## 4 Hydrodynamic Modeling and Analysis of the Robot
### Hydrodynamic Modeling
\(V_{T}\) is the relative velocity between the water with the paddle which is given by  
\[V_{T}=\omega y\cos\theta-V \tag{5}\]  
where \(V\) is the speed of the robot, \(|V_{T}|\) is the absolute value of \(V_{T}\) and it ensures that when \(V_{T}\) is negative, and the force provided by paddle for the robot is the resistance. By replacing \(V_{T}\) with Equation (5), Equation (4) can be rewritten as  
\[F_{T}=\int_{e}^{f}\frac{1}{2}K_{T}H(t)(\omega y\cos\theta-V)|\omega y\cos \theta-V|dy \tag{6}\]  
Similarly, the speed of the robot in the lateral direction was zero and the lift force \(F_{L}\) at each paddle can be calculated by  
\[F_{L}=\int_{e}^{f}\frac{1}{2}K_{L}H(t)(\omega y\sin\theta)^{2}dy \tag{7}\]  
where \(K_{L}\) is a custom lift force coefficient and expressed as \(K_{L}=C_{L}\rho\sin\ \theta\), \(C_{L}\) is the dimensionless coefficient in the lift calculation which is related to \(\theta\). Due to the symmetry of the robot structure, when the robot moves forward, it only produces three kinds of motions: surge motion in the forward direction, heave motion in the vertical direction, and pitch motion. Since the supporting mechanisms spread out on the water surface to make the robot stand stalely, the pitch motion of the robot can be ignored. The paddle is always perpendicular to the water surface which is considered to not produce vertical force, so the heave motion of the robot is mainly affected by the supporting mechanisms. Similar to the hydrodynamics of a ship or an AUV, and under the assumption that the water is always calm, the drag force \(F_{R}\) in the forward direction provided by the supporting mechanisms is  
\[F_{R}=F_{D}+F_{M} \tag{8}\]  
\[F_{D}=K_{D}(V-U_{0})^{2}+K_{D1}(V-U0)+K_{D2}+X_{W}W+X_{WW}WW\approx K_{D}V^{2} +X_{W}W+X_{WW}W^{2} \tag{9}\]\[F_{M}=M_{V}\frac{dV}{dt} \tag{10}\]  
where \(F_{D}\) is the viscous hydrodynamic force, \(F_{M}\) is the inertia hydrodynamic force related to the added mass, \(K_{D}\), \(K_{D1}\), \(K_{D2}\), \(X_{W}\), and \(X_{WW}\) are viscous hydrodynamic coefficients, \(W\) is the vertical velocity of the robot, \(U_{0}\) is the work point of the robot, \(M_{V}\) is the added mass of the robot. The work point \(U_{0}\) is the speed of the robot we specify, which affects values of some viscous hydrodynamic coefficients. In this paper, the relevant viscous hydrodynamic coefficients are measured under the drive of a motor in Section 4.2.1 and the work point \(U_{0}\) is the speed of the motor. The force \(F_{B}\) in the vertical direction provided by the supporting mechanisms is  
\[F_{B}=Z_{0}+M_{W}\frac{dW}{dt}+Z_{V}(V-U_{0})+Z_{W}W+Z_{VV}(V-U_{0})^{2}+Z_{WW} W^{2}+Z_{\xi}\xi+Z_{\xi\xi}\xi^{2} \tag{11}\]  
where \(Z_{0}\), \(Z_{V}\), \(Z_{W}\), \(Z_{VV}\), and \(Z_{WW}\) are the viscous hydrodynamic coefficients in the vertical direction, \(M_{W}\) is the added mass of the robot in the vertical direction, \(\xi\) is the displacement of the robot in the vertical direction, \(Z_{\xi}\) and \(Z_{\xi\xi}\) are the hydrodynamic coefficients related to the gravity and buoyancy of the robot. The water surface was near the largest cross-section of the ellipsoid floats which had a greater damping effect on the heave motion of the robot. In order to simplify the analysis, the effect of the heave motion on the surge motion of robot were ignored (i.e., the terms \(X_{W}W\) and \(X_{WW}WW\) in Equation (9) are ignored). Since the heave motion is not the focus of this article and requires lots of work to obtain the relevant hydrodynamic confidents, we will not further analyze the vertical motion of the robot. Based on [PERSON]'s second law, Equations (6) and (8)-(10), the general hydrodynamic model of the robot in the forward direction is  
\[F=\frac{1}{2}C\rho AV_{0}{}^{2} \tag{3}\]0  
where m is the weight of the robot. Equation (12) represents the relationship between the hydrodynamics and motion of the robot. The general hydrodynamic model can represent all forward motions of the robot.