## 3 Device and Experiment  
According to the above-mentioned mathematical model, the step gauge calibration system is developed, as shown in Figure 3.  
The trigger signal of the probe of CMM is used to determine the position of the measuring surface of the step gauge. Before the experiment, it is necessary to calibrate the trigger accuracy of CMM probe. In this system, we adopt the trigger probe with a parallel leaf spring structure, and the displacement change of the probe can be measured by a Linear Variable Differential Transformer (LVDT). The measuring principle of the probe is shown in Figure 4. Given the U\({}_{\rm prim}\) signal at the input end of LVDT, when the probe moves to the left side, it drives the core to move to the left, which  
Figure 3: The step gauge measurement system; (**a**) actual calibration scenario of step gauge measurement system; (**b**) the critical optical system. Among this, 1 is the refractive index tracker to compensate the laser wavelength change, 2 is the plane interferometer, 3, 4 and 5 are the three-way length measuring interferometers from a single-frequency laser. Figure 2: Simulation distribution of measurement length deviation of different distances of step gauge with different attitude angle \(\theta\); (**a**). \(\theta\) is a negative angle, which means the normal direction of the measured surface and the starting zero plane of the step gauge is opposite; (**b**) when \(\theta\) is a positive angle, the normal direction of the measured surface and the starting zero plane of the step gauge are the same. leads to the greater voltage U\({}_{\text{left}}\) in the left than the voltage U\({}_{\text{right}}\) in the right, so the output voltage U\({}_{\text{sec}}\) is displayed as a sinusoidal positive signal; otherwise, it is a sinusoidal negative signal. After shaping, subdividing and amplifying the output voltage U\({}_{\text{sec}}\), it is converted into digital voltage signal via an 18-bit analog-to-digital conversion(AD) card first, and then converted into the moved displacement value. In this system, the trigger sampling interval of the probe is in the linear region of the falling edge of the probe voltage signal shown in Figure 4c. The key technology of the step gauge measurement system in realizing the high measurement accuracy is to realize the synchronous sampling and signal synthesis of the probe voltage signal of the CMM and the displacement measurement data of the laser interferometer. The probe of CMM produces the voltage signal with linear change in the measurement process, as shown in Figure 5a. Through synchronous dynamic sampling, the corresponding laser interferometer displacement measurement signal is obtained, as shown in Figure 5b. After the two signals are synthesized, the constant coordinates of the measuring points at the time of triggering are obtained, as shown in Figure 5c. Before calibration, we first use the known high-level standard ball (its sphericity is 0.03 \(\upmu\)m, its diameter is 29.98604 mm) to calibrate the diameter of the CMM probe. The two-side measurement points of the standard ball are measured 10 times, and the measurement results are shown in Table 1 below. Figure 4: Measuring principle diagram of precision displacement probe for Coordinate Measuring Machine (CMM). (**a**) Mechanical schematic diagram of the CMM probe; (**b**) circuit schematic diagram of the CMM probe; (**c**) schematic diagram of the voltage of the probe signal changing with time during the contact measurement of the probe. Figure 5: Signal synthetic diagram of probe voltage and displacement of laser interferometer: (**a**) the sampled voltage value via AD card after triggering during measurement; (**b**) the sampled laser value after triggering; (**c**) the combined value of laser value and voltage value obtained by synchronous sampling after triggering. Then, the calibrated diameter of the probe can be calculated by Equation (8).