## 3 Results  
### Wing and Body Kinematics of Butterfly Forward Flight  
Our aim in this paper is to present the clearest forward flight and reveal the special flying skills of the butterfly (_Chilasa clytia_). Ten videos of the butterfly free flying about \(300\) ms long were filmed. Two of them were selected as the two cases (denoted as VC1 and VC2) of forward flight because the displacement of the butterfly centroid in the \(X\)-axis and \(Y\)-axis directions satisfied \(|dy/dx|<0.15\). For the two cases, the analyzed kinematics results of the butterfly are shown in Table 4, including the total time of forward flight (\(t_{\mathrm{total}}\)), the average stroke amplitude (\(\bar{A}_{f}\)), the average flapping frequency (\(\bar{f}\)), the average pitch angle (\(\bar{\theta}_{p}\)), the maximum pitch angle  
Figure 3: The real motion (top) and prescribed motion (bottom) of the butterfly during a stroke: (**a**,**e**) start of downstroke; (**b**,**f**) mid-downstroke; (**c**,**g**) start of upstroke; and (**d**,**h**) mid-upstroke. \((\theta_{p\,\text{max}})\), the minimum pitch angle \((\theta_{p\,\text{min}})\), the average abdominal swing angle \((\bar{\theta}_{a})\), the maximum abdominal swing angle \((\theta_{a\,\text{max}})\), and the minimum abdominal swing angle \((\theta_{a\,\text{min}})\). Two complete flapping cycles were intercepted from the two videos, named Stroke #1 and Stroke #2, which correspond to the subscript numbers, respectively. For each of the flights, the recordings of 2-3 wing flapping cycles were digitized. Snapshots of a butterfly in forward flight VC1 and VC2 from the front-view camera are shown in Figure 4a,b, respectively. Because the displacement along the \(Z\)-axis is very close to zero throughout the entire process of forward flight, only the displacements along the \(X\)-axis and \(Y\)-axis are plotted. The coordinates at 0 ms are set to (0, 0, 0). Figure 4: Snapshots from the front-view camera of a butterfly in forward flight: VC1 (**a**); and VC2 (**b**). Trajectories of the butterfly feature points in the upward direction (along the \(Y\)-axis) and the forward direction (along the \(X\)-axis) during forward flight: VC1 (**c**); and VC2 (**d**). \begin{table}
\begin{tabular}{l c c c c c c c c c c c c c c} \hline \hline
**ID** & \(\theta_{1p\,\text{max}}\) & \(\theta_{1p\,\text{min}}\) & \(\bar{\theta}_{1p}\) & \(\theta_{1a\,\text{max}}\) & \(\theta_{1a\,\text{min}}\) & \(\bar{\theta}_{1a}\) & \(\theta_{2p\,\text{max}}\) & \(\theta_{2p\,\text{min}}\) & \(\bar{\theta}_{2p}\) & \(\theta_{2a\,\text{max}}\) & \(\theta_{2a\,\text{min}}\) & \(\bar{\theta}_{2a}\) & \(t_{\text{total}}\) & \(\bar{\mathbf{A}}_{f}\) & \(\bar{\mathbf{f}}\) \\ \hline VC1 & 62.7 & 19.2 & 38.5 & \(-11.9\) & \(-72.8\) & \(-33.7\) & 79.3 & 15.2 & 42.2 & \(-17.8\) & \(-89.4\) & \(-47.7\) & 290 & 120 & 10.6 \\ VC2 & 59.1 & \(-0.1\) & 29.2 & 13.3 & \(-46.9\) & \(-13.8\) & 89.7 & 23.5 & 60.8 & \(-17.3\) & \(-88.6\) & \(-42.2\) & 297 & 135 & 9.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Kinematics parameters of the butterfly obtained by experimental observation. Figure 4c,d shows the trajectory of the butterfly feature points during the forward flight (space trajectory curve and body trajectory curve) recorded using high-speed photography, respectively. It can be seen that the body and wings of the butterfly have obvious oscillations during flight. After tracking all the feature points based on all recorded frames, the displacements and kinematic curves of the butterfly in the relative coordinate system were obtained from VC1 and VC2 and plotted in Figures 5 and 6, respectively. All the stroke moments were also labeled. Grey shading indicates the DS, while white shading denotes the US. The purple shading indicates the gliding flight from the end of the US to the beginning of the next flapping cycle.