# Design and Verification of a Digital Controller for a 2-Piece Hemispherical Resonator Gyroscope

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Electromechanical Modeling of a HRG with Switched Harmonic Excitations

#### 2.1. Full Equations of Motion with Harmonic Excitations

_{11}= (2/τ) + Δ(1/τ)cos2θ

_{τ}, C

_{12}= C

_{21}= Δ(1/τ)sin2θ

_{τ}, C

_{22}= (2/τ) − Δ(1/τ)cos2θ

_{τ}, K

_{11}= ω

^{2}− ωΔωcos2θ

_{ω}, K

_{12}= K

_{21}= − ωΔωsin2θ

_{ω}, K

_{22}= ω

^{2}+ ωΔωcos2θ

_{ω}, ω

^{2}= (ω

^{2}

_{1}+ ω

^{2}

_{2})/2, 1/τ = ½(1/τ

_{1}+1/τ

_{2}), ωΔω = (ω

^{2}

_{1}− ω

^{2}

_{2})/2, Δ(1/τ) = (1/τ

_{1}) − (1/τ

_{2}), ${\theta}_{\omega}$ is the angle of the unbalance between the x-axis and the major axis of resonance mode, ${\theta}_{\tau}$ is the angle between the x-axis and the major axis of the linear damper (readers should refer to Appendix A for further details).

_{n}= 2Q/ω, $\mathrm{Q}$ is the quality factor. In this moment, the excitation force to be applied continuously can be modeled with the harmonic function as follows:

#### 2.2. The Nominal Amplitude and Time Constant of HRG with Switched Harmonic Excitations

- Case a.
- ${C}_{1}{T}_{0}\left(n-1\right)<t\le {C}_{2}{T}_{0}+{C}_{1}{T}_{0}\left(n-1\right),n=1,2,\cdots ,\infty $$$\mathrm{x}\left(t\right)={\displaystyle \sum}_{k=1}^{k=n-1}\left[{{\displaystyle \int}}_{{C}_{1}{T}_{0}\left(k-1\right)}^{{C}_{2}{T}_{0}+{C}_{1}{T}_{0}\left(k-1\right)}f\left(\tau \right)h\left(t-\tau \right)d\tau \right]+{{\displaystyle \int}}_{{C}_{1}{T}_{0}\left(n-1\right)}^{t}f\left(\tau \right)h\left(t-\tau \right)d\tau $$
- Case b.
- ${C}_{2}{T}_{0}\left(n-1\right)+{C}_{1}{T}_{0}<t\le 2{C}_{1}{T}_{0}n,n=1,2,\cdots ,\infty $$$\mathrm{x}\left(t\right)={\displaystyle \sum}_{k=1}^{k=n}\left[{{\displaystyle \int}}_{{C}_{1}{T}_{0}\left(k-1\right)}^{{C}_{2}{T}_{0}+{C}_{1}{T}_{0}\left(k-1\right)}f\left(\tau \right)h\left(t-\tau \right)d\tau \right]$$

- Case a.
- ${C}_{1}{T}_{0}\left(n-1\right)<t\le {C}_{2}{T}_{0}+{C}_{1}{T}_{0}\left(n-1\right),\mathrm{n}=1,2,\cdots ,\infty $$$x\left(t\right)=\frac{\frac{{f}_{0}}{{m}_{n}{\omega}_{d}}}{{\left(\frac{\omega}{2Q}\right)}^{2}+4{\omega}^{2}}\{{\displaystyle \sum}_{k=1}^{k=n-1}{e}^{\frac{\pi {C}_{1}\left(k-1\right)}{Q}}\left({e}^{\frac{2\pi {C}_{2}-\omega t}{2Q}}-{e}^{-\frac{\omega t}{2Q}}\right)\left[\left(\frac{\omega}{2Q}+4\omega Q\right)\mathrm{sin}{\omega}_{d}t+\omega \mathrm{cos}{\omega}_{d}t\right]\phantom{\rule{0ex}{0ex}}+\left[\left(4\omega Q\mathrm{sin}\omega t+\omega \mathrm{cos}\omega t\right)\left(1-{e}^{\frac{2\pi {C}_{1}\left(n-1\right)-\omega t}{2Q}}\right)-\frac{\omega}{2Q}\mathrm{sin}\omega t\text{}{e}^{\frac{2\pi {C}_{1}\left(n-1\right)-\omega t}{2Q}}\right]\}$$
- Case b.
- ${C}_{2}{T}_{0}\left(n-1\right)+{C}_{1}{T}_{0}<t\le 2{C}_{1}{T}_{0}n,\mathrm{n}=1,2,\cdots ,\infty $$$x\left(t\right)=\frac{\frac{{f}_{0}}{{m}_{n}{\omega}_{d}}}{{\left(\frac{\omega}{2Q}\right)}^{2}+4{\omega}^{2}}{\displaystyle \sum}_{k=1}^{k=n}{e}^{\frac{\pi {C}_{1}\left(k-1\right)}{Q}}\left({e}^{\frac{2\pi {C}_{2}-\omega t}{2Q}}-{e}^{-\frac{\omega t}{2Q}}\right)\left[\left(\frac{\omega}{2Q}+4\omega Q\right)\mathrm{sin}{\omega}_{d}t+\omega \mathrm{cos}{\omega}_{d}t\right]$$

