Research presents the development of computational technology for prediction and control by orbital spacecraft magnetic cleanliness based on geometric inverse magneto static problem solution with consideration of magnetic characteristics uncertainty to improve the spacecraft magnetic cleanliness and its controllability in orbit. Geometric inverse problem solution reduced to vector game solution with COMSOL Muliphysics software calculated payof vector. Game solution calculated based on heuristic optimization algorithms from Pareto-optimal solutions taking into account binary preference relations local games solutions. Based on developed computational technology results of prediction and control by «Sich-2» microsatellite family for ensuring microsatellite magnetic cleanliness presented.
The urgency of the work is due to the need to ensure high accuracy of the magnetic control system for
the angular orientation of the spacecraft in near-Earth space [
Modern requirements [
Technologies for ensuring the spacecraft "magnetic cleanliness" managed by NASA and ESA include
interrelated works of an organizational, technical and metrological nature. The foundation of this
technology is the calculation models of the spacecraft, which allow analytical or numerical prediction of
the spacecraft magnetic characteristics based on the knowledge of the magnetic field of its constituent
parts [
One of the fundamental components of this technology is the organization of metrological support
for determining in the multiple magnetic dipole models format the magnetic characteristics of the
component parts of the spacecraft on specialized magnetic measuring stands [
Anatolii Pidhornyi Institute of Power Machines and Systems of the National Academy of Sciences of Ukraine (IPMash of the National Academy of Sciences of Ukraine) has a powerful specialized experimental base, the basis of which is the unique and unique Magnetodynamic complex in Ukraine, which is included in the list of scientific objects that constitute the national property of Ukraine. At the IPMash magnetodynamic complex the experimental part of the fundamental studies of the magnetism of various technical objects and their physical models (spacecraft, ships, electric power equipment, building structures, pipelines) is carried out, the analysis of the spatio-temporal structure of their magnetic field (including ultra-small level - with induction less than 10-8 T). At the IPMash stand experimental studies of developed methods and means of targeted change of magnetic characteristics of various technical objects are carried out. Since 2003 year, the magnetometer stand has been providing tests of all orbital spacecraft launched into near-Earth orbit, namely "Microsat" (2003), "EgiptSat-1" (2007), "Sich-2" (2011), "Sich-2-30" (2022).
During space engineering testing in accordance to latest standards of the European Space Agency
ECSS-E-HB-20-07A [
This work is devoted to the development of computational technology for control by orbital spacecraft magnetic cleanliness based on geometric inverse magneto static problem solution with consideration of magnetic characteristics uncertainty to improve the spacecraft magnetic cleanliness and its controllability in orbit.
When designing a mathematical model of the spacecraft magnetic field the Gauss equation [
∞ 1 1 ∑︁ ( + 1) +2 ∑︁ ( cos() + ℎ sin())(cos ); 4 =1 =0 (2) = = 1 ∑∞︁ 1 ∑︁ ( sin() − ℎ cos() (cos ) 4 =1 +2 =0 sin 41 ∑∞=︁1 1+2 ∑=︁0 ( cos() + ℎ sin()) sin1 · . . . ; ·( − + 1) +1(cos ) − ( + 1) cos (cos ).
The solution of equation (2) with respect to the coeficients , ℎ represents spacecraft multipole mathematical model.
The mathematical model of spacecraft magnetic field in the form of a multipole model (2), proposed
by K. Gauss in the study of Earth magnetism [
As follows from the analysis of expressions (2) the spacecraft magnetic field at distances is greater
than three of its maximum overall dimensions are determined mainly by members of the first degree
series, i.e. the first three multipole coeficients [
All spacecraft elements undergo strict control for magnetic cleanliness and, as a rule, their preliminary demagnetization is performed. The components , , – of the spacecraft magnetic moment elements are measured before installation and meet the stringent requirements of magnetic cleanliness.
