A motion characteristics modeled angular position sensor by nonlinear transfer of differential capacitance for miniaturized scanning mirrors

Liu et al. Microsystems & Nanoengineering (2023)9:148 https://doi.org/10.1038/s41378-023-00619-8 ARTICLE Microsystems & Nanoengineering www.nature.com/micronano Open Access A motion characteristics modeled angular position sensor by nonlinear transfer of differential capacitance for miniaturized scanning mirrors 1234567890():,; 1234567890():,; 1234567890():,; 1234567890():,; Songtao Liu1,2, Gaofei Zhang1,2,3 ✉, Lingyun Zhang1,2, Junya Wang4, Minghao Gong1,2 and Zheng You1,4 Abstract In this paper, an angular position sensor (APS) designed for a resonant miniaturized scanning mirror (M-SM) is presented. The APS operates based on the principle of differential variable capacitance, significantly expanding the detectable bandwidth from a few hertz to several kilohertz. By modeling the motion characteristics, the sampling rates of the biaxial scanning angles are 1473.6 times and 539.4 times higher than those of conventional sensors. Initially, the motion characteristics model is presented as a simple harmonic motion, converting sampled capacitance into continuous capacitance. Subsequently, the nonparallel state of the M-SM and sensor is transformed into a parallel state through the space coordinate system transformation. Furthermore, a 2D nonlinear angle transfer function is developed to convert the differential capacitance into an angle, thereby mitigating the nonlinear errors resulting from large angles. Achieving an accuracy better than 0.014°, the measuring range expands from ±0.5729° (±10 mrad) to ±5.026° ( ± 87 mrad). Additionally, the capturing mode and tracking mode are proposed to monitor real-time angular changes of the M-SM with an accuracy of 0.017°. High-precision APSs have enhanced beam pointing accuracy and resolution and can thereby be used to advance the development of laser components, including light detection and ranging (LiDAR). Introduction Angular position sensors (APSs) are in high demand across various industrial, automotive, and robotic applications1. Recent advancements in micro-electromechanical system scanning mirror (MEMS-SM)-based LiDAR have resulted in new APSs that are capable of measuring the out-of-plane scanning angle. APSs for MEMS-SM have been developed based on piezoresistivity2–8, piezoelectricity9–11, electromagnetic induction12–15, active detection16–21, etc. The piezoresistive APS features simple manufacturing processes and high integration2,3. The angle estimation algorithm based on the unscented Kalman filter (UKF) Correspondence: Gaofei Zhang () 1 Department of Precision Instrument, Tsinghua University, Beijing 100084, China 2 Key Laboratory of Smart Microsystem (Tsinghua University), Ministry of Education, Beijing 100084, China Full list of author information is available at the end of the article was employed to reduce noise4. The piezoresistive coefficient was estimated through master-mode frequency response function (FRF) curve analysis5, and closed-loop control using Wheatstone bridges and diffusion piezoresistive sensors was implemented6. A calibrated scale is utilized to accurately calibrate the voltage signal of the piezoresistive sensor, ensuring a precision of 0.5° or less7. A temperature compensation scheme is proposed to accommodate temperature noise8. The piezoelectric APS is fabricated along with the aluminum nitride (AIN) driving film, and a maximum shape error of 25% may be caused9. The driving voltage of the aluminum scandium nitrogen (AlScN) film is calibrated using a position sensitive detector (PSD) to determine the angle; however, motion nonlinearity can lead to calibration failure10. An accuracy of less than 0.1° is achieved by precise position feedback control11. The electromagnetic APS comprises an integrated permanent magnet and an energized coil on the MEMS-SM. © The Author(s) 2023 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/. Liu et al. Microsystems & Nanoengineering (2023)9:148 The accuracy of 0.067° is obtained by establishing the equations of the induced electromotive force and scanning angle12. Accuracy is further improved by optimizing the coil position13. Typically, the generated induced electromotive force is extremely weak and requires amplification by several hundred times for detection14. Without utilizing the electromagnetic driving principle, additional coils and permanent magnets would need to be assembled, resulting in increased volume loads within the system15. The active detection APS comprises a VCSEL and three photodiodes (PDs) that receive optical power proportional to the small scanning angle of 5°16. The scanning angle is determined by measuring the differential optical voltage of the quadrant photodetectors (QPDs), achieving an accuracy of 0.1°17. Laser diodes, QPDs, and readout circuits can be integrated onto a single chip18. The configuration incorporating nine PDs with high-resolution optical sensors and calculations ensured a goniometric accuracy better than 0.1°19. The PD was replaced with a PSD, reducing the measurement uncertainty to approximately 0.026°. Furthermore, the main sources of error were analyzed20. The size of the PSD sensor and the laser module significantly exceeds that of MEMS-SM, resulting in increased volume load21. However, the constrained mirror size and limited angular measurement accuracy of MEMS-SM limit the ability of LIDAR to detect long-range targets. Nonsilicon-based miniaturized scanning mirrors (M-SMs) have been proposed for obtaining larger apertures, and APS schemes with higher accuracy have been explored. Differential capacitors provide excellent accuracy performance22,23, and associated multi-segment electrodes are commonly employed in in-plane angle measurements, characterized by uniform spacing between the capacitive electrodes. The straightforwardness of this principle has spurred significant research endeavors, particularly in applications such as measuring the rotational angles of motors or gyroscopes1,24,25. Nonetheless, when the capacitor pole-plate spacing dynamically changes with the angle, the introduction of numerous nonlinear calculation terms complicates the theoretical model describing the relationship between the angle and capacita (...truncated)