#### 2.3. Verification of the Analytic Results through Simulations

#### 2.4. Electromechanical Modeling between Resonator and Electrodes

## 3. Design of the Signal Processing and Control Algorithm

#### 3.1. Design of the Signal Processing Algorithm

#### 3.2. Design of the Control Algorithm

#### 3.3. Numerical Verification of the Algorithm through Simulations

#### 3.4. Experimental Verification of the Algorithm

## 4. Conclusions

## Author Contributions

## Conflicts of Interest

## Appendix A

## Appendix B

## Appendix C

- Case a.
- ${C}_{1}{T}_{0}\left(n-1\right)<t\le {C}_{2}{T}_{0}+{C}_{1}{T}_{0}\left(n-1\right),n=1,2,\cdots ,\infty $$$x\left(t\right)=\frac{\frac{{f}_{0}}{{m}_{n}{\omega}_{d}}}{{\left(\frac{\omega}{2Q}\right)}^{2}+4{\omega}^{2}}\{{\displaystyle \sum}_{k=1}^{k=n-1}{e}^{\frac{\pi {C}_{1}\left(k-1\right)}{Q}}\left({e}^{\frac{2\pi {C}_{2}-\omega t}{2Q}}-{e}^{-\frac{\omega t}{2Q}}\right)\left[\left(\frac{\omega}{2Q}+4\omega Q\right)\mathrm{sin}{\omega}_{d}t+\omega \mathrm{cos}{\omega}_{d}t\right]\phantom{\rule{0ex}{0ex}}+\left[\left(4\omega Q\mathrm{sin}\omega t+\omega \mathrm{cos}\omega t\right)\left(1-{e}^{\frac{2\pi {C}_{1}\left(n-1\right)-\omega t}{2Q}}\right)-\frac{\omega}{2Q}\mathrm{sin}\omega t{e}^{\frac{2\pi {C}_{1}\left(n-1\right)-\omega t}{2Q}}\right]\}$$
- Case b.
- ${C}_{2}{T}_{0}\left(n-1\right)+{C}_{1}{T}_{0}<t\le 2{C}_{1}{T}_{0}n,n=1,2,\cdots ,\infty $$$x\left(t\right)=\frac{\frac{{f}_{0}}{{m}_{n}{\omega}_{d}}}{{\left(\frac{\omega}{2Q}\right)}^{2}+4{\omega}^{2}}{\displaystyle \sum}_{k=1}^{k=n}{e}^{\frac{\pi {C}_{1}\left(k-1\right)}{Q}}\left({e}^{\frac{2\pi {C}_{2}-\omega t}{2Q}}-{e}^{-\frac{\omega t}{2Q}}\right)\left[\left(\frac{\omega}{2Q}+4\omega Q\right)\mathrm{sin}{\omega}_{d}t+\omega \mathrm{cos}{\omega}_{d}t\right]$$

