## 4 Hydrodynamic Modeling and Analysis of the Robot
### Hydrodynamic Analysis for the Robot  
In the previous two sections, the general hydrodynamic model of the robot is established and the coefficients in the model are solved. In this section, the hydrodynamic characteristics of the robot when it performs the ellipse-like spatial stroke trajectory is studied. Below we derive the hydrodynamic model for when the robot performs the ellipse-like spatial stroke trajectory from the general hydrodynamic  
Figure 12: The force acting on the four ellipsoid floats in the water as a function of \(t\) at two different mesh numbers. model. According to the kinematics analysis of the paddle in Section 3, The water line \(H(t)\) in Equation (12) when the robot performs the ellipse-like spatial stroke trajectory becomes  
\[\left\{ \begin{array}{ll}H(t)=0&b_{1}\leq 0.024\\ H(t)=\int\omega_{1}\mathrm{P}_{1}dt-0.024&b_{1}>0.024\end{array} \right. \tag{25}\]  
\(\omega\) and \(\theta\) in Equation (12) should be replaced by \(\omega_{2}\) and \(\int\omega_{2}dt\), respectively. Therefore, the hydrodynamic model when the robot performs the ellipse-like spatial stroke trajectory is  
\[\left\{ \begin{array}{ll}-K_{D}V^{2}-M_{V}\frac{dV}{dt}=m\frac{dV}{dt}&b_{1} \leq 0.024\\ \int^{t}Kr(\int\omega_{1}\mathrm{P}_{1}dt-0.024)[\omega_{2}y\cos(\int\omega_{ 2}dt)-V]\|\omega_{2}y\cos(\int\omega_{2}dt)-V]dy-K_{D}V^{2}-M_{V}\frac{dV}{dt}=m \frac{dV}{dt}&b_{1}>0.024\end{array} \right. \tag{26}\]  
From the numerical simulation of Equation (26) calculated by the MATLAB software, the speed of the robot in different stroke cycles \(T\) is shown in Figure 13. The result denotes that the robot reaches the dynamic balance after going through three stroke cycles. The final speed of the robot fluctuates periodically around a constant value just as we predicted. The average speeds of the robot in the stroke cycles of 1.6, 2.0, 2.4, 2.8, and 3.2 s were 0.108, 0.087, 0.075, 0.059, and 0.050 m/s, respectively, when it reached a dynamic balance. As the stroke cycle \(T\) increases, the speed of the robot decreases. In order to verify the effect of the rectangular corner on the paddle and the stroke efficiency of the paddle in the following content, the movement of the paddle with the stroke cycle \(T=1.6\) s was used as the research object. In Section 4.1, a rectangle corner near the shaft of the paddle was cut to reduce the resistance generated by the paddle. In order to prove this, a case in which the robot has two intact paddles was conducted as a numerical simulation. The intact paddles performed the same movement as the paddles with a rectangular piece cut out of the corner and the stroke cycle was \(T=1.6\) s. In this case, the hydrodynamic model of the robot can be easily derived from Equation (12) and is omitted here. The speed of the robot with two intact paddles is shown in Figure 14. It can be found that the speed of the robot with two intact paddles was lower than the robot with the paddles with a rectangular piece cut out of the corner. This explains that cutting off a rectangular piece from the corner near the shaft from the paddles can increase the thrust of the robot. Figure 13: The speeds of the robot as a function of time \(t\) in different stroke cycles \(T\). The blue, pink gray, red, and black lines are the speeds of the robot when the stroke cycles are 1.6, 2, 2.4, 2.8 and 3.2 s, respectively. In order to evaluate the stroke efficiency of the robot when it performs the ellipse-like spatial stroke trajectory, another case was set up for comparison. In this case, the vertical translation and rotational movements of the paddle were alternating, but the vertical translation speed and rotational speed of the paddle were the same as the robot when it performs the ellipse-like spatial stroke trajectory.