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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

To investigate the drive-mode resonance frequency of a micromachined vibratory gyroscope (MVG), one needs to measure it accurately and efficiently. The conventional approach to measure the resonant frequency is by performing a sweep frequency test and spectrum analysis. The method is time-consuming and inconvenient because of the requirements of many test points, a lot of data storage and off-line analyses. In this paper, we propose two novel measurement methods, the search method and track method, respectively. The former is based on the magnitude-frequency characteristics of the drive mode, utilizing a one-dimensional search technique. The latter is based on the phase-frequency characteristics, applying a feedback control loop. Their performances in precision, noise resistivity and efficiency are analyzed through detailed simulations. A test system is implemented based on a field programmable gate array (FPGA) and experiments are carried out. By comparing with the common approach, feasibility and superiorities of the proposed methods are validated. In particular, significant efficiency improvements are achieved whereby the conventional frequency method consumes nearly 5,000 s to finish a measurement, while only 5 s is needed for the track method and 1 s for the search method.

With their advantages of low power dissipation, compact bulk, and low weight, MVGs have broad applications in military and commercial areas, such as navigation assistance, vehicle platform stabilization, consumer electronics and automation. The operation of an MVG based on the Coriolis force. First, a vibration is generated and maintained along the drive direction. If there exists an angular movement around the input axis, a Coriolis force will be formed. Then, the force causes a vibration in the sense direction. One can obtain the input angular rate by detecting the vibration along the sense axis [

Commonly, the drive mode of an MVG is excited at resonance with constant amplitude. In this case, the sense-mode vibration frequency equals the drive-mode resonant frequency [

The conventional approach to measure the resonant frequency is by performing sweep frequency tests and spectrum analyses [

In this paper, two novel resonance frequency measurement methods for the drive mode of MVGs are introduced. One, called the search method, is based on the magnitude-frequency characteristics, and utilizes a one-dimensional search technique. The other, called the track method, is based on the phase-frequency characteristics, and applies a feedback control loop. The proposed measurements can be run on-line and perform efficiently and accurately.

In the next section, we present and analyze dynamics of the MVG. Section 3 introduces basic fundamentals of the proposed methods. Section 4 discusses the performance in simulation. Section 5 presents the implemented system and experimental results. Finally, conclusions are provided in Section 6.

A typical MVG includes a vibration structure supported by suspensions and some electrodes. Normally, the structure oscillates freely along two orthogonal axes: the drive and sense axis. The overall system can be modeled as a mass-spring-damper structure having two degrees of freedom (2-DOF), as shown in

In the drive direction, a controlled sinusoidal force is generated to make the mass vibrate at the drive-mode resonant frequency and achieve stable amplitude by the use of the automatic gain control (AGC) method [_{x}_{y}_{x}_{y}_{d}

From _{d}

According to the transfer function in

As an example, an MVG with the parameters listed in

Traditionally, the sweep frequency method is adopted to measure the resonant frequency of the MVG. In this method, the frequency of the input signal increases or decreases by a set step (the stepped frequency) in a given range (sweep range), and all of the response signals should be collected and recorded. In order to get a more accurate result, the stepped frequency should be small enough, and the dwell time at each frequency point should be long enough to avoid the influence of the transient. Generally, a sweep frequency test requires more than a few minutes. A problem arises that the obtained resonant frequency may not be the real frequency because the dynamics of the system may have changed substantially during the course of the measurement. In this section, two efficient methods, the search method and the track method, are introduced and their fundamentals are presented below.

Differentiating

By setting

In practice, the MVG is normally packaged in specific vacuum level and _{m}_{r}_{r}_{m}_{r}

The synchronous demodulation module is adopted to obtain the amplitude of the response signal, whose principle diagram is presented in

A search controller is used to control search process. It is realized based on the one-dimensional search technique, whose flowchart is shown in _{l}_{h}

The oscillator is used to generate sine signal for a given frequency. Direct Digital Synthesizer (DDS) is a candidate to achieve the function [

According to

Defining a parameter Δ

Generally, _{r}_{r}

It is implied that cot_{r}

According to

Based on the above analyses, a feedback control system is proposed to measure the resonance frequency _{r}

The synchronous demodulation module here is the same as that in the search method.

The track controller is a key module in the system, which is used to smooth the value of cot

The oscillator module here is the same as that in Section 3.1, which is also used to generate sine signals.

The closed-loop control system is similar to the phase locked loop (PLL) [_{r}

When the loop reaches stability, the frequency _{r}

In this section, simulation systems are built using Simulink to investigate the performance of the proposed methods, including the efficiency, precision, and noise resistivity. MVG parameters used here are same as those in

First, the precision and efficiency are studied by assigning six different resonant frequencies in the range of 4–5 kHz. The results are presented in ^{−8} and a measurement process requires only 44 search steps.

Then, the performance comparison is simulated at different noise levels. In the simulation, the resonant frequency to be measured is set as 4,000 Hz and random noise with different signal-noise-ratio (SNR) is injected into the detected amplitude of the response signal. Simulation results are plotted in ^{−5}.

