# Digital Control and Demodulation Algorithm for Compact Open-Loop Fiber-Optic Gyroscope

^{1}

^{2}

^{*}

## Abstract

**:**

^{3}achieves a bias instability of less than 0.15°/h, an angle random walk (ARW) of less than 0.015°/√h, a start-up time of less than 1 s, and a 3 dB bandwidth beyond 160 Hz. This low-cost, compact, and high-performance gyro is sufficient to satisfy the requirements of applications in the navigation and control fields such as unmanned driving.

## 1. Introduction

^{3}, which is about 50% of the size of similar gyros, e.g., the DSP17XX series from KVH Inc. with a dimension of ϕ45.7 × 22.9 mm

^{2}. At the same time, the use of a low-arithmetic-power processor significantly reduces the cost of our gyro. This kind of cost-effective and compact gyro provides high performances in bias instability of less than 0.15°/h, angle random walk (ARW) of less than 0.015°/√h, start-up time of less than 1 s, and 3 dB bandwidth of more than 160 Hz. These characteristics promise that the open-loop FOG with our new demodulation algorithm can find potential applications in many fields, such as automation and control.

## 2. Demodulation and Control Algorithm

#### 2.1. Demodulation Algorithm

_{s}of the two counterpropagating beams in the fiber coil is proportional to the rotational angular velocity Ω [14],

_{0}is half of the maximum light intensity detected and w is the angular frequency of modulation. In order to use a low voltage to excite its phase-modulating signal, the PZT is driven close to its resonance frequency. The modulator can only operate efficiently in a small frequency bandwidth of about 5 kHz around its resonance and driving frequencies deviating from the resonance will result in a sharp decay in the PZT oscillation amplitude. Therefore, the effect of multiple harmonics of the PZT is negligible. τ represents the phase delay, which corresponds to the sampling delay for this digital gyro, ϕ

_{m}is the amplitude of phase modulation, and φ

_{sq}= φ

_{s}+ φ

_{q}, where φ

_{s}is the Sagnac phase shift and φ

_{q}represents the polarization error that cannot be decoupled from the Sagnac phase shift and can only be suppressed with optimized optical devices [17,18].

_{0}, settled modulation amplitude ϕ

_{m}, and smallest possible sampling delay τ are all necessary to precisely obtain the Sagnac phase shift. The control algorithm is described as follows.

#### 2.2. Control of the Synchronous Sampling Delay

_{sq}and ξ range from −π/2 to π/2, Δ = 0 only happens when τ = 0. Therefore, the synchronous sampling delay τ could be controlled with feedback to make τ = 0 by measuring Δ. Choosing the appropriate picking interval ξ can maximize Γ = sin ξ sin (ϕ

_{m}cos ξ) so as to obtain an optimum sensitivity of feedback control for Δ. With different values of ϕ

_{m}, the variation in Γ as a function of the picking interval is shown in Figure 3. Note that the optimum ξ is between 0.8 and 1, and considering the discreteness of sampling, ξ is set to a value of 5π/18. Thus, τ can be controlled with calculations of the 4 sampling points at 2π/9, 7π/9, 11π/9, and 16π/9. When τ approaches zero, Equation (10) then reduces to

#### 2.3. Control of the Modulation Amplitude

_{1}(ϕ

_{m}) takes the maximum value, which means that ϕ

_{m}≈ 1.84 and J

_{1}(ϕ

_{m}) ≈ 0.58. Define that

_{sq}) > 0. Based on Equations (15)–(17), we can determine that

_{1}(err

_{1}is a predefined acceptable control error for PZT, which is 0.0003 in this work), ϕ

_{m}could be controlled to (1.84 ± 0.005) rad.

#### 2.4. Control of Optical Intensity

_{0}should be settled to a predefined constant value (i.e., the first derivative equals zero) in the demodulation control process. Consequently, the gyro can realize a stable output as long as the I

_{0}is closed-loop controlled. Since the SNR of the FOG mainly depends on the shot noise that is proportional to √I

_{0}[18], the gyro’s input optical intensity should be increased as much as possible in order to obtain more accurate demodulation results. However, a higher optical intensity of the light source leads to its reduced lifetime and increased relative intensity noise [19] and thus decreased SNR. Therefore, considering both factors, in this work, a moderate optical intensity I

_{0}corresponding to an optical power of 10 μW is chosen for our gyro. The control process is then given as follows. With ϕ

_{m}= 1.84 rad, substitute ϕ

_{m}into Equations (13) and (16), and thus,

_{0}− T| ≤ err

_{2}(err

_{2}is the predetermined acceptable optical power control error depending on the control accuracy of the SLED, which is 0.005 μW in this work), where T is the target value of the optical power, i.e., 10 μW in this work.

