## 2 Relative Position and Attitude Measurement of Maglev Vibration Isolation System
### Relative Position and Attitude Measurement Based on Eddy Current  
A three-dimensional position measurement model is established, as shown in Figure 3. Sensors X1 and X2 can measure relative positional changes along the x-axis and angular changes around the x-axis; sensors Y1 and Y2 can measure relative positional changes along the y-axis and angular changes around the y-axis; and sensors Z1, Z2, Z3, and Z4 can measure relative positional changes along the z-axis and angular changes around the z-axis. In order to analyze the relative motion between the floating platform and the base, the base coordinate system S and the floating platform coordinate system F are established. The XOZ plane is made coplanar with the reflective surfaces of sensors Y1 and Y2, the YOZ plane is made coplanar with the reflective surfaces of sensors X1 and X2, and the XOY plane is made coplanar with the reflective surfaces of sensors Z1, Z2, Z3, and Z4. The sensor coordinate systems C\({}_{X1}\), C\({}_{X2}\), C\({}_{Y1}\), C\({}_{Y2}\), C\({}_{Z1}\), C\({}_{Z2}\), C\({}_{Z3}\), and C\({}_{Z4}\), respectively, are established at the centroid of the reflection surface of each eddy current sensor, and the z-axis direction of sensors X1, X2, Y1, Y2, Z1, Z2, Z3, and Z4 is the same as the z-axis direction of the platform coordinate system. The transformation matrix between the base coordinate system and the floating platform coordinate system is defined as C, and the Euler angle of the coordinate transformation are assumed as \(\theta_{x}\), \(\theta_{y}\), \(\theta_{z}\). According to [PERSON]'s equation, there is:  
\[ \begin{split}\mathbf{C}&=R_{\theta_{x}}R_{\theta_{y}}R_{ \theta_{z}}\\ &=\left[\begin{array}{ccc}\cos\theta_{y}\cos\theta_{z}&-\cos \theta_{y}\sin\theta_{z}&\sin\theta_{y}\\ \sin\theta_{x}\sin\theta_{y}\cos\theta_{z}+\cos\theta_{x}\sin\theta_{z}&-\sin \theta_{x}\sin\theta_{y}\sin\theta_{z}+\cos\theta_{x}\cos\theta_{z}&-\sin \theta_{x}\cos\theta_{y}\\ -\cos\theta_{x}\sin\theta_{y}\cos\theta_{z}+\sin\theta_{x}\sin\theta_{z}& \cos\theta_{x}\sin\theta_{y}\sin\theta_{z}+\sin\theta_{x}\cos\theta_{z}&\cos \theta_{x}\cos\theta_{y}\end{array}\right]\\ \\ R_{\theta_{x}}=\begin{bmatrix}1&0&0\\ 0&\cos\theta_{x}&-\sin\theta_{x}\\ 0&\sin\theta_{x}&\cos\theta_{x}\end{bmatrix},R_{\theta_{y}}=\begin{bmatrix} \cos\theta_{y}&0&\sin\theta_{y}\\ 0&1&0\\ -\sin\theta_{y}&0&\cos\theta_{y}\end{bmatrix},R_{\theta_{z}}=\begin{bmatrix} \cos\theta_{z}&-\sin\theta_{z}&0\\ \sin\theta_{z}&\cos\theta_{z}&0\\ 0&0&1\end{bmatrix}\end{split}  \tag{2}\]  
Considering the small-angle rotation, the transformation matrix C can be simplified to:  
\[\mathbf{C}= \begin{bmatrix}1&-\theta_{z}&\theta_{y}\\ \theta_{z}&1&-\theta_{x}\\ -\theta_{y}&\theta_{x}&1\end{bmatrix}  \tag{3}\]  
The transformation matrix between the reflective surface coordinate system and the base coordinate system is defined as C\({}_{\rm p}\). According to the installation position of the eddy current sensor, the following is obtained:  
\[ \begin{split}\mathbf{C}_{pX1}=\left[\begin{array}{ccc}-1&0&0\\ 0&1&0\\ 0&0&1\end{array}\right],\mathbf{C}_{pX2}=\left[\begin{array}{ccc}1&0&0\\ 0&-1&0\\ 0&0&1\end{array}\right],\mathbf{C}_{pY1}=\left[\begin{array}{ccc}-1&0&0\\ 0&-1&0\\ 0&0&1\end{array}\right],\mathbf{C}_{pY2}=\left[\begin{array}{ccc}1&0&0\\ 0&1&0\\ 0&0&1\end{array}\right]\\ \\ \mathbf{C}_{pZ1}=\left[\begin{array}{ccc}-1&0&0\\ 0&1&0\\ 0&0&1\end{array}\right],\mathbf{C}_{pZ2}=\left[\begin{array}{ccc}1&0&0\\ 0&-1&0\\ 0&0&1\end{array}\right],\mathbf{C}_{pZ3}=\left[\begin{array}{ccc}-1&0&0\\ 0&-1&0\\ 0&0&1\end{array}\right],\mathbf{C}_{pZ4}=\left[\begin{array}{ccc}1&0&0\\ 0&1&0\\ 0&0&1\end{array}\right]\\ \end{split}  \tag{4}\]  
Assuming that in the initial state, the floating platform coordinate system coincides with the base coordinate system. (1) In the base coordinate system, the position vector of the base deviating from the floating platform after disturbance is defined as r = [x y z]T.  
Figure 3: Model of relative position and attitude measurement based on eddy current.