# Estimation of inertial sensors error impact on platformless inertial navigation system accuracy

### Abstract

#### For citation:

Andreyev S.V., Ilinykh V.V., Ilinykh O.A., Chertkov M.S., Klyuchnikov A.V. Estimation of inertial sensors error impact on platformless inertial navigation system accuracy. *Journal of «Almaz – Antey» Air and Space Defence Corporation*. 2018;(2):29-34.
https://doi.org/10.38013/2542-0542-2018-2-29-34

## Introduction

Platformless inertial navigation systems (PINS) are the important components of modern aircraft (A/C), which allow autonomous determination of current motion parameters. The PINS instrumentation includes apparent linear acceleration sensors (LAS) and angular velocity sensors (AVS) precisely oriented relative to the bound coordinate system. The moving object state vector determined by the PINS is sufficient for solving the problems related to aircraft stabilization and flight path control. A growing number of errors in measurement of current motion parameters is a well-known feature of the PINS. Error values depend on instrumental errors of inertial sensors, the accuracy of sensors installation inside the sensor element unit (SEU), and the quality of PINS initial alignment [1]. Estimation of the effect of instrumental errors on motion parameters measurement accuracy allows to correctly specify the requirements for the LAS and AVS in order to assure the desired performance quality of the whole navigation system. Sometimes, the known methods of random process research such as sampling variance and spectral density cannot help identify the source of error and its contribution to the resultant noise signal [2]. To solve such problems, researchers often apply a popular method of analysis of inertial sensor measurement errors using the Allan variance. This paper focuses on errors of inertial sensors that belong to the specific type of the sensor assembly developed by Arzamas Research and Development Enterprise ANPP “Temp-Avia” (Arzamas). We conducted mathematical simulation of a positioning error dynamic pattern regarding instrumental error estimates obtained with the help of the Allan variance such as zero signal bias instability of the LAS and AVS, angle random walk and velocity random work (ARW/VRW).

## Description of mathematical error model for motion parameter determination

Using initial data received from the LAS and AVS, inertial navigation systems determine current motion parameters. To ensure high-quality navigation management in automatic control systems being developed, it is necessary to understand how errors of inertial sensors affect the accuracy of measured motion parameters of a controlled object. The research papers [2, 3] lay emphasis on zero signal bias (both constant and random bias), errors of scale factor conversion, and on random components such as white noise.

Zero signal bias is an additive error component of sensor’s primary measurements. Even the most accurate calibration and adjustment allows for some residual bias. Particular attention is paid to the stability of zero signal bias between starts and to the initial bias. The level of bias which can be estimated is excluded from sensor measurements.

The scale factor error becomes evident when the tilt angle of the best straight line of transformation of physical measurements into a sensor output parameter fails to match the tilt angle of its real line of transformation. The scale factor error is a multiplicative component of the primary measurement error and often manifests itself when sensors are exposed to impacts being near the limits of the sensor measurement range.

Random noise as a stochastic process is the integral part of sensor output signals. Its integration with measurements inevitably leads to generation of an additive stochastic noise component in motion parameters to be determined.

Below is a system of differential equations which describe a dynamic pattern of the measurement error of motion parameters such as velocity, attitude, and angular orientation. Errors of the LAS and AVS described above are taken into account:

where ∆V – object velocity measurement error vector in geographic coordinate system;

α – matrix of minute errors of orientation angles of the bound coordinate system relative to the geographic coordinate system (minute errors of roll, heading, and pitch angles);

A _{g}_{1} – matrix of transition from the bound coordinate system to the geographic coordinate system;

a _{k}_{1} – acceleration vector directed along the axes of the bound coordinate system;

∆K, ∆С – diagonal matrices of scale factor errors of linear acceleration sensors and angular velocity sensors;

da, dω – vectors consisting of instabilities of zero signals from linear and angular velocity sensors;

na, nω – vectors consisting of velocity and angle random walk (VRW, ARW);

t – current time;

∆S – positioning error vector in the geographic coordinate system;

ω1 – object angular velocity vector in the bound coordinate system;

∆α – angular orientation measurement error vector.

## Estimation of inertial sensor instrumental errors

To estimate parameters da, dω, na, nω of the specified system of equations, we can use the Allan variance as a method of time sequence analysis to determine noise characteristics as an averaged time function. Today, the Allan variation determination method is widely represented in research papers [4, 5]. To analyse errors of the sensor under study, the square root of the calculated value of Allan variances (Allan variance or deviation) is used. Fig. 1 shows a typical graph representing the Allan deviation in the logarithmic scale.

