## 3 Geometric Error Measurement Algorithm
### Set-Up Error Identification  
As expected, the set-up errors of the reflector standard fixture greatly affect the measurement accuracy of the proposed measurement method. In this subsection, the mathematical measurement equations and estimation algorithm are proposed by using HTMs and the least squares method. In the ideal case, a rotary table operates along the prospective path. As a result, the ideal position of each reflector at each prospective measurement point can be estimated by the first measurement position. However, in a real case, the set-up errors of the reflector standard fixture always exist and influence the measurement values when the proposed measurement method is executed for measuring the rotary table geometric errors. According to the results mentioned above, the real positions of the reflectors at each  
Figure 1: Schematic diagram of the proposed measurement method. measurement point are obtained. Consequently, the difference between the ideal and real positions of the reflector at each measurement point is formulated with Equation (2):  
\[{}^{real}P(i,m)={}^{ideal}P(i,m)+dP^{m}_{k,i}\quad k\in x,y,z, \tag{2}\]  
where \({}^{real}P(i,m)\) and \({}^{ideal}P(i,m)\) denote the real and ideal positions of the \(m\)-th reflector at the \(i\)-th measurement point, respectively. Moreover, \(dP^{m}_{k,i}\) represents the difference between the ideal and real positions of the \(m\)-th reflector at the \(i\)-th measurement point in the \(x\), \(y\), and \(z\) directions. Furthermore, since the higher term has a small impact on the results of Equation (2), a first-order Taylor series expansion method is applied to calculate the difference between the ideal and real positions of the reflector at the \(i\)-th measurement point and can be expressed as follows:  
\[dP^{m}_{k,i}=\sum_{t=1}^{4}\frac{\partial P^{m}_{k,i}}{\partial E_{t}}\Delta E _{t}, \tag{3}\]  
where \(E_{t}\) represents the four set-up errors of the rotary table depicted above. When conducting the measurements of the rotary table, the rotary table is driven to the angular position \(\theta_{i}\) at the \(i\)-th measurement point to identify the real center positions of the reflector. Therefore, the relative equations for determining the difference between the ideal and real positions of the \(m\)-th reflector at the \(i\)-th measurement point in the \(x\), \(y\), and \(z\) directions and the set-up errors of the reflector standard fixture are given as follows:  
\[dP^{m}_{k,i}=\left[ \begin{array}{c}dP^{m}_{x,1}\\ dP^{m}_{y,1}\\ dP^{m}_{z,1}\\ \vdots\\ dP^{m}_{y,i}\\ dP^{m}_{z,i}\\ \end{array} \right]_{i}=A_{3i\times 4}E_{4\times 1}\] \[=\left[ \begin{array}{cccc}0&z_{1}^{m}-z_{1}^{m}\cos\theta_{1}- y_{1}^{m}\sin\theta_{1}&-y_{1}^{m}+y_{1}^{m}\cos\theta_{1}-z_{1}^{m}\sin\theta_{1} \\ 1-\cos\theta_{1}&\sin\theta_{1}&0&0\\ \sin\theta_{1}&-1+\cos\theta_{1}&0&0\\ \vdots&\vdots&\vdots&\vdots\\ 0&0&z_{i}^{m}-z_{i}^{m}\cos\theta_{i}-y_{i}^{m}\sin\theta_{i}&-y_{i}^{m}+y_{ i}^{m}\cos\theta_{i}-z_{i}^{m}\sin\theta_{i}\\ 1-\cos\theta_{i}&\sin\theta_{i}&0&0\\ \sin\theta_{i}&-1+\cos\theta_{i}&0&0\\ \end{array} \right]_{3i\times 4}\left[ \begin{array}{c}O_{ya}\\ O_{za}\\ S_{ya}\\ S_{za}\\ \end{array} \right]\]  
where the parameters \(x_{i}^{m},y_{i}^{m}\) and \(z_{i}^{m}\) represent the spatial coordinate positions of the \(m\)-th reflector, respectively. As shown, Equation (4) is an overdetermined system. As a result, the least squares method is used for estimating the set-up errors of the reflector standard fixture. For approximating the solution of overdetermined systems, a standard mathematical optimization method is built by using the least squares method.