# High-Accuracy Calibration Based on Linearity Adjustment for Eddy Current Displacement Sensor
## 1 Introduction  
Online precise displacement measurement and positioning are important parts of the manufacturing process quality inspection system [1]. Traditional methods are usually based on electronic theodolites, iGPS and laser trackers [2]. For electronic theodolites, the accuracy of angle measurement is up to 0.5\({}^{\circ}\); however, the accuracy of distance measurement only reaches millimeter level due to the introduction of the sight distance method [3]. The precision of iGPS and laser trackers are very high in wide range measurements. However, due to the limitation of equipment response frequency and manual auxiliary operation, they are only suitable for low dynamic measurements [4]. Eddy current displacement sensors (ECDSs), due to the micron resolution and the high dynamic response up to 20 kHz even in harsh environments and their low cost [5; 6; 7; 8], are widely used in measurement of displacement or movement in aviation, shipbuilding and military and many other fields [9; 10; 11]. [PERSON] and [PERSON] demonstrated resonant inductive coupling-based eddy current sensing as a promising piston position sensing mechanism for large vertical displacement micromirrors and presented a frequency detection method to simultaneously sense the piston position and tilt angle of the mirror plate of large vertical displacement micromirrors with nanometer resolution [12; 13]. Moreover, in order to achieve closed-loop control of the inclinedangle of the mirrors on the satellite communication equipment, Micro-Epsilon (Ortenburg, Germany) provided a high-accuracy ECDS system to accurately measure the mirror rotation angle with angular resolutions better than one microrad [14]. Meanwhile, due to the miniaturized design of the ECDS and its advantages of the resistance to oil contamination and high temperature, it was also used in turbocharger test benches for the detection of turbocharger performance under different stress states [15]. The resolution of the ECDS can reach micron level or even below [16; 17; 18]. However, the relatively low sensitivity and linearity seriously restrict the detection accuracy because it is usually calibrated by a linear model before industrial use, neglecting the nonlinearity. In order to obtain better performance of the sensor, much research has been conducted for improving the sensitivity, optimizing the circuit design and high-accuracy calibration. From the aspect of optimizing the circuit design, [PERSON] explored the relationship between the measurement sensitivity and sensor parameters using an equivalent circuit model, and conducted experiments with a series of Cu films of different thickness, which showed that the sensor made of a lower resistant multi-wire Cu line had better sensitivity [19]. [PERSON] discussed key design aspects and their interfaces for ECDSs, and employed second-order LC cross-coupled oscillators as the front-end stage to improve the precision during the measurements [20]. [PERSON] and [PERSON] analyzed the optimal operating frequency for minimum influence of temperature (20-600 \({}^{\circ}\)C) on displacement measurement to reduce the displacement measurement ambiguity [21]. Many other research works have also been conducted, such as using a small flat sensing coil for mechanical stability and compactness [22], increasing the excitation current to enhance detection sensitivity of amplitude parameter [23], etc. However, in addition to complex workload, the ECDSs still suffer from a certain degree of nonlinearity due to the output characteristics of the designed circuit. Thus, subsequent high-accuracy calibration of ECDS is necessary. [PERSON] proposed a piecewise linear fitting method that increased the linearity to above 99%, while the accuracy reached within 30 \(\upmu\)m in the range of 6 mm [24]. [PERSON] constructed the inverse model of the ECDS using the radial basis function (RBF) neural network to achieve the high-precision calibration with an error less than 0.7%, which satisfied the requirement of the maglev system control [25].