## 3 Design of Concise Robust Path-Following Control with RRR
### Control Design
#### 3.2.2 Concise RRR Control Based on Backstepping  
**Theorem 2**.: _Considering the ship RRR system in Equation (6), the proposed algorithm in Equation (22) based on the Backstepping method is capable of stabilizing the roll motion and guaranteeing the uniformly asymptotic stability of the closed-loop RRR system._  
\[u_{r}=-\frac{1}{K_{\phi}\omega_{\phi}^{2}}\left[-\left(2\xi\omega_{\phi}p+ \omega_{\phi}^{2}\int pdt\right)+(1+k_{r1}k_{r2})e_{r}+(k_{r1}+k_{r2})\dot{e}_{ r}\right] \tag{22}\]  
_where the roll angle deviation \(e_{r}=\phi-\phi_{d}\), \(k_{r1}\), \(k_{r2}\) are positive design parameters._  
**Proof.** Set \(x_{r1}=\phi\), \(x_{r2}=p\), and then the roll motion model in Equation (6) is rewritten as follows  
\[\left\{\begin{array}{l}\dot{x}_{r1}=x_{r2}\\ \dot{x}_{r2}=g(x_{r2})+b_{r}u_{r}\\ y_{r}=x_{r1}\end{array}\right. \tag{23}\]  
where \(y_{r}\in R\), \(b_{r}=K_{\phi}\omega_{\phi}^{2}\), \(g(x_{r2})=-\left(2\xi\omega_{\phi}x_{r2}+\omega_{\phi}^{2}\int x_{r2}dt\right)\), which is a nonlinear function of \(x_{r2}\).  
Define \(z_{r1}=x_{r1}-\phi_{d}\) and \(z_{r2}=x_{r2}-\beta(z_{r1})\), and then one can get  
\[\dot{z}_{r1}=\dot{x}_{r1}-\dot{\phi}_{d}=x_{r2}-\dot{\phi}_{d}=z_{r2}+\beta(z _{r1})-\dot{\phi}_{d} \tag{24}\]  
\(\beta(z_{r1})\) is chosen as an intermediate control function defined as  
\[\beta(z_{r1})=\dot{\phi}_{d}-k_{r1}z_{r1} \tag{25}\]  
and  
\[\dot{\beta}(z_{r1})=\ddot{\phi}_{d}-k_{r1}\dot{z}_{r1} \tag{26}\]  
Substituting Equation (25) into Equation (24), one can get  
\[\dot{z}_{r1}=-k_{r1}z_{r1}+z_{r2} \tag{27}\]  
Considering the following Lyapunov function candidate  
\[V_{r}=\frac{1}{2}z_{r1}^{2}+\frac{1}{2}z_{r2}^{2} \tag{28}\]  
Then, one can get the time derivative of \(V_{r}\)  
\[\dot{V}_{r}=-k_{r1}z_{r1}^{2}+z_{r2}\Big{[}g(x_{r2})-\dot{\beta}(z_{r1})+z_{r1 }+b_{r}u_{r}\Big{]} \tag{29}\]To guarantee \(\dot{V}_{r}\leq 0\), actual control function is chosen as follows  
\[u_{r}=\frac{1}{b_{r}}\Big{(}\dot{\beta}(z_{r1})-g(x_{r2})-z_{r1}-k_{r2}z_{r2} \Big{)} \tag{30}\]  
Substituting Equation (30) into Equation (29), one can get  
\[\dot{V}_{r}=-k_{r1}z_{r1}^{2}-k_{r2}z_{r2}^{2}<0,\ \forall z_{r1}\
eq 0,z_{r2}\
eq 0 \tag{31}\]  
In accordance with [PERSON] stability theory, the RRR control system can be stabilized by the control law in Equation (30), and all the variables in RRR control loop are of uniformly asymptotic stability with an equilibrium point \([x_{r1},x_{r2}]=[\phi_{d},\dot{\phi}_{d}]\).  
Substituting Equation (26) into Equation (30), the control law is transformed into Equation (32).  
\[\begin{array}{l}u_{r}=-\frac{1}{b_{r}}\Big{[}g(x_{r2})+(1+k_{r1}k_{r2})(x_{r 1}-\phi_{d})+(k_{r1}+k_{r2})\Big{(}\dot{x}_{r1}-\dot{\phi}_{d}\Big{)}\Big{]}\\ =-\frac{1}{K_{\phi}\omega_{\phi}^{2}}\left[-\Big{(}2\xi\omega_{\phi}p+\omega_ {\phi}^{2}\int pd\Big{)}+(1+k_{r1}k_{r2})e_{r}+(k_{r1}+k_{r2})\dot{e}_{r}\right] \\ =-\frac{1}{K_{\phi}\omega_{\phi}^{2}}\left[g(x_{r2})+v_{r}\right]\end{array} \tag{32}\]  
This ends the proof of Theorem 2. \(\square\)  
**Remark 3**.: _The essences of the separate control laws in Equations (21) and (32) based on Backstepping are to compensate the system's linearity or nonlinearity and to stabilize the control loops by PD controllers \(v_{c}\) and \(v_{r}\). Furthermore, the aforementioned consideration leads to a control law with concise form which is applicable for linear and nonlinear systems. However, the design parameters of \(k_{c1},k_{c2},k_{r1},k_{r2}\) can only be obtained by trial and error, which are of little engineering significance and difficult to be determined for robust and optimal performances._