# A Miniature Flapping Mechanism Using an Origami-Based Spherical Six-Bar Pattern
## 3 Mechanism Analysis
### Kinematic Modeling  
For three-dimensional analysis, the spherical coordinate system is employed as shown in Figure 6. Based on the coordinate, a vector loop formula is derived as follows:  
\[\overset{\rightarrow}{r}+\overset{\rightarrow}{L}_{3}+\overset{\rightarrow}{L }_{2}+\overset{\rightarrow}{L}_{1}=\overset{\rightarrow}{G} \tag{1}\]  
\[r\left[ \begin{array}{c}\cos\theta_{1}\\ \sin\theta_{1}\\ 0\end{array} \right]+L_{3}\left[ \begin{array}{c}\cos\theta_{2}\\ \sin\theta_{2}\\ 0\end{array} \right]+L_{2}\left[ \begin{array}{c}\sin\phi_{3}\cos\theta_{2}\\ \sin\phi_{3}\sin\theta_{2}\\ \cos\phi_{3}\end{array} \right]+L_{1}\left[ \begin{array}{c}\sin\phi_{4}\\ 0\\ \cos\phi_{4}\end{array} \right]=\left[ \begin{array}{c}G\\ 0\\ -d\end{array} \right] \tag{2}\]  
where \(r\), \(L_{1}\), \(L_{2}\) and \(L_{3}\) are the length of the crank, the output link, the connecting rod, and the spherical bar, respectively. The angle parameters such as \(\theta_{1}\), \(\phi_{4}\), \(\theta_{2}\), and \(\phi_{3}\) are indicated in Figure 6.  
For the first condition, the transmission needs to be designed for the wings to have a symmetric flapping angle, \(\pm 40^{\circ}\), from the neutral position. When applied to this kinematic modeling, the highest and lowest values of \(\phi_{4}\) based on the neutral point (\(\phi_{4}\) =180\({}^{\circ}\)) are generated as 180\({}^{\circ}\) + 40\({}^{\circ}\) and 180\({}^{\circ}\) - 40\({}^{\circ}\), respectively. \[r\left[ \begin{array}{c}1\\ 0\\ 0\end{array} \right]+L_{3}\left[ \begin{array}{c}1\\ 0\\ 0\end{array} \right]+L_{2}\left[ \begin{array}{c}\sin\phi_{3}\\ 0\\ \cos\phi_{3}\end{array} \right]+L_{1}\left[ \begin{array}{c}\sin(180^{\circ}+ 40^{\circ})\\ 0\\ \cos(180^{\circ}+40^{\circ})\end{array} \right]=\left[ \begin{array}{c}G\\ 0\\ -d\end{array} \right] \tag{3}\]  
\[\left[ \begin{array}{c}-1\\ 0\\ 0\end{array} \right]+L_{3}\left[ \begin{array}{c}1\\ 0\\ 0\end{array} \right]+L_{2}\left[ \begin{array}{c}\sin\phi_{3}\\ 0\\ \cos\phi_{3}\end{array} \right]+L_{1}\left[ \begin{array}{c}\sin(180^{\circ}- 40^{\circ})\\ 0\\ \cos(180^{\circ}-40^{\circ})\end{array} \right]=\left[ \begin{array}{c}G\\ 0\\ -d\end{array} \right] \tag{4}\]  
Based on the Equations (3) and (4), the relation between \(L_{2}\) and \(L_{3}\) is given as follows:  
\[{L_{2}}^{2}=\left(G-L_{3}\right)^{2}+\left(\frac{r}{\tan 40^{\circ}}-d\right)^{2} \tag{5}\]  
The secondary condition is to make the neutral of the crank and the neutral of the wing equal. In order to get the thrust during a wing-beat cycle, the key is to maintain the angle of attack (AoA) of the wing. Theoretically, when the equal shape and speed of the wings are given, 45\({}^{\circ}\) is the proper AoA for the maximum thrust [34]. In order to maintain the AoA, the wing angular velocities of upstroke and downstroke need to be as symmetrical and constant as possible. This condition is indicated in Figure 6, which shows the relation between the flapping angle and the crank angle. When the curve  
Figure 6: The vector loop of the constrained spherical six-bar transmission for the kinematic analysis. Diagonal view (**A,B**) and x-z plane view (**C**). corresponds to the ideal cosine curve, the condition is satisfied. In Figure 6, the output flapping angle is \(\pi\) when the crank input angle is \(\pi/2\) and \(3\pi/2\). That is, it is preferable that the path length is the same as the path length between the highest point and the neutral point (from the point \(a\) and \(b\)) and the path length between the neutral point and the lowest point (from the point \(b\) to \(c\)). \[r\left[ \begin{array}{c}0\\ 1\\ 0\end{array} \right]+L_{3}\left[ \begin{array}{c}\cos\theta_{2b}\\ \sin\theta_{2b}\\ 0\end{array} \right]+L_{2}\left[ \begin{array}{c}\sin\phi_{3b}\cos\theta_{2b} \\ \sin\phi_{3b}\sin\theta_{2b}\\ \cos\phi_{3b}\end{array} \right]+L_{1}\left[ \begin{array}{c}0\\ 0\\ -1\end{array} \right]=\left[ \begin{array}{c}G\\ 0\\ -d\end{array} \right] \tag{6}\]  
\[r\left[ \begin{array}{c}0\\ -1\\ 0\end{array} \right]+L_{3}\left[