Then, the components , , of technical object magnetic field at any point of space
with coordinates , , in the form of the multiple magnetic dipole models (MDM) of the spacecraft
with the magnetic moment , , of dipole located at the points of the space of the
spacecraft with coordinates (, , ) calculated [
1 4 ′5 ⎣ ⎡2′2 − ′2 − ′2 3′′ 3′′ ... 3′′ · · · 2 ′2 − ′2 − ′2 ... 3′′ 3′′ 3′′ 2′2 − ′2 − ′2 ⎤ ⎡⎤ ⎦ · ⎣⎦ (3) Here the designations are introduced
= − , = − , = − , ′ = (( − )2 + ( − )2 + ( − )2) 21 .
All Ukrainian spacecraft after installing all the elements are examined for magnetic cleanliness at the magnetic measuring stand Anatolii Pidhornyi Institute of Mechanical Engineering Problems of the National Academy of Sciences of Ukraine. According to real measurements, the spacecraft magnetic moment and the magnetic field at the installation point of the onboard magnetometer are calculated. For this purpose, according to the data of measurements of the magnetic field in the near zone of the spacecraft, the real values of the moment vectors of the dipoles of the received are restored. In this case, it is assumed that the coordinates of the location of the dipoles in the space of the spacecraft remain unchanged.
If the spacecraft multiple magnetic dipole model (3) obtained based on the vector of measured magnetic field is too rough, then on the basis of the vector of the measured magnetic field, not only the magnetic moments , , of the dipoles, but also their position in the space of the technical object with coordinates , , can be calculated.
Let us consider the design of the technical object multiple magnetic dipole models only based on the vector of the measured magnetic field. Let us introduce the vector of desired parameters , the components of which are the desired values magnetic moments , , of the dipoles and coordinates , , of their position in the space of the spacecraft.
We also introduce the vector of uncertainty parameters of the magnetic moments of the spacecraft
the components of which are the deviations during the operation of the spacecraft of the magnetic
moments of the spacecraft elements from their central values, taken in the design of the control system
for the magnetic field of the spacecraft [
= (, ). (4)
Here, the vector nonlinear function (, ) obtained on the basis of expression (3) with respect to the vector of unknown variables, whose components are the desired values magnetic moments , , of the dipoles and coordinates , , of their position in the space of the spacecraft and the vector of the parameters of the uncertainties of the magnetic moments of the elements of the technical object.
In nonlinear equation (4) the number of unknown components of the vector equal to six times the number of dipoles, and the number of equations is equal to three times the number of measurement points.
Let us introduce the vector of the discrepancy between the vector of the measured magnetic ifeld and the vector of the predicted by model (4) magnetic field
(, ) = − (, ) = − (, ).
We write the objective nonlinear function as the weighted sum of squared residuals between the measured and predicted by the model (4) values of the magnetic field (, ) = ((, )) (, ). (5) (6)
The nonlinear objective function (6) obtained on the basis of expression (2) with respect to the vector of unknown variables, whose components are the desired values magnetic moments , , of the dipoles and coordinates , , of their position in the space of the spacecraft and the vector of the parameters of the magnetic moments uncertainties of the spacecraft elements.
As a rule, when optimizing the nonlinear objective function (6) · = (, ), · = (, ), (7) it is necessary to take into account restrictions on the values of magnetic moments , , of the dipoles and coordinates , , of their position in the space of the spacecraft.
Let’s consider another approach to the design of spacecraft multiple magnetic dipole models. Usually the designer of the spacecraft knows the of the elements of the spacecraft, which are the main sources of the initial magnetic field of the technical object. These are polarization relays, batteries and highfrequency valves. The technical object designer knows the number of these elements, the coordinates , , of their location in the spacecraft space, as well as the nominal values , , of their magnetic moments. Then the vector of the magnetic field components , , at the given points of the space with coordinates , , can be calculated based on spacecraft multiple magnetic dipole model (3).
