# Pitch Channel Control of a REMUS AUV with Input Saturation and Coupling Disturbances
## 2 Model
### Underwater Vehicle Model  
For underwater vehicle, two coordinate systems are usually used for convenience [1]. As shown in Figure 1, \(O-XYZ\) is inertial frames in earth (\(E\)-frame), and \(o-xyz\) is the coordinate system for a vehicle, which is defined as \(B\)-frame. In addition, REMUS 100 vehicle's states are drawn in Figure 1 and reflected in Table 1. The vehicle has only one propeller fixed in the tail, and four control fins are mounted in the form of a across shape near the tail of the vehicle. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Coordinate** & **Position \& Angles \(E\)-Frame** & **Linear \& Angular Velocities \(B\)-Frame** & **Forces \& Moments \(B\)-Frame** \\ \hline \hline Surge & \(X\) & \(u\) & \(F_{X}\) \\ Sway & \(Y\) & \(v\) & \(F_{Y}\) \\ Heave & \(Z\) & \(w\) & \(F_{Z}\) \\ Roll & \(\phi\) & \(p\) & \(M_{K}\) \\ Pitch & \(\theta\) & \(q\) & \(M_{M}\) \\ Yaw & \(\psi\) & \(r\) & \(M_{N}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations for Underwater vehicle. Figure 1: REMUS body frame and inertial reference frame. The kinematics transformation between vehicle's velocity in B-frame and position in E-frame is listed as Equations (1) and (28,1,30). \[\left[ \begin{array}{c}\dot{X}\\ \dot{Y}\\ \dot{Z}\end{array} \right]=\left[ \begin{array}{cccc}c\left(\psi\right)c\left( \theta\right)&-s\left(\psi\right)c\left(\phi\right)+c\left(\psi\right)s\left( \phi\right)s\left(\theta\right)&s\left(\psi\right)s\left(\phi\right)+c\left( \psi\right)c\left(\phi\right)s\left(\theta\right)\\ s\left(\psi\right)c\left(\theta\right)&c\left(\psi\right)c\left(\phi\right)+s \left(\psi\right)s\left(\phi\right)s\left(\theta\right)&-c\left(\psi\right)s \left(\phi\right)+s\left(\psi\right)c\left(\phi\right)s\left(\theta\right)\\ -s\left(\theta\right)&c\left(\theta\right)s\left(\phi\right)&c\left(\theta \right)c\left(\phi\right)\end{array} \right]\left[ \begin{array}{c}u\\ v\\ w\end{array} \right] \tag{1}\]  
\[\left[ \begin{array}{c}\phi\\ \theta\\ \psi\end{array} \right]=\left[ \begin{array}{cccc}1&s\left(\phi\right)t \left(\theta\right)&c\left(\phi\right)t\left(\theta\right)\\ 0&c\left(\phi\right)&-s\left(\phi\right)\\ 0&s\left(\phi\right)/c\left(\theta\right)&c\left(\phi\right)/c\left(\theta \right)\end{array} \right]\left[ \begin{array}{c}p\\ q\\ r\end{array} \right] \tag{2}\]  
where \(c(*)=cos(*),s(*)=sin(*),t(*)=tan(*)\). According to the states shown in Table 1 as well as assumptions that the origin of the body frame is at the REMUS vehicle center of buoyancy, the vehicle is treated as a rigid body. The general mathematical descriptions for the motion of the vehicle are listed in Equation (3) [28,30]. \[ \begin{array}{l}m\left[\dot{u}-vr+wq-x_{G}(q^{2}+r^{2})+y_{G}(pq-\dot{r})+z _{G}(pr+\dot{q})\right]=\sum F_{X}\\ m\left[\dot{v}-wp+ur-y_{G}(r^{2}+p^{2})+z_{G}(qr-\dot{p})+x_{G}(qp+\dot{r}) \right]=\sum F_{Y}\\ m\left[\dot{w}-uq+vp-z_{G}(q^{2}+p^{2})+x_{G}(rp-\dot{q})+y_{G}(rq+\dot{p}) \right]=\sum F_{Z}\\ l_{x}\dot{p}+\left(l_{z}-I_{y}\right)qr+m\left[y_{G}(\dot{w}-uq+vp)-z_{G}( \dot{v}-wp+ur)\right]=\sum M_{K}\\ l_{y}\dot{q}+\left(l_{x}-I_{z}\right)rp+m\left[z_{G}(\dot{u}-vr+wq)-x_{G}( \dot{w}-uq+vp)\right]=\sum M_{M}\\ l_{z}\dot{r}+\left(l_{y}-I_{x}\right)pq+m\left[x_{G}(\dot{v}-wp+ur)-y_{G}( \dot{u}-vr+qw)\right]=\sum M_{N}\end{array}  \tag{3}\]  
In Equation (3), \(r_{G}=\left[x_{G},y_{G},z_{G}\right]\) and \(r_{B}=\left[x_{B},y_{B},z_{B}\right]\) stand for the gravity and buoyancy center coordinates in vehicle's \(B\)-frame, respectively. \(m\) represents the mass of REMUS vehicle. Hydrostatic force brought by gravity and buoyancy of vehicle, control force of actuator, added mass, hydrodynamic force as well as force and torque generated by the thruster are included in the six DOF nonlinear model of REMUS vehicle. Considering the external forces and torques, we can obtain the exact mathematical expression listed in Equation (4) [30]. \[