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This paper presents a signal processing technique to improve angular rate accuracy of the gyroscope by combining the outputs of an array of MEMS gyroscope. A mathematical model for the accuracy improvement was described and a Kalman filter (KF) was designed to obtain optimal rate estimates. Especially, the rate signal was modeled by a first-order Markov process instead of a random walk to improve overall performance. The accuracy of the combined rate signal and affecting factors were analyzed using a steady-state covariance. A system comprising a six-gyroscope array was developed to test the presented KF. Experimental tests proved that the presented model was effective at improving the gyroscope accuracy. The experimental results indicated that six identical gyroscopes with an ARW noise of 6.2 °/√h and a bias drift of 54.14 °/h could be combined into a rate signal with an ARW noise of 1.8 °/√h and a bias drift of 16.3 °/h, while the estimated rate signal by the random walk model has an ARW noise of 2.4 °/√h and a bias drift of 20.6 °/h. It revealed that both models could improve the angular rate accuracy and have a similar performance in static condition. In dynamic condition, the test results showed that the first-order Markov process model could reduce the dynamic errors 20% more than the random walk model.

Micro Electromechanical System (MEMS) gyroscopes have been used for measuring rate or angle of rotation in various inertial measurement fields thanks to their attractive advantages such as small size, low cost, possible batch fabrication and low power consumption [

In recent years, the redundant MEMS inertial sensors have been utilized integrated with GPS to improve navigation performance. Numerous studies and researches have been undertaken on redundant inertial measurement unit (IMU) integration, whereby the measurements of multiple IMUs are fused [

The key of combining multiple gyroscopes for accuracy improvement lies in rate signal modeling and the optimal filter design. Therefore, how to model the rate signal is a prerequisite for constructing a virtual gyroscope system. Through analyzing the current approaches [

Markov process modeling is a powerful and commonly used technique that has been introduced for modeling inertial sensors errors for several years, but it has not been used for modeling the virtual gyroscope before. In most applications, the maneuverability characteristics of aircrafts can be regarded as contained in a certain frequency bandwidth and magnitude [_{k+1}_{k+1}_{k}_{k}

Although the dynamic characteristics of the rate signal can be more accurately represented by using an Autoregressive (AR) model of orders higher than one to model the rate signal, it would result in estimating more system parameters and increasing the complexity of filter. On the other hand, the first-order Markov process satisfies the requirement for establishing the system state-space model. Therefore, in this paper the first-order Markov process is used to model the rate signal, and then a complete KF is designed for obtaining the optimal rate estimate. Furthermore, the factors that affect the system performance have been analyzed. Lastly, the hardware of the virtual gyroscope system is implemented and the performance of the virtual gyroscope with two different rate signal models are tested and compared.

The structure and principle of the virtual gyroscope is shown in

The true rate signal can be modeled directly to improve accuracy. For most applications of gyroscope, the rate signal is propagated according to an approximate power spectral density function in which the three degree-of-freedom motion is contained. In other words, this motion is expected to be within a certain frequency bandwidth and magnitude [_{ω}_{ω}_{ω}_{ω}_{ω}^{T}_{ω}δ

With such direct modeling of the rate signal, the rate signal can be estimated and obtained directly using a KF. Furthermore, the accuracy of the combined rate signal can be analyzed by the KF covariance; this can provide a reference for system improvement and parameters adjustment. In particular, it is suitable for the individual gyroscope which only includes the random noise of the white noise.

The measurement errors of MEMS gyroscope is usually composed of the known errors and random errors. The known errors can be eliminated by the testing procedure, thus only the random errors are discussed for the gyroscope. A common model for the MEMS gyroscope is widely used in many application which mainly includes the white noise denoted as angular random walk (ARW) and bias drift due to rate random walk (RRW) [^{T}

The matrix

Based on the aforementioned system state-space model of

_{∞}^{T}^{−1}_{∞}

Inserting _{∞}_{∞}

The discrete KF can be derived by directly discretizing the continuous KF of _{ω}_{∞}

From the above description of state-space model and KF, _{ω}

The bandwidth is an important parameter for the KF, it is related to the structure and parameters of the KF such as variance _{ω}

Due to a input rate signal is detected by multiple gyroscopes, the outputs of the gyroscope array can be expressed as _{s}_{s}

The frequency response of the KF can be obtained as:

The −3 dB standard is used to define the KF bandwidth, using KF frequency response, the bandwidth can be expressed as:

With the same derivation and analysis, the bandwidth of KF modeled by the random walk can be expressed as:

Given the identical system parameters, it is obvious that _{rw}

On the other hand, the performance of the KF and accuracy of the combined rate signal can be evaluated by the steady-state covariance. It can be seen from _{ω}_{ω}_{∞}^{−1}. Substituting _{ω}

It shows that _{∞}^{−1} = 0, for this case the KF reaches the best performance and the combined rate signal with the best accuracy could be obtained. The ^{−1} is related to the number of individual gyroscopes, noise statistical quantities of gyroscopes and correlation factors ^{T}^{−1}

Therefore a minimum _{∞} can be achieved through adjusting the factors in _{∞} will approach zero as _{ω}

In addition, _{∞} will decrease when _{ω}_{ω}_{ω}_{ω}_{ω}

On the other hand, different values of _{ω}_{ω}_{ω}_{ω}_{ω}_{ω}

The performance of the KF was analyzed in Section 2.4. It showed that the accuracy could be considerably improved when the gyroscope array has a negative correlation. However, in a practical implementation, the gyroscope array with an expected correlation factor has so far been hard to artificially design and fabricate. Additionally, usually the performance can be further improved through increasing the number N of the individual sensors in the array (

The correlation matrix can be used to indicate noise correlation between the multiple gyroscopes. Since the individual gyroscopes are described by a simple model of _{i}_{j}

The hardware of the virtual gyroscope is mainly composed of sensor array, central processing unit, A/D data acquisition unit, serial communication unit and FLASH memory.

