## 4 Hydrodynamic Modeling and Analysis of the Robot
### Coefficients Evaluation of the Hydrodynamic Model
#### 4.2.1 \(K_{t}\), \(K_{l}\) and \(K_{d}\) Measurement  
A quasi-static approach [27; 29] was adopted to determine the relationship between \(K_{T}\) with \(\theta\) and the relationship between \(K_{L}\) with \(\theta\). According to Equation (6), when the paddle moves straight on the water at a fixed angle and a fixed water line \(H(t)=\mathrm{h}\), the generated force \(F_{e}\) can be expressed as  
\[F_{e}=\frac{1}{2}K_{T}\mathrm{x}_{ef}\mathrm{h}V_{m}^{2} \tag{17}\]  
where \(V_{m}\) is the speed of the paddle, \(\mathrm{x}_{ef}\) is the distance of point \(e\) to point \(f\). We can measure different values of \(F_{e}\) when setting the immobile paddle at different angles \(\theta\). From Equation (17), we can get different values of \(K_{T}\) when the immobile paddle is at different angles \(\theta\). Using this static measurement relationship between \(K_{T}\) and \(\theta\) to represent the relationship between the dynamics of \(K_{T}\) and \(\theta\) when the paddle rotates is a quasi-static approach. The relationship between \(K_{T}\) and \(\theta\) is not continuous because the data \(F_{e}\) corresponding to \(\theta\) obtained by the experiment are not continuous and it cannot represent a real condition when the paddle rotates on the water. Therefore, we used these data to fit a continuous appropriate function \(K_{T}=f(\theta)\) to indicate the relationship between \(K_{T}\) and \(\theta\). The \(K_{T}\) measured when the immobile paddle was set at angle \(\theta\) was equal to the \(K_{L}\) measured when the immobile paddle was set at angle \(\pi/2-\theta\). The function relationship between \(K_{L}\) and \(\theta\) can be expressed as \(K_{L}=f\left(\pi/2-\theta\right)\). So we can get the relationship between \(K_{L}\) and \(\theta\) according to the function relationship between \(K_{T}\) and \(\theta\). \(K_{D}=F_{D}/V^{2}\), which can be obtained by Equation (9) and \(K_{D}\) can be determined when \(F_{D}\) and \(V\) are measured according to the experiment. In order to get the function relationship between \(K_{T}\) and \(\theta\), a stent was fixed on the slider and the stent was connected to the paddle by a 3-axis force sensor (MX020-10N, NMB, Japan) (Figure 8a). When the motor drove the slider, the paddle moved with the slider and slid on the water. The generated force was measured by the force sensor. The speed of the dragging motor was set to 26 cm/s considering the length of the guide (1.5 m) and the force generated by the paddle. Because the paddle was always perpendicular to the horizontal plane, the force sensor measured the horizontal force of the paddle in the moving direction. The sensor measured the average forces of paddle \(F_{e}\) when it slid with different angles \(\theta\) in a sink. A total of 19 groups of \(F_{e}\) and \(\theta\) combinations were tested when \(\theta\) increased from 0\({}^{\circ}\) to 90\({}^{\circ}\) with the angular interval of 5\({}^{\circ}\). The data recorded different \(\theta\) corresponding to \(F_{e}\) measured in the experiment are shown in Figure 9. According to Equation (17), the \(K_{T}\) corresponding to different angles \(\theta\) has been calculated (i.e., points in Figure 10). A quadratic polynomial function was used to fit these points to get a continuous relationship between \(K_{T}\) and \(\theta\) because the quadratic polynomial function is simple and its function curve conforms to the image formed by the data. The fitted curve is shown in Figure 10 and Equation (18) is the fitting function:  
\[K_{T}=-2796(\frac{\pi}{2}-\theta)^{2}+6918(\frac{\pi}{2}-\theta) \tag{18}\]  
The function relationship between \(K_{L}\) and \(\theta\) is expressed as \(K_{L}=\)\(f\) (\(\pi/2-\theta\)). Through Equation (18), we can get the relationship between \(K_{L}\) and \(\theta\) as  
\[K_{L}=-2796\theta^{2}+6918\theta \tag{19}\]  
Figure 8b shows the experiment for measuring the viscous hydrodynamic force \(F_{D}\) when it was dragged moving on the water by a motor at a speed of \(V_{D}\) which was similar to the previous experimental method.