The study describes a mathematical error model of a platformless inertial navigation system and focuses on using Allan variance as a method for estimating such instrumental errors of sensors, such as zero signal bias instability, angle random walk and rate random walk. The paper shows the results of the work of the mathematical error model, the model being constructed using the estimated instrumental errors of a sample of sensor assembly which consists of three ring laser gyroscopes and a three-axis accelerometer unit.

Описана математическая модель погрешностей бесплатформенной инерциальной навигационной системы. Рассмотрен способ оценки с помощью вариации Аллана таких инструментальных погрешностей датчиков, как нестабильности смещений нулевых сигналов, случайные уходы углов и скоростей. Представлены результаты работы математической модели погрешностей, построенной с использованием оцененных инструментальных погрешностей образца блока чувствительных элементов, состоящего из трех кольцевых лазерных гироскопов и трехосевого блока акселерометров.

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 [

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 g1 – matrix of transition from the bound coordinate system to the geographic coordinate system;

a k1 – 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.

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.

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.

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 [

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 [

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 [

Table 3

Positioning errors

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.

The authors declare that there are no conflicts of interest present.