- Case a.
- ${C}_{1}{T}_{0}\left(n-1\right)<t\le {C}_{2}{T}_{0}+{C}_{1}{T}_{0}\left(n-1\right),n=1,2,\cdots ,\infty $$$\mathrm{A}\left(t\right)={C}_{x}\omega \sqrt{16{Q}^{2}+1}\left[{\displaystyle \sum}_{k=1}^{k=n-1}\left({e}^{\frac{\pi {C}_{2}}{Q}}-1\right){e}^{\frac{2\pi {C}_{1}\left(k-1\right)-\omega t}{2Q}}+\left(1-{e}^{\frac{2\pi {C}_{1}\left(n-1\right)-\omega t}{2Q}}\right)\right]$$
- Case b.
- ${C}_{2}{T}_{0}\left(n-1\right)+{C}_{1}{T}_{0}<t\le 2{C}_{1}{T}_{0}n,\mathrm{n}=1,2,\cdots ,\infty $$$\mathrm{A}\left(t\right)={C}_{x}\omega \sqrt{16{Q}^{2}+1}{\displaystyle \sum}_{k=1}^{k=n}\left({e}^{\frac{\pi {C}_{2}}{Q}}-1\right){e}^{\frac{2\pi {C}_{1}\left(k-1\right)-\omega t}{2Q}}$$

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**Figure 1.**The displacement output by continuous harmonic excitation: (

**a**) The control force input (N); (

**b**) The displacement output ($t\le 40\xb7\frac{2\mathsf{\pi}}{\mathsf{\omega}}\text{}\mathrm{s}$); (

**c**) The displacement output ($t\le 3000\text{}\mathrm{s}$).

**Figure 2.**Comparison of the switched control force $\mathrm{f}\left(\tau \right)$ and impulse response $\mathrm{h}\left(t-\tau \right)$: (

**a**) case of ${C}_{1}{T}_{0}<t\le {C}_{2}{T}_{0}$; (

**b**) case of ${C}_{2}{T}_{0}+{C}_{1}{T}_{0}<t\le 2{C}_{1}{T}_{0}$.

**Figure 3.**The displacement output by switched harmonic excitation: (

**a**) The control force input (N); (

**b**) The displacement output ($t\le 40\xb7\frac{2\mathsf{\pi}}{\mathsf{\omega}}\text{}\mathrm{s}$); (

**c**) The displacement output ($t\le 3000\text{}\mathrm{s}$).

**Figure 5.**The displacement output by switched harmonic excitation [Matlab/Simulink]: (

**a**) The control force input (N); (

**b**) The displacement output ($t\le 40\xb7\frac{2\mathsf{\pi}}{\mathsf{\omega}}\text{}\mathrm{s}$); (

**c**) The displacement output ($t\le 3000\text{}\mathrm{s}$) (yellow: continuous/pink: switched excitation).

**Figure 6.**The position of a point on the shell with the angular coordinates $\mathsf{\alpha},\mathsf{\phi}$ in a Cartesian coordinate frame fixed to the resonator.

**Figure 12.**Matlab/Simulink simulation program to verify the signal processing and control algorithm.

**Figure 13.**Matlab/Simulink simulation results; control varialbes: (

**a**) the amplitude and quadrature control variables; (

**b**) the rate and phase control variables.

**Figure 14.**Matlab/Simulink simulation results: (

**a**) the digital optputs to control; (

**b**) an estimate of the input rate.

**Figure 16.**A estimate of the Q, ΔQ and Δf by using the controllers: (

**a**) Measured Q-factor in according to target angle θ; (

**b**) Vibration of the target angle θ.

**Figure 17.**Test results to measure the bias instability and ARW: (

**a**) Gyro rate output for 15 h in the sound absorbing chamber; (

**b**) Allan variance analysis result for the estimate of bias instability and ARW.

Design Variables | Values |
---|---|

Resonant Frequency | $\omega =7.0\text{}\mathrm{kHz}\times 2\mathsf{\pi}\left(\mathrm{rad}/\mathrm{s}\right)$ |

Quality factor | $\mathrm{Q}=7\times {10}^{6}$ |

Modal mass | ${m}_{n}=0.85\times {10}^{-3}\text{}\mathrm{kg}$ |

The change of control force per unit voltage | $df/dv=2.78\times {10}^{-6}\text{}\mathrm{N}/\mathrm{V}\text{}$ |

