# Design and Motion Simulation of an Underwater Glider in the Vertical Plane
## 2 UG Design  
To achieve better mechanical properties for resisting seawater pressure and to facilitate fabrication, the shape of the underwater glider was designed by applying MYRING hull profile equations [10]. The MYRING hull profile is illustrated in Figure 1.  
The head and stern sections are expressed as shown in Equations (1)-(3):  
\[r_{\text{head}}\left(x\right)=\frac{1}{2}d\left[1-\left(\frac{x-a}{a}\right)^{ 2}\right]^{\frac{1}{2}},0\leq x\leq a \tag{1}\]  
Figure 1: MYRING hull profile. \[r_{\text{mid}}(x)=\frac{1}{2}d,a\leq x\leq a+b \tag{2}\] \[r_{\text{stem}}(x)=\frac{1}{2}d-\Big{(}\frac{3d}{2c^{2}}-\frac{ \tan\delta}{c}\Big{)}(x-a-b)^{2}+\Big{(}\frac{d}{c^{3}}-\frac{\tan\delta}{c^{2}} \Big{)}(x-a-b)^{3},\] (3) \[a+b\leq x\leq a+b+c\]  
Here, \(d\) represents the body diameter, \(a\) represents the head length, \(n\) can be used to determine the shape of the head section, \(c\) represents the length of the stern section, and \(\theta\) represents the tail wrap angle, which determines the shape of the stern. The greater the angle of \(\delta\), the smoother is the shape change at the stern. The parameters of the MYRING hull profile equations in this study were as follows: \(a\) = 280 mm, \(b\) = 1530 mm, \(c\) = 400 mm, \(d\) = 260 mm, \(n\) = 2, and tan \(\delta\) = 0.8. Net buoyancy is the main driving force for the motion of an underwater glider, and it determines the performance and reliability of the glider. The underwater glider changes its buoyancy by drawing and discharging water through a built-in pump. When the net buoyancy is positive, the glider floats upward, and when the net buoyancy is negative, the glider moves downward. Furthermore, the glider is subject to hydrodynamic forces from water when moving underwater. The force analysis of the glider during stable gliding is presented in Figure 2.  
The drag force can be expressed by  
\[D=\frac{1}{2}\rho C_{D}A_{D}V^{2} \tag{4}\]  
Here, \(\rho\) is the density of the fluid, \(C_{D}\) the drag coefficient of the glider, \(A_{D}\) the cross-sectional area of the glider, and \(V\) the gliding speed. During steady gliding in the water, the force will maintain balance on the glider; therefore, the net buoyancy \(\Delta B\) and hydrodynamic drag force \(D\) are correlated as follows:  
\[D=\Delta Bsin\zeta \tag{5}\]  
where \(\zeta\) is the gliding angle. Combining Equations (4) and (5), the net buoyancy is expressed as  
\[\Delta B=\frac{1}{2 sin\zeta}\rho C_{D}A_{D}V^{2} \tag{6}\]  
The net buoyancy of the UG is expressed as  
\[r_{\text{head}}\left(x\right)=\frac{1}{2}d\left[1-\left(\frac{x-a}{a}\right)^{ 2}\right]^{\frac{1}{2}},0\leq x\leq a \tag{1}\]0  
\[r_{\text{head}}\left(x\right)=\frac{1}{2}d\left[1-\left(\frac{x-a}{a}\right)^{ 2}\right]^{\frac{1}{2}},0\leq x\leq a \tag{1}\]1 Force analysis of the UG. Here, \(V_{b}\) is the volume of drainage of the buoyancy engine, and Equations (6) and (7) are merged to obtain the equation for the displacement of the buoyancy engine of the UG, which occurs as the gliding angle is 35\({}^{\circ}\)[11]. \[r_{\text{head}}\left(x\right)=\frac{1}{2}d\left[1-\left(\frac{x-a}{a}\right)^{ 2}\right]^{\frac{1}{2}},0\leq x\leq a \tag{1}\]2  
Based on the design specifications, an underwater glider that can reach a maximum path speed of 2 knots was designed. [PERSON] et al. [11] present a similar underwater glider with a diameter of 0.22 m, whose total drag coefficient is 0.2629. The water volume was obtained using the equations above. Considering the internal space of the waterproof housing and the compactness of the structure, the inner diameter of the cylinder of the buoyancy engine was set to 0.15 m. From the known conditions, the displacement of the piston can be calculated to be approximately 74.3 mm when the glider reaches the maximum speed. Taking into account the error in the theoretical calculation and the difference in the hydrodynamic drag coefficient, the effective stroke of the piston was set to 220 mm.