Note that the values , , of the magnetic moments of these main elements of the spacecraft can be refined on the basis of the vector of the measured magnetic field.
As a rule, the spacecraft multiple magnetic dipole models obtained in this way is a rather rough approximation to the actual magnetic range of the spacecraft. To refine this model, consider the following approach. Let’s introduce more dipoles wits magnetic moment , , located at the points of the spacecraft with coordinates , , . Let us introduce the vector of desired parameters , the components of which are the desired values magnetic moments , , of the dipoles and coordinates , , of their position in the space of the spacecraft. We also introduce the vector of uncertainty parameters of the magnetic moments of the spacecraft. Then, based on the spacecraft multiple magnetic dipole models (4) calculated the vector (, ) of additional magnetic field, generated by only additional dipoles at the measurement points. (, ) = (, ). (8)
We introduce the vector of the initial magnetic field of the spacecraft, the components of which are the components of the magnetic field of the spacecraft calculated in this way at the measurement points generated by the main elements of the technical object with known values of the magnetic moments and the coordinates of their location in the space of the spacecraft.
Then one can calculate the vector of resulting magnetic field generated by dipoles with known magnetic moments nominal values , , and coordinates , , of their location in the technical object space and generated by additional dipoles with unknown magnetic moments , , and unknown coordinates , , of their location in the spacecraft space (, ) = + (, ). (9)
Then the problem (7) of calculated the vectors of unknown parameters of additionally introduced dipoles solved similarly to the problem (9) of calculated the vector of unknown parameters of dipoles for design of the spacecraft multiple magnetic dipole model.
Usually, the spacecraft magnetic cleanliness requirements presented in the form of restrictions on
the total magnetic moment of the spacecraft and the magnitude of the magnetic field at the installation
point of the onboard magnetometer [
To compensate for the initial magnetic field of the spacecraft we introduce magnetic dipoles with unknown magnetic moments , , located at points with unknown coordinates , , .
Let us introduce the vector of the desired parameters for solving the problem of compensating the
initial magnetic field of the technical object, whose components are the oblique values of the magnetic
moments , , and coordinates , , of the location of the compensating magnetic
dipoles in the spacecraft space. Then, for a given value of the vector of the desired parameters
of the compensating dipoles, the vector () of the compensating magnetic field generated by all
compensating dipoles at the installation point of the onboard magnetometer and the vector () of
the compensating magnetic moment generated by all compensating dipoles calculated [
Then we calculated the vector (, ) of resulting magnetic moment and vector (, ) of resulting magnetic field generated at the installation point of the onboard magnetometer by the spacecraft elements and all compensating dipoles
(, ) = () + (), (, ) = () + ().
Then the problem of calculated the coordinates , , of spatial arrangement and magnetic moments , , of the compensating dipoles can be reduced to solving the problem of vector minimax optimization of resulting magnetic moment of the spacecraft and the resulting magnetic field at the installation point of the onboard magnetometer (10) (11) (12) (13) (14)
Note that this approach is standard when designing of robust control, when the coordinates of the spatial arrangement and the magnitudes of the magnetic moments of the compensating dipoles are found from the conditions for minimizing the modulus of spacecraft magnetic field at the magnetometer installation point for the “worst” values of the magnetic moments of the elements of the spacecraft [9, 10, 11, 12].
The solution of the vector minimax optimization problem wits vector objective function (11) and (12) is the set of unimprovable solutions – the Pareto set of optimal solutions if only one vector objective function given [13, 14, 15]. Such a statement of the optimization problem is an ill-posed problem, since the solution in the form of a Pareto optimal set of unimprovable solutions is devoid of engineering sense from the point of view of practical application [16]. In addition to the vector optimization criterion (10), it is also necessary to have information about the binary relations of preference of local solutions to each other in order to correctly solve the problem of multi-criteria optimization. This approach makes it possible to significantly narrow the range of possible optimal solutions to the original multi-criteria optimization problem [17, 18, 19].
