Error Analysis and Modeling for an Absolute Capacitive Displacement Measuring System with High Accuracy and Long Range

We proposed a novel kind of absolute capacitive grating displacement measuring system with both high accuracy and long range in a previous article. The measuring system includes both a MOVER and a STATOR, the contact surfaces of which are coated by a thin layer of dielectric film with a low friction coefficient and high hardness. The measuring system works in contact mode to minimize the gap changes. This paper presents a theoretical analysis of the influence of some factors, including fabrication errors, installation errors, and environment disturbance, on measurement signals. The measuring signal model was modified according to the analysis. The signal processing methods were investigated to improve the signal sensitivity and signal-to-noise ratio (SNR). The displacement calculation model shows that the design of orthogonal signals can solve the dead-zone problem. Absolute displacement was obtained by a simple method using two coarse signals and highly accurate displacement was further obtained while using two fine signals with the help of absolute information. According to the displacement calculation model and error analysis, the error in fine calculation functions mainly determines the model’s accuracy and is locally affected by coarse calculation functions. It was also determined that amplitude differences, non-orthogonality, and signal offsets are not related to the accuracy of the displacement calculation model. The experiments were carried out to confirm the abovementioned theoretical analysis. The experimental results show that the displacement resolution and error in the displacement calculation model reach ±4.8 nm and ±34 nm, respectively, in the displacement range of 5 mm. The experiments and the theoretical analyses both indicate that our proposed measuring system has great potential for achieving an accuracy of tens of nanometers and a range of hundreds of millimeters.


Signal Functions Modified with Non-Orthogonality
As known from basic measuring principles, theoretically, and are orthogonal to and ( ) , respectively, and ( ) and ( ) are orthogonal to ( ) and ( ) , respectively. However, orthogonality can be affected by fabrication errors and installation errors. The signal functions for fine measurement and coarse measurement can be, respectively, rewritten as: where and are very small constants.

Scaling Coefficient Functions
As described above, scaling coefficient functions were introduced to modify the signal model due to changes in amplitude and nonlinearity and can be expressed simply by polynomials. The scaling coefficient functions are described as follows: In Equation (s1.3), n is the degree of the polynomials, and its value is determined according to the specific circumstances. The higher the degree of the polynomials is, the more complicated the variation it can express. The scaling coefficient function where ξ is a very small constant.

High-Frequency Noises
In practical applications, interference from electromagnetic signals and mechanical vibrations can lead to high-frequency noises that influence the measuring system's performance. For convenience, four of the same high-frequency noises, labeled , were introduced into the theoretical functions for fine measurement, and another four of the same high-frequency noises, labeled , were introduced into the theoretical functions for coarse measurement. High-frequency noises have a variety of frequency components and can be described as:

White Noises
In practical applications, white noise is ubiquitous in signals and affects the SNR; thus, it was introduced into the measuring signal model. For convenience, four different white noises of the same level, all labeled , were introduced into the four theoretical functions for fine measurement. Four different white noises of the same level, labeled , were also applied to the other four theoretical functions for coarse measurement. Both and were generated by the Gauss white noise equation.

S2.1. Differential Method
The differential method is defined as follows: According to Equation (10), Equation (s2.1) can be rewritten as: where and are fine signals, and and are coarse signals. Simulated curves of fine signals and coarse signals are shown in Figure 4a.
The displacement resolution of fine signals and coarse signals is

S2.2. Ratio Method
The ratio algorithm is defined as follows: According to Equation (10), Equation (s2.5) can be rewritten as: . (s2.6) Similarly, and are called fine signals, and and are called coarse signals. Simulated curves of fine signals and coarse signals are shown in Figure 4b.
In Equation (s2.6), take the expression of as an example for analysis. Since + ≪ + + + and + ≪ − + + + , the term + can be neglected. Besides this, there is ≈ 1, = , , so: The maximum variable signal of is: The other expressions of Equation (s2.6) can be analyzed using the same method, so The noise in the fine signals and coarse signals cannot be directly obtained from Equation (s2.6). However, we can calculate it by taking the following approach: (s2.11) The displacement resolution of fine signals and coarse signals is:

S2.3. Differential-Ratio Method
The differential-ratio approach is defined as follows: (s2.13) Similarly, and are called fine signals, and and are called coarse signals. Simulated curves of fine signals and coarse signals are shown in Figure 4c.
According to Equation (10), Equation (s2.13) can be rewritten as: . (s2.14) Using the same analytical method as for the ratio approach, we can obtain the maximum variable value of fine signals and coarse signals: The noise in the fine signals and coarse signals is calculated as follows: (s2.17) The displacement resolution of fine signals and coarse signals is:

S3. Analysis of the Error Components of the Entire Experimental System
To understand different types of errors, we performed an analysis of the error components of the entire experimental system. In the experimental setup, a HEIDENHAIN-CERTO length gauge (H.) was used to calibrate the capacitive displacement system (CDS), and the characteristics of H. are shown in Table S1. The accuracy of H. is less than ±0.03 μm in a short-range measurement, and the repeatability/precision is also less than ±0.03 μm. As shown in Figure S1, there is uncertainty in the CDS, which refers to the difference between the H. displacement and the CDS displacement. This can be caused by two types of errors: (1) the first type of error is a connection error between the CDS and H., which is not a characteristic of the CDS itself; (2) the second type of error is a systematic error in the CDS itself, including the error analyzed in Section 6, which can be compensated for.
The first type of error certainly influences the accuracy of the CDS ( Figure S1). However, such error is not a characteristic of the CDS itself. The error can be very small if we improve the accuracy of the calibration system. This type of error does not fall within the scope of our analysis. The second type of error, which is also called displacement calculation model error, is analyzed in Section 6. The uncertainty (±40 nm) in the displacement calculation model error was obtained according to the difference between the calibrated displacement and the calculated displacement. The uncertainty (±40 nm) is not a component of the first type of error. Thus, the uncertainty may be close to that of H.  Figure S1. Accuracy of the entire experimental system.