})\lvert\theta_{2}\rvert^{\varphi_{1}} \theta_{6}-\kappa_{2}\phi_{5}\) \(\omega_{2}=-\kappa_{3}\theta_{3}-\kappa_{4}\theta_{6}+\lambda\lvert\theta_{2 }\rvert^{\varphi_{2}}\sin t\) Step 6. Take measurements \(m_{11}\), \(m_{22}\), \(m_{23}\), \(m_{32}\), \(m_{33}\), \(d_{11}\), \(d_{22}\), \(d_{23}\), \(d_{32}\), \(d_{33}\) of the USV. Step 7. Compute virtual control inputs \(\tau_{1}\) and \(\tau_{2}\) \(\tau_{1}=d_{22}m_{11}^{-1}\Big{[}m_{11}d_{22}^{-1}\big{(}\theta_{1}+\theta_{5} \big{)}+m_{22}d_{22}^{-1}\theta_{4}\theta_{6}-\theta_{2}\theta_{6}-\alpha_{1}\) \(-\left(d_{22}m_{33}-m_{22}d_{23}\right)d_{22}^{-1}m_{22}^{-1}\theta_{3}\theta_ {6}\Big{]}\), \(\tau_{2}=\omega_{2}\). Step 8.