## 5 Comparison between Model and Experimental Results
### Simulation Based on the Proposed Model  
As in the experiment, we simulated the response to a step input. Since the derived models are for rotational speed, they were transformed into models for the angle of rotation as follows:  
\[\frac{F(s)}{\theta(s)}=\frac{Ks(s^{2}+as+b)}{(s^{2}+cs+d)(s^{2}+es+g)} \tag{7}\]  
\[\frac{M(s)}{\theta(s)}=\frac{Ks^{3}}{s^{4}+as^{3}+bs^{2}+cs+d} \tag{8}\]  
Assuming that the forces due to the rotation of each axis are independent of each other, the longitudinal thrust was calculated by superposing the change in axial force of each axis onto the steady-state force as follows:  
\[f_{x}(t)=f_{ss}+\left\{f_{1x}(t)+f_{2x}(t)\right\} \tag{9}\]  
Figure 15: Angular position input for the verification test.  
The sum of the models of the derived \(x\)-axis force variation allowed us to obtain the attenuated \(x\)-axis thrust force. The rotation of the coordinate system by two axes is expressed in a matrix as follows:  
\[R(\theta_{1},\theta_{2})=R_{z}(\theta_{1})R_{y}(\theta_{2})=\left[\begin{array}[ ]{ccc}\cos\theta_{1}\cos\theta_{2}&-\sin\theta_{1}&\cos\theta_{1}\sin\theta_{ 2}\\ \sin\theta_{1}\cos\theta_{2}&\cos\theta_{1}&\sin\theta_{1}\sin\theta_{2}\\ -\sin\theta_{2}&0&\cos\theta_{2}\end{array}\right] \tag{10}\]  
The results of force simulation in moving coordinates can be transformed into forces in fixed coordinates as follows:  
\[\left[\begin{array}{c}f_{X}\\ f_{Y}\\ f_{Z}\end{array}\right]=R^{T}(\theta_{1},\theta_{2})\left[\begin{array}{c}f_ {x}\\ f_{1y}\\ f_{2z}\end{array}\right] \tag{11}\]  
The experimental results and simulated values are compared in Figure 16. The desired value is the force in the fixed coordinate system for the actual rotation angle of the two axes when there is only ideal longitudinal propulsion. In other words, the desired forces are the decomposition force to the rotation angle measured by the encoder values of the two servo motors, based on the propulsion model of the fixed thruster. These are ideal values that do not consider force attenuation or tangential force due to the rotation of the thruster. In the initial position, the propulsion occurs in only the \(X\)-axis direction, but as the two axes rotate simultaneously, the direction of the propulsion changes. For each force and moment, the steady-state error and root mean square error (RMSE) are defined as performance indices and are shown in Table 6.  
Figure 16: Comparison between experimental results and model under the rotational input with 65% PWM: (a) \(f_{X}(t)\), (b) \(f_{Y}(t)\), (c) \(f_{Z}(t)\) and (d) \(M_{Z}(t)\).  
The simulation results from the derived model followed the waveform changes in the experimental results well. The change in force on the rotation of the two axes was modeled, but the model did not include high-frequency oscillations from the rotation of the propeller. As the motor encoders measuring the rotation angle are very precise, the steady-state error observed in the \(X\)- and \(Z\)-axial forces can be considered as interference from forces from the water surface and water tanks. Overshoots in the experimental values are caused by the vibration of the gimbal structure. It was observed that more vibrations caused by step inputs occurred than in the modeling experiments.