The problem of finding a local minimum at one point of the considered space is, as a rule, multiextreme, containing local minima and maxima, therefore, for its solution, it is advisable to use algorithms of stochastic. Currently, the most widely used are multi-agent stochastic optimization methods that use only the speed of particles. To find the solution of minimax vector optimization problem (11) from Paretooptimal decisions [20, 21, 22] taking into account the preference relations, we used special nonlinear algorithms of stochastic multi-agent optimization. First-order methods have good convergence in the region far from the local optimum, when the first derivative has significant values [23].
The main disadvantage of first-order search methods, which use only the first derivative – the speed of particles, is their low eficiency of the search and the possibility of getting stuck in the search near the local minimum, where the value of the rate of change of the objective function tends to zero. The advantage of second-order algorithms is the ability to determine not only the direction of movement, but also the size of the movement step to the optimum, so that with a quadratic approximation of the objective function, the optimum found in one iteration [24, 25, 26].
To search the components () optimal values of the vector of the desired parameters minimizing vector optimization criterion (11) and (12) for calculating velocities () and accelerations () of particle of swarm using the following steps [27, 28, 29, 30] · = (, ), · = (, ), · = (, ), · = (, ), ( + 1) = 1 () + 1 1 ()(1 () − 1 ()) [ () − ()] +
2 2 ()(1 () − 2 ()) [︀ * () − ()]︀ . ( + 1) = 2 () + 3 3 ()(3 () − 3 ()) [ () − ()] +
4 4 ()(4 () − 4 ()) [︀ * () − ()]︀ .
Here, () and * () – the best–local and global positions (), () and * () – the best-local and global velocity () of the –th particle, found respectively by only one –th particle and all the particles of swarm.
Random numbers (), () and constants , , are tuning parameters, – Heaviside function.
To search the components () optimal values of the vector of the parameters of the uncertainty of the control object of the system of active silencing maximizing the same vector optimization criterion (10) for calculating velocities () and accelerations () of particle of swarm using the steps similarly (13), (14). However, unlike (13), (14), the best local and global position and velocity components are those that lead not to a decrease in the corresponding components of the vector objective function (10), but vice versa to their increase. This is where the “malicious” behavior of the vector of uncertainties of the designed system manifested.
The use of the Archimedes algorithm for calculating minimax vector optimization problem (10) solutions with binary preference relations it possible to reduce the calculating time.
Anatolii Pidhornyi Institute of Power Machines and Systems of the National Academy of Sciences of Ukraine (IPMash of the National Academy of Sciences of Ukraine) has the magnetodynamic complex includes premises with a total area of about 20002 and consists of a magnet measuring stand with an area of 4502 equipped with unique world-class magnetometric equipment, a technological plot of land with an area of 5.3ℎ, which ensures a "magnetic silence" mode, power supply systems with a capacity of 800 and others engineering systems and structures.
The measurements procedure of components of the spacecraft magnetic moment involves compensation of the Earth magnetic field in three orthogonal directions , and . The procedure for determining the components of the magnetic moment of the test object involves the alternating influence on it of the magnetic field of the windings in three orthogonal directions , and with induction 40 . Measurements of the spacecraft magnetic moment according to the existing technology, involves the stationary installation of twelve magnetometer "Magnetomat 1.782" at the corresponding points of the sphere with a radius of 2 m. For measuring the magnetic moment of the component parts (blocks) of the spacecraft magnetometer "Magnetoscop 1.069" used.
Figure 1 shows the spacecraft "SICH-2-1" on the measuring stand.
Ensuring the high accuracy of the MSU operation, with modern trends in the reduction of spacecraft mass, requires the minimization of its magnetic moment and high accuracy of its determination.