Three simulations are carried out to investigate the performance of the track method. In the first simulation, the resonant frequency is set as a constant (_{r}

The second simulation is carried out to evaluate the performance for tracking varying frequency, where the resonant frequency changes linearly with a 10 Hz/s slope from 4,000 Hz to 4,050 Hz. The results are plotted in

Finally, the performance of noise resistivity is simulated. The resonant frequency here is still set as a constant (_{r}

In order to study the presented methods experimentally, a measurement system is designed and implemented, as shown in

Two experiments are carried out to assess the actual performance of the proposed measurement methods. One is for normal temperature test where the resonant frequency varies slightly. The other is under different temperature conditions where the resonant frequency changes significantly. In each experiment, three methods are used in turn to facilitate comparison. The common parameters in the experiments are presented in

In normal temperature experiment, five repeated tests are carried out.

The second experiment is carried out to measure the varying resonant frequency. The system is placed in a the temperature chamber. We make the temperature change from 5 °C to 50 °C with each temperature point kept for 1 h. The measurement results are plotted in

It should be noticed that these experimental results are inadequate to evaluate the measurement precision directly. The reason is twofold. First, the true value of the drive-mode resonant frequency is unknown and variable. Second, for each experiment, three methods are conducted at different time, meaning that their true values may be different.

This paper focuses on the measurement for the drive-mode resonant frequency of a MVG. Two novel methods, the search method and track method, are proposed. The search method is based on the magnitude-frequency characteristics of the drive mode, and utilizes a one-dimensional search technique, while the track method is based on the phase-frequency characteristics, and applies a closed-loop control technique.

The feasibilities of the two measurement methods are validated by simulations and experiments. The simulations results show that they behave well in both measurement accuracy and noise resistivity. When the SNR of a detected signal is 50 dB, the relative error of the measurement value using the search method is only on the order of 10^{−5}, and 10^{−6} for the track method.

Significant improvements in measurement efficiency are achieved by the proposed methods. Experimental results show that the traditional sweep frequency method consumes nearly 5,000 s to finish one test, while only 5 s are needed for the track method and 1 s for the search method.

Additionally, the proposed methods are easy to implement on-line because they require only a small amount of data storage. They are also applicable for resonators similar to MVGs. It should be noticed that these two novel methods fail to provide information about other parameters, for example, the Q-factor, so they are suitable for users who only want to obtain the resonant frequency of a MVG.

This work was supported by the National Natural Science Foundation of China (Grant No. 61104201) and National University of Defense Technology Innovation Foundation For Postgraduate (Grant No. 4345111141N).

The authors declare no conflict of interest.

A simplified model of a MVG.

The magnitude and phase transfer function plot, with the inset showing a detailed view around the resonant frequency.

The block diagram of the proposed search method.

The principle diagram of the synchronous demodulation technique.

The flowchart of the applied one dimension search technique.

The block diagram of the track method.

The comparison of the proposed describer (cot

Measurement results for different resonant frequencies using the search method.

Measurement results of the search method under different level noise.

Measurement value (left) and error (right) for a constant resonant frequency (_{r}

Track results when resonant frequency changes linearly from 4,000 Hz to 4,050 Hz using the track method.

Errors of the track method under different level noise (_{r}

The implemented system for experiments. (

Experimental results of three methods in normal temperature. (

Experimental results of three methods for varying resonant frequency in different temperature.

Example parameters for an MVG.

_{m} |
_{r}_{r} |
||
---|---|---|---|

Value | 5 μg | 4KHz | 2,000 |

Measurement errors and search steps for different resonant frequencies using the search method.

Measurement value(Hz) | 3999.9997 | 4199.9997 | 4399.9997 | 4599.9997 | 4799.9997 | 4999.9997 |

Relative Error | −6.2479e-8 | −6.2470e-8 | −6.2466e-8 | −6.2501e-8 | −6.2530e-8 | −6.2548e-8 |

Search steps | 44 | 44 | 44 | 44 | 44 | 44 |

Measurement results statistics of the search method under different level noise.

Measurement value (Hz) | 3999.9997 | 3999.9886 | 4000.0786 | 4000.1357 | 3852.6087 |

Relative error | −6.248e-8 | −2.8500e-6 | 1.9650e-5 | 3.3925e-5 | −0.0368 |

Search steps | 45 | 45 | 45 | 45 | 45 |

Parameters in the experiments.

| |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Value | 3.5–4.5 KHz | 0.001 | 0.618 | 50 ms | 32 | 256 | 0 | 50 ms | 3.5–4.5 KHz | 0.01 Hz | 50 ms |

Statistics of experimental results in normal temperature.

Sweep frequency |
4156.44 | 4155.34 | 4155.74 | 4155.53 | 4155.83 | 4155.78 | 0.4170 |

Track method(Hz) | 4155.72 | 4155.69 | 4155.67 | 4155.65 | 4155.67 | 4155.68 | 0.0265 |

Search method(Hz) | 4155.51 | 4155.63 | 4155.72 | 4155.75 | 4155.55 | 4155.63 | 0.1040 |