# Precision Analysis and Error Compensation of a Telescope Truss Structure Based on Robotics
## 2 Structural Analysis of the Telescope Manipulator Based on Stiffness Performance Index
### Optimization Principle  
Assuming that the connecting rod of the robot arm is a rigid body, the main reason for the deformation of the robot arm comes from the joints. Each joint can be regarded as a linear spring body. Thus, by connecting the SM to the end of the robot arm, the displacement of the SM can be passed through the robot arm. The displacement at the end is as follows:  
\[\Delta x_{g}=Cw, \tag{1}\]  
where \(\Delta x_{g}\) and \(Cw\) represent the generalized displacement and generalized force, respectively. \(C\) is a \(6\times 6\) flexibility matrix,  
\[C=JK_{\theta}^{-1}J^{T}=\left[ \begin{array}{cc}C_{tt}&C_{tr}\\ C_{tr}^{T}&C_{rr}\end{array} \right], \tag{2}\]  
and \(K_{\theta}\) is the joint stiffness matrix of the robotic arm, defined as  
\[K_{\theta}=\left[ \begin{array}{cccc}k_{1}&0&\cdots&0\\ 0&k_{2}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&k_{6}\end{array} \right]. \tag{3}\]  
Owing to the constraints of the distance between the PM and SM, we selected the FANUC Robot M-20 iA/12L as the research object. The corresponding \(K_{\theta}\) can be expressed as diagonal matrix \((5.88\times 10^{8},5.36\times 10^{8},2.71\times 10^{8},2.12\times 10^{8},1.22 \times 10^{8},1.06\times 10^{8})\) (\(N\cdot mm/rad\)). \(J\) is the Jacobian matrix of the robotic manipulator. We established the coordinate system of each joint for the selected manipulator, as shown in Figure 2. Then, the Denavit-Hartenberg parameters (DHm) can be obtained, as shown in Table 1.  
Figure 2: Kinematics model. (**a**) The coordinate system establishment. (**b**) The Kinematic parameters. Considering the slow motion of the telescope during observation, it can be regarded as a quasi-static process. The acceleration can be ignored, and only the influence of gravity is considered. Therefore, the force at the end of the truss can ignore the influence of the moment on the system, and only the translation needs to be considered. Therefore, the displacement of the SM can be expressed as:  
\[\left[ \begin{array}{c}\Delta X\\ 0\end{array} \right]=\left[ \begin{array}{cc}C_{tt}&C_{tr}\\ C_{tr}^{T}&C_{rr}\end{array} \right]\left[ \begin{array}{c}F\\ 0\end{array} \right]=\left[ \begin{array}{c}C_{tt}F\\ 0\end{array} \right], \tag{4}\]  
where \(F=\left[ \begin{array}{cc}F_{x}&F_{y}&F_{z}\end{array} \right]^{T}\) represents the force applied to the end of the robotic arm. Under the condition of pitching motion, \(F\) is composed of the components of gravity of the SM in different directions at different pitch angles of the telescope system. \(\Delta X=\left[ \begin{array}{cc}\Delta x&\Delta y&\Delta z\end{array}  \right]^{T}\) indicates the displacement at the end of the robotic arm. The misalignment of the secondary mirror can be calculated and verified using Equation (4). For a robotic system, the optimization of the posture can be regarded as the maximization of the stiffness coefficient:  
\[k_{stif}=\frac{1}{\sqrt[3]{\det(C_{tt})}}=\frac{1}{\sqrt[3]{\det(J_{11}K_{11}^ {-1})_{11}^{T}+J_{12}K_{22}^{-1})_{12}^{T}}}. \tag{5}\]  
Among them, \(J_{11}\), \(J_{12}\) is the \(3\times 3\) sub-matrix of \(J\):  
\[J=\left[ \begin{array}{cc}J_{11}&J_{12}\\ J_{21}&J_{22}\end{array} \right]. \tag{6}\]  
\(k_{stif}\) is affected by \(\theta_{i}\), \(a_{i}\), and \(\alpha_{i}\), where \(a_{i}\) and \(\alpha_{i}\) are values determined after establishing the coordinate system, and \(\theta_{i}\) is the angle between \(J_{i}\) and \(J_{i+1}\). The stiffness coefficient has the invariance of coordinate system transformation, and therefore, the optimization model can be expressed as:  
\[\left\{ \begin{array}{c}max_{stiff}\\ s.t.T(q)=T(P,N,\theta)\\ q_{inin}\leq q_{i}\leq q_{inin},1\leq i\leq 6\end{array} \right., \tag{7}\]  
where \(q=\left[ \begin{array}{cccc}q_{1}&q_{2}&q_{3}&q_{4}&q_{5}&q_{6}\end{array} \right]\) is a vector composed of the angles, and \(T(P,N,\theta)\) is the pose matrix connected with the SM that contains the elements \(P,\ N,\ \text{and}\ \theta\).