Let us consider the application of the developed method for solving the problem of ensuring the magnetic cleanliness of the «Sich-2-1» spacecraft. Experimental measurements of the magnetic characteristics of the «Sich-2-1» spacecraft carried out at the Anatolii Pidhornyi Institute of Power Machines and Systems of the National Academy of Sciences of Ukraine stand. Based on these measurements, we calculate the model of the «Sich-2-1» spacecraft. The experimentally measured value of the total magnetic moment of spacecraft is equal = [0.24, 0.5, 0.4]. The dispersion of the magnetic field prediction error in this case is = 7560.6. The value of the experimentally measured magnetic moment of the spacecraft implies the presence of several dipoles located in the space of the spacecraft. In the calculation it is assumed that the model of the magnetic field of the spacecraft represents one dipole located at the origin of the spacecraft.
Based on the experimental measured magnetic field at first the spacecraft magnetic field model presented as a single dipole located in the center of the spacecraft. Based on the vector of the measured magnetic field of the spacecraft , the moments of this single dipole = [0.24, 0.5, 0.4] were calculated. The dispersion of the magnetic field prediction error in this case is = 7272.7.
Let us now consider the mathematical model of the magnetic field of the spacecraft in the form of a single dipole, the location coordinates of which in the space of the spacecraft also need to be calculated. For the calculated value of the moment = [0.2664, 0.1641, 0.1434] and coordinates = [0.2158, -0.4136, 0.0859] of the location of such a single dipole, the prediction error variance is = 3239.8. Note that the location of the only dipole not at the origin of the coordinates, but at the point with the optimal coordinates made it possible to reduce the dispersion of the magnetic field prediction by a factor of 2.3337.
If, when solving the problem of optimizing the values of the magnetic moments and the coordinates of the location of one dipole, we introduce restrictions on the magnitude of the dipole moments in the form of restrictions [-0.8, -0.8, -0.8] ≤ ≤ [0.8, 0.8, 0.8], optimal values of the moments = [0.2388, 0.1921, 0.1258] and coordinates = [0.2056, -0.4146, 0] of the location of such a single dipole, the prediction error variance is = 3325.1. Thus, under restrictions on the magnitude of the dipole moments, the optimum values of the magnetic moments are at the limits and, in this case, the dispersion increases by a factor of 2.2738.
Let us now consider the model of the spacecraft magnetic field in the form of two dipoles. If, when solving the problem of optimizing the values of the magnetic moments and the coordinates of the location of two dipoles, we introduce restrictions on the magnitude of the dipole moments in the form of restrictions [-0.8, -0.8, -0.8] ≤ ≤ [0.8, 0.8, 0.8], optimal values of the moments 1 = [0.3538, -0.0326, -0.0345] and 2 = [-0.6137, 0.6695, -0.2802] and the coordinates 1 = [0.3090, -0.3080, 0.0867] and 2 = [-0.0657, -0.0789, -0.3908] of the location of two dipoles, the dispersion the prediction error is = 1203.4. Thus, under restrictions on the magnitude of the two dipoles moments, the optimum values of the magnetic moments are at the limits and, in this case, the dispersion increases by a factor of 6.2827.
The simplest satellite multiple magnetic dipole models is a model consisting of two dipoles with magnetic dipoles and coordinates 1 = [︀ 1.8404 −0.4147 0.1316 ]︀ , 2 = [︀ −1.9641 1.3347 −0.3664 1 = [︀ 0.2173 −0.1625 −0.0366 ︀] , 2 = [︀ 0.1609 −0.0306 −0.1397 ︀] , ︀] . deviations for spacecraft mathematical model in the form of two dipoles.