Bias voltage | ${V}_{B}=200\text{}\mathrm{V}$ |

Nominal control voltage | $\overline{{v}_{c}}=420\text{}\mathrm{mV}$ |

The change of capacitance per unit displacement | $dC/dx=13.9\times {10}^{-9}\text{}\mathrm{F}/\mathrm{m}$ |

Design Variables | Values |
---|---|

Radius of Resonator | $\mathrm{R}=15.3\mathrm{mm}$ |

Nominal gap between resonator and electrode block | ${d}_{0}=120\mathsf{\mu}\mathrm{m}$ |

Dielectric permittivity | $\epsilon =8.85\times {10}^{-12}\text{}\mathrm{F}/\mathrm{m}$ |

Electrodes angles in azimuth | ${\phi}_{1}=-18\xb0,\text{}{\phi}_{2}=18\xb0$ |

Electrodes angles in elevation | ${\alpha}_{1}=78.2\xb0,\text{}{\alpha}_{2}=90\xb0$ |

Bias voltage | ${V}_{bias}=200\text{}\mathrm{V}$ |

Quality factor | $\mathrm{Q}=7\times {10}^{6}$ |

Modal mass | ${m}_{n}=0.85\times {10}^{-3}\text{}\mathrm{kg}$ |

Resonant frequency | $\mathsf{\omega}=4.4\times {10}^{4}\text{}\mathrm{rad}/\mathrm{s}$ |

Feedback capacitance of the charge amp | ${C}_{f}=22\text{}\mathrm{pF}$ |

Case of the Parallel Plate Capacitor | Case of Considering the Mode Shape of Resonator | |
---|---|---|

${C}_{0}$ | $3.8\mathrm{pF}$ | $3.8\mathrm{pF}$ |

$\frac{dC}{dx}$ | $17.5\times {10}^{-9}\mathrm{F}/\mathrm{m}$ | $13.9\times {10}^{-9}\mathrm{F}/\mathrm{m}$ |

$\frac{df}{d{v}_{ac}}$ | $3.5\times {10}^{-6\text{}}\mathrm{N}/\mathrm{V}$ | $2.78\times {10}^{-6}\mathrm{N}/\mathrm{V}$ |

$\frac{dx}{df}$ | $4.26\mathrm{m}/\mathrm{N}$ | $4.26\mathrm{m}/\mathrm{N}$ |

$\frac{d{v}_{ac}}{dC}$ | $9.1\times {10}^{12}\mathrm{V}/\mathrm{F}$ | $9.1\times {10}^{12}\mathrm{V}/\mathrm{F}$ |

Control Command | Designation | The Period |
---|---|---|

${N}_{\varphi}$ | Phase control | 1 operation period |

${N}_{a}$ | Amplitude control | X-control: update after X-sensing ${N}_{x}=f\left({c}_{x}\left[i\right],\text{}{s}_{x}\left[i\right],{c}_{y}\left[i-1\right],\text{}{s}_{y}\left[i-1\right]\right)$ Y-control: update after Y-sensing ${N}_{y}=f\left({c}_{x}\left[i\right],\text{}{s}_{x}\left[i\right],{c}_{y}\left[i\right],\text{}{s}_{y}\left[i\right]\right)$ |

${N}_{q}$ | Quadrature control | |

${N}_{r}$ | Rate control |

Control variable | Formula | Designation | Control Command | Target Value |
---|---|---|---|---|

$E$ | ${a}^{2}+{q}^{2}$ | Amplitude | ${N}_{a}$ | ${E}_{0}=1\text{}{\mathsf{\mu}\mathrm{m}}^{2}\text{}$ |

$Q$ | $2aq$ | Quadrature | ${N}_{q}$ | ${Q}_{0}=0\text{}{\mathsf{\mu}\mathrm{m}}^{2}$ |

$S$ | $\left({a}^{2}-{q}^{2}\right)\mathrm{sin}2\theta $ | Rate | ${N}_{r}$ | ${S}_{0}=1\text{}{\mathsf{\mu}\mathrm{m}}^{2}$ |

$L$ | $-\left({a}^{2}-{q}^{2}\right)\mathrm{sin}2{\varphi}^{\prime}$ | Phase | ${N}_{\varphi}$ | ${L}_{0}=0\text{}{\mathsf{\mu}\mathrm{m}}^{2}$ |