#### 2.5. Suppression of the Bias with a Periodic Flip of Modulation

_{1}(ϕ

_{m}) is an odd function, the error can be easily eliminated by flipping the modulation polarity regularly and changing the output polarity simultaneously. The detailed modulation process is as follows.

_{sq}(ϕ

_{m}) and φ

_{sq}(−ϕ

_{m}) are the demodulated phases with positive and negative modulation, respectively, and ε is the error due to electric crosstalk. Hence, the bias due to the crosstalk can be eliminated, and the accurate gyro output can be achieved by synchronously flipping the polarity of the gyro’s output according to

## 3. Software, Hardware Implementation, and System Construction

^{2}. The hardware diagram and software process are shown in Figure 4. The FPGA controls the analog-to-digital converter (ADC) to sample the interference light and the digital-to-analog converter (DAC) to achieve the closed-loop feedback of the SLED output power and the phase of the PZT modulator. The computation required by the digital demodulation of the Sagnac phase with bias suppression is also conducted in the FPGA. The timing of the above processes is precisely controlled with a crystal oscillator. The initial values of the control parameters as a function of start-up temperature are determined during the calibration process after the gyro is assembled. They are tabulated and pre-stored in the memory of the gyro’s circuit board for the fast retrieval of multiple control parameters at the start-up within the full temperature range [24].

^{3}using a quadrupole symmetrical winding method. An in-house packaged miniaturized GaAs SLED light source of a center wavelength of 830 nm was adopted. It was mounted on a heat sink that had thermal contact with the mechanical structure of our open-loop FOG, and no active cooling was applied to it. The output power of the light source can be more than 500 μW under a driving current of 60 mA with a fiber-coupled power of 50 μW. The 2 couplers were in-house developed fused taper-type couplers using the 40 μm polarization-maintaining fiber. The beam-splitting ratio error was less than 5%, and the polarization-maintaining capability was greater than 20 dB. The polarizer was also an in-house developed 45° tiled fiber Bragg grating whose extinct ratio is more than 35 dB [25,26]. A 100 kHz sinusoidal-modulated PZT phase modulator with about 1 m long wrapped fiber was integrated into the system to reduce the size and cost of the FOG. All the optical components and streamlined circuit board with our new algorithm fit into a volume of only 25 × 20 × 40 mm

^{3}to construct the miniaturized open-loop fiber-optic gyroscope, which is shown in Figure 5d.

## 4. Determinations of Gyroscope Performance

_{I}) [27]. As shown in Figure 7b, on the Allan variance curve, ARW can be estimated by dividing the value at 1 s by 60 to be 0.012°/√h and B

_{I}can be calculated by dividing the bottom of the curve by 0.664 to be 0.11°/h. To verify the dynamic performance of the gyro, we tested the amplitude–frequency characteristics of the gyroscope with a swing table under different swing frequencies [28]. The FOG had excellent dynamic characteristics with a 3 dB bandwidth of 160 Hz, as shown in Figure 7c, which can meet the application requirements in the field of automatic driving, robot control, etc. [29]. The Gaussian-like decay of the amplitude–frequency curve indicates that the bandwidth of the gyro was mainly limited by the low-pass filter in the algorithm, which can be further improved by optimizing the filter. The effectiveness of the bias elimination algorithm was also verified, as shown in Figure 7d, and after turning on the periodic flip, the gyro’s bias decreases from −55.2°/h to −8.7°/h, which is close to the celestial component of the earth’s rotation speed of −8.4 °/h at the test location (Xi’an, Shaanxi Province, China, with its latitude of 34° N). We placed the gyro in reverse (the direction of the sensitive axis changed from pointing to the ground to the sky) and repeated the above experiment. The output of the gyro changed from −37.5°/h to 7.9°/h after turning on the periodic flip. The earth’s rotation speed was eliminated by adding the forward and reverse output together and dividing this sum by two. By periodically flipping its sensitive axis, the gyro gives an output of −0.25°/h, compared with −46.4°/h obtained without periodic flip.

