Stochastic error simulation method of fiber optic gyros based on performance indicators

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Abstract

Gyro simulation is an important process of inertial navigation theory research, with the major difficulty being the stochastic error modeling. One commonly used stochastic model for a fiber optic gyro (FOG) is a Gaussian white (GW) noise plus a first order Markov process. The model parameters are usually obtained by using time series analysis methods or the Allan variance method through FOG static experiment. However, in a real life situation, a FOG may not be used. In this paper, a simulation method is proposed for estimating the stochastic errors of FOG. When using this method, the model parameters are set based on performance indicators, which are chosen as the angle random walk (ARW) and bias stability. During the research, the ARW and bias stability indicators of the GW noise and the first order Markov process are analyzed separately. Their analytical expressions are derived to reveal the relation between the model parameters and performance indicators. In order to verify the theory, a large number of simulations were carried out. The results show that the statistical performance indicators of the simulated signals are consistent with the theory. Furthermore, a simulation of a VG951 FOG is designed in this research. The Allan variance curve of the simulated signal is in agreement with the real one.

Introduction

FOG is a kind of optical gyros and its principle is based on the Sagnac effect [1]. It can provide accurate angular velocity information partially because of its insensitivity to vibration, shock and acceleration. FOGs are widely used in inertial navigation systems (INSs), which are applied in many fields such as aviation, spaceflight and navigation [2].

When verifying a new INS theory or program, numerical simulation is always the initial step [3]. An INS is a dead-reckoning system, as a result, the position, the velocity and the attitude errors produced by gyros' errors spread over time [4]. Therefore, the simulation quality of a FOG based INS mainly depends on the fidelity of FOG simulation. A FOG signal includes the nominal value and errors. In the INS simulation process, the nominal value of the FOG signal can be derived from the vehicle trajectory, which is easy to get. The simulation of FOG's errors needs error modeling and parameters identification [5], which is relatively difficult. So the simulation of FOG errors is the key process.

The FOG errors mainly consist of two parts, the deterministic and the stochastic [6]. Major sources of the deterministic error include bias and scale factor errors, their models are definite and the parameters are easy to calculate [7]. However, the models of the stochastic errors are more complex than the deterministic errors and the model parameters are relatively difficult to obtain. The approach for solving the parameters of the stochastic model is discussed in this paper.

At present there is no unifying model for describing FOG stochastic errors. Five stochastic models are built by IEEE standard [8]. They are angle random walk, bias instability, rate random walk, rate ramp and quantization noise. Each of the five model terms relates to some physical phenomena and corresponds to some errors of the components inside FOG. Those model parameters can be extended to INS filter equation as state variables in a real-time INS navigation process, and then they can be estimated and compensated [9]. However, if the simulation purpose is only to explore the basic characteristics of a FOG, complicated models are not needed. A gyro's stochastic models can be classified into two parts for simplicity, a high frequency component and a low frequency component [6]. In many applications, both of the two parts can be well described by simple models. The GW noise is a commonly used model for the high frequency component, while the first order Markov process is for the low frequency component [10], [11].

The characteristics of a FOG are reflected by its performance indicators, which are usually provided by the producers [15]. The ARW and bias stability are two main specifications that evaluate FOG's stochastic errors [16]. When simulating a FOG's signal, it is expected that the characteristics of the simulated FOG signal can be consistent with the wanted one, which means that the parameters of the stochastic models can be obtained according to the wanted performance indicators. But the relation between the two performance indicators and the parameters of FOG's stochastic models is not known.

The previous approaches to obtaining the parameters of the stochastic models are to carry out FOG static experiments by using the Allan variance method [12], [13] or some other improved methods [14], which need real FOGs. When FOG simulations are carried out for INS theory verification, a number of FOGs with different specifications are needed. However, the corresponding real equipments are not always available. In this situation, the parameters are usually got by the trial and error method [7], which requires time and experience. As a result, the simulated signal may not accord with the wanted one, and then the inappropriate parameters may produce improper navigation errors and decrease the authenticity of the INS simulation. The main contribution of this paper is to establish a bridge from FOG's performance indicators to its stochastic model parameters, so as to guide the parameters setting in FOG's simulations.

In the subsequent sections, we first introduce the calculation procedure of the ARW and bias stability, and explain them from the perspectives of stochastic signal processing. Then the GW noise and the first order Markov process are analyzed as the model of FOG's stochastic errors, and formulae are derived to establish a bridge between the two performance indicators and the two stochastic models' parameters. In Section 4, numerous simulations are devised to verify the proposed theory. The simulation results show high consistency with the theoretical values.

Section snippets

FOG performance indicators

Angle random walk (ARW) and bias stability are two most important performance indicators for a FOG, as they are related to its stochastic errors therefore can directly reflect its precision. ARW mainly represents characteristics of high-frequency noises [8], of which the correlation time is much shorter than the sample time. Bias stability refers to low-frequency noises, of which the correlation time is much longer than the sample time.

To estimate a FOG's ARW and bias stability, the Allan

Relation between model parameters and performance indicators

The models of FOGs' stochastic errors are complex and generally not unique [21], but some typical models usually are used. A simple model is assumed to consist of GW noise and the first order Markov process. The GW noise is supposed to reflect the high-frequency characteristics of FOG's stochastic errors, while the first order Markov process shows the low-frequency characteristics. In discrete time, the model can be expressed asgk=wk+mkwhere gk is a gyro's discrete stochastic error, wk is the

Simulation and analysis

In order to verify the derived relation between FOG's model parameters and performance indicators, some simulations are conducted. The verification is divided into two parts:

  • (1)

    One is to verify the relation between ARW and parameters of GW noise, and the relation between bias stability and parameters of GW noise plus the first order Markov process. Several typical conditions are designed, and 100 simulation experiments are carried out for each condition. The statistics results are compared with

Conclusions

In this paper, a stochastic error simulation method of FOGs was proposed. The stochastic errors were modeled as a GW noise plus a first order Markov process, and their parameters were set based on two FOGs' performance indicators, which were adopted as ARW and bias stability. Using signal processing techniques, the two performance indicators were linked with stochastic models' parameters, and the corresponding formulae were derived.

Since there are two known performance indicators and three

Acknowledgment

This work was supported by the National Natural Science Foundation of China (91016019, 61174197), Funding of Jiangsu Innovation Program for Graduate Education (CXLX11_0201), Funding for Outstanding Doctoral Dissertation in NUAA (BCXJ11-04), and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

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