## 1 Introduction  
Absolute position displacement measurement sensors play an irreplaceable role in the field of high-precision measurements and as a component in the closed loop feedback control of computer numerical control (CNC) machine tool systems. The rapid development of high-precision positioning technology in CNC machine tool systems has generated an increasing demand for robust absolute displacement measurement sensors with high positioning accuracy and excellent repetitive positioning precision in harsh environments, such as under the conditions of electromagnetic interference, mechanical vibration, extreme temperature variation, and dust contamination [1; 2; 3; 4; 5; 6; 7]. Meanwhile, many types of absolute grating displacement measurement sensors with high precision and high resolution have been developed, including capacitive grating sensors [1; 2], magnetic grating sensors [8], and optical grating sensors [9]. Absolute optical grating (AOG) sensors are based on an evaluation of the intensity of incident light from a known illumination source, and represent an efficient means for conducting closed loop position control in CNC machine tool systems [10; 11; 12; 13]. The measurement error of AOG sensors has been reduced to values as small as \(\pm\)0.275 \(\upmu\)m over a 10 mm range and \(\pm\)5 \(\upmu\)m over a full 3040 mm range. However, mechanical vibration, temperature variation, and dust contamination affect the light intensity and other characteristics of the illumination source incident on the sensor, all of which detract from the measurement performance of AOG sensors. It has a weak anti-interference capability, particularly in the harsh environments associated with many CNC machining applications [13; 14]. In contrast to AOG sensors, both magnetic and capacitive grating sensors have high tolerance for mechanical vibration and contamination due to dust and oil and, therefore, are commonly employed in servo drive systems [2; 8; 15]. The measurement error of absolute magnetic grating sensors can achieve values as low as \(\pm\)5 \(\upmu\)m over a 1000 mm range and can obtain a measurement resolution of 0.75 \(\upmu\)m [15; 16; 17]. However, the large size of the magnetic head and poor resistance to electromagnetic interference restrict the use of such sensors for high accuracy applications. In comparison, capacitive grating sensors provide a fuller range of environmental advantages, which make them ideally suited to harsh machining environments [2; 18; 19; 20; 21]. Moreover, the measurement error of absolute capacitive grating sensors can achieve values as low as \(\pm\)0.3 \(\upmu\)m over a 6 mm range, with an error of less than 0.01% over the full measurement range [2]. However, the measurement range of absolute capacitive grating sensors is limited, which restricts their use in practical applications [2]. The present work addresses the limitations of absolute capacitive grating sensors by developing a novel capacitive absolute linear displacement sensor that employs a vernier-type structure with time-grating based on a previously proposed single-row capacitive sensing structure [22]. The proposed vernier-type absolute time-grating (VATG) sensor employs two capacitor rows. According to the design of vernier calipers, the auxiliary scale, or vernier scale, includes \(n\) gradations over the total caliper length, while the main scale includes \(n-1\) gradations over the same total caliper length. Similarly, one capacitor row of the VATG sensor is composed of \(n\) periods over the total sensor length and serves as a fine measurement scale to ensure high measurement resolution, which is analogous to the auxiliary scale of vernier calipers. The other row of the VATG sensor is composed of \(n-1\) periods over the total sensor length as a coarse measurement scale used to find the phase difference of two rows to realize absolute positioning, which is analogous to the main scale of vernier calipers.