## 4 Hydrodynamic Modeling and Analysis of the Robot
### Hydrodynamic Modeling  
The water strider robot mainly focuses on a forward motion in practical applications, therefore, only the hydrodynamic model of the robot when the robot performs a forward motion has been studied. Because it would be complicated to consider all the circumstances, this study intended to start with a simplification. We assumed that the flow direction of the water was consistent with the motion direction of the robot and the rate of the water was steady. The propulsion was produced by the interaction between the paddles and the water when the paddles rotate. This process produced some resistance when the linear velocity of some parts of the paddles near the shaft were slower than the speed of the robot. The other resistance came from the four floats on the supporting mechanisms contacting the water surface when the robot moved. To reduce the resistance generated by a paddle, a rectangle corner near the shaft (i.e., the \(r\), \(u\), \(q\), \(e\) area in Figure 6) was cut from the paddle. This ensures the linear velocity of the rest paddle immersed in the water will be greater than the speed of the robot. Meanwhile, a rib was added to enhance the stiffness of the paddle. In Section 4.3, it is proved that cutting the rectangle corner can increase the speed of the robot. The force generated by the paddle can be divided into a thrust and a lift (Figure 7) [27]. For the forward motion of the robot, the driving force of the robot was determined by the resultant force of the two paddles immersed in the water. In the direction perpendicular to the forward direction of the robot, the two paddles generating the lift forces were equal in magnitude and opposite in direction, so they cancel each other out. In the forward direction of the robot, the two paddles generating the thrust forces were equal in the magnitude and same in the direction, so the sum  
Figure 5: The trajectory curves of points \(e,f\) and \(c\). The blue, green and black lines represent the curves of points \(f,e\), and \(c\), respectively. The red points and arrows, respectively, are the initial positions and initial directions of the points. of the thrust forces was the driving force of the robot. The thrust force can be calculated approximately by a formula that is used to calculate the drag force of the external flow past an object [23; 24; 28]. It is  
\[F=\frac{1}{2}C\rho AV_{0}{}^{2} \tag{3}\]  
where \(F\) is the resistance, \(C\) is a dimensionless coefficient which is determined by the shape of the object and the surface condition, \(\rho\) is the fluid density, \(A\) is the part of area that the object immerses in water and projects in the plane perpendicular to the forward direction of the object, and \(V_{0}\) is the relative velocity between the object and the water. This formula cannot be used directly to calculate the thrust of the paddle, because the linear velocity of each point on the paddle is different. To get around this limitation, a differential method was adopted to divide the entire paddle into many infinite sections along the radial direction of the shaft, and the integration of the force of each infinite section was used to approximate the thrust. The thrust of the paddle \(F_{T}\) can be approximated as  
\[F_{T}=\int_{\rm e}^{f}\frac{1}{2}K_{T}H(t)V_{T}|V_{T}|dy \tag{4}\]  
where \(K_{T}\) is a customized thrust coefficient and expressed as \(K_{T}=C_{T}\rho\cos\theta\), \(C_{T}\) is the dimensionless coefficient in the thrust calculation which is related to \(\theta\), \(\theta\) is the rotational angle of the paddle (Figure 7), \(\omega\) is the rotational speed of the paddle, \(t\) is the time that the paddle is rowing, \(H(t)\) is the water line (i.e., dashed blue line in Figure 6) which represents the depth of the paddle immersed in the water, \(y\) is coordinate in the Y axis of the coordinate system A, and \(dy\) is the infinitesimal increase in the Y axis. Here, the lower and upper limit of integration (i.e., \(e\) and \(f\)) are determined by the shape of the paddle in the local coordinate system A.