Relative motion parameters estimation of a non-cooperative spacecraft from visual information
Heading:
| 1Salnikov, NN, 1Melnychuk, SV, 1Gubarev, VF, 1Maksymyuk, LV, 1Shevchenko, VM 1Space Research Institute of the National Academy of Sciences of Ukraine and the State Space Agency of Ukraine, Kyiv, Ukraine |
| Space Sci. & Technol. 2023, 29 ;(3):16-23 |
| https://doi.org/10.15407/knit2023.03.016 |
| Publication Language: English |
Abstract: In this work, we consider the problem of determining parameters of the relative motion of a non-cooperative spacecraft (NSC), which is in free uncontrolled motion, based on the results of measuring the distance to this vehicle and its attitude quaternion. The measurements are assumed to be made by some computer vision system (CVS). A specific type of СVS is not considered. It is supposed the CVS measures the distance and attitude of the so-called graphical reference frame rigidly fixed on the NSC. The parameters of relative motion include the distance vector to the center of mass (c.m.) of the NSC, the attitude quaternion of the principal inertia axes of the NSC relative to the CVS reference frame, the attitude quaternion of the graphical reference frame relative to the NSC principal reference frame, the ratio of the inertia moments, the position vector of the c.m. in the graphical reference frame.
The problem is solved using a dynamic filter based on the ellipsoidal estimation method. The method implies knowledge of the maximum values of the measurement noise only, the stochastic noise characteristics are not assumed to be known and therefore are not used. The properties of the proposed algorithm have been demonstrated using numerical simulations. The results obtained are supposed to be used in the development, creation, and testing of a navigation system for the rendezvous and docking of a service spacecraft, developed by a group of enterprises in the space industry of Ukraine under the leadership of the Limited Liability Company “Kurs–Orbital”.
|
| Keywords: estimation, relative motion parameters, spacecraft, video image |
References:
1. Moghaddam B.M., Chhabra R. On the guidance, navigation and control of in-orbit space robotic missions: A survey and prospective vision. Acta Astronautica, vol. 184, pp. 70-100 (2021).
https://doi.org/10.1016/j.actaastro.2021.03.029
2. Savchuk A.P., Fokov A.A. Determination of non-cooperative object parameters in orbital service tasks. Technical Mechanics, № 4, pp. 30-45. (2018).[in Russian].
https://doi.org/10.15407/itm2018.04.030
3. Opromolla R. et. al. A review of cooperative and uncooperative spacecraft pose determination techniques for close proximity operations. Progress in Aerospace Sciences, vol. 93, pp.53-72. (2017).
https://doi.org/10.1016/j.paerosci.2017.07.001
4. Balahoncev V.G., Ivanov V.A., Shabanov V.I. Sblizhenie v kosmose[Approaching in space]. Moscow: Voenizdat, 1973, 240 p. [in Russian].
5. Bragazin A.F. Control of spacecraft approaching (navigation, guidance, motion correction. Koroljov: RKK "Jenergija", 2018. 470 p. [in Russian].
6. Farrell J.A. Aided navigation. GPS with High Rate Sensors. New York: The McGraw-Hill Companies, 2008. 553 p.
7. Fehse W. Automated Rendezvous and Docking of Spacecraft. Cambridge: Cambridge University Press, 2003. 517 p.
https://doi.org/10.1017/CBO9780511543388
8. Aghili F. Automated Rendezvous & Docking without Impact Using a Reliable 3D Vision System. Guidance, Navigation, and Control Conference, 2-5 August (2010), Toronto, Ontario Canada.
https://doi.org/10.2514/6.2010-7602
9. Opromolla R., Nocerino A. Uncooperative Spacecraft Relative Navigation With LIDAR-Based Unscented Kalman Filter. IEEE Access, vol. 7, pp. 180012-180026. (2019).
https://doi.org/10.1109/ACCESS.2019.2959438
10. Nocerino A., Opromolla R., Fasano G., Grassi M. LIDAR-based multi-step approach for relative state and inertia parameters determination of an uncooperative target. Acta Astronautica, vol. 181, pp. 662-678, (2021).
https://doi.org/10.1016/j.actaastro.2021.02.019
11. Aghili F., Kuryllo M., Okouneva G., English Ch. Fault-Tolerant Position/Attitude Estimation of Free-Floating Space Objects Using a Laser Range Sensor. IEEE Sensors Journal. (2011). vol. 11, No. 1, pp. 176-185.
https://doi.org/10.1109/JSEN.2010.2056365
12. Aghili F., Su C. Robust Relative Navigation by Integration of ICP and Adaptive Kalman Filter Using Laser Scanner and IMU. IEEE/ASME Transactions on Mechatronics. (2016). vol. 21. no. 4, pp. 2015-2026.
https://doi.org/10.1109/TMECH.2016.2547905
13. Kelsey J. M., Byrne J., Cosgrove M., Seereeram S., Mehra R. K. Vision-based relative pose estimation for autonomous rendezvous and docking. 2006 IEEE Aerospace Conference. (2006), pp. 20-39.
https://doi.org/10.1109/AERO.2006.1655916.
14. Segal S., Carmi A., Gurfil P. Vision-Based Relative State Estimation of Non-Cooperative Spacecraft Under Modeling Uncertainty. Aerospace Conference. Piscataway(NJ): IEEE Publ. (2011). pp. 1-8.
https://doi.org/10.1109/AERO.2011.5747479
15. Oumer N.W., Panin G. Tracking and pose estimation of non-cooperative satellite for on-orbit servicing. Proseedings of the conference i-SAIRAS 2012. European Space Agency (ESA). i-SAIRAS. (2012), 4-6 Sep 2012, Turin, Italy.