**. Typical view of Allan deviation curve**

Fig. 1

Fig. 1

The Allan deviation curve is divided into sections defining a certain error of the sensor under study. The Allan variance is the sum of squares of different noise components, which can be represented as the polynomial:

Coefficients R, K, B, N, and Q define the intensity of individual noise components.

Values of variance coefficients and respective slopes of the Allan variance curve are given in Table 1.

**Table 1**

**Allan variance noise components**

As the object for studying noise components, we used the SEU based on three LG-2 ring laser gyroscopes used as the LAS and the BA-24 accelerometer unit, which together form a triad of LASs. Fig. 2 shows a SEU prototype with designated sensitive axes.

**. SEU appearance**

Fig. 2

Fig. 2

The SEU under study has the following characteristics. Relative error of scale factor of each angular velocity measuring channel is 0.1 % maximum. Systematic component of zero signal drift in each angular velocity measuring channel is 0.5 deg/h maximum. Relative error of scale factor of each apparent linear acceleration measuring channel is 0.1 % maximum. Systematic component of zero signal drift in apparent linear acceleration measuring channels is 2 ∙ 10^{–2} m/s² maximum. Angular velocity measurement range is ±350 deg/s. Apparent linear acceleration measurement range is ±30g. Non-orthogonality of sensitive axes is 5′ maximum. The SEU under study allows to transform the angular velocity vector and the apparent linear acceleration vector into numeric code corresponding to projections of the vectors to the axes of the orthogonal coordinate system bound with the SEU (BCS). The SEU transmits data represented as a serial numeric code to the consumer via a multiplex data transmission channel as per State Standard GOST R 52070–2003 [6]. The unit is a terminal device. Rate of data exchange with the SEU is 100 Hz. The SEU measured output parameters are the current angle, rate of angle increase per clock cycle, rate of angular velocity increase per clock cycle. Clock cycle is a period of data transmission to the consumer equal to 10 ms.

To process the SEU data using the Allan variance, we recorded SEU measurement data for 4 h in normal climatic conditions. Fig. 3 shows the obtained Allan deviation data for the LAS and AVS of the SEU prototype under study.

Based on the obtained Alan deviation data, we applied the method described in [7] to determine the zero signal bias instability upon start for the LAS and AVS:

as well as angle random walk (ARW) and velocity random walk using the following formula

where τ is selected in the section of the Allan deviation curve with the slope –1/2 plotted in the logarithmic scale.

Calculated values of the zero signal bias instability, angle random walk and velocity random walk are given in Table 2.

**Table 2**

**Errors of LAS and AVS determined by Allan variance**

For rapid assessment of accuracy of the PINS based on the SEU described above, we conducted mathematical simulation in order to estimate accumulated errors by coordinates (ranges) and velocities for the period of PINS operation equal to 600 s. Mathematical simulation was carried out by means of numerical integration of the system of differential equations (1) using the Runge – Kutta method of the 4th order [8]. We simulated an abstract trajectory. The velocity along x axis was assumed to be a ramping-down parameter from 350 m/s to 50 m/s, the velocity along y axis was ramping up from 10 m/s to 70 m/s; the velocity along z axis was equal to zero. Positioning errors caused by measurement errors of the SEU under study were not greater than 1500 m for 600s of autonomous operation of the inertial system (Table 3).

**Table 3**

**Positioning errors**

## Conclusion

As a result of the research, we obtained estimation characteristics of LAS and AVS zero signal bias instability, angle random walk and velocity random walk (ARW/VRW) for a particular SEU prototype. We proposed a mathematical model, which allowed to estimate the effect of instrumental errors on accuracy of the moving object’s motion parameters measurement. In the future, we are planning to improve the represented mathematical model, taking into account instrumental errors that have not been discussed herein (quantisation noise, trend, etc.). The developed mathematical model is used for estimating accuracy characteristics of the PINS during semi-realistic simulation of moving object control systems [9, 10], as well as during verification of full-scale test results.

## References

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### About the Authors

**S. V. Andreyev**Russian Federation

**V. V. Ilinykh**Russian Federation

**O. A. Ilinykh**Russian Federation

**M. S. Chertkov**Russian Federation

**A. V. Klyuchnikov**Russian Federation

### Review

#### For citation:

Andreyev S.V., Ilinykh V.V., Ilinykh O.A., Chertkov M.S., Klyuchnikov A.V. Estimation of inertial sensors error impact on platformless inertial navigation system accuracy. *Journal of «Almaz – Antey» Air and Space Defence Corporation*. 2018;(2):29-34.
https://doi.org/10.38013/2542-0542-2018-2-29-34