## 3 Experiment and Results
### Microsphere Measurement  
As the same measurement procedure of the maximal cross-sectional profile mentioned above, the other five maximal cross-sectional profiles are measured. The intersection angle of any two adjacent sections is 30\({}^{\circ}\), and the six maximal cross-sectional profiles are evenly distributed on the microsphere. Figure 14 shows the measuring data (red solid triangle) and the fitted data (blue solid dot) of the cross-sectional circles. Figure 14: The measured and fitted data of the six cross-section profiles. Figure 13: Repeatability measurement of a cross section profile. As mentioned in Section 2.1, the yaw error is eliminated by measuring the two opposite cross-sectional profiles. The yaw error is combined of the linear error and the nonlinear error. The linear error is caused by the non-perpendicularity between the motion direction of the microsphere and the measuring line of the probes, and its value is about 16.5\({}^{\prime}\). The nonlinear error is caused by the microstage-3, and its value fluctuates within \(-\)20-20 nm. Measurement results before and after eliminating the yaw are listed in Table 1. The dimensions of the six cross-sectional profiles are estimated. For each cross section of the microsphere, 20 points are sampled to fit the dimensions of the profile circle by the least-squares method. The overall microsphere is constructed with six profiles with 120 points, and this amount of data is sufficient to calculate the sphericity. The diameter and sphericity of the microsphere are fitted by using the least-squares method. As demonstrated in Table 1, after eliminating the yaw error, the sphericity error of the ruby microsphere is reduced. The measured ruby microsphere in this study is the tip of a Renishaw commercial stylus. The type of the Renishaw commercial stylus is A-5003-5201, the nominal diameter of the ruby microsphere is 300 \(\upmu\)m. The ruby microsphere meets grade 5 in the standard of DIN 5401:2002, the dimensional tolerance of the diameter is \(\pm\)5.63 \(\upmu\)m, the spherical dimension deviation of a single circle (roundness: the deviation from sphere form) is 0.13 \(\upmu\)m, and the diameter deviation of every circle is less than 0.13 \(\upmu\)m.  
Table 1 demonstrates that the measured diameters of the six angular cross-sectional circles conform to the diameter tolerance range, the maximum diameter difference of the measured circles is 0.123 \(\upmu\)m, and the roundness of the cross section circle is less than 0.13 \(\upmu\)m. The measurement results are all consistent with the DIN 5401:2002 standard, and the measurement accuracy of the developed method can be guaranteed. In this system, after eliminating the yaw error, there are mainly four errors exist in the process of measurement. They are the alignment error of the two probes, the repeatability of the measurement position, the linearity of micro stage-3 and the residual yaw error. The principle of the alignment error is demonstrated in Figure 9, where the offset distance between the ideal maximum cross section and the actual cross section can be controlled within about 1 \(\upmu\)m. The measured diameter is 303.643 \(\upmu\)m. According to the principle in Figure 9, the standard deviation of the calculated alignment error is 2.2 nm. The linearity of micro stage-3 is 0.007 percent of the motion displacement estimated based on the technological manual, which is about 9.5 nm. After eliminating the yaw error, the residual errors of the measured six cross-sectional profiles are shown in Figure 15.