The main objective of the paper is to study the system errors of azimuth determination in the dynamic scheme of north finding on the base of the molecular-electronic sensitive angular motion sensor. Introduced theoretical and experimental study of some error compensation methods. Investigated the most significant system inaccuracies of azimuth determination depended on MET sensor g-sensitivity factor and the occurrence of rotation uneven in the system and as a result of tiny angular accelerations which appeared. Methods and algorithms of error reduce are experimentally verified.

The crucial task of the modern stage of the development of technology is the problem of the exact binding of various measuring and navigation systems to the true north heading. The importance of high-precision determination of true azimuth is emphasized by a number of practical applications in registration, navigation, targeting problem solving and object direction determination, high-precision orientation of communication systems, geodesy, and so on.

The traditional way to determine azimuth is to measure the direction with the help of a magnetic compass, considering (or not) the terrain magnetic declination. Although this method has been known for a long time, noncorrespondence of the magnetic and geographic poles, as well as the presence of magnetic anomalies or magnetized objects nearby, makes the compass usage inaccurate for the solution of the specified tasks of geographical north finding [

The solution of the problem of positioning and determining the motion direction in a number of cases can be achieved with the help of modern satellite systems, such as GPS/GLONASS [

Modern methods of high-precision orientation determination include a wide usage of various gyrocompass devices [

An alternative to the established trend of various gyroscopic systems development to solve the task of direction determination is an autonomous method of object azimuth finding based on finding of Earth angular velocity vector by measuring the Coriolis forces with a linear accelerometer rotating around an axis parallel to the sensitivity axis of that accelerometer [

From the point of view of technical implementation and high accuracy attainability in finding the geographic north heading, a better option is a well-known method based on the direct finding of Earth angular velocity vector by means of an angular accelerometer rotating about the axis orthogonal to the sensitivity axis of that accelerometer. The principle of the method is the modulation of the Earth rotation rate signal by forced mechanical rotation of the angular accelerometer. In case the rotation axis is not parallel to

In this paper, an autonomous system of finding the azimuth by a dynamic method based on a molecular-electronic angular velocity sensor will be studied in detail. The basic scheme of the assembly operation is shown in detail in [

Schematic design of the device to measure the geographical north heading.

The rotating platform 1 is equipped with a molecular-electronic angular motion sensor 2 (SA: sensitive axis) and an inclinometer 3 detecting the deviation of the site from the horizontal position. The platform is driven by motors 4. At the site, next to the rotating platform, a magnetic encoder 5 is installed. The encoder registers the position of the platform in time with respect to the start position. Signals from the sensors are registered in the data acquisition unit 6, processed in block 7, and visualized on the computer real time, which allows to control sensor signals clearly, including nonuniformity of the platform rotation. Figure

User window and sensor signals at the platform rotation real time. White is the signal of the molecular-electronic sensor, red is the signal of the inclinometer, green is the signal of the magnetic encoder, and blue is the differential of the encoder signal.

General view of the device for the geographical north finding.

The accuracy of the azimuth determination by the developed device and the measurement method generally depends on a number of factors, in particular, on the orthogonality degree of the platform rotation axis of and the angular motion sensor sensitivity axis and rotation axis beatings, which change the slope of the angular motion sensor sensitivity axis in relation to the acceleration of gravity, which will also be seen in the appearance of a periodic signal at the sensor output in case of corresponding sensitivity to linear accelerations [

In this paper, we will consider two factors that determine the special aspects of the dynamic method of azimuth finding with the help of the latest molecular-electronic angular motion sensors.

The signal of the molecular-electronic angular motion sensor after being brought to the rotation in accordance with the dynamic method algorithm and calculation of the Fourier transform at the platform rotation frequency is as follows:

The first term represents a useful signal. The second term represents errors due to the influence of the linear acceleration (gravity). The third term is the parasitic signal conditioned by nonuniformity of the platform rotation. The fourth term describes errors associated with the molecular-electronic sensor self-noise. The fourth term was studied in detail in [

The present study and the second and third terms from (

One of the reasons for the error in north finding by the proposed above dynamic scheme is the effect of linear accelerations, the second term of (

Now we determine how the errors related to the change in the sensor position to gravity can be corrected if the platform rotation axis is not vertical in a small range of angles. To do this, consider a sensor rotating relative to one of the axes located in the plane of the toroidal channel of its body (Figure

Schematic view of the molecular-electronic sensor of angular motions.

