# Stabilization of Underactuated Surface Vessels: A Continuous Fractional Power Control Method
## 2 System Modeling and the Objective
### Model Transformation  
For reasons mentioned above, to design a stabilizing controller for an underactuated vessel with off-diagonal dynamics is rather difficult. A necessary step toward a feasible solution is to reduce the complexity of system model in a certain way. Here in this paper, we introduce the following coordinate transformation [24],  
\[\tilde{x}=x+\epsilon(\cos\psi-1),\quad\tilde{y}=y+\epsilon\sin\psi,\quad \tilde{v}=v+\epsilon r, \tag{3}\]  
where \(\epsilon=m_{23}m_{22}^{-1}\), and input transformation  
\[\begin{cases}\tau_{1}=&\frac{1}{m_{11}}\Big{[}\tau_{u}+m_{22}vr+ \frac{(m_{23}+m_{32})}{2}r^{2}-d_{11}u\Big{]},\\ \tau_{2}=&\frac{m_{22}}{m_{22}m_{33}-m_{32}m_{23}}\bigg{\{}\tau_{r}+ \frac{uv}{m_{22}}(m_{11}m_{22}-m_{22}^{2})\\ &\quad+\frac{ur}{m_{22}}\Big{[}m_{11}m_{32}-0.5m_{22}(m_{23}-m_{32})\Big{]}\\ &\quad+(m_{32}d_{22}-m_{22}d_{32})\frac{v}{m_{22}}\\ &\quad-(m_{22}d_{33}-m_{32}d_{23})\frac{r}{m_{22}}\bigg{\}}.\end{cases} \tag{4}\]  
Hereinafter, we will consider \((\tilde{x},\tilde{y},\psi)\) as the output variables of the USV instead of \((x,y,\psi)\). Using the above change of coordinates, the USV dynamics in (1) can be written as  
\[\begin{cases}\hat{x}=u\cos\psi-\bar{v}\sin\psi,\\ \hat{y}=u\sin\psi+\bar{v}\cos\psi,\\ \hat{\psi}=r,\quad\hat{v}=-\gamma_{1}\bar{v}-\gamma_{2}ur+\gamma_{3}r,\\ \hat{u}=\tau_{1},\quad\hat{r}=\tau_{2},\end{cases} \tag{5}\]  
Figure 1: Earth-fixed and body-fixed coordinate systems.  
where \(\gamma_{1}=\frac{d_{22}}{m_{22}}\), \(\gamma_{2}=\frac{m_{11}}{m_{22}}\) and \(\gamma_{3}=\frac{d_{22}m_{23}}{m_{22}^{2}}-\frac{d_{23}}{m_{22}}\).  
To simplify the forthcoming analysis and design, we further introduce the following global diffeomorphism and input transformation [10],  
\[\left\{\begin{aligned} \theta_{1}&=\tilde{x}\cos \psi+\tilde{y}\sin\psi,\\ \theta_{2}&=-\tilde{x}\sin\psi+\tilde{y}\cos\psi+ \frac{1}{\gamma_{1}}\tilde{v}-\frac{\gamma_{3}}{\gamma_{1}}\psi,\\ \theta_{3}&=\psi,\qquad\qquad\theta_{4}=\tilde{v},\\ \theta_{5}&=-\frac{\gamma_{2}}{\gamma_{1}}u-\theta_{ 1}\qquad\theta_{6}=r,\end{aligned}\right. \tag{6}\]  
and  
\[\left\{\begin{aligned} \varpi_{1}=&\frac{\gamma_{1}}{ \gamma_{2}}(\theta_{1}+\theta_{5})-\frac{\gamma_{3}}{\gamma_{1}}\theta_{3} \theta_{6}-\frac{\gamma_{2}}{\gamma_{1}}\tau_{1}+\frac{1}{\gamma_{1}}\theta_ {4}\theta_{6}\\ &-\theta_{2}\theta_{6},\\ \varpi_{2}=&\tau_{2}.\end{aligned}\right. \tag{7}\]  
Then, combining the derivative of Equation (6) with Equations (5) and (7) yields that  
\[\left\{\begin{aligned} \dot{\theta}_{1}&=-\frac{ \gamma_{1}}{\gamma_{2}}\theta_{1}-\frac{\gamma_{1}}{\gamma_{2}}\theta_{5}+ \theta_{2}\theta_{6}-\frac{1}{\gamma_{1}}\theta_{4}\theta_{6}+\frac{\gamma_{3 }}{\gamma_{1}}\theta_{3}\theta_{6},\\ \dot{\theta}_{4}&=-\gamma_{1}\theta_{4}+\gamma_{1} \theta_{6}(\theta_{1}+\theta_{5})+\gamma_{3}\theta_{6},\end{aligned}\right. \tag{8}\]  
\[\left\{\begin{aligned} \dot{\theta}_{2}&=\theta_{5} \theta_{6},\quad\dot{\theta}_{5}=\varpi_{1},\\ \dot{\theta}_{3}&=\theta_{6},\qquad\dot{\theta}_{6}= \varpi_{2},\end{aligned}\right. \tag{9}\]  
where \(\theta_{1}\), \(\theta_{2}\), \(\theta_{3}\), \(\theta_{4}\), \(\theta_{5}\) and \(\theta_{6}\) are transformed states, \(\varpi_{1}\) and \(\varpi_{2}\) are the inputs of transformed model.  
Please note that with the above model transformations, the USV system is decomposed into a cascade system comprising (8) and (9). For the system (8) and (9), we introduced following results from [8].  
**Lemma 1** (Ref. [8]).: _Stabilization of system (1) is equivalent to that of system (8) and (9)._  
**Lemma 2** (Ref. [8]).: _If subsystem (9) is globally asymptotically stable, then systems (8) and (9) are globally asymptotically stable._  
According to Lemmas 1 and 2, for the system in (1) to be stabilized globally and asymptotically, we need only to design a controller that can globally asymptotically stabilize subsystem (9).