## 5. Conclusions

^{3}, and the comprehensive testing reveals its bias instability of less than 0.15°/h, an ARW of less than 0.015°/√h, a start-up time of less than 1 s, and a 3 dB bandwidth beyond 160 Hz. We believe that this kind of performance can meet the navigation needs of autonomous driving vehicles [30] and the control and automation needs of many other applications such as androids.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Clivati, C.; Calonico, D.; Costanzo, G.A.; Mura, A.; Pizzocaro, M.; Levi, F. Large-area fiber-optic gyroscope on a multiplexed fiber network. Opt. Lett.
**2013**, 38, 1092–1094. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Yao, X.S.; Xuan, H.F.; Chen, X.J.; Zou, H.H.; Liu, X.; Zhao, X. Polarimetry fiber optic gyroscope. Opt. Express
**2019**, 27, 19984–19995. [Google Scholar] [CrossRef] [PubMed] - Jin, J.; He, J.; Song, N.; Ma, K.; Kong, L. A compact four-axis interferometric fiber optic gyroscope based on multiplexing for space application. J. Light. Technol.
**2020**, 38, 6655–6663. [Google Scholar] [CrossRef] - Udd, E.; Digonnet, M.J.F. Design and Development of Fiber Optic Gyroscopes; SPIE Press: Washington, DC, USA, 2019; pp. 35–50. [Google Scholar]
- KVH Industries Inc. Guide to comparing gyro and IMU technologies-micro-electro-mechanical systems and fiber optic gyros. KVH White Pap.
**2014**, 3–5. [Google Scholar] - Zhang, D.; Liang, C.; Li, N. A novel approach to double the sensitivity of polarization maintaining interferometric fiber optic gyroscope. Sensors
**2020**, 20, 3762. [Google Scholar] [CrossRef] - Wu, W.; Zhou, K.; Lu, C.; Xian, T. Open-loop fiber-optic gyroscope with a double sensitivity employing a polarization splitter and Faraday rotator mirror. Opt. Lett.
**2018**, 43, 5861–5864. [Google Scholar] [CrossRef] - Medjadba, H.; Lecler, S.; Simohamed, L.M.; Fontaine, J.; Kiefer, R. An optimal open-loop multimode fiber gyroscope for rate-grade performance applications. Opt. Fiber Technol.
**2011**, 17, 546–553. [Google Scholar] [CrossRef] - Wang, X.; He, C.; Wang, Z. Method for suppressing the bias drift of interferometric all-fiber optic gyroscopes. Opt. Lett.
**2011**, 36, 1191–1193. [Google Scholar] [CrossRef] - Wang, Q.; Yang, C.; Wang, X.; Wang, Z. All-digital signal-processing open-loop fiber-optic gyroscope with enlarged dynamic range. Opt. Lett.
**2013**, 38, 5422–5425. [Google Scholar] [CrossRef] - Bacurau, R.M.; Spengler, A.W.; Dante, A.; Morais, F.J.O.; Duarte, L.F.C.; Ribeiro, L.E.B.; Ferreira, E.C. Technique for suppressing the electronic offset drift of interferometric open-loop fiber optic gyroscopes. Opt. Lett.
**2016**, 41, 5186–5189. [Google Scholar] [CrossRef] - Lu, C.; Zhou, K. A low-cost all-digital demodulation method for fiber-optic interferometric sensors. Meas Sci Technol.
**2018**, 29, 105103. [Google Scholar] [CrossRef] - Gronau, Y.; Tur, M. Digital signal processing for an open-loop fiber-optic gyroscope. Appl Opt.
**1995**, 34, 5849–5853. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lefèvre, H.C. The fiber-optic gyroscope: Challenges to become the ultimate rotation-sensing technology. Opt. Fiber Technol.
**2013**, 19(6), 828–838. [Google Scholar] [CrossRef] - Bergh, R.A. All-single-mode fiber-optic gyroscope with long-term stability. Opt. Lett.
**1981**, 6, 502–504. [Google Scholar] [CrossRef] [PubMed] - Song, N.; Cai, W.; Song, J.; Jin, J.; Wu, C. Structure optimization of small-diameter polarization-maintaining photonic crystal fiber for mini coil of spaceborne miniature fiber-optic gyroscope. Appl. Opt.
**2015**, 54, 9831–9838. [Google Scholar] [CrossRef] [PubMed] - Chamoun, J.N.; Digonnet, M.J.F. Noise and bias error due to polarization coupling in a fiber optic gyroscope. J. Light. Technol.
**2015**, 33, 2839–2847. [Google Scholar] [CrossRef] - Ulrich, R. Fiber-optic rotation sensing with low drift. Opt. Lett.
**1980**, 5, 173–175. [Google Scholar] [CrossRef] [PubMed] - Li, Y.; Ben, F.; Luo, R.; Peng, C.; Li, Z. Excess relative intensity noise suppression in depolarized interferometric fiber optic gyroscopes. Opt. Commun.
**2019**, 440, 83–88. [Google Scholar] [CrossRef] - Zhang, Y.; Guo, Y.; Li, C.; Wang, Y.; Wang, Z. A new open-loop fiber optic gyro error compensation method based on angular velocity error modeling. Sensors
**2015**, 15, 4899–4912. [Google Scholar] [CrossRef] [Green Version] - Li, Z.; Dong, L.; Li, H.; Zhang, J.; Wang, X.; Zhang, H. An analog readout circuit with a noise-reduction input buffer for MEMS microphone. IEEE Trans. Circuits Syst.
**2022**, 69, 438–443. [Google Scholar] [CrossRef] - Bednar, T.; Babusiak, B.; Labuda, M.; Smetana, M.; Borik, S. Common-mode voltage reduction in capacitive sensing of biosignal using capacitive grounding and DRL electrode. Sensors
**2021**, 21, 2568. [Google Scholar] [CrossRef] [PubMed] - Napoli, J.; Ward, R. Two decades of KVH fiber optic gyro technology: From large, low performance FOGs to compact, precise FOGs and FOG-based inertial systems. DGON ISS
**2016**, 1–19. [Google Scholar] - Din, H.; Iqbal, F.; Park, J.; Lee, B. Bias-repeatability analysis of vacuum-packaged 3-Axis MEMS gyroscope using oven-controlled system. Sensors
**2023**, 23, 256. [Google Scholar] [CrossRef] [PubMed] - Jiang, B.; Hou, Y.; Wang, H.; Gan, X.; Li, A.; Hao, Z.; Zhou, K.; Zhang, L.; Zhao, J. Few-layer graphene integrated tilted fiber grating for all-optical switching. J. Light. Technol.
**2020**, 39, 1477–1482. [Google Scholar] [CrossRef] - Jiang, B.; Zhao, J. Nanomaterial-functionalized tilted fiber gratings for optical modulation and sensing. J. Light. Technol.
**2022**, 1, 1558–2213. [Google Scholar] [CrossRef] - Wang, L.; Zhang, C.; Gao, S.; Wang, T.; Lin, T.; Li, X. Application of fast dynamic Allan variance for the characterization of FOGs-based measurement while drilling. Sensors
**2016**, 16, 2078. [Google Scholar] [CrossRef] [Green Version] - Ren, W.; Luo, Y.; He, Q.; Zhou, X.; Deng, C.; Mao, Y.; Ren, G. Stabilization control of electro-optical tracking system with fiber-optic gyroscope based on modified smith predictor control scheme. IEEE Sens. Lett.
**2018**, 18, 8172–8178. [Google Scholar] [CrossRef] - Passaro, V.M.N.; Cuccovillo, A.; Vaiani, L.; De Carlo, M.; Campanella, C.E. Gyroscope technology and applications: A review in the industrial perspective. Sensors
**2017**, 17, 2284. [Google Scholar] [CrossRef] [Green Version] - Vivacqua, R.; Vassallo, R.; Martins, F. A low cost sensors approach for accurate vehicle localization and autonomous driving application. Sensors
**2017**, 17, 2359. [Google Scholar] [CrossRef]

