# Multi-Axis Laser Interferometer Not Affected by Installation Errors Based on Nonlinear Computation
## 2 Interferometer Measurement Model
### 6-DOF Displacement Measurement Model  
In the 3-DOF displacement measurement of the X-Y stage, the assumption that the stage only moves in one plane is adopted. Although this assumption holds approximately in the case that the assembly is ideal, there is still deviation from the actual situation. When it comes to having sufficient computing resources, it is necessary to discuss and apply the 6-DOF displacement measurement to reduce the deviation between the model and the actual situation.  
The 6-DOF displacement measurement scheme of the X-Y-Z stage using a nine-axis interferometer is proposed and shown in Figure 4. There are three interferometers that measure each one of the three adjacent surfaces of the stage. Similar to the 3-DOF displacement measurement of the X-Y stage, the mathematical model of the readings of the nine interferometers can be deduced, which is too complex because it needs to calculate the laser path in 3-dimensional space. Since the 6-DOF model is only more complex than the 3-DOF model, and there is no difference in the modeling principle, the detailed expression of the 6-DOF model is not given in this section, but the function heading is given.  
The same as in the 3-DOF measurement, the O-XYZ coordinate axis is set to ensure that the origin coincides with the center of the stage at the reference position. Let \((x,y,z,\theta,\varphi,\omega)\) be the 6-DOF displacement, then the readings of interferometers at the measurement position are  
\[V_{i}=V_{i}(x,y,z,\theta,\varphi,\omega,x_{di},y_{di},z_{di},\alpha_{i},\beta_ {i}),i=1,2,\ldots,9 \tag{7}\]  
where \(V_{i}\) is the reading of the \(i^{\text{th}}\) interferometer, \(V_{i}(\cdot)\) is the function of the length variation of laser path from the reference position to the measurement position, \((x_{di},y_{di},z_{di})\) is the installation position of the \(i^{\text{th}}\) interferometer, and \((\alpha_{i},\beta_{i})\) is the angle of the \(i^{\text{th}}\) laser beam.  
Also, to make the model solvable, the number of equations is supposed to be larger than the number of unknowns. Hence, all the interferometers' readings are recorded at every 20 distinct measurement positions for one displacement computation. Let \((x_{n},y_{n},z_{n},\theta_{n},\varphi_{n},\omega_{n})\) be the displacement at the \(n^{\text{th}}\) measurement positions, \(n=1,2,\ldots,20\), the nine interferometers' reading at 20 measurement positions are  
\[V_{ni}=V_{ni}(x_{n},y_{n},z_{n},\theta_{n},\varphi_{n},\omega_{n},x_{di},y_{di},z_{di},\alpha_{i},\beta_{i}),\;n=1,2,\ldots,20,\;i=1,2,\ldots,9 \tag{8}\]  
where \(V_{ni}\) is the reading of the \(i^{\text{th}}\) interferometer at the \(n^{\text{th}}\) measurement position, \(n=1,\,2,\ldots,\,20,\;i=1,\,2,\ldots,\,9\).  
Since there are a total of 180 equations and 165 unknowns in Equation (8), the 6-DOF displacement measurement model (8) is also theoretically solvable. Similar to the analysis in Section 2.2, the installation errors do not affect the displacement computation, and the nonlinear model without any approximation provides high accuracy.  
As we can see, both the 3-DOF model and the 6-DOF model are nonlinear equation systems with a large number of unknowns. In this case, the existence and uniqueness of the solution, as well as the solution algorithm, are all key factors affecting the accurate displacement computation, which are discussed in Section 3.