## 2 Materials and Methods
### Measurement and Calculation of Kinematics Parameters  
#### 2.2.1 Coordinate System of the Flight Kinematics  
In this study, the Cartesian coordinate system is introduced to describe the wing and body kinematics of a butterfly. Three coordinate systems are applied to define the axis, the plane, and the characteristic angles related to the butterfly motion in Figure 1e: The first is the inertial coordinate system \((X,Y,Z)\); its origin \((O)\) is set at the initial center of gravity (COG), which coincides with initial release position. The \(X\)-axis is defined therein as the direction in which a butterfly moves forward, while the \(Y\)-axis is the direction opposite to that of gravity. The thorax rotation around the \(Z\)-axis is described with pitch motion, which is defined as the body pitch angle between the centerline of the thorax and horizontal plane  
Figure 1: Observation experiment for butterfly flight based on high-speed camera: **(a)** geometry of a _Chilasa clytia_ and the computational model with mesh, butterfly body mesh model; **(b)** live butterfly with the forewing (FW) and hindwing (HW) in natural environment; **(c)** schematic diagram of butterfly observation feature points and body kinematic parameters; **(d)** experiment system; and **(e)** three coordinate systems to describe the motion of a butterfly, namely the inertial coordinate system \((X,Y,Z)\), the body-fixed coordinate system \((X_{b},Y_{b},Z_{b})\) and wing-fixed coordinate system \((X_{av},Y_{iw},Z_{w})\).  
(X-\(Z\) plane). For instance, if the pitch angle is zero, the body will lie in horizontal plane; when it is 90\({}^{\circ}\), the body will be vertical.  
The body-fixed coordinate system \((X_{b},Y_{b},Z_{b})\) is adopted to describe the body motion. The origin of the body-fixed coordinate system \((O_{b})\) is set at the COG of the butterfly, which translates and rotates with the body. During the forward flight, the flight velocity of the butterfly is expressed as \(V\), and the butterfly's displacement is decomposable into two directions in the inertial coordinate system, the \(X\)- and \(Y\)-axis. The velocities in the \(X\) and \(Y\) directions are horizontal velocity \(v_{x}\) and vertical velocity \(v_{y}\), respectively. The symmetry axis of the butterfly thorax is defined as the thorax axis, which coincides with \(X_{b}\)-axis. The symmetry axis of the butterfly abdomen is defined as the abdomen axis. \(X_{b}\)-Z\({}_{b}\) and \(X_{b}\)-\(Y_{b}\) denote the horizontal and vertical body planes, respectively. The abdomen rotates around the \(Z_{b}\)-axis, and the angle is defined as abdominal swing angle.  
The wing-fixed coordinate system \((X_{w},Y_{w},Z_{w})\) is adopted to describe the wing-flapping motion. The motion of the wings is connected and simultaneous with the butterfly body in real flight. The origin of the wing-fixed coordinate system \((O_{w})\) is set at the wing root and translates and rotates with the body. The butterfly flaps its wings around \(X_{w}\)-axis, which is simplified to be parallel to the thoracic axis in this paper; the flapping angle is 90\({}^{\circ}\) when the two wings clap together.  
The plane swept by the connection between the wing tip and the wing root during the fight is defined as the stroke plane of the butterfly wing. The angle between the stroke plane and the horizontal \(X\)-\(Z\) plane is defined as \(\beta\). In this study, the thoracic pitch angle, the abdominal swing angle, and the approximate flapping angle are denoted by \(\theta_{p}\), \(\theta_{a}\), and \(\theta_{f}\), respectively. This paper mainly considers the longitudinal flight of butterflies, so the roll and yaw motions are ignored.