Mathematical Analysis and Experimental Implementation of a Self-Calibration Method for Ultra-Precision Displacement Sensors

This paper starts from a revisit to the self-calibration concept of displacement sensors proposed previously by Kiyono et al. In this self-calibration scheme, two displacement sensors are used to perform a mutual calibration operation to determine their linearity error components without using a higher-accuracy reference, while their sensitivity coefficients are assumed to be known and are determined separately. A set of mathematical equations are newly established in this paper to provide a mathematical basis for the self-calibration concept. Based on the established mathematical equations, the parameters influencing the calibration results are investigated through mathematical analysis and simulation. As new findings of this paper, it is verified that the polynomial fitting errors of the discrete calibration data and the thermal drifts of the sensor outputs during the sampling process have significant impacts on the calibration results and must be carefully suppressed in practical operations. Based on these findings, a calibration system is designed and constructed for calibration experiments in which motor-driven stages are employed to shorten the calibration time and to reduce the influences of thermal drifts. The experimental results are compared with those obtained using a laser interferometer, from which the effectiveness of the constructed calibration system is verified. An Artificial Neural Network (ANN) is also introduced in this paper for fitting the calibration data containing high-frequency components instead of the conventional least-squares polynomial fitting method, which had been employed in the previous work. The self-calibration results show better repeatability than the interferometer-based calibration results. The uncertainty associated with the proposed self-calibration method was preliminarily evaluated using a Monte Carlo analysis based on experimental data.