Based on this model, we calculate the spacecraft magnetic moment as the sum (8) of the magnetic moments of the dipoles and magnetic field = [︀ −0.1237 at the installation point of the onboard magnetometer. The simplest solution to the problem of compensating the initial satellite magnetic moment is to place this one compensating dipole with a magnetic moment = [︀ 0.1237 −0.92 opposite to the satellite moment. Even with the location of this compensating dipole at the origin of the spacecraft coordinates, the magnetic field induction = [︀ 1.6155 at the point of installation of the magnetometer decreases by a factor 6.23. The location coordinates of this compensating dipole calculated from the condition of minimizing the resulting magnetic field at magnetometer installed point. Optimal position coordinates of the compensating dipole and at the same time the induction level at the installation point of the magnetometer 1 = [︀ 0.5
0.5 −0.5 ]︀ = [︀ 0.8903
Thus, due to the installation of a compensating dipole at the optimal point, the magnetic field induction module at the magnetometer installation point decreased by 3.15 times compared with the magnetometer installation in the middle of the spacecraft and decreased by 19.8 times compared to the satellite without compensation. A more accurate compensation of the spacecraft magnetic moment is achieved by placing two compensation dipoles with magnetic moments 1 = [︀ −1.8404 0.4147 −0.1316
︀] ,
2 = [︀ 1.9641 −1.3347 0.3664 ]︀ ,
whose location coordinates in the spacecraft space are calculated by minimizing the magnetic field at
onboard magnetometer installation point taking into account design constraints. Note that in the work
[
Computational technology for prediction and control by orbital spacecraft magnetic cleanliness based on geometric inverse magneto static problem solution with consideration of magnetic characteristics uncertainty to improve the spacecraft magnetic cleanliness and its controllability in orbit developed.
Geometric inverse problem solution reduced to vector game solution with COMSOL Muliphysics software calculated payof vector. Game solution calculated based on heuristic optimization algorithms from Pareto-optimal solutions taking into account binary preference relations local games solutions.
Results of prediction and control by «Sich-2» microsatellite family for ensuring microsatellite magnetic cleanliness based on developed computational technology presented.
The author(s) have not employed any Generative AI tools. [7] A. Popov, E. Tserne, V. Volosyuk, S. Zhyla, V. Pavlikov, N. Ruzhentsev, et al., Invariant polarization signatures for recognition of hydrometeors by airborne weather radars, in: O. Gervasi, B. Murgante, D. Taniar, B. O. Apduhan, A. C. Braga, C. Garau, A. Stratigea (Eds.), Computational Science and Its Applications – ICCSA 2023. ICCSA 2023. Lecture Notes in Computer Science, vol. 13956, Springer Nature Switzerland, Cham, 2023, pp. 201–217. doi:10.1007/978-3-031-36805-9_14. [8] T. Nikitina, B. Kuznetsov, Y. Averyanova, O. Sushchenko, I. Ostroumov, N. Kuzmenko, et al., Method for design of magnetic field active silencing system based on robust meta model, in: S. Shukla, H. Sayama, J. V. Kureethara, D. K. Mishra (Eds.), Data Science and Security. IDSCS 2023. Lecture Notes in Networks and Systems, vol. 922, Springer Nature Singapore, Singapore, 2024, pp. 103–111. doi:10.1007/978-981-97-0975-5_9. [9] T. Nikitina, B. Kuznetsov, N. Ruzhentsev, O. Havrylenko, K. Dergachov, V. Volosyuk, et al., Algorithm of robust control