# Design and Modelling of an Amphibious Spherical Robot Attached with Assistant Fins
## 3 Dynamic Model of the Robot
### Generalized Forces
\(S_{w}\) is the wetted surface area, which can be written as  
\[S_{w}=\int_{0}^{H_{0}}2\pi\sqrt{R^{2}-(h-R)^{2}}.\sqrt{1+\frac{(h-R)^{2}}{R^{2 }-(h-R)^{2}}}dh=2\pi\text{R}H_{0}. \tag{22}\]  
The frictional resistance coefficient is determined by the Reynolds number Re, namely,  
\[\text{Re}=\frac{V_{equ}.L_{wl}}{\
u}, \tag{23}\]  
where \(\
u\) is the kinematic viscosity of water; \(L_{wl}\) represents the waterline length of the spherical shell,  
\[L_{wl}=2\times\sqrt{2 RH_{0}-H_{0}^{2}}. \tag{24}\]  
An estimated value of Re for the amphibious spherical robot described in this paper is about \(1.45\times 10^{5}\) given that \(V_{equ}=0.5\) m/s, \(L_{wl}=0.33\) m, the kinematic viscosity \(\
u=1.139\times 10^{-6}\) m\({}^{2}\)/s, and the temperature of the water is \(15\)\({}^{\circ}\)C. This Re value is less than the critical Reynolds number of a flat plate, which is approximately \(1.0\times 10^{6}\). Therefore, the water flow relative to the spherical shell can be presumed as laminar flow, which is characterized by smooth, constant fluid motion. The frictional resistance coefficient \(C_{f}\) is then given as [29]  
\[C_{f}=1.328\text{Re}^{-\frac{1}{2}}. \tag{25}\]As the outside surface of the spherical shell is smooth, the roughness of the spherical shell barely contributes to the total resistance, so the roughness coefficient \(\Delta C_{f}\) is set as zero in the calculation. Solving Equations (21)-(25), we have  
\[R_{f}=1.328\pi\sqrt{\frac{vV_{cap}^{3}}{L_{val}}}\rho_{w}RH_{0}. \tag{26}\]  
Using Equation (26), we can obtain a roughly estimate of the frictional resistance of the spherical shell in terms of its specifications. To get an accurate value of the frictional resistance, experiment or Computational fluid dynamics (CFD) methods are necessary, which, however, are not the emphasis here. (c) Pressure Resistance  
The hydrodynamic force exerted on the spherical shell consists of the frictional resistance and the pressure resistance. The pressure resistance is due to the decrease of pressure between the front of the spherical shell and its rear. It is difficult to separate the frictional resistance and pressure resistance theoretically. However, researchers have carried out CFD simulations and experiments to measure the resultant of those two resistances [30; 31]. So, it will be possible to evaluate the pressure resistance approximately by subtracting the frictional resistance from the resultant resistance in a somewhat simple way. The hydrodynamic force exerted on the spherical shell then can be presented in the \(O\)-\(XYZ\) frame as \({}^{O}\mathbf{F}_{hyd}=(F_{hX},F_{hY},F_{hZ})\), its corresponding torque relative to the center of sphere is \({}^{O}\mathbf{\eta}_{hyd}=(\eta_{hX},\eta_{hY},\eta_{hZ})\). (d) Resultant Force Exerted on the Fins  
According to [PERSON] et al. [32], when a fin moves relative to the water, a drag force and a lift force are exerted on the fin. The drag force \(D\) is in line with the inlet water, whose velocity is \({}^{f}V_{w}\), and the lift force \(L\) is perpendicular to the inlet water. They can be calculated as follows [32]  
Figure 6:0  
Figure 6:1  
where \(S_{f}\) is the cross-section area of the fin submerged in the water; the angle of attack \(\alpha_{e}\) is defined as the angle between the inflow water's velocity and the cross-section plane of the fin. The maximum lift coefficient \(C_{\rm{Lmax}}\) is determined by the shape of the fin. \(C_{\rm{Dmax}}\) is the maximum drag coefficient. Both these two coefficients can be acquired by experiments. The components of the drag force and the lift force projected in the horizontal direction and vertical direction can be calculated as:  
Figure 6:2  
Figure 6:3  
where \(F_{iX}\) denotes the components of the \(i\)th fin in the \(X\) direction, \(F_{iZ}\) denotes the components of the \(i\)th fin in the \(Z\) direction, \(\varphi\) is the phase angle of the fin in its rotation cycle.