## 5 Nonlinear Observer  
### Nonlinear Observer Design  
The ROV is a highly nonlinear and time-varying system with coupling between the degrees of freedom. For cases of weak maneuvering, the ROV's motion can be divided into horizontal and vertical motion, which can obtain satisfactory results. The control system of the ROV is often the integration of the multiple function modules, including the observer, controller units, and dead reckoning units. Because the number of sensors that the ROV carries onboard is limited, only some of the state information can be measured. The nonlinear observer uses the measurable variables to estimate the velocities and the cable disturbing force information that cannot be measured online and in real-time. The structure diagram of the control system is shown below (Figure 7). The observer structure (Figure 8) includes two parts: (1) The ROV dynamic model represents the dynamic and kinematic characteristics of the ROV in the water, including the ideal ROV model and the external disturbing cable force model, which is generated by using the lumped parameter method; (2) The observer model includes the ideal ROV model and a simplified model of the cable disturbing force. The observer inputs are the thruster outputs \(F_{Tx},F_{Ty},F_{Tz},M_{Tz}\) in the body-fixed coordinates. The observer also needs measurements \(u_{em},v_{em},w_{em},r_{em}\) provided by the dead reckoning units. \(\dot{a},\dot{v},\dot{w},\dot{r}\) represent the observed values of \(u_{e},v_{e},w_{e},r_{e}\). \(e_{u},e_{v},e_{w},e_{r}\) represent the observation  
Figure 6: Comparison of the simulation results with the experimental results. Figure 7: Control system structure of the ROV. error. The observer outputs are the velocity estimation \(\hat{u}_{c},\hat{v}_{c},\hat{w}_{c}\) and the cable disturbing force \(F_{cablex},F_{cabley},F_{cablez}\). Different types of sensors are installed on the ROV, such as the gyroscope, three-axis accelerometer, compass, Doppler velocity log, and the baseline system. Considering the equipment cost and the installation space, the number of sensors installed on ROV is limited. Some information cannot be obtained with sensors, such as the flow rate and the cable disturbing force. The nonlinear observer uses the measurable information to estimate the unknown state variables without any increase in the peripheral sensor equipment, and improves the overall performance of the system effectively. An ROV always has a bilateral symmetrical structure, and its rolling and pitching motion velocities are small. Therefore, the influence of the rolling and pitching motion terms is often ignored. For the convenience of discussion, the ROV model can be simplified into a 4-DOF expression as follows:  
\[\left\{ \begin{array}{l}\dot{u}_{r}=\frac{1}{m-X_{\psi}}\Big{[}(m-Y_{\psi})v_{ r}r+X_{u}u_{r}+X_{u|u|}u_{r}|u_{r}|+(F_{Tx}+F_{bcathex})\Big{]}\\ \dot{v}_{r}=\frac{1}{m-Y_{\psi}}\Big{[}-(m-X_{ii})u_{r}r+Y_{\psi}v_{r}+Y_{\psi| \psi|}v_{r}|v_{r}|+\Big{(}F_{Ty}+F_{bcathkey}\Big{)}\Big{]}\\ \dot{w}_{r}=\frac{1}{m-Z_{\psi}}\Big{[}Z_{\psi}w_{r}+Z_{w|\psi|}|w_{r}|w_{r}+( W-B+F_{Tz}+F_{bcathez})\Big{]}\\ \dot{r}=\frac{1}{l_{zz}-N_{\psi}}\Big{[}(Y_{\psi}-X_{ii})u_{r}v_{r}+N_{r}r+N_{ r|r|}r|+(M_{Tz}+M_{bcathez})\Big{]}\end{array} \right. \tag{41}\]  
The motion function can be written as follows:  
\[\left\{ \begin{array}{l}\dot{X}=u_{r}\cos\psi-v_{r}\sin\psi+u_{c}\\ \dot{Y}=u_{r}\sin\psi+v_{r}\cos\psi+v_{c}\\ \dot{Z}=w_{r}+w_{c}\\ \dot{\psi}=r\end{array} \right. \tag{42}\]  
The current model can be expressed as follows:  
\[\left\{ \begin{array}{l}\dot{u}_{c}=-\mu_{cx}u_{c}+\omega_{cx}\\ \dot{v}_{c}=-\mu_{cx}v_{c}+\omega_{cy}\\ \dot{w}_{c}=-\mu_{cx}w_{c}+\omega_{cz}\end{array} \right. \tag{43}\]  
where \(\mu_{cx},\mu_{cy},\mu_{cz}>0,\omega_{cx},\omega_{cy},\omega_{cz}\) are the white noise with a mean value of zero. (41)-(43) express the 4-DOF model. Next, a reasonable state observer is designed to observe the unmeasurable parameters with sensors such as \(u_{c},v_{c},w_{c}\).