## 4 Hydrodynamic Modeling and Analysis of the Robot
### Coefficients Evaluation of the Hydrodynamic Model
#### 4.2.2 \(M_{v}\) Measurement  
\(M_{V}\) is the added mass of the robot and was measured by a CFD simulation here. In the CFD simulation, the hydrodynamic coefficient measurement of an underwater robot was mainly carried out by planar motion mechanism (PMM) tests. A PMM test can obtain the functional relationship between the drag force and time of a robot by forcing the robot to perform periodic oscillations in a fixed direction. By performing a Fourier transform on the functional relationship, the hydrodynamic coefficient parameters such as the acceleration term and velocity term can be obtained [30]. In order to obtain the added mass of the robot in the forward direction, the robot was forced to perform a periodic  
Figure 10: The value of \(K_{T}\) as a function of the angle of the paddle \(\theta\). Points are the values of \(K_{T}\) by calculation and cure is the fitted cure of \(K_{T}\). Figure 9: The measured \(F_{e}\) as a function of the angle of the paddle \(\theta\). pure surge motion in the robot forward direction. The displacement \(X\), velocity \(\overset{\bullet}{X}\), and acceleration \(\overset{\bullet\bullet\bullet}{X}\) of the robot are given by following equations:  
\[X=\alpha\sin(\beta t) \tag{20}\]  
\[\overset{\bullet}{X}=\alpha\beta\cos(\beta t) \tag{21}\]  
\[\overset{\bullet\bullet\bullet}{X}=-\alpha\beta^{2}\sin(\beta t) \tag{22}\]  
where \(\alpha\) is the amplitude and \(\beta\) is the circular frequency of the surge motion. Here \(\alpha\) is set as 0.04 m and \(\beta\) is set as 0.8\(\pi\). Ignoring the effect of air on the robot and the four ellipsoid floats immersed in the water, we can solve the added mass by calculating the hydrodynamic force of the four ellipsoid floats. Unlike the underwater robot which was completely in the water, the four ellipsoid floats were approximately half in the water and half in the air. A two-phase flow CFD model was used to study their hydrodynamics. Since the effect of air on the robot was neglected, the air domain was added on the CFD model only to simulate the conditions of the four floats on the gas-liquid surface and the force of the floats acted by the air was not calculated. The two-phase flow CFD model in the COMSOL software (COMSOL Inc., Stockholm, Sweden) is shown in Figure 11.  
The model uses an unstructured mesh method. The upper part of the model is the air domain (\(3.4\text{ m}\times 1.8\text{ m}\times 0.2\text{ m}\)) and the lower part is the water domain (\(3.4\text{ m}\times 1.8\text{ m}\times 0.2\text{ m}\)). The left boundary is the velocity inlet and the inflow velocity \(U\) is set as 0.1 m/s. The right boundary is the pressure outlet and pressure is set as 0. The other boundaries of the air domain and the water domain  
Figure 11: The computational fluid dynamics (CFD) two-phase flow model in COMSOL software: (**a**) overall of the model; (**b**) the area around the floats. were set as no slip wall. The four floats were in the gas-liquid surface and exhibited a periodic surge motion in a direction perpendicular to the inflow direction. The motion of four floats were defined by the moving mesh method in the COMSOL software. The Reynolds number of the CFD model Re can be calculated by  
\[\text{Re}=\frac{\rho\underline{U}d}{\mu} \tag{23}\]  
where \(d\) is the long diameter of the float and \(\mu\) is the viscosity of the water. By calculation, the Reynolds number is 10,500 and a RANS \(k\)-\(\varepsilon\) turbulent model was used in the CFD simulation. The force of the four floats in the robot forward direction can be exported from the COMSOL software and is shown in Figure 12. The model was calculated by using 384,965 meshes and 239,690 meshes. It was found that the results of the double calculations were similar and the results were independent of the number of meshes.