# High-Efficiency Closed-Loop Control of a Robotic Fish via Virtual Musculoskeletal Methodology
## 4 Performance Evaluation
### In Steady Flow  
The robotic fish swims in a steady flow. The given speed is \(v^{*}=0.4\,\mathrm{m/s}\) at the beginning, and it steps to 0.5 m/s at 15 s. How the feedback of the speed in the control structure stabilizes the swimming speed is illustrated in Figures 5 and 6. In Figure 4, the relationships between the forward speed \(v\) and the frequency \(f\) and the amplitude \(A\) of the oscillation are depicted. When we set the frequency and the amplitude for desired  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Parameters** & \(\mathbf{A_{0}}\) & \(\mathbf{A_{1}}\) & \(\mathbf{A_{2}}\) & \(\mathbf{L_{c1}}\) & \(\mathbf{L_{d1}}\) & \(\mathbf{L_{c2}}\) & \(\mathbf{L_{d2}}\) & \(\mathbf{m_{0}}\) & \(\mathbf{m_{1}}\) & \(\mathbf{m_{2}}\) \\ \hline
**Value** & 0.03 m\({}^{2}\) & 0.05 m\({}^{2}\) & 0.05 m\({}^{2}\) & 0.12 m & 0.12 m & 0.12 m & 0.12 m & 0.8 kg & 0.4 kg & 0.5 kg \\ \hline \hline \end{tabular}
\end{table}
Table 1: The specific values of the parameters in Figure 1.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Parameters** & \(\mathbf{C_{0}}\) & \(\mathbf{C_{1}}\) & \(\mathbf{C_{2}}\) & \(\mathbf{\rho}\) & \(\mathbf{K_{1}}\) & \(\mathbf{K_{2}}\) & \(\mathbf{H_{1}}\) & \(\mathbf{H_{2}}\) & \(\mathbf{K_{v1}}\) & \(\mathbf{k_{v2}}\) \\ \hline
**Value** & 0.9 & 1.54 & 1.54 & 1000 kg/m\({}^{3}\) & 5 & 16 & 1 & 9 & 0.1 & 0.1 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The specific values of the hydrodynamic and controller parameters.  
Figure 4: Waveform parameters to the swimming speed.  
swimming speed, such as 0.4 m/s and 0.5 m/s, Figure 5 shows that if the speed feedback is not added to form the closed-loop control, there will be a deviation between the real swimming speed \(v\_\)without feedback and the given speed \(v^{*}\).  
Because the robotic fish model is not exactly the same as the live fish, the function derived from the data of the fish in nature does not fit the robotic fish perfectly. Additionally, the speed cannot follow the given value if there is a disturbance in the water.  
Figure 6 sketches the speed following when the robotic fish is controlled with speed feedback. The speed of the PM(\(v\_\)PM) and the VMM(\(v\_\)VMM) both follow the given speed. After the step at 15 s, the response time of the VMM and that of the PM is very close.  
Figure 7 shows the angle of the joint between the head and the body. Both curves of the two control methods are stable sinusoidal waveforms that follow the given \(\theta_{10}\) with a \(30^{\circ}\) delay. This delay is caused by the calculation time, the response time of the dynamic model, and the hydrodynamic model. The curve of VMM has an offset of the amplitude because of the VMM complaints to the force from the water. As the hydrodynamic forces on the robotic fish are not completely symmetrical, the offset of the amplitude is added in the wave.  
Figure 5: The forward speed of robotic fish without speed feedback.  
Figure 6: The forward speed of robotic fish with speed feedback of two methods.  
Figure 11 gives the results of how much the VMM increases the propulsive efficiency of the robotic fish. Before 2.5 s, in the start-up phase, these two methods have the same efficiency. When the robotic fish reaches the speed of 0.4 m/s, the efficiency of the VMM (\(\eta\)_vMM) is one percentage point higher than that of the PM (\(\eta\)_PM). The given speed jumps to 0.5 m/s at 15 s, and both efficiencies decrease with increasing speed. At 0.5 m/s, the VMM plays a more significant role in improving the propulsive efficiency. The efficiency of VMM is 32%, which is 3.4 percentage points higher than that of the PM.  
Figure 11: The efficiency of robotic fish in steady flow.  
Figure 10: The feedback of the virtual spindle.  
Figure 9: The output torque of the second joint.