# Wing Geometry and Kinematic Parameters Optimization of Flapping Wing Hovering Flight
## 3 Wing Kinematics  
In this paper, the right wing Euler angles relative to stroke plane in right wing root frame of reference (\(x_{rr}y_{rr}z_{rr}\)) are defined by three angles (flapping angle (\(\varphi\)), stroke deviation angle (\(\theta\)) and pitch angle (\(\psi\)), Figure 3). The determination of direction of three angles obeys a right-hand rule. Moreover, the body frame of reference (\(X_{\text{b}}Y_{\text{b}}Z_{\text{b}}\)) and lab frame of reference (\(X_{\text{lab}}Y_{\text{lab}}Z_{\text{lab}}\)) for fruit fly are also illustrated for clarity. The stroke plane is the horizontal plane of the lab frame of reference (\(X_{\text{lab}}O_{\text{b}}Z_{\text{lab}}\)). The radius vector of \(O_{\text{root}}\) relative to \(O_{\text{b}}\) is neglected in posterior torques calculation. The inclined angle of longitudinal body axis (\(Z_{\text{b}}\)) relative to \(Z_{\text{lab}}\) axis (\(\beta\)) is \(47.5^{\circ}\) for hovering fruit fly [27]. Take a concise hovering design and simplifying optimization analysis for FWMAV in engineering into consideration [2; 12; 33], we assume the stoke deviation is absent. Using the coordination systems shown in Figure 3, we write the complete transformation matrix from the right wing root frame of reference (\(x_{rr}y_{rr}z_{rr}\)) to the right wing planform fixed frame (\(x_{rr}y_{rr}z_{rr}\)) as  
\[ \begin{array}{c}r\overset{rw}{r}R=\left[\begin{array}{ccc}\cos\phi&\sin \phi&0\\ -\cos\psi\sin\phi&\cos\psi\cos\phi&\sin\psi\\ \sin\psi\sin\phi&\sin\psi\cos\phi&\cos\psi\end{array}\right].\end{array}  \tag{7}\]  
The converse transformation matrix from the wing planform fixed frame to the right wing root frame of reference can be given by \(\overset{rr}{r_{rw}}R=\overset{rw}{rr}R^{-1}\). According to above transformation and wing kinematics differentiating to time, the wing angular velocity vector in right wing planform fixed frame  
Figure 3: Coordinate systems and definition of right wing Euler angles. can be acquired from the transformation of the middle coordination frame (mid) to the right wing planform fixed frame due to flapping and pitch motion, namely, \({}^{rw}\omega=\overset{rw}{mial}R_{x,-y}\dot{\phi}\overset{r}{k}_{mid}+\dot{ \psi}\overset{r}{i}\overset{r}{i}\overset{r}{r_{w}}\), where \(i\) and \(k\) denote unit vectors along \(x\) and \(z\) directions of local frame, respectively, and the right superscript of \({}^{rw}\omega\) is the abbreviation of the right wing. Thus \({}^{rw}\omega\) can be extended as  
\[{}^{rw}\omega=\left[ \begin{array}{c}\omega_{x}\\ \omega_{y}\\ \omega_{z}\end{array} \right]=\left[ \begin{array}{ccc}1&0&0\\ 0&\cos\psi&\sin\psi\\ 0&-\sin\psi&\cos\psi\end{array} \right]\left[ \begin{array}{c}0\\ 0\\ \dot{\phi}\end{array} \right]+\left[ \begin{array}{c}\dot{\psi}\\ 0\\ 0\end{array} \right]=\left[ \begin{array}{c}\dot{\psi}\\ \dot{\phi}\mathrm{sin}\psi\\ \dot{\phi}\mathrm{cos}\psi\end{array} \right], \tag{8}\]  
where the left superscript of \({}^{rw}\omega\) is the abbreviation of the right wing. The wing angular acceleration velocity vector \({}^{rw}\dot{\omega}\) in the right wing planform fixed frame can be expressed as  
\[{}^{rw}\dot{\omega}=\left[ \begin{array}{c}\dot{\omega}_{x}\\ \dot{\omega}_{y}\\ \dot{\omega}_{z}\end{array} \right]=\left[ \begin{array}{c}\ddot{\psi}\\ \ddot{\phi}\mathrm{sin}\psi+\dot{\phi}\dot{\psi}\mathrm{cos}\psi\\ \ddot{\phi}\mathrm{cos}\psi-\dot{\phi}\dot{\psi}\mathrm{sin}\psi\end{array}  \right]. \tag{9}\]  
Here, in order to additionally consider the effect of time scale of wing stroke reversal and wing pitch reversal on optimization including WKP, we adopt the wing kinematic pattern of [PERSON] and [PERSON] [8], but overlook the effect of stroke plane deviation angle on optimization because of its low amplitude [12; 21].