Let, in addition,

If the liquid in the toroidal channel of the angular motion sensor was strictly homogeneous, then the moment of inertia forces creating the liquid circulation in the channel would be

The coefficients

Suppose the platform comes into rotation counterclockwise with angular velocity

Here

Denote the complex coefficient of the transformation of the moment of forces

Here,

Similarly, for the rotation in the opposite direction, it is as follows:

Calculate the ratio (

Here,

Take into account that

Thus, if the direction of the rotation axis inclination (the angle

In turn, to find the angle

The method of platform rotation at two different angular velocities: suppose that two series of experiments are consistently carried out. These experiments consist of successive clockwise and counterclockwise rotation of the platform, with the described above two different rotational speeds

Denote the amplitude and relationship phase of the of the sensor signals when rotating counterclockwise and clockwise by

Then, in addition to (

The last expression with allowance for (

Simultaneously with the angular motion sensor, a gravity-sensitive sensor (sensitive to the inclination angle), for example, an accelerometer, is placed on the platform: let, for simplicity, the accelerometer sensitivity axis is directed along the OY axis, that is, it lies in the toroid plane. Then the accelerometer output signals at rotation counterclockwise or clockwise measured in the experiment are given by the following expressions:

For the output signals ratio:

From here,

Consider (

The axes nonorthogonality lead to the fact that, apart from the actual angular velocity of the Earth rotation, the signal received from the angular motion sensor has a projection to the sensor sensitivity axis, in general, the nonconstant rotation speed of the platform. Considering the fact that the unevenness of the platform rotation, as a rule, has a periodicity corresponding to the rotation period, that parasitic signal, caused by the unevenness of rotation, is at the same frequency as the measured Earth rotation speed modulated by the platform rotation. Such parasitic signal cannot be eliminated by frequency filtering and by an increase in the averaging signal time. As a result, the accuracy of north finding is significantly reduced.

Consider such situation in more detail. Figure

Molecular-electronic sensor of angular motion on a rotating platform. 1: mobile platform; 2: molecular-electronic sensor.

As observed in practice, the fact that the rotation of the platform can be uneven should be considered. Consider how much the nonorthogonality of the position of the sensor sensitivity axis to the platform rotation axis affects the errors in the device readings. If the sensitivity axis is located at some angle to the platform rotation axis different from 90 degrees, then the molecular-electronic sensor, in addition to the Earth rotation, feels the unevenness of the platform rotation, which significantly affects the accuracy of north finding.

To study the effect of the described mechanism on the error in North finding, the data from a magnetic encoder is used. The encoder can measure the platform rotation speed in real time. Let us recall that the magnetic encoder is installed next to the rotating platform (product 5 in Figure

In order to reduce the influence of the parasitic signal on the readings of the molecular-electronic sensor, it is necessary to bring the angular motion sensor into a position where the SA (axis of sensitivity of a sensor) is perpendicular to the platform rotation axis or to subtract the additive from the measured signal due to the discussed effect.

At the controller command, the platform rotation is defined. It is characterized by a strong unevenness (e.g., the platform evenly increases its rotational speed with some constant angular acceleration). In that case, the third term in (

Figure

Amplitudes of the signal spectra of a molecular-electronic sensor reduced to the dimension rad/s. The blue and red curves show different angles of the sensitivity axis of the molecular-electronic sensor to the platform rotation.

Amplitudes of the signal spectra of the platform rotation speed sensor (encoder).

From the analysis of Figures

In addition, the determination of the angle of inclination allows to take into account the parasitic signal during further data processing. To do this, the platform is brought into deliberately uneven rotation straight before the measurements (Figure

Profile of the change in speed with obviously uneven rotation (the ordinate axis shows rad/sec and the abscissa axis shows time in seconds).

Signal spectra of a molecular-electronic sensor and a signal of the encoder’s rotational speed with obviously uneven rotation. (The abscissa axis shows frequency in Hz and the ordinate axis shows relative units).

Calibration of the molecular-electronic sensor with an uneven rotation signal near the rotation frequency, the sensitivity

Based on the assumption that there is a parasitic signal associated with the unevenness of the platform rotation in the signal of the molecular-electronic sensor, the following formula can be used

Equation (

Schematically, the operation of the algorithm can be represented in the following block diagram in Figure

The algorithm for error compensation associated with uneven platform rotation.

The following symbols are introduced in the diagram:

The result of the practical application of this algorithm in an experiment is shown in Figure

Spectrum of the MEP signal before (blue) and after (red) the application of the algorithm with even rotation at a frequency of 0.137 Hz (blue is before correction and red is after correction).

The contribution of the component conditioned by the unevenness of the platform rotation and the corresponding extra-axial sensitivity of the molecular-electronic sensor was about 10–12% of the total signal amplitude.

Thus, the general algorithm for control of the analyzed measurement errors in the device for determining the North heading using the dynamic method of the azimuth determination consists in setting the sensitivity axis of the molecular-electronic sensor perpendicular to the platform angular velocity vector, basing on the readings of a sensor measuring the platform rotational speed in time. The errors that remain after this procedure are taken into account by subtracting the corresponding parasitic signal from the signal of the molecular-electronic angular motion sensor. Then, the measurements are carried out with the account of the algorithm of the sensor sensitivity to linear acceleration, the North heading is calculated by (

Two significant mechanisms of system errors for a dynamic scheme of azimuth determination with the base of MET angular motion sensor have been studied. A theoretical model of MET g-sensitivity error factor has been developed, and an appropriate compensate algorithm was calculated. There have been experimentally studied system errors due to tiny nonoptionality of sensor sensitive axis and the main system axis with the presence of system axis angular motion speed fluctuations (parasite angular acceleration). An appropriate compensation algorithm was developed and approved.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the Russian Ministry of Education and Science state assignment under Grant 3.3197.2017/ПЧ.