## 3 Results
### Wing and Body Kinematics of Butterfly Forward Flight
After being released from the initial release position, the butterfly first flaps its wings slightly for one or two strokes with a small reduction in flight altitude, and its average flight speed increases accordingly; then, the average pitch angle increases, which is equivalent to an increase in the angle of attack of the body. The butterfly starts to climb, and its average flight speed decreases accordingly. The vertical acceleration \((a_{y})\) reaches the maximum value at the mid-downstroke of the DS, at which time a lift peak appears, and almost no lift is generated during the US. For almost the entire downstroke, \(a_{y}\) is positive because the instantaneous vertical aerodynamic force of the butterfly is lift, while the horizontal acceleration \((a_{x})\) is positive because the instantaneous horizontal aerodynamic force is drag. \(a_{x}\) reaches the maximum value at near the mid-upstroke of the US, at which time a peak of the thrust appears. The kinematics analysis result shows that the phase difference between butterfly flapping motion and pitching motion is close to \(\pi/2\). Under different flight conditions, the phase difference between the pitch angle and the swing angle varies greatly, which is about \(\pi\) in our observation. Figure 5: Time variations of body kinematics of VC1: (**a**) displacement of the joint point during forward flight; (**b**) velocity of the butterfly centroid (joint point); (**c**) acceleration of the butterfly centroid and FFT smoothing for the second flapping cycle; and (**d**) oscillation of thoracic pitch angle \((\theta_{p})\) and abdominal swing angle \((\theta_{a})\). Grey shading indicates the DS, purple shading denotes the gliding moment, and white shading denotes the US. The observation analysis of wing tip kinematics is shown in Figure 7. The maximum velocity of the wing tip is reached when the wings reach mid-downstroke or mid-upstroke position during flapping downward or upward, respectively. \(v_{x}^{wt}\) and \(v_{y}^{wt}\) change periodically with flapping motion. The maximum and minimum values of the resultant velocity of the wingtip in VC1 are 2.6 and 0.6 m/s, respectively. The average resultant velocity of the wingtip is 1.6 m/s. Similarly, the average, maximum, and minimum resultant velocity of the wingtip in VC2 are 1.4, 2.1, and 0.6 m/s, respectively. Figure 6: Time variations of body kinematics of VC2: (**a**) displacement of the joint point during forward flight; (**b**) velocity of the butterfly centroid (joint point); (**c**) acceleration of the butterfly centroid and FFT smoothing for the second flapping cycle; and (**d**) oscillation of thoracic pitch angle (\(\theta_{p}\)) and abdominal swing angle (\(\theta_{a}\)). Grey shading indicates the DS, purple shading denotes the gliding moment, and white shading denotes the US. Figure 7: Time variations of wing tip kinematics of: VC1 (**a**); and VC2 (**b**). During the flight, the glide flight mode is used to reduce flight energy consumption. This mode often appears between the end of the upper flap and the beginning of the next flapping cycle. At this time, the wings are almost closed on the back, the projection area in the flight direction is the smallest, and the air resistance is also small. Therefore, the main body has a certain height drop during the gliding process due to its own inertia, and the flight velocity will increase accordingly. Due to the unstable flight of the butterfly, it is difficult to capture the state of the butterfly's continuous forward flight under free flight conditions. In this experiment, the butterfly is released from a fixed position to guarantee that the butterfly's head faces the \(-X\) direction of the space. For VC1 and VC2, the flying behaviors of the experimental butterfly are similar. After the butterfly is released, it accelerates to fly forward in the initial stage, and the velocity and flight trajectory have fluctuation characteristics with the same frequency as the flapping motion.