# Design and Kinematic Analysis of Deployable Antenna for Bionic Bird Tail Feather
## 4 Kinematic Analysis  
### Kinematic Modeling  
Due to the constraint of the synchronous slider mechanism, the 18 ribs have the same angle of spread between ribs at all times. To simplify the analysis, only the position velocity calculation of the driving part components is needed to launch the motion velocity of all ribs, as shown in Figure 17.  
The structure contains six movable members, seven low and zero high subs, according to the formula of degrees of freedom:  
\[F=3n-2p_{l}-p_{h} \tag{1}\]  
The system is calculated as a two-degree-of-freedom system.  
\(F\) is the degree of freedom of the mechanism; \(n\) is the number of moving members of the mechanism; \(p_{l}\) is the number of low subs; and \(p_{h}\) is the number of high subs.  
Figure 16: Schematic diagram of the second section of the mechanism.  
Figure 14: Schematic diagram of mechanism folding.  
Figure 15: Schematic diagram of the first section of mechanism expansion.  
The driving part of connection position 1 is divided into the driving linkage part and the synchronous mechanism part. First of all, the driving linkage part is analyzed: the auxiliary line is \(AD\), \(DI\), over \(A\) for the x-axis vertical line, extended \(DE\); the auxiliary angle is \(\alpha_{1}\), \(\alpha_{2}\), \(\alpha_{3}\), \(\alpha_{4}\), where \(CD=DE\), \(DE\bot EI\).  
The connecting rod is divided into two parts: the driving linkage part and the synchronous mechanism part, as shown in Figures 18 and 19. Through the solution of the geometric position of each linkage, Equations (2) and (3), respectively, were solved.  
\[\left\{\begin{array}{l}l_{\rm AD}=\sqrt{l_{\rm OC}^{2}+\left(x_{\rm A}-x_{ \rm D}\right)^{2}}\\ \theta_{1}+\alpha_{1}=\arccos[(l_{AD}^{2}+l_{AI}^{2}-l_{DI}^{2})/(2l_{\rm AD}l _{\rm AI})]\\ \alpha_{1}=\arctan[(x_{\rm A}-x_{\rm D})/l_{\rm OC}]\\ \alpha_{2}=\arccos[(l_{AD}^{2}+l_{DI}^{2}-l_{AI}^{2})/(2l_{\rm AD}l_{\rm DI})] \\ \alpha_{3}=\pi/2-\alpha_{1}-\alpha_{2}\\ \theta_{1}=(\theta_{1}+\alpha_{1})-\alpha_{1}\\ \theta_{2}=\alpha_{4}-\alpha_{3}\\ \alpha_{4}=\arctan(l_{\rm EI}/l_{\rm DE})\\ \end{array}\right. \tag{2}\]  
\[\left\{\begin{array}{l}l_{\rm CH}=l_{\rm CD}\cot(\theta_{2}/2)=l_{\rm HE}\\ l_{\rm BH}=l_{\rm BC}+l_{\rm CH}\\ l_{\rm HJ}=l_{\rm HE}+l_{\rm EJ}\\ \theta_{2}=2\pi-\arccos[(l_{\rm BH}^{2}+l_{\rm HJ}^{2}-l_{\rm B}^{2})/2l_{\rm BH }l_{\rm HJ})]\\ l_{\rm BJ}\sin\theta_{3}=l_{\rm HJ}\sin\theta_{2}\\ \end{array}\right. \tag{3}\]  
Figure 17: Schematic diagram of the drive part of connection position 1.  
Figure 18: Schematic diagram of driving connecting rod.  
The union of Equations (2) and (3) can be solved using  
\[l_{\rm HJ}=\frac{2l_{\rm BH}\cos\theta_{2}\pm\sqrt{\left(2l_{\rm BH}\cos\theta_{2} \right)^{2}-4l_{\rm BH}^{\,2}+4l_{\rm BJ}^{\,2}}}{2} \tag{4}\]  
This leads to the following:  
\[\left\{\begin{array}{l}l_{\rm EJ}=l_{\rm HJ}-l_{\rm HE}\\ \theta_{3}=\arcsin(l_{\rm HJ}\sin\theta_{2}/l_{\rm BJ})\end{array}\right. \tag{5}\]  
where \(l_{\rm xy}\) denotes the distance between the points x and y, and \(\theta_{\rm n}\) denotes the angle between the two line segments.  
The variation law of (3) is also known, so the expression of time t can be substituted into the angle of rotation of the connecting rod BJ and the displacement of the slider on the antenna rib.  
There are variables in the equation:  
\[l_{\rm OC}=y_{\rm C}-vt \tag{6}\]  
where the variables are shown in Table 1.