The purpose of the study was to consider an algorithm for obtaining the measurement information from a dynamically tuned gyroscope in the mode of an angular velocity sensor and output signal noise component estimate, the algorithm being based on the Allan variance method. The results obtained were evaluated

Рассмотрен алгоритм получения измерительной информации с динамически настраиваемого гироскопа в режиме датчика угловых скоростей и оценки шумовых составляющих выходного сигнала, основанный на методе вариации Аллана. Проведена оценка полученных результатов

The source of data in a navigation system is a gyro inertial unit (GIU), comprising gyroscopes and accelerometers. The components and characteristic features of the initial data sensors (IDS) are determined by the requirements of a particular system. Navigation system accuracy will depend on the accuracy of the sensing elements (SE) proper and that of the processing channel. The IDS used for measuring angular motion parameters in serially produced strap- down inertial navigation systems (SINS) and advanced systems manufactured at JSC Arzamas Instrument-Building Plant named after P. I. Plandin are dynamically tuned gyroscopes (DTG) in the mode of angular velocity sensors (AVS).

To improve SINS navigation accuracy, development of new approaches and methods is required for perfection of those technical characteristics that provide control of the output parameters capable of resolving the output signal into components for subsequent analysis of each one of them.

This paper offers a new algorithm and software for processing DTG output signal in the AVS mode, developed by M. A. Sharova under supervision of S. S. Diadin at JSC Arzamas Instrument-Building Plant named after P. I. Plan- din. Numerical values of errors present in the data are determined and analysed.

At present, widely used is the classical method of gyroscope output signal evaluation, according to which the total error of the output signal additive component, which can be found in each output response measurement, is the sum of random (Δ0) and systematic (Δс) components:

Δ = Δс +Δ0. (1)

An output response measuring model is shown in Fig. 1, with the following designations introduced:

ω - true value of angular velocity;

- angular velocity with account of systematic and random components;

Ω - angular increment with account of systematic and random components.

Fig. 1. Model for measuring angular velocity with account of systematic and random components

However, electronic equipment and errors in the gyroscope output signal produce additional error, commensurate with the error of instrument itself; such error cannot be detected by the classical method, which is the latter’s drawback.

A magnitude varying randomly under the influence of multiple factors which affect the instrument considering measurement of a single parameter is called a random error.

A magnitude which is under the influence of several factors and whose value remains constant over time is a systematic error.

The classical method of DTG drift errors analysis, applied in angular velocity measurement, consists in determination of the systematic and random components by each of the sensitivity axes, since the DTG has two measurement channels, implemented according to the following algorithm.

The many known methods of stochastic noise analysis in the inertial sensors’ output data include a method of spectral power density (SPD) in the frequency domain, a method using correlation function, as well as dispersion methods. However, the most known and used one is the Allan variance method.

Lying in the base of the Allan variance method is the method of cluster analysis, which implies splitting of the whole dataset into groups of certain length.

We divide the entire volume of data subject to analysis, as shown in Fig. 2, into a certain number of groups of equal length, with averaging interval

T = nt0,

where n - number of samples in the measurement interval (n = 1,2, ... < N/2);

t0 - meter sampling increment.

Fig. 2. Data structure diagram used in Allan variance calculation

The number of groups is determined by formula

Here, N – the number of measurements.

Let us calculate the mean data value for each separate group:

Here, Ωk (t) - mean output velocity value in a group, starting from the k-th point and comprising n data points;

tk - meter sampling increment in the k-th point.

The next mean value of the group’s output velocity can be expressed as follows:

where tk +1 = tk + T

Having calculated the difference between the mean values of each pair of the adjacent groups, we have:

Set ξ, determined by equation (4) for each group with averaging interval T, forms an aggregate of random magnitudes. In this case, it is ξ dispersion value for all groups of the same sizethat is of interest.

Having calculated mean square of these differences, we divide it by a certain coefficient and obtain:

Brackets in equation (5) determine the averaging operation for a set of groups.

A logarithmic plot of Allan variance σ(Τ) dependence on averaging time T is a direct indication of the types of random processes present in the gyroscope output data, since the value of Allan variance σ2(Τ) is a measurable magnitude.

The Allan variance method is a technique for classifying and assessing the level of noise in the output signal associated with statistical properties of intrinsic random processes affecting gyroscope operation. As applies to DTG, its inherent errors when operating in the AVS mode, which can be analysed by means of the Allan variance method, are determined.

The most part of errors as per the Allan variance method contained in a single angular velocity measurement can be represented as a model

Iω=ω + Rω + Κω + Bω + Nω+ Qω. (6)

Here Iω - measurements (deg/h);

Rω − angular velocity trend;Kω − angular velocity random walk;Bω − zero instability;Nω − angular random walk;Qω − quantisation noise.

Given in Fig. 3 is Allan deviation curve for a signal changing under the influence of various stochastic processes which correspond to the gyroscope errors.

Fig. 3. Correspondence of Allan variance segments to typical processes

Angular velocity trend Rω is a systematic error representing determinate deviation of an output signal received from the considered sensor over large periods of time. Gyroscope angular random walk Nω is an additive white noise component, manifesting itself as noise or slow variation of output signals over time [

Resulting from experimental setup (Fig. 4) operation, a DTG output signal in the AVS mode was obtained.

