Mathematical model of random sphere absolute measurement and the theory of multiple spheres random

Random sphere absolute measurement is a simple and efficient method. It assumes that the mean of the surface in different regions of the test sphere necessarily converge to the ideal plane. Although a large number of experiments have verified this assumption, the relevant theory has not yet been perfected. In this study, we develop a mathematical model for the random sphere absolute measurement from the perspective of moments of random variables. It shows that the random range of the test spheres significantly affects the repeatability of measurement. Meanwhile, based on this mathematical model, we proposed the multiple sphere random theory. In the experiment, the RMS repeatability for single sphere random absolute measurements is 0.00045λ under the condition of large random range and 0.00017λ under the condition of small random range. The experiment verifies the influence of the random range on the repeatability of the measurement. Meanwhile, a multiple spheres random absolute measurement experiment was completed, which achieved a similar repeatability to the single sphere random under the condition of poorer surface quality of the test sphere and fewer averaging times. The RMS repeatability is 0.00022λ. This method greatly reduced the cost of the sphere absolute measurement.