# An Online Simultaneous Measurement of the Dual-Axis Straightness Error for Machine Tools
## 1 Introduction  
Straightness, linear positioning and angular errors exist in every linear movement system [1], and the straightness of a moving platform can be measured by using a displacement indicator according to ISO230-1. The industry compensation method employed for straightness error measurements is usually conducted by using a single-axis mechanism with a laser interferometer and an optical mirror/prism that moves the other axis to the center of the measured axis, and then establishes the compensation data for vertical straightness with the measured \(X\)- and \(Y\)-axis straightness errors.  
[PERSON] et al. [2] proposed a low-cost straightness measurement system using a laser source and a quadrant detector. [PERSON] et al. [3] integrated a dual-linear reflector, a Wollaston prism, and a corner cube to establish a measurement system of repeatability (3\(\sigma\)) of 1 \(\upmu\)m. In a similar way, [PERSON] [4] and [PERSON] [5] utilized a single-mode fiber-coupled laser module as the reference line for straightness measurements. [PERSON] et al. [6] proposed a three-probe system for measuring the parallelism and straightness of a pair of rails for ultra-precision guideways. [PERSON] et al. [7] first described how randomly arising illumination noises produce unpredicted variations in the position sensing detector (PSD) output. [PERSON] et al. [8] proposed a straightness error measurement approach based on common-path compensation in order to eliminate the laser beam drift by utilizing an optical module with a PSD and a quadrant detector (QD).  
[PERSON] et al. [9] developed a new measurement method based on a phase sensitive detection technique, which consists of a single-mode fiber pigtailed laser diode, a collimator, a PSD, and a corner cube. To eliminate the influence of background light, the laser diode was modulated by a sinusoidal wave current and the PSD signal phase was demodulated by using the sensitive detectiontechnique. [PERSON] et al. (2018) replaced the laser interferometer with a customized taut-wire module and successfully developed a straightness measurement system with a standard deviation of 1.5 \(\upmu\)m over a measuring range of 1.5 m.  
[PERSON] et al. (2018) also replaced the laser interferometer with a high-power fiber-coupled light emitting diode (LED) as a light source for measuring straightness errors within a limited detecting range. [PERSON] et al. (2018) developed a fast and accurate system for gantry type milling centers by real-time modeling one serial of straightness errors using B-spline curves.  
The \(X\) and \(Y\) axes of a machine tool need to be moved during the machining process; however, the vertical straightness errors are compensated by using the individually measured vertical straightness errors of \(X\) or \(Y\) axis. The single-axis mechanism utilized in above mentioned literature may result in compensation error for vertical straightness due to the variation of center-of-gravity, especially for machine tools of long-travelling distance or high-loading operation conditions. In this study, a dual-axis measurement system based on a single-axis mechanism via the least squares method, is developed. Its advantage is that the system can simultaneously measure and compensate both vertical straightness and flatness errors for the moving working table of the machine tool.