# Bio-Inspired Morphing Tail for Flapping-Wings Aerial Robots Using Macro Fiber Composites
## 2 MFC Tail Actuator Characterization
### MFC Theoretical Displacement Model
Therefore, the total potential energy of the system can be expressed as:  
\[U_{TOTAL}=U_{HOST}(c_{1},c_{2},c_{3},c_{4})+U_{MFC,Actuated}(c_{1},c_{2},c_{3},c_{4}) \tag{3}\]where the potential energy of the actuator and the host lamina can be separately calculated by the following expressions:  
\[ \begin{split} U_{HOST}=\int_{-\frac{L_{x,bat}}{2}}^{\frac{L_{x,bat}}{2 }}\int_{-\frac{L_{y,bat}}{2}}^{\frac{L_{y,bat}}{2}}\int_{-\frac{L_{x,bat}}{2}}^{ \frac{H_{x,bat}}{2}}[\sigma_{bust,x}e_{host,x}+\sigma_{host,y}e_{host,y}\\ +\sigma_{host,xy}\gamma_{host,xy}]dzdydx\end{split}  \tag{4}\]  
\[ \begin{split} U_{MFC,Actuated}=\int_{-\frac{L_{x,MFC}}{2}}^{\frac{L_{ x,MFC}}{2}}\int_{-\frac{L_{y,MFC}}{2}}^{\frac{H_{x,MFC}}{2}}[(\sigma_{MFC,x}- \sigma_{PZ,x})\epsilon_{MFC,x}\\ +(\sigma_{MFC,y}-\sigma_{PZ,y})\epsilon_{MFC,y}+(\sigma_{MFC,xy}- \sigma_{PZ,xy})\gamma_{MFC,xy})]dzdydx\end{split}  \tag{5}\]  
where \(\sigma_{i,j}\) are the stresses of the \(i\) element in \(j\) direction, \(e_{i,j}\) and \(\gamma_{i},j\) are the strains in the same nomenclature that \(\sigma_{i,j}\), and \(L_{k,m}\) are the longitude of the material in k direction. The notation of the elements is \(HOST\) for host material, \(MFC\) for the MFC complete actuator, and \(PZ\) for the piezoelectric material of the MFC. The process to obtain the strains and stresses in the function of the unknown coefficients of the different elements are well detailed in [28]. Finally, the \(U_{TOTAL}\) can be expressed as function of \(c_{1}\), \(c_{2}\), \(c_{3}\), and \(c_{4}\). The next step is to apply the Rayleigh-Ritz method as:  
\[\frac{\partial U_{TOTAL}}{\partial c_{i}}=0\hskip 28.452756 pt\text{i}=1,2,3,4 \tag{6}\]  
Regarding this equation, the values of \(c_{1}\), \(c_{2}\), \(c_{3}\), and \(c_{4}\) are known and the displacement equation can be obtained by replacing the values in (2). Figure 3 shows the theoretical model deformation of a MFC actuated by 1500 V. The colors represent the deformation in the Z-axis, where the red color is the major deformation. This deformation is consistent with the real deformations in the experiments that will be detailed in further sections. Tail Actuators Comparison  
MFC present some features that can improve the behavior of these as actuators when compared with traditional solutions. This subsection will focus on explaining the advantages that are offered by MFC when compared with the traditional way of actuating a flapping-wing tail, the servomotor. Figure 3: Theoretical Model Representation. The comparison will be done between a commercial SAVOX SH-0255 MG servomotor, which is used in the same project for the traditional approach of the tail, and the previously described P1-8514 MFC. The geometry of both actuators is the first main difference between both systems. The servomotor has a cubic shape of \(22.8\times 12\times 29.4\) mm. This is a bulky shape in all directions. In comparison, the MFC has a shape of \(101\times 20\times 0.5\) mm. As can be seen, MFC has a very low height. This characteristic allows the integration of these actuators in much more restrictive geometries, allowing it to merge with the desired surface. Additionally, it is remarkable to mention the energy efficiency. The one of servomotors is high, around 85 percent. However, the one of the MFC is virtually 100 percent, as the energy consumption of these actuators is almost zero. The weight is also reduced when using the MFC. A main reason for this is that MFC are used as a structural part of the design. They add stiffness to the system and allow for reducing in other materials, so the design is more optimized in terms of weight when compared with the external actuation performed by the servomotor. The speed is also an important factor, affecting the capability of controlling the tail. The mechanical reaction of the servomotor is acceptable, but not instantaneous, while the MFC have an almost instant reaction when tension is applied to them, as a result of the properties of piezoelectric to deform when a electrical field is applied.