## 4 Robot Modeling
### Discussion of the Results  
According to the above results, it can be stated that the links have similar dynamics regardless of their position. However, there are differences in the gain, which means a greater displacement as the closer to the end of the flagellum is measured. Specifically, the tip displacement of the second link shows an increase in gain, whereas the tip displacement of the third link also presents a lag with respect to the first link as a consequence of the length, the flexible behavior of IPMC and the interaction with the environment. These conclusions are illustrated in Figure 8, which schematizes the relation between the first and the rest of the links. (At this point, it should be recalled that the first link is active, while the other two behave as passive links). The values of the gains and the phase lag of the third link can be directly deducted from Table 2 as follows: \(K_{2}=3.72\), \(K_{3}=7.37\), and \(\phi=4^{\circ}\). It should be noted that the gain does not increase linearly due to the fact that the IPMC bends into a C-shape, so the closer to the end is measured, the higher the displacement. Hence, it can be concluded that the displacement of the end of each flagellum segment, or of any IPMC point, is characterized by the same dynamics, which remains unchanged along with the link with mere variations of the gain for low-frequency applications. This conclusion is also consistent with other studies reported in the literature, where IPMCs show changes in model gain when the links do not have the same length or are manufactured from different substrates (see e.g., [31; 43; 55; 41]). Moreover, from the above conclusion, it can be stated that a non-reciprocal motion cannot be achieved with a single actuation if the geometrical and mechanical design of \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & & \(\boldsymbol{\alpha}\) & \(\boldsymbol{a_{0}\) (\(\times 10^{-3}\))} & \(\boldsymbol{b_{2}\) (\(\times 10^{-3}\))} & \(\boldsymbol{b_{1}\) (\(\times 10^{-3}\))} \\ \hline \multirow{2}{*}{First link} & \(G_{1}\) & - & 13.70 & 1.62 & 12.60 \\  & \(G_{2}\) & 1.79 & 13.33 & - & 3.30 \\ \hline \multirow{2}{*}{Second link} & \(G_{1}\) & - & 51.00 & 1.61 & 12.72 \\  & \(G_{2}\) & 1.81 & 48.10 & - & 3.10 \\ \hline \multirow{2}{*}{Third link} & \(G_{1}\) & - & 376.09 & 2.13 & 23.31 \\  & \(G_{2}\) & 1.71 & 352.80 & - & 5.80 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Model parameters obtained from identification procedure. Figure 8: Diagram block of swimming robot model for single actuation. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Model** & & \(\boldsymbol{J\) (\(\times 10^{-5}\))} & MSE (\(\times 10^{-7}\)) & MAD (\(\times 10^{-4}\)) & MD (\(\times 10^{-3}\)) & \(\mathbf{R^{2}}\) \\ \hline \multirow{2}{*}{First link} & \(P_{1}\) & \(\boldsymbol{2.4884}\) & \(\boldsymbol{4.5210}\) & \(\boldsymbol{5.4392}\) & \(\boldsymbol{2.800}\) & \(\boldsymbol{0.9153}\) \\  & \(P_{2}\) & 3.4992 & 5.0188 & 5.7149 & 3.3000 & 0.9059 \\ \hline \multirow{2}{*}{Second link} & \(P_{1}\) & \(\boldsymbol{4.9163}\) & \(\boldsymbol{59.7630}\) & \(\boldsymbol{13.0000}\) & \(\boldsymbol{24.0000}\) & \(\boldsymbol{0.9630}\) \\  & \(P_{2}\) & 12.3680 & 104.2300 & 19.0000 & 28.5000 & 0.9354 \\ \hline \multirow{2}{*}{Third link} & \(P_{1}\) & \(\boldsymbol{530.0000}\) & 1654.7000 & 69.0000 & \(\boldsymbol{126.5000}\) & 0.9796 \\  & \(P_{2}\) & 550.0000 & \(\boldsymbol{1392.7200}\) & \(\boldsymbol{45.000}\) & 186.4000 & \(\boldsymbol{0.9828}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance indices after fitting models \(P_{1}(s)\) and \(P_{2}(s)\). Numbers in bold correspond to the best fit. the swimming robot has not been analyzed previously and designed for that purpose, otherwise the dynamics of consecutive links are subjected to the active link.