## 2 Theoretical Analysis  
### UAM Method  
The platform and geometry of the wing considered in the present study is shown in Figure 2a. It should be noted that the wings are assumed to be rigid and the deformation of wings are not considered, which serves a limitation of the study. A 2D airfoil of the rigid wing defined in an inertial coordinate system is illustrated in Figure 2b. The UAM developed in [20] is used to calculate the  
Figure 1: The landing trajectories of a bird and fixed-wing aircraft. aerodynamic forces of the airfoil. The inertial coordinate system (\(\widetilde{\xi}-\overline{\eta}\)) remains fixed with respect to the Earth. The airfoil coordinate system (\(\xi-\eta\)) is fixed at the center of the airfoil and moves with it during flapping motion. The 2D airfoil has three degrees of freedom as illustrated in Figure 2, the pitching (\(\alpha,\alpha\)), lunging (\(l,l\)), and heaving (\(h,h\)), where the dot () means the differentiation with respect to time. As shown in Figure 3, the classical Joukowski transformation [11; 12] can be used to uniquely map a circle of radius \(R\) in the \(z\)-plane into an airfoil in the \(\zeta\)-plane. The transformation can be expressed in \(\zeta\)-plane as follows:  
\[\xi=2 Rcos\theta+\sigma\Big{(}sin\theta-\frac{1}{2}sin2\theta\Big{)}-\tau\Big{(} cos\theta-\frac{1}{2}cos2\theta\Big{)} \tag{1}\]  
\[\eta=\tau\Big{(}sin\theta-\frac{1}{2}sin2\theta\Big{)}+\sigma\Big{(}cos\theta- \frac{1}{2}cos2\theta\Big{)} \tag{2}\]  
where \(\zeta\) and \(z\) are the complex coordinates in the \(\zeta\)-plane and the \(z\)-plane respectively; \(\tau\) and \(\sigma\) are non-dimensional factors governing the thickness and the camber of the airfoil. \(\theta\) is the angular displacement about the origin of the \(z\)-plane. According to the UAM [20], the aerodynamic force dF of a 2D airfoil of unit span \(dx\) can be expressed:  
\[\mathrm{dF}=(F_{0}+F_{1}+F_{2}+F_{3})dx \tag{3}\]  
Figure 3: Joukowski transformation of a circle to an airfoil. Figure 2: (**a**) The 3D wing platform and dimension; (**b**) a 2D airfoil in inertial coordinate system. \[F_{0}=\rho\overline{U_{0}}\Gamma_{0} \tag{4}\]  
\[F_{1}=\rho\frac{d}{dt}\int_{a}\xi\Upsilon_{0}(\xi)d\xi \tag{5}\]  
\[F_{2}=\rho\frac{d}{dt}\int_{a}\xi\Upsilon_{1}(\xi)d\xi \tag{6}\]  
\[F_{3}=-\rho\overline{U_{0}}\Gamma_{0}+\rho\frac{d}{dt}\int_{\
u k}\xi\Upsilon_ {\
u k}(\xi)d\xi+\rho\frac{d}{dt}\int_{l_{0}}\xi\Upsilon_{l_{0}}(\xi)d\xi \tag{7}\]  
\[\xi=2 Rcos\theta+\sigma\Big{(}sin\theta-\frac{1}{2}sin2\theta\Big{)}-\tau\Big{(} cos\theta-\frac{1}{2}cos2\theta\Big{)} \tag{1}\]0  
\[\xi=2 Rcos\theta+\sigma\Big{(}sin\theta-\frac{1}{2}sin2\theta\Big{)}-\tau\Big{(} cos\theta-\frac{1}{2}cos2\theta\Big{)} \tag{1}\]1  
where \(\overline{U_{0}}\) refers to the complex conjugate of the incident velocity \(U_{0}\). \(\rho\) is the air density. \(\Gamma_{0}\) is the total quasi-steady wake-free bound circulation. \(\Upsilon\) is the vorticity in the aerofoil-wake system which can be decoupled into the aerofoil-bound component \(\Upsilon_{a}\) and the shed component \(\Upsilon_{s}\). \(\Upsilon_{0}\) is the wake-free component of \(\Upsilon_{a}\) and \(\Upsilon_{1}\) is the wake-induced component. \(\Upsilon_{wk}\) and \(\Upsilon_{l_{0}}\) are the trailing edge wake and the leading edge wake component of \(\Upsilon_{s}\). \(\mathrm{R}\) is the wing dimension as in \[\xi=2 Rcos\theta+\sigma\Big{(}sin\theta-\frac{1}{2}sin2\theta\Big{)}-\tau\Big{(} cos\theta-\frac{1}{2}cos2\theta\Big{)} \tag{1}\]2 In addition to the above parameters, the wing motion is represented by the pitching angle \((\alpha,\alpha)\), lumging \((l,l)\) having amplitude \((h,h)\) and incident velocity \(U_{0}\) that are dimensional parameters in Equations (4)-(9) and illustrated in Figure 2b. The UAM takes into account the shedding process of the leading edge and trailing edge vortices considering the stall case in a very large AOA.