for multi-stand rolling mill strip based on stochastic multi-swarm multi-agent optimization, in: S. Shukla, H. Sayama, J. V. Kureethara, D. K. Mishra (Eds.), Data Science and Security. IDSCS 2023. Lecture Notes in Networks and Systems, vol. 922, Springer Nature Singapore, Singapore, 2024, pp. 247–255. doi:10.1007/978-981-97-0975-5_22. [10] M. Ostroverkhov, V. Chumack, E. Monakhov, Ouput voltage stabilization process simulation in generator with hybrid excitation at variable drive speed, in: 2nd Ukraine Conference on Electrical and Computer Engineering (UKRCON), IEEE, Lviv, Ukraine, 2019, pp. 310–313. doi:10.1109/ ukrcon.2019.8879781. [11] O. Sushchenko, Y. Bezkorovainyi, O. Solomentsev, M. Zaliskyi, O. Holubnychyi, I. Ostroumov, et al., Algorithm of determining errors of gimballed inertial navigation system, in: O. Gervasi, B. Murgante, C. Garau, D. Taniar, A. M. A. C. Rocha, M. N. Faginas Lago (Eds.), Computational Science and Its Applications – ICCSA 2024 Workshops. ICCSA 2024. Lecture Notes in Computer Science, vol. 14816, Springer Nature Switzerland, Cham, 2024, pp. 206–218. doi:10.1007/978-3-031-65223-3_14. [12] O. Chornyi, S. Serhiienko, A virtual complex with the parametric adjustment to electromechanical system parameters, Technical Electrodynamics 1 (2019) 38–41. doi:10.15407/techned2019. 01.038. [13] I. Shchur, L. Kasha, M. Bukavyn, Eficiency evaluation of single and modular cascade machines operation in electric vehicle, in: 15th International Conference on Advanced Trends in Radioelectronics, Telecommunications and Computer Engineering (TCSET), IEEE, Lviv-Slavske, Ukraine, 2020, pp. 156–161. doi:10.1109/tcset49122.2020.235413. [14] I. Shchur, V. Turkovskyi, Comparative study of brushless DC motor drives with diferent conifgurations of modular multilevel cascaded converters, in: 15th International Conference on Advanced Trends in Radioelectronics, Telecommunications and Computer Engineering (TCSET), IEEE, Lviv-Slavske, Ukraine, 2020, pp. 447–451. doi:10.1109/tcset49122.2020.235473. [15] B. Kuznetsov, I. Bovdui, T. Nikitina, R. Voliansky, O. Sushchenko, Y. Bezkorovainyi, et al., Eficiency of overhead power lines magnetic field combined electromagnetic silence, in: M. Aghaei, X. Zhang, H. Ren (Eds.), Fifth International Conference on Green Energy, Environment, and Sustainable Development (GEESD 2024), volume 13279, International Society for Optics and Photonics, SPIE, 2024, p. 1327907. URL: https://doi.org/10.1117/12.3043769. doi:10.1117/12.3043769. [16] M. Zagirnyak, O. Bisikalo, O. Chorna, O. Chornyi, A model of the assessment of an induction motor condition and operation life, based on the measurement of the external magnetic field, in: 3rd International Conference on Intelligent Energy and Power Systems (IEPS), IEEE, Kharkiv, Ukraine, 2018, pp. 316–321. doi:10.1109/ieps.2018.8559564. [17] V. Halchenko, R. Trembovetska, C. Bazilo, N. Tychkova, Computer simulation of the process of profiles measuring of objects electrophysical parameters by surface eddy current probes, in: Lecture Notes on Data Engineering and Communications Technologies, vol. 178, Springer Nature Switzerland, Cham, 2023, pp. 411–424. doi:10.1007/978-3-031-35467-0_25. [18] V. Halchenko, D. Bacherikov, S. Filimonov, N. Filimonova, Improvement of a linear screw piezo motor design for use in accurate liquid dosing assembly, in: Smart Technologies in Urban Engineering. STUE 2022. Lecture Notes in Networks and Systems, vol. 536„ Springer Nature Switzerland, Cham, 2023, pp. 237–247. doi:10.1007/978-3-031-20141-7_22. [19] R. Voliansky, B. Kuznetsov, I. Bovdui, Y. Averyanova, I. Ostroumov, O. Sushchenko, et al., Variablestructure interval-based dufing oscillator, in: 2024 IEEE 42nd International Conference on Electronics and Nanotechnology (ELNANO), 2024, pp. 581–586. doi:10.1109/ELNANO63394. 