Solution to the problem of detecting a malfunction of the position sensor of small spacecraft

The classic scheme for constructing a modern system for controlling the parameters of the angular motion of a spacecraft involves the use of high-precision gyroscopes, periodically adjusted according to the readings of optoelectronic devices. The most widely used in orientation and navigation systems are precision angular velocity sensors and gyroscopes on an electrostatic suspension, which determine the angles of rotation of the aircraft around its center of mass; Angular and linear accelerometers are also used, mounted in a certain way on the body of the aircraft.


Introduction
Let us consider the problem of diagnosing the state of an object as a problem of pattern recognition [1,2].The state of an object is described by many parameters (signs) that define it.Recognizing the state of an object consists of assigning it to one of the possible classes.The number of classes depends on the characteristics of the problem and the goals of the study.The recognition problem is that for a given object ω and a set of classes Ω1, Ω2, …, Ωm according to educational information I0(Ω1, Ω2, …, Ωm) about classes and description I(ω) calculate predicate values Pi(Ω) -ω ∈ Ωi, i = 1, 2, …, m.Object Occurrence Information ω to class Ωi encoded by characters «1» (ω ∈ Ωi), «0» (ω ∉ Ωi), Δ -it is unknown whether ω belongs to the class Ωi or not, and is written in the form of a so-called information vector  (ω)=( 1 (ω),…,  (ω)), где   ∈ {0, 1, Δ}.
Information coming from a small spacecraft (SSV) is processed by coordination and computing centers, which provide the necessary data to the Flight Control Center [3,4].The solution to the problem of determining the loss of communication was considered on the basis of the method of group accounting of arguments (MGUA) and on the basis of various neural networks.The Subscriber is in constant readiness to establish communication with the Mission Control Center (MCC), which establishes communication by sending Z directives.Immediately after the connection is established, the Subscriber begins to issue an initiative message to the MCC with information about functional telemetry control (IFCT).Initiative messages and receipts are issued by the Subscriber against the background of transmitted functional control information with a tact of 1 second.
The receipt code is entered in the receipt header field Code, and the number of the last received packet is entered in the NPrm field.In the structure of the reverse channel information (RCI), the Code field contains one of three values: 1, 2 or 0. The purpose of control is to determine an emergency situation when executing a set of rules.If within three or more cycles the control center does not receive any messages from the Subscriber, then the issuance of directives stops.At the same time, the control center considers that the connection has been interrupted and proceeds to attempts to re-establish it.
If the message transmission was successful, the MCC begins to transmit receipts with the directive number Nz and reverse channel information (RCI).In the PKI structure, the Code field contains one of three values: 1 -writing to a dynamic storage device (DRAM) is in progress and the file is being written to disk; 2 -reading from DZU is in progress; 0 -when receiving directives, as well as after completing the execution of selected directives.
If there is no information in the future, we assume that the value of the attribute is 0.5.

Detection of abnormal situations using PNN network
To solve the problem of detecting emergency situations, it is proposed to use a probabilistic neural network (PNN -Probabilistic Neural Network) [5].The ANN learns to estimate the probability of a situation belonging to one of three classes.The probabilistic network has three layers: an input layer, a sample layer and an output layer, in which each class has its own output.The output values are determined by the Gaussian function: where    -weighting coefficients equal to the elements of the corresponding sample vector    -th class;   -elements of the unknown input vector; σ -standard deviation value,  2 -dispersion.
In accordance with (1), to determine the output of the ANN we use as    values of mathematical expectation of signs   and dispersion for the vectors of the training set (see table 1 and see table 2).  3.
Table 3. Fragment of the result of situation recognition using the PNN network.

Detecting a malfunction of the MKA position sensor. Correlation approach
Angular velocity sensors are one of the main and most advanced sensitive elements of control, stabilization and navigation systems, used to measure the angular velocity of a spacecraft from 0,001 before 10 s -1 in inertial space [6].Very stringent requirements are imposed on the characteristics of angular velocity sensors (ARS).Thus, the upper range of speeds measured by modern CRS corresponds to tens and hundreds of degrees per second.The upper range of input influences, in which the DUS is required to provide measurements of angular velocity, reaches 100 Hz [7].
The study performed a complete analysis of pairwise correlation of sensor readings in order to identify an informative feature that serves to identify failures and malfunctions (see table 4).Let us estimate the degree of connection between these parameters based on a sample of values using the pair correlation coefficient [8]: Where  and  -means of the corresponding samples.The pairwise correlation coefficient varies from -1 to +1.The closer it is in absolute value to unity, the closer the statistical relationship between x and y to linear functional.The following indirect qualitative interpretation of the possible values of the correlation coefficient can be given: if |  | < 0,3 -there is practically no connection; 0,3 ≤ |  | < 0,7 -connection is average; 0,7 ≤ |  | < 0,9 -the connection is strong; 0,9 ≤ |  | < 0,99 -the connection is very strong.
To assess the multicollinearity of the readings of the position sensors of the small spacecraft, we use a matrix of pair correlation coefficients.The matrix (table 5) contains an interfactor correlation coefficient      > 0,7, therefore, there is multicollinearity in this multiple regression model: Because Since the parameters D and INTV correlate with each other, the value of the correlation coefficient is partially affected by the influence of other variables.
The degree of joint influence of factors on the result is assessed by the multiple correlation index, which, with a linear relationship, can be calculated using the formula: Where Δ  -determinant of the matrix of pair correlation coefficients, Δ 11 -algebraic complement of an element   correlation matrix.
Let's calculate the multiple correlation coefficient for three characteristics: We get The calculation results are shown in table 6.
Table 6.Fragment of the calculation of the multiple correlation coefficient for three position sensors of the small spacecraft.
Comparing the values given in this table, we come to the conclusion that a very strong relationship has been found between the following indicators: Thus, an analysis of possible options for grouping sensor readings was carried out and, based on correlation analysis methods, the most appropriate combinations of readings were found for detecting malfunctions of the MCA position sensor.
To detect a failed sensor, we will calculate the correlation between sensor readings in a sliding "window" mode.The width of the time window is 200 samples, which corresponds to a 40-second time interval.For each triplets of readings (i, j, k) pairwise correlation coefficients are calculated and reduced to the values "0" and "1" using the specified threshold.When rij = 1 and rik = 1 sensor i is normal, but when rij = 0 and rik = 0 there is a failure.As a result of the experiment, it was possible to identify failures of sensors D and INTV (figure 1).

Conclusion
This work demonstrates the possibility of using the apparatus of probabilistic neural networks.The problem of diagnosing the state of an object is considered as a problem of pattern recognition.Using correlation analysis methods, it was possible to identify faults in the position sensors of the small spacecraft.

Table 4 .
Position sensor system.D -range, km; V -speed, m/sec; UKP -heading angle along the communication line; UTP -pitch angle along the communication line; UKA -heading angle along the communication line; UTA -pitch angle along the communication line; B1 -roll angle -the angle of rotation around the X axis of the ship; INTV -integral of speed, m.

Figure 1 .
Figure 1.Results of identifying malfunctions of the position sensors of the small spacecraft.

Table 1 .
Mathematical expectation of features for training sample vectors.

Table 2 .
Variance of features for vectors of the training set.
The activation function of the output layer generates the probability value of the defined class:  () =    (),  -class number.Values f (k) can be interpreted as estimates of the probability of an element belonging to a certain class.After training, control vectors, both reference and those not involved in training, were fed to the network input. The results of the experiment on recognizing situations are shown in table

Table 5 .
Fragment of the correlation matrix of readings of the position sensors of the small spacecraft.