## 4 Calibration Method
### Detection of Installation Eccentricity Error of Test Rods  
In an ideal case, the two test rods need to be coaxially assembled on the end of the transit axis to extend the hidden transit axis. However, the test rods may be eccentrically mounted; we refer to the eccentricity error as the installation eccentricity error, which may result in errors in the transit offset and tilt error calibration process. If the test rod is coaxial with the transit axis, after the transit axis rotates 360 degrees, the trajectory of the central axis is one point at \(O\), while the trajectory of the central axis is a circle due to the installation eccentricity of test rods. The installation eccentricity of the test rod \(OO^{\prime}\) can therefore be resolved into \(X\) and \(Y\) components (\(OM\) and \(ON\), respectively) in the transit axis's coordinate system; it can be shown that \(OM=OO^{\prime}\cdot cos\theta\) and \(ON=OO^{\prime}\cdot sin\theta\), as shown in Figure 3.  
Figure 3: The installation eccentricity error model of test rod. Figure 2: The transit offset and tilt errors model. We use the telecentric measurement system to detect _OM_ and _ON_. The telecentric measurement system consists of two sensor heads (transmitter and receiver), a controller, and a monitor. The transmitter emits collimated lighting with a green LED to project a shadow on the CMOS sensor in the receiver; measurement is then performed using this captured image, and the projection of the test rods can be recorded and displayed on the monitor. This is done in order to detect the installation eccentricity error in the \(X\) component and \(Y\) component (_OM_ and _ON_, respectively). The telecentric measurement system is established at two positions, as shown in Figure 4.  
In order to obtain the installation eccentricity error along the \(X\) component (_OM_), the telecentric measurement system is established at position 1, as shown in Figure 4. The transmitter and the receiver are vertically placed, and the green LED light emitted from the transmitter is projected to test rod A; the image of test rod A will be recorded and displayed on the monitor. The initial position of the central axis of the test rod A is recorded as \(Ax_{0}\); then, the transit axis rotates 360 degrees at equal intervals of \(N\) degrees. Thus, the offset \(Ax_{i}\) (\(i=1,2\ldots 360/N\)) with respect to the initial position at different angles can be obtained, as shown in Figure 5. The standing axis rotates 180 degrees, and the green LED light emitted from the transmitter is projected to test rod B. A similar measurement is repeated. The initial position \(Bx_{0}\) and offset \(Bx_{i}\) (\(i=1,2\ldots 360/N\)) can also be obtained. In order to measure the installation eccentricity error along the \(Y\) component (_ON_), the telecentric measurement system is established at position 2, as shown in Figure 4. The transmitter and the receiver are horizontally placed, and the green LED light emitted from the transmitter is projected to test rod A again; the image of test rod A will be recorded and displayed on the monitor. The initial position of the central axis of the test rod A is recorded as \(Ay_{0}\); then, the transit axis rotates 360 degrees at equal intervals of \(N\) degrees. Thus, the offset \(Ay_{i}\) (\(i=1,2\ldots 360/N\)) with respect to the initial position at different angles can be obtained, as shown in Figure 5. The standing axis rotates 180 degrees, and the green LED light emitted from the transmitter is projected to test rod B, wherein a similar measurement  
Figure 4: The position of the telecentric measurement system. is repeated. The initial position \(By_{0}\) and the offset \(By_{i}\) (\(i=1,2\ldots 360/N\)) can also be obtained. Finally, the coordinates \(A(x_{i},y_{i})\) and \(B(x_{i},y_{i})\) of the central axis of test rod A and B, respectively, are obtained.