Control Variable | Control Force (Digital Control Command) | Input Phase | ${N}_{x}$ Component | ${N}_{y}$ Component |
---|---|---|---|---|

$E$ | ${F}_{a}\left({N}_{a}\right)$ | $-\mathrm{sin}\left(\omega t+{\varphi}_{corr}\right)$ | ${N}_{a}\mathrm{cos}\theta $ | ${N}_{a}\mathrm{sin}\theta $ |

$Q$ | ${F}_{q}\left({N}_{q}\right)$ | $\mathrm{cos}\left(\omega t+{\varphi}_{corr}\right)$ | $-{N}_{q}\mathrm{sin}\theta $ | ${N}_{q}\mathrm{cos}\theta $ |

$S$ | ${F}_{r}\left({N}_{r}\right)$ | $-\mathrm{sin}\left(\omega t+{\varphi}_{corr}\right)$ | $-{N}_{r}\mathrm{sin}\theta $ | ${N}_{r}\mathrm{cos}\theta $ |

Control Variable | Error | Command | PI Control Output |
---|---|---|---|

E | ${e}_{a}={E}_{0}-E$ | ${N}_{a}$ | ${N}_{k}={N}_{k-1}+{K}_{P}\left({e}_{k}-{e}_{k-1}\right)+\frac{{T}_{s}}{2{T}_{I}}\left({e}_{k}+{e}_{k-1}\right)$ |

Q | ${e}_{q}=-Q$ | ${N}_{a}$ | where ${K}_{P}$ is the Proportional gain ${T}_{I}$ is the Integral time ${T}_{s}$ is sampling time (=$1/{f}_{s}$) |

S | ${e}_{r}={S}_{0}-S$ | ${N}_{a}$ | |

L | ${e}_{\varphi}=L$ | ${N}_{a}$ |

Control Loop | Phase (PLL) | Amplitude | Quadrature | Rate |
---|---|---|---|---|

Bandwidth (Hz) | 7.5–12.5 | 1–5 | 1–5 | 7.5–12.5 |

Design Variables of the Controller | Matlab/Simulink Simulation Ver. | DSP Uploading Program Ver. |
---|---|---|

Frequency | 7.1 kHz | 7.12605 kHz |

Target Amp.(${E}_{0}$) | 512 bits (scaling) | 540 bits (scaling) |

Phase Delay | 1820 bits | 1682 bits |

Phase control P gain | 6000 bits | 4000 bits |

Phase control I gain | 10 bits | 25 bits |

Amplitude control P gain | 725 bits | 700 bits |

Amplitude control I gain | 1 bits | 1 bits |

Quadrature control P gain | 800 bits | 700 bits |

Quadrature control I gain | 6 bits | 1 bits |

Rate control P gain | 400,000 bits | 400,000 bits |

Rate control I gain | 300 bits | 300 bits |

Target Azimuth Angle θ | Time Constant τ | Target Value |
---|---|---|

–90° | 387 s | 8.6635 × 10^{6} |

−60° | 424 s | 9.4987 × 10^{6} |

−30° | 324 s | 7.2620 × 10^{6} |

0° | 281 s | 6.2853 × 10^{6} |

30° | 328 s | 7.3470 × 10^{6} |

60° | 374 s | 8.3804 × 10^{6} |

90° | 335 s | 7.4885 × 10^{6} |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lee, J.; Yun, S.W.; Rhim, J.
Design and Verification of a Digital Controller for a 2-Piece Hemispherical Resonator Gyroscope. *Sensors* **2016**, *16*, 555.
https://doi.org/10.3390/s16040555

**AMA Style**

Lee J, Yun SW, Rhim J.
Design and Verification of a Digital Controller for a 2-Piece Hemispherical Resonator Gyroscope. *Sensors*. 2016; 16(4):555.
https://doi.org/10.3390/s16040555

**Chicago/Turabian Style**

Lee, Jungshin, Sung Wook Yun, and Jaewook Rhim.
2016. "Design and Verification of a Digital Controller for a 2-Piece Hemispherical Resonator Gyroscope" *Sensors* 16, no. 4: 555.
https://doi.org/10.3390/s16040555