## 3 Errors Identification Principle
### Calculation of the 6 Degrees-of-Freedom Errors  
By the method mentioned in Sections 3.1 and 3.2, the relative angles \(\theta_{xi}\) and \(\theta_{yi}\) and position \(X_{Fi}\) and \(Y_{Fi}\) of the \(i\)th target were calibrated. To ensure repeatability of the measurement system, the angle difference between the first and the last plane mirror should be, where possible, the same as the angle difference of calibration. The alignment can be conducted based on the spot position in the image, which can reduce the cosine error of the measurement effectively. When the 1-D artifact was located on the \(X\)-axis of the machine tool horizontally, the measurement value of the relative angles \(\theta^{\prime}{}_{xi}\) and \(\theta^{\prime}{}_{yi}\) and relative position \(X^{\prime}{}_{Fi}\) and \(Y_{Fi}\) of the \(i\)th target could be obtained by the machine tool to be tested. In the process of measurement, \(lx_{MiM0}\) and \(ly_{MiM0}\) should be computed by the feedback value of the machine tool. That means that \(X^{\prime}{}_{Fi}\), \(Y^{\prime}{}_{Fi}\) are obtained by the following equations:  
\[X^{\prime}{}_{Fi}=k(x^{\prime}{}_{M0}-x^{\prime}{}_{F0})+\left(X^{\prime}{}_{ Mi}-X^{\prime}{}_{M0}\right)+k(x^{\prime}{}_{Fi}-x^{\prime}{}_{Mi}), \tag{7}\]  
\[Y^{\prime}{}_{Fi}=k\Big{(}y^{\prime}{}_{M0}-y^{\prime}{}_{F0})+\Big{(}Y^{ \prime}{}_{Mi}-Y^{\prime}{}_{M0}\Big{)}+k\Big{(}y^{\prime}{}_{Fi}-y^{\prime}{ }_{Mi}\Big{)}, \tag{8}\]  
where \(x^{\prime}{}_{M0}\), \(y^{\prime}{}_{M0}\), \(x^{\prime}{}_{Mi}\), \(y^{\prime}{}_{Mi}\), \(x^{\prime}{}_{F0}\), \(y^{\prime}{}_{F0}\), \(x^{\prime}{}_{Fi}\), and \(y^{\prime}{}_{Fi}\) are the coordinates of the spot in the image system when measuring horizontally, and \(X^{\prime}{}_{M0}\), \(X^{\prime}{}_{Mi}\), \(Y^{\prime}{}_{M0}\), and \(Y^{\prime}{}_{Mi}\) are the measured coordinates of the measurement point in the machine coordinate system. Due to geometric errors in the linear motion of the machine tool, the relative angle and position measured by the machine tool are not the same as the calibration values. The roll error \(\varepsilon_{x}(x)\) and pitch error \(\varepsilon_{y}(x)\) are caused by deviation in the relative angles and the position error \(\delta_{x}(x)\) and horizontal straightness error \(\delta_{y}(x)\) are caused by deviation in the relative position. Thus, by calculating the deviation between the measurement values and calibration values, these errors at the target positions can be identified. The roll error \(\varepsilon_{x}(x)\) and the pitch error \(\varepsilon_{y}(x)\) at the target position can be identified by the following equations:  
\[\varepsilon_{x}(x)=\theta^{\prime}{}_{xi}-\theta_{xi}, \tag{9}\]  
\[\varepsilon_{y}(x)=\theta^{\prime}{}_{yi}-\theta_{yi}. \tag{10}\]  
Due to the character of a revolutionary paraboloid, the displacement difference measured by the machine tool includes coupling of angle errors. Hence, the position error \(\delta_{x}(x)\) at the target position can be identified by the following equation:  
\[\delta_{x}(x)=X^{\prime}{}_{Fi}-X_{Fi}-km\varepsilon_{y}(x_{Fi}), \tag{11}\]  
where \(k\) and \(m\) represent the conversion coefficients from spot displacement in the image to actual displacement and from angle variation to spot displacement in the image, respectively, and \(\varepsilon_{y}(x_{Fi})\) is the pitch error at the target position.