# Development of an Insect-like Flapping-Wing Micro Air Vehicle with Parallel Control Mechanism
## 2 Prototype Design
### Control Mechanism  
The parallel mechanism is suitable for FMAV control due to its compact construction and high stiffness. A 2 RSS/U parallel mechanism was adopted to control the stroke plane of FMAV. The stroke plane determines the direction of aerodynamic forces and moments generated by the flapping wing mechanism. Controlling the orientation of the stroke plane can produce the required control torque. The schematic of the control mechanism is shown in Figure 6, where the plane \(AOD\) is the controlled plane with two degrees of freedom of rotation under the constraints of a Hooke hinge at point \(O\). Point \(C\) and \(F\) are plane hinges around the \(y\)-axis. Points \(A,\ B,\ D\), and \(E\) are spherical hinges. Point \(P\) is the projection of point \(O\) in the plane \(CPF\). The outputs are the angles of rotation \(\alpha,\ \beta\) around the x and y axes and the inputs are two angles of rotation \(\xi_{1},\ \xi_{2}\) that by the servo. Figure 5: The physical drawing of the flapping wing mechanism. Figure 6: The schematic of pitch and roll control. Plane \(AOD\) indicates the stroke plane. Rod \(BC\) and \(EF\) indicate the servo rocker. The control chain of the parallel mechanism can be described by Equation (4). \[\left\{ \begin{array}{l}\overset{\rightarrow}{OC}=O\alpha+\overset{\rightarrow}{ AB}+\overset{\rightarrow}{BC}=[-d_{1},d_{2},-h_{1}]^{T}\\ \overset{\rightarrow}{OF}=\overset{\rightarrow}{OD}+\overset{\rightarrow}{ DE}+\overset{\rightarrow}{EF}=[d_{1},d_{2},-h_{1}]^{T}\end{array} \right. \tag{4}\]  
The position of rocker \(BC\) and \(EF\) is defined by \(\xi_{1},\ \xi_{2}\) as Equation (5). \[\left\{ \begin{array}{l}\overset{\rightarrow}{BC}=[-l_{1}\cos\xi_{1},0,-l_{1 }\sin\xi_{1}]^{T}\\ \overset{\rightarrow}{EF}=[l_{1}\cos\xi_{2},0,-l_{1}\sin\xi_{2}]^{T}\end{array} \right. \tag{5}\]  
The attitude of the controlled plane can be defined by \(\alpha,\ \beta\) as Equation (6). \[\left\{ \begin{array}{l}\overset{\rightarrow}{OA}=\mathbf{Q}_{x}\mathbf{Q}_ {y}[-d_{3},d_{4},0]^{T}\\ \overset{\rightarrow}{OD}=\mathbf{Q}_{x}\mathbf{Q}_{y}[d_{3},d_{4},0]^{T} \end{array} \right. \tag{6}\]  
\[\text{where}\ \mathbf{Q}_{x}=\left[ \begin{array}{cccc}1&&\\ \cos\alpha&-\sin\alpha\\ \sin\alpha&\cos\alpha\end{array} \right],\mathbf{Q}_{y}=\left[ \begin{array}{ ccccc}\cos\beta&&\sin\beta\\ &1&&\\ -\sin\beta&&\cos\beta\end{array} \right].\]  
Therefore, we can establish the correlation equation for inputs \(\xi_{1},\ \xi_{2}\) and outputs \(\alpha,\ \beta\) by Equation(7). \[\left| \begin{array}{l}\overset{\rightarrow}{AB}\\ \overset{\rightarrow}{DE}\end{array} \right|=\left|\overset{\rightarrow}{OC}- \overset{\rightarrow}{OA}-\overset{\rightarrow}{BC}\right|=l \tag{7}\]  
The required drive force can be calculated by the principle of virtual work, as shown in Equation (8). The loads on the control plane are defined as \(M_{x},\ M_{y}\), and the drive torque required by servos are defined as \(M_{1},\ M_{2}\). \(\varphi\) is the generalized coordinate. After obtaining the relationship of \(\xi_{1},\ \xi_{2}\), and \(\alpha,\ \beta\), we can calculate the partial differential numerically to obtain the required drive torque of the servos under different servo angles, which helps to avoid overload. \[\sum M=M_{x}\frac{\partial\alpha}{\partial\varphi}+M_{y}\frac{\partial\beta}{ \partial\varphi}+M_{1}\frac{\partial\xi_{1}}{\partial\varphi}+M_{2}\frac{ \partial\xi_{2}}{\partial\varphi} \tag{8}\]  
In order to determine the appropriate mechanism parameters, we need to take into account the relationship between the dimensions of the components, the maximum control angle required, the working angle of the servo, and the maximum output torque. This set of parameters is chosen as the design parameters for the parallel mechanism, as shown in Table 1.  
Considering the input and output variables of this mechanism, the numerical method is effective to obtained the forward and inverse solutions. The different drive platform angles corresponded to the output angles of the left and right servos, as shown in Figure 7.