2024.10756860. [20] K. Maksymenko-Sheiko, T. Sheiko, D. Lisin, N. Petrenko, Mathematical and computer modeling of the forms of multi-zone fuel elements with plates, Journal of Mechanical Engineering 25 (2022) 32–38. doi:10.15407/pmach2022.04.032. [21] P. Hontarovskyi, N. Smetankina, S. Ugrimov, N. Garmash, I. Melezhyk, Computational studies of the thermal stress state of multilayer glazing with electric heating, Journal of Mechanical Engineering 25 (2022) 14–21. doi:10.15407/pmach2022.02.014. [22] A. Kostikov, L. Zevin, H. Krol, A. Vorontsova, The optimal correcting the power value of a nuclear power plant power unit reactor in the event of equipment failures, Journal of Mechanical Engineering 25 (2022) 40–45. doi:10.15407/pmach2022.03.040. [23] A. Rusanov, V. Subotin, O. Khoryev, Y. Bykov, P. Korotaiev, Y. Ahibalov, Efect of 3D shape of pump-turbine runner blade on flow characteristics in turbine mode, Journal of Mechanical Engineering 25 (2022) 6–14. doi:10.15407/pmach2022.04.006. [24] S. Kurennov, N. Smetankina, V. Pavlikov, D. Dvoretskaya, V. Radchenko, Mathematical model of the stress state of the antenna radome joint with the load-bearing edging of the skin cutout, in: D. Cioboată (Ed.), International Conference on Reliable Systems Engineering (ICoRSE) - 2021. ICoRSE 2021. Lecture Notes in Networks and Systems, vol 305, Springer, Cham, 2022, pp. 287–295. doi:10.1007/978-3-030-83368-8_28. [25] S. Kurennov, N. Smetankina, Stress-strain state of a double lap joint of circular form. axisymmetric model, in: M. Nechyporuk, V. Pavlikov, D. Kritskiy (Eds.), Integrated Computer Technologies in Mechanical Engineering - 2021. ICTM 2021. Lecture Notes in Networks and Systems, vol. 367, Springer, Cham, 2022, pp. 36–46. doi:10.1007/978-3-030-94259-5_4. [26] O. Solomentsev, M. Zaliskyi, O. Holubnychyi, I. Ostroumov, O. Sushchenko, Y. Bezkorovainyi, et al., Eficiency analysis of current repair procedures for aviation radio equipment, in: I. Ostroumov, M. Zaliskyi (Eds.), Proceedings of the 2nd International Workshop on Advances in Civil Aviation Systems Development. ACASD 2024. Lecture Notes in Networks and Systems, vol. 992, Springer Nature Switzerland, Cham, 2024, pp. 281–295. doi:10.1007/978-3-031-60196-5_21. [27] M. Ummels, Stochastic Multiplayer Games Theory and Algorithms, Amsterdam University Press,
Amsterdam, The Netherlands, 2010. [28] X.-S. Yang, Z. Cui, R. Xiao, A. H. Gandomi, M. Karamanoglu, Swarm Intelligence and Bio-Inspired
Computation: Theory and Applications, Elsevier Inc., 2013. [29] M. Zaliskyi, O. Solomentsev, O. Holubnychyi, I. Ostroumov, O. Sushchenko, Y. Averyanova, et al., Methodology for substantiating the infrastructure of aviation radio equipment repair centers, CEUR Workshop Proceedings 3732 (2024) 136–148. [30] O. Holubnychyi, M. Zaliskyi, I. Ostroumov, O. Sushchenko, O. Solomentsev, Y. Averyanova, et al., Self-organization technique with a norm transformation based filtering for sustainable infocommunications within cns/atm systems, in: I. Ostroumov, M. Zaliskyi (Eds.), Proceedings of the 2nd International Workshop on Advances in Civil Aviation Systems Development. ACASD 2024. Lecture Notes in Networks and Systems, vol. 992, Springer Nature Switzerland, Cham, 2024, pp. 262–278. doi:10.1007/978-3-031-60196-5_20.