# Multi-Axis Laser Interferometer Not Affected by Installation Errors Based on Nonlinear Computation
## 2 Interferometer Measurement Model
### 3-DOF Displacement Measurement Model  
To avoid measurement errors caused by installation errors, the installation parameters such as laser beam angles and emission point positions are added to the model as unknowns. Then, it is necessary to add constraints to make the model solvable, and one way is to increase the redundant measurement information by using more interferometers. In this paper, a 3-DOF displacement measurement of the X-Y stage using a six-axis interferometer is proposed. And the measurement scheme is shown in Figure 2. There are six interferometers for measuring the two adjacent sides of the X-Y stage. Assuming that the X-Y stage only moves in the X-O-Y plane, and all the laser beams are supposed to be in the X-O-Y plane, the measurement scheme is also expressed in Figure 3.  
Figure 1: (**a**) Interferometer and stage at reference position. (**b**) Interferometer and stage at measurement position. Figure 2: Scheme of 3-DOF displacement measurement. In Figure 3, the reference position of the stage is shown as the dotted line, and the measurement position is shown as the true line; the O-XYZ axis is set so that the coordinates of the stage center and the angle of the stage are both equal to 0 at the reference position. Due to the installation errors, there is an angle of each laser beam, which is set as \(\alpha_{i},i=1,\;2,\dots,\;6\), and the unknown installation position of each interferometer is set as \((x_{di},y_{di}),i=1,\;2,\dots,\;6\). Based on the single interferometer laser path model proposed in Section 2.1, the readings of the six interferometers can be modeled as  
\[\left\{ \begin{array}{l}V_{1}=\frac{2\cos(\theta-\alpha_{1})}{\cos(2\theta-2 \alpha_{1})}[(x-x_{d1})\cos\theta+(y-y_{d1})\sin\theta-b/2]+\frac{2\cos\alpha_{ 1}}{\cos 2\alpha_{1}}(x_{d1}+b/2)\\ V_{2}=\frac{2\cos(\theta-\alpha_{2})}{\cos(2\theta-2\alpha_{2})}[(x-x_{d2}) \cos\theta+(y-y_{d2})\sin\theta-b/2]+\frac{2\cos\alpha_{2}}{\cos 2\alpha_{2}}(x_{d2}+ b/2)\\ V_{3}=\frac{2\cos(\theta-\alpha_{3})}{\cos(2\theta-2\alpha_{3})}[(x-x_{d3}) \cos\theta+(y-y_{d3})\sin\theta-b/2]+\frac{2\cos\alpha_{3}}{\cos 2\alpha_{3}}(x_{d3}+ b/2)\\ V_{4}=\frac{2\cos(\theta-\alpha_{4})}{\cos(2\theta-2\alpha_{4})}[(y-y_{d4}) \cos\theta+(x-x_{d4})\sin\theta-h/2]+\frac{2\cos\alpha_{5}}{\cos 2\alpha_{4}}(y_{d4}+ h/2)\\ V_{5}=\frac{2\cos(\theta-\alpha_{5})}{\cos(2\theta-2\alpha_{5})}[(y-y_{d5}) \cos\theta+(x-x_{d5})\sin\theta-h/2]+\frac{2\cos\alpha_{5}}{\cos 2\alpha_{5}}(y_{d5}+ h/2)\\ V_{6}=\frac{2\cos(\theta-\alpha_{6})}{\cos(2\theta-2\alpha_{6})}[(y-y_{d6}) \cos\theta+(x-x_{d6})\sin\theta-h/2]+\frac{2\cos\alpha_{6}}{\cos 2\alpha_{6}}(y_{d6}+ h/2)\end{array} \right. \tag{3}\]  
where \(V_{i}\) is the reading of the \(i\)th interferometer, \(i=1,\;2,\dots,\;6\), and \((x,y,\theta)\) is the 3-DOF displacement of the stage at the measurement position since the coordinate of the stage center is equal to 0 at the reference position. According to Equation (3), after obtaining the value of \(V_{i}\) in practice, there are a total of 21 unknowns including \((x_{di},y_{di},\alpha_{i}),i=1,\;2,\dots,\;6\), and \((x,y,\theta)\) in the nonlinear equation system. Since the number of unknowns is greater than the number of constraints, it is unrealistic to solve the displacement \((x,y,\theta)\) from Equation (3). Therefore, the number of measurements is increased to gain more redundant information; that is, all the interferometers' readings are recorded at every 10 distinct measurement positions for one displacement computation. Let \((x_{n},y_{n},\theta_{n})\) be the displacement at \(n\)th measurement posi  
Figure 3: Measurement principle of 3-DOF displacement of X-Y stage with a 6-axis interferometer. tions, \(n=1,\ 2,\dots,\ 10\), the six interferometers' readings at 10 measurement positions can be modeled as  
\[\left\{