The experimental setup, whose block diagram is given in Fig. 4, comprises the following elements:

Fig. 4. Block diagram of experimental setup for conducting tests: HCC – heat and cold chamber; DTG – dynamically tuned gyroscope; RCB – remotely controlled bracket; PS – power source; CP – control panel

For correct operation of the Advantech PCI-1710 card, it is necessary to install on the user’s computer the software supplied in a set with the card. This software enables to set up the equipment and save settings in Windows registry, as well as use the settings in calling up API functions for working with the driver. The use of this driver makes accessible a wide range of functions in various programming environments for enhancing hardware performance, such as the event handler function, which enables to find out the status of data transmission or reception.

Fig. 5 illustrates interaction between hardware and software.

Fig. 5. Hardware and software interaction

Operation with the analogue-to-digital converter (ADC) card starts with its initialisation. First, it is necessary to determine the constants, command words, card control parameters, declare structures, and allocate memory for data arrays. Then check the card for errors and create stream for reading the incoming data. This done, the card is ready for operation and waiting for the input data in the standby mode.

The procedure of ADC module preparing for operation can be represented in the form of a block diagram (Fig. 6).

Fig. 6. Block diagram of ADC preparation for work

Operation of the setup (see Fig. 4) consists in simulation of gyro inertial unit operation, whose measurement time we take equal to 10 min.

Implementing in action the setup given in Fig. 4, we collect experimental data, write them down, save, and export to files *.txt. A fragment of the experimental data is given in the table.

Experimental data fragment

Measurement channel, deg/h | |
---|---|

X | Y |

91,8 | -139,00 |

То же | -116,13 |

68,38 | -139,93 |

... | ... |

114,28 | -103,76 |

57,14 | -127,55 |

To have an initial notion of the data, it is necessary to convert them [

Fig. 7. Graph of gyroscope output signal by axes X and Y under normal climatic conditions (22 °С)

To calculate Allan variance σ2(Τ), we shall select data averaging interval T divisible by the meter sampling increment:

Here, θk – angle accumulated as a result of integration of the output velocity values, calculated by the formula

A block diagram of an algorithm implementing the Allan variance method is given in Fig. 8.

Fig. 8. Block diagram of algorithm based on the Allan variance method

Allan variance σ2(Τ) based on the known noise components [

Here, R, K, B, N, Q - coefficients of polynomial p(Tk).

Each one of the obtained coefficients R, K, B, N, Q of the polynomial (8) is a separate noise component present in the gyroscope output signal [

The essence of this method consists in finding coefficients R, K, B, N, Q that would conform to the condition of the minimum square sum of Allan variance σ 2(T) deviations from the approximating surface:

where σ(Τk) - Allan variance;p(Tk)– polynomial approximant.

For expression (9) to be true, all partial derivatives of functional Φ by all {R,K,B,N,Q} must be equal to zero:

Now we have a system of equations for finding 5 indeterminates {R, K, B, N, Q}:

which can be more conveniently written in the matrix form as

Here, [A] – time matrix of data averaging;

[C] – vector designating the sought coefficients;

[B] – vector reflecting the obtained Allan variance value.

The coefficient matrix, in accordance with the least squares algorithm can be resolved as follows:

[C ] = -( AT А )-1 ATB, (12)

where АT − transposed matrix A.

A block diagram of the algorithm implementing the least squares method for obtaining the coefficients is given in Fig. 9.

Fig. 9. Block diagram of algorithm of the least squares method

For the convenience of work in studying DTG output signal for the presence of noise components and for displaying respective information and graphs, a software package in the C++ Builder 6 programming environment was developed. The runtime program includes the following functions:

After program startup, the main window “Allan variance calculation” opens, serving for central control of the program module. In the right-hand part of the window is a field for displaying a graph of the values obtained from all calculations. By pressing button “Measure”, an additional module with the initial data graphs is called up (see Fig. 7). In the left-hand part of the window is a control field, where it is possible to enter climatic effects, meter sampling rate, and measurement time. Field “Coefficients”, located below, is intended for displaying the final results of all calculations. Besides, the main window features buttons by means of which sampling of the output signals received from DTG and writing them to file is performed, followed by displaying the received signals in a graph in the additional module, running calculation of Allan variance deviations and approximation procedure with subsequent building of a graph of the results obtained.

The result of module operation is finding respective coefficients and outputting an additional graph with consideration of the coefficients obtained (Fig. 10).

Fig. 10. Window with approximating curve graph

Resulting from application of a new algorithm for the analysis of the additive component of DTG output signal, an algorithm and software were developed, enabling to calculate 4 basic noise components: zero instability, angular velocity random walk, angular random walk, and angular velocity trend, as well as to determine quantisation noise present in the DTG output signal and occurring as a result of analogue signal conversion into digital one.

Practical application of a new algorithm based on the Allan variance method in serial production on a lot of DTG samples made it possible to single out and analyse random components of errors. The results obtained can be used at the stage of GIU calibration.

Notably, the obtained coefficient values remain stable and do not change from start to start, which testifies to reliability of the obtained results and adequacy of both the algorithm and the developed SW.

The developed SW is the first and indispensable step in preparing GIU, which use DTG as their part, for manufacturing application in SINS of advanced products. Resulting from trials of the developed SW in serial production of DTG in amount of two month’s worth of the output, the random component was reduced from 30 to 50 % as compared with the classical method of assessing sensor output signals. Further trials of the method in serial production at the stage of GIU calibration are anticipated.

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