Six ADXRS300 MEMS gyroscopes are utilized to form a gyroscope array. A TMS320VC5416 DSP chip is chosen as core processor for the virtual gyroscope. The A/D data acquisition unit uses a 16-bit ADS7807 to collect voltage signals from gyroscope array. The system operates in serial mode and connects with DSP directly through the 16-bit data bus. FLASH memory unit uses 4Mbit Flash chip AM29LV400 to provide storage space for external system program. The virtual gyroscope signal is exported by RS-232. The experimental results will be presented to quantify the performance of the virtual gyroscope. The geometry configuration of the sensors array should be considered to improve the estimation process. It had been shown in [

The ARW, bias drift and noise density of the virtual gyroscope with two KF models are tested and compared. The performances of the virtual gyroscope are evaluated using FFT analysis and root Allan variance of a zero rate output recorded for 1 hour. Due to the 40 Hz bandwidth of an individual gyroscope, and the bandwidth of the virtual gyroscope will not be higher than 40 Hz, thus the sampling rate was set to 200 Hz to satisfy the Nyquist theorem. The system parameters are chosen as _{ω}_{ω}^{2}/s^{3}. The comparisons of FFT plot and Allan variance measurement between the virtual gyroscope and single gyroscope are shown in

From the FFT plot, the noise level indicates noise floor of ∼0.11°/s/√Hz for the single gyroscope and 0.03°/s/√Hz for the virtual gyroscope modeled by the first-order Markov process, whereas the corresponding value estimated by the random walk model is 0.04°/s/√Hz. It also suggests a low-pass characteristic of the KF. In addition, it indicates a noise floor of ∼0.05°/s/√Hz for the rate signal obtained by averaging outputs of the six individual gyroscopes.

From the Allan variance plot, both the ARW noise and bias drift are reduced by fusing multiple measurements from gyroscope array.

The dynamic tests are carried out on a horizontal turntable (_{ω}^{2}/s^{3}.

Three kinds of dynamic condition are chosen to test the presented model. Firstly, the turntable is controlled to rotate in the horizontal plane with a 40 °/s constant rate, the outputs of the individual gyroscopes and virtual gyroscope are shown in

In the random rate test, the virtual gyroscope signals could well reflect the dynamic characteristic of the input angular rate (

In the swing test, the amplitude of combined rate signal estimated by the first-order Markov model reaches to 62.29 °/s, which is basically in accordance with the experimental setting, meanwhile, the 1σ errors are reduced to 0.16 °/s that is much smaller than 1.8 °/s estimated by the random walk model. Furthermore,

It would be useful to verify the presented KF to expand the experiments to different axes and rotations sensed by several axes in addition to the experiments. However, such experiment is difficult to conduct because of the limitation in testing condition and all the individual gyroscopes are oriented along the same axis. Therefore, the verification in the paper was only implemented through such experiments.

In this paper, the first-order Markov process was used to model the rate signal for fusing multiple MEMS gyroscopes to improve the overall accuracy. It indicated that the six-gyroscope array with an ARW noise of 6.17 °/√h and a bias drift of 54.11 °/h were combined into a rate signal having an ARW noise of 1.83 °/√h and a bias drift of 16.32 °/h. The presented KF also reduced the dynamic errors by over 20% compared to the KF modeled by the random walk. It proved that the first-order Markov process is efficient for modeling rate signal to improve the system overall performance.

In the future fabrication of a number of integrated MEMS gyroscope arrays on a single chip would enhance the uniformity between the gyroscopes and the correlation between the gyroscope array still need to be further researched and explored.

The authors gratefully acknowledge Chinese New Century Excellent Talents in University (NCET), and Shaanxi Province overall planning innovative project (Contract No. 2011KTCQ01-26). Finally, the authors wish to thank referees for their valuable comments that improved this paper.

Structure and principle of the virtual gyroscope.

Virtual gyroscope implementation using a discrete KF.

Relationship between the gyroscope noise reduction and correlation factor

A prototype of the virtual gyroscope system.

FFT plot of the virtual gyroscope compared to the single gyroscope and averaging outputs of the gyroscope array.

Allan variance results of the virtual gyroscope compared to the single gyroscope and averaging outputs of the gyroscope array.

Constant rate test of the virtual gyroscope. (

Random rate test of the virtual gyroscope. (

Sinusoidal rate test of the virtual gyroscope. (

Correlation matrix of noises for six-gyro array.

1.000 | 0.021 | 0.007 | 0.005 | 0.005 | 0.004 | |

0.021 | 1.000 | 0.038 | 0.016 | 0.007 | 0.009 | |

0.007 | 0.038 | 1.000 | 0.036 | 0.015 | 0.009 | |

0.005 | 0.016 | 0.036 | 1.000 | 0.038 | 0.017 | |

0.005 | 0.007 | 0.015 | 0.038 | 1.000 | 0.032 | |

0.004 | 0.009 | 0.009 | 0.017 | 0.032 | 1.000 |

Static test results of the virtual gyroscope.

0.11 | 0.05 | 0.04 | 0.03 | |

6.17 | 2.73 | 2.35 | 1.83 | |

294.28 | 161.21 | 125.83 | 120.61 | |

54.11 | 22.72 | 20.64 | 16.32 |

Dynamic test results of the virtual gyroscope (unit: °/s).

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

40.15 | 1.45 | 1.61 | 62.64 | 0.79 | |

40.14 | 0.06 | 0.47 | 60.47 | 1.80 | |

40.09 | 0.05 | 0.29 | 62.29 | 0.16 |