## 6 Conclusions  
In this paper, an artificial eukaryotic flagellum (ABF) swimming robot made up of ionic polymer-metal composite (IPMC) has been studied for the generation of planar traveling waves through single and distributed actuation in low Reynolds number (Re) environments. The robot comprised three independent and electrically isolated actuators manufactured over the same 10 mm long IPMC sheet. Firstly, the swimming robot dynamics was identified for control purposes through its frequency response. It was obtained experimentally in LabVIEW by attaching a couple of gold electrodes to the first actuator and measuring the tip deflection by means of a laser distance meter. The experiments were performed considering the first segment as actuator and the rest acting as passive flexible links, and the deflection of each of the links was measured. Models consisting of a fractional integrator in series with a resonant system of both fractional and integer order were identified in the frequency range \([0.1,50]\) rad/s for each link. The results showed that both models can be adequate for control purposes. Furthermore, the identification process led to the conclusion that the tip displacement and of any IPMC point are characterized by the same dynamic, which remains unchanged along with the link with mere variations of the gain for low-frequency applications. From this conclusion, it was also deduced that single actuation of the robot does not allow to produce a non-reciprocal motion, since the dynamics of the second and third links are subject to the dynamics of the first link. There are only differences in the gain and a slight phase lag, which are not sufficient to generate a non-reciprocal motion. Secondly, a controller robust to gain variations was designed to control link deflection regardless of link length allowing to address a distributed actuation with the same controller design. It was tuned based on the methodology of the reference model and two control  
Figure 13: Three-dimensional representation of non-reciprocal motion for: (**a**) ideal waveform, (**b**) simulated motion with \(C_{\text{PID}}(s)\), (**c**) simulated motion with \(C_{\text{I}}^{\ast}(s)\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Link** & \multicolumn{2}{c}{**IAE**} & \multicolumn{2}{c}{**ITAE**} \\ \hline  & **C\({}_{\text{I}}\)(s)** & **C\({}_{\text{PID}}\)(s)** & **C\({}_{\text{I}}\)(s)** & **C\({}_{\text{PID}}\)(s)** \\ \hline First link & 4.67 & 0.68 & 434.90 & 42 \\ Second link & 13.87 & 2.28 & 1283.50 & 140.75 \\ Third link & 18.04 & 3.97 & 1861.70 & 269.74 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Tracking performance indices. strategies were proposed: a classical proportional-integral-derivative (PID) controller and a non-integer order integrator. The results showed that the PID offers a better response to tracking control as a consequence of the increased system speed, but does not meet the design specifications. In contrast, the non-integer order integrator meets the step response specifications but provides worse results for the tracking problem. Finally, the motions of the AEF swimming robot composed of three links of the same length were analyzed to validate that the robot performs a non-reciprocal motion, determining that a distributed actuation generates a planar wave motion. Therefore, it was also concluded that a distributed actuation (active or passive) is necessary to develop a non-reciprocal motion. Our future works will focus on: (1) validating the distributed actuation method in the real prototype; and (2) studying the control approach for a swimming robot with segments of different lengths designed to optimize robot propulsion. This work involved all coauthors. Conceptualization, I.T. and B.M.V. ; Formal analysis, J.E.T. ; Investigation, J.E.T. and P.R. ; Project administration, I.T. and B.M.V. ; Writing--original draft, J.E.T. ; Writing--review and editing, C.N.-G., I.T. and B.M.V. All authors have read and agreed to the published version of the manuscript.