# Dynamics Modelling and Simulation for Deployment Characteristics of Mesh Reflector Antennas
## 2 Dynamic Modelling of the Truss Antenna  
### Dynamic Equations of Truss  
The deployable truss consists of a certain amount of parallelogram mechanisms named truss cells, which provide a smooth deployment for the antenna. As illustrated in Figure 2, two adjacent cells comprise of three synchronous joints (point \(B_{i},D_{i},C_{i+1}\)), three T-shaped joints (Points \(A_{i},C_{i},D_{i+1}\)), horizontal rods, vertical rods, and diagonal sleeves. The truss can be deployed by pulling the cables through the diagonal sleeve after a pre-deployed phase provided by the torsional spring.  
To describe the motion of the truss, as illustrated in Figure 2, a body coordinate frame of reference \(O_{i}x_{i}y_{i}z_{i}\) is attached on the truss cell \(i(i=1\sim n)\), in which axis \(O_{i}y_{i}\) coincides with the rod \(A_{i}B_{i}\). For a deployable truss consisting of \(n\) cells, define \(\varphi=2\pi/n\) as the angle between the coordinate systems \(O_{i}x_{i}y_{i}z_{i}\) and \(O_{i+1}x_{i+1}y_{i+1}z_{i+1}\). The configuration of truss cell is determined by the deployment angle \(\theta\), which is the angle between \(A_{i}D_{i}\) and the axis \(x_{i}\). Assuming the global coordinate reference is attached to the body frame \(O_{1}x_{1}y_{1}z_{1}\), the transformation matrix from the frame \(O_{i}x_{i}y_{i}z_{i}\) to \(O_{i+1}x_{i+1}y_{i+1}z_{i+1}\) can be formulated as  
\[\mathbf{T}_{i+1}=\left[\begin{array}{cccc}\cos\varphi&0&\sin\varphi&X_{Di}\\ 0&1&0&Y_{Di}\\ -\sin\varphi&0&\cos\varphi&0\\ 0&0&0&1\end{array}\right] \tag{1}\]  
where \(L_{2}\) is the length of the horizontal rod. Note that the deployment angle \(\theta\) is positive when the cell \(i\) is odd, while \(\theta\) is negative when \(i\) is even. \(X_{Di}\) and \(Y_{Di}\), the translational displacement from \(O_{i}\) to \(O_{i+1}\), can be obtained by  
\[\left\{\begin{array}{l}X_{Di}=L_{2}\cos\theta,Y_{Di}=L_{2}\sin\theta,\quad \text{when $i$ is odd number}\\ X_{Di}=L_{2}\cos\theta,Y_{Di}=-L_{2}\sin\theta,\quad\text{when $i$ is even number}\end{array}\right. \tag{2}\]  
Then, the global position vector of an arbitrary point on the truss cell \(i\) can be obtained by transforming the body frame into the global frame and it can be expressed as  
Figure 2: Deployment of the deployable truss.  
\[\mathbf{s}=\left[\begin{array}{c}x\\ y\\ z\\ 1\end{array}\right]=\left[\begin{array}{ccc}\cos[(i-1)\varphi]&0&\sin[(i-1) \varphi]&\{\sum\limits_{k=2}^{i}\cos[(k-2)\varphi]|L_{2}\cos\theta\\ 0&1&0&0.5[1-(-1)^{i-1}]L_{2}\sin\theta\\ -\sin[(i-1)\varphi]&0&\cos[(i-1)\varphi]&-\sum\limits_{k=2}^{i}\sin[(k-2) \varphi]|L_{2}\cos\theta\\ 0&0&0&1\end{array}\right]\mathbf{P}_{i} \tag{3}\]  
where \(\mathbf{s}\) is the global position vector of the truss and \(\mathbf{P}_{i}=[P_{\mathrm{i}x},P_{\mathrm{i}y},P_{\mathrm{i}x},1]^{\mathrm{T}}\) is the position vector of the point defined in the body coordinate system.  
Velocity vector of the point on the truss can be derived as  
\[\mathbf{v}=\left[\begin{array}{c}v_{x}\\ v_{y}\\ v_{z}\end{array}\right]=\left[\begin{array}{c}-\mathrm{i}\sum\limits_{k=2} ^{i}\cos[(k-2)\varphi]|L_{2}\dot{\partial}\sin\theta\\ 0.5[1-(-1)^{i-1}]L_{2}\partial\cos\theta\\ \{\sum\limits_{k=2}^{i}\sin[(k-2)\varphi]|L_{2}\partial\sin\theta\end{array} \right] \tag{4}\]  
Mass of the rod can be equivalent to the nodes of the rod, and ignoring the geopotential energy of the deployable truss considering the working condition, the kinetic energy can be expressed as  
\[T_{t}=m_{t}\sum\limits_{j=2}^{n}\left(v_{x}^{2}+v_{y}^{2}+v_{z}^{2}\right) \tag{5}\]  
where \(m_{t}\) is the equivalent mass of the rod.