**Figure 3.**Optimized picking interval for the elimination of the synchronous sampling delay, τ, at different modulation amplitudes.

**Figure 4.**(

**a**) Diagram of PZT control; (

**b**) periodically flipped driving pattern, the modulation generated by the driving, and the corresponding interference signal of the gyro (assuming that the gyro has a Sagnac phase shift of 0.05 rad for clarity).

**Figure 5.**(

**a**) Circuit hardware diagram, (

**b**) picture of the circuit hardware, (

**c**) software process of FOG, and (

**d**) picture of the FOG.

**Figure 6.**Dynamic control results of (

**a**) SLED, (

**b**) PZT, and (

**c**) sampling delay during the startup process.

**Figure 7.**The measured results for our digital open-loop fiber-optic gyroscope. (

**a**) Gyro output at different angular rates, (

**b**) results for Allan variance testing, (

**c**) gyro amplitude-frequency characteristics, and (

**d**) bias comparison with the different algorithms.

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**MDPI and ACS Style**

Chen, L.; Huang, Z.; Mao, Y.; Jiang, B.; Zhao, J.
Digital Control and Demodulation Algorithm for Compact Open-Loop Fiber-Optic Gyroscope. *Sensors* **2023**, *23*, 1473.
https://doi.org/10.3390/s23031473

**AMA Style**

Chen L, Huang Z, Mao Y, Jiang B, Zhao J.
Digital Control and Demodulation Algorithm for Compact Open-Loop Fiber-Optic Gyroscope. *Sensors*. 2023; 23(3):1473.
https://doi.org/10.3390/s23031473

**Chicago/Turabian Style**

Chen, Lin, Zhao Huang, Yuzheng Mao, Biqiang Jiang, and Jianlin Zhao.
2023. "Digital Control and Demodulation Algorithm for Compact Open-Loop Fiber-Optic Gyroscope" *Sensors* 23, no. 3: 1473.
https://doi.org/10.3390/s23031473