## 7 Simulation Results
### Dynamic Control and Simulation of Marine Vehicles  
AUG is selected as a special example to discuss the vehicles dynamic and control, for AUG's maneuverability is low compared to other kinds of vehicles. The glider prototype was designed from 2003, and the engineering versions were completed in 2007. The glider of engineering versions can dive down to 800 meters depth, with durance of 500 kilometers. The dynamic model of AUG that will be utilized in the observing systems has been discussed in [26] as:  
\[ \begin{split}\dot{\mathrm{v}}&=\mathrm{M}^{-1} \left\{-\dot{\mathrm{M}}\mathrm{v}+\left[\begin{matrix}\mathrm{P}\times \Omega\\ \mathrm{\Pi}\times\Omega+\mathrm{P}\times\mathrm{V}\end{matrix}\right]+ \left[\begin{matrix}\mathrm{m}_{\mathrm{b}}\mathrm{g}\mathrm{\left(R_{EB}^{T} k\right)}\\ \left(\mathrm{m}_{\mathrm{m}rr}\mathrm{r}_{\mathrm{m}rr}+\mathrm{m}_{\mathrm{r}rb} \mathrm{r}_{\mathrm{r}b}+\mathrm{m}_{\mathrm{b}}\mathrm{r}_{\mathrm{b}} \right)\times\left(R_{EB}^{T}k\right)\end{matrix}\right]+\left[\begin{matrix} \mathrm{F}\\ \mathrm{T}\end{matrix}\right]\right\}\\ \dot{\mathrm{m}_{\mathrm{b}}}&=\mathrm{u}_{\mathrm{b}}\end{split}  \tag{3}\]  
Glider motions mainly include steady vertical motion and three dimensional spiraling gliding motions. With the assumption that the position of \(\mathrm{m}_{\mathrm{m}rr}\) is fixed at \(\mathrm{r}_{\mathrm{m}rr}\), and with a constant net buoyancy which implies that the control input \(\dot{\mathrm{m}_{\mathrm{b}}}=0\), glider's equilibrium equations in three dimension are:  
\[ \begin{split}\left\{\begin{matrix}\mathrm{P}\times\Omega+\mathrm{m }_{\mathrm{b}}\mathrm{g}\mathrm{\left(R_{EB}^{T}k\right)+F=0}\\ \left(\mathrm{\Pi}\times\Omega+\mathrm{P}\times\mathrm{V}+\left(\mathrm{m}_{ \mathrm{m}rr}\mathrm{r}_{\mathrm{m}rr}+\mathrm{m}_{\mathrm{r}rb}\mathrm{r}_{ \mathrm{r}b}+\mathrm{m}_{\mathrm{b}}\mathrm{r}_{\mathrm{b}}\right)\times\left( R_{EB}^{T}k\right)+T=0\end{matrix}\right.\end{split}  \tag{4}\]  
When a glider glides in the spiraling motion with a constant velocity, the pitch and roll angles are constant [27]. This situation implies that \(\mathrm{R_{EB}^{T}k}\) is constant. By taking the derivative of \(\mathrm{R_{EB}^{T}k}\) with respect to time [28], we get:  
\[\Omega\times\left(R_{EB}^{T}k\right)=0 \tag{5}\]  
From Equation (5), it is known that the underwater glider glides with constant angular velocities along a circular helix which is aligned with the direction of gravity, and the angular velocity is \(\Omega=\mathrm{R_{EB}^{T}k}\omega_{3}\). We select the net buoyancy mass \(\mathrm{m_{b}}\), position \(\mathrm{r}_{\mathrm{m}rr}\) and rotational angle \(\gamma\) of \(\mathrm{r}_{\mathrm{m}rr}\) as the control input parameters, and give a description of a glider experiment performed during May 2011 in Songhua Pool, Jilin Province, China. CTD is used to get glider depth information and the digital compass module TCM3 (3-Axis Compass) to get glider attitude angles. Due to the depth limitation of the pool, the glider gliding depth is restricted within approximately 40 m. The sampling periods of TCM3 and CTD are approximately 6 seconds. In the experiment, the position and rotational angle (pitch angle \(\theta\), roll angle \(\phi\)) of \(\mathrm{m}_{\mathrm{m}rr}\) are adjusted intermittently with TCM3 feedback information, and in a Matlab simulation, the open loop control is utilized to verify the glider's hydrodynamic properties. The equilibrium point of buoyancy and position of mass \(\mathrm{m}_{\mathrm{m}rr}\) are \(\mathrm{m_{b}}=0\) and \(\mathrm{r}_{\mathrm{m}rr}=0.4016\) m. In the experiment, the gliding control mission in one gliding period is given as buoyancy \(\mathrm{m_{b}}=0.2\) kg, and offset of the linearly and rotationally movable mass as \(\mathrm{\parallel\!\Delta}\)\(\mathrm{r}_{\mathrm{m}rr}\)\(\mathrm{\parallel\!=0.01\,m}\) and pitch angle \(\mathrm{\parallel\!\theta}\)\(\mathrm{\parallel\!\!=20^{\circ}}\). Figure 8(a) shows glider depth information in the vertical plane.