16. D'Amico S., Benn M., Jørgensen J.L. Pose estimation of an uncooperative spacecraft from actual space imagery. International Journal of Space Science and Engineering. (2014). Vol.2. No.2, pp.171 - 189.
https://doi.org/10.1504/IJSPACESE.2014.060600
17. Yu X., Yu F., He Z. Stereo vision based relative state estimation for non-cooperative spacecraft with outliers. Proceedings of the 33rd Chinese Control Conference. (2014). pp. 763-769,
https://doi.org/10.1109/ChiCC.2014.6896723
18. Volpe R., Sabatini M., Palmerini G.B. Pose and Shape Reconstruction of a Noncooperative Spacecraft Using Camera and Range Measurements. International Journal of Aerospace Engineering. (2017). vol. 2017, Article ID 4535316, 13 p.
https://doi.org/10.1155/2017/4535316
19. Dementhon D.F., Davis L.S. Model-based object pose in 25 lines of code. International Journal of Computer Vision. (1995). vol. 15, pp.123-141.
https://doi.org/10.1007/BF01450852
20. Masutani Y., Iwatsu T., Miyazaki F. Motion estimation of unknown rigid body under no external forces and moments. Proceedings of the 1994 IEEE International Conference on Robotics and Automation (1994). vol.2, pp. 1066-1072.
https://doi.org/10.1109/ROBOT.1994.351227
21. Shijie et.al. Monocular Vision-based Two-stage Iterative Algorithm for Relative Position and Attitude Estimation of Docking Spacecraft. Chinese Journal of Aeronautics. (2010). vol. 23, issue 2, pp. 204-210.
https://doi.org/10.1016/S1000-9361(09)60206-5
22. Capuano V., Kim K., Hu J., Harvard A., Chung S. Monocular-Based Pose Determination of Uncooperative Known and Unknown Space Objects. Proceedings of the 69th International Astronautical Congress (IAC).(2018). Bremen, Germany, 1-5 October 2018.
23. Espinoza A.T., Setterfield T.P. Point-to-CAD 3D Registration Algorithm for Relative Navigation Using Depth-Based Maps.(2019). 2019 IEEE Aerospace Conference, pp. 1-7,
https://doi.org/10.1109/AERO.2019.8742148
24. Liang H., Wang J., Wang Y., Huo W. Monocular-vision-based spacecraft relative state estimation under dual number algebra. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering. (2020). vol. 234, pp. 221 - 235.
https://doi.org/10.1177/0954410019864754
25. Ivanov D. S., Karpenko S. O., Ovchinnikov M. Ju., Sakovich M. A. Relative motion determination of satellites at their separation on results of videoimages processing, Preprinty IPM im. M. V. Keldysha[Preprints of Keldysh IAM]. (2012), No. 057, 24 p.
26. Salnikov N.N., Melnychuk S.V., Gubarev V.F. Ellipsoidal Pose Estimation of an Uncooperative Spacecraft from Video Image Data. in book "Control Systems: Theory and Applications". River Publishers Series in Automation, Control and Robotics, 2018. pp. 169-195.
27. Grishin V.A., Zhukov B.S. Peculiarities of image recognition at its application to relative navigation tasks at spacecraft docking. Modern problems of Earth Remote Control. (2020). Vol. 17, No 7. pp. 58-66.
https://doi.org/10.21046/2070-7401-2020-17-7-58-66
28. Molina Saqui J.C., Tkachev S.S. Kalman filter application for the angular motion estimation by video processing. Keldysh IAM Preprints. (2021). No. 27. 27 p.
https://doi.org/10.20948/prepr-2021-27-e
29. J.-F. Shi, S. Ulrich, S. Ruel, Spacecraft pose estimation using principal component analysis and a monocular camera. AIAA Guidance, Navigation, and Control Conference. (2017). Grapevine, Texas, 9 - 13 January 2017, p. 1034.
30. Gubarev V., Salnikov N., Melnychuk S., Shevchenko V., Maksymyuk L. Special Cases in Determining the Spacecraft Position and Attitude Using Computer Vision System. Chapter 10 in the book "Advanced Control Systems: Theory and Applications". (2021). River Publishers Series in Automation, Control and Robotics. pp. 289-316.
https://doi.org/10.1201/9781003337010-12
31. Volosov V., Salnikov N., Melnychuk S., Shevchenko V. Control Synthesis of Rotational and Spatial Spacecraft Motion at Approaching Stage of Docking. Chapter 12 in the book "Advanced Control Systems: Theory and Applications". (2021). River Publishers Series in Automation, Control and Robotics. pp.331-364.
https://doi.org/10.1201/9781003337010-14
32. Koshkin N., Melikyants S., Korobeinikova E., Shakun L., Strakhova S., Kashuba V., Romanyuk Ya., Terpan S. Simulation of the orbiting spacecraft to analysis and understand their rotation based on photometry. Odessa Astronomical Publications. (2019). vol. 32, pp.158-161.
https://doi.org/10.18524/1810-4215.2019.32.183899
33. Koshkin N., Shakun L., Kozhukhov O., Kozhukhov D., Mamarev V., Prysiaznyi V., Ozeryan A., Bilinsky A., Kudak V., Neubauer I. Simultaneous multi-site photometry of LEO satellites for rotation characterization. Proceedings of the 8th European Conference on Space Debris. (2021). Darmstadt, Germany, 20-23 April 2021.
http://conference.sdo.esoc.esa.int.
34. Sarychev V.A., Paglione P., Guerman A.D. Stability of Equilibria for a Satellite Subject to Gravitational and Constant Torques. Journal of Guidance Control and Dynamics. (2008). vol. 31. № 2. P. 386-394.
https://doi.org/10.2514/1.28753
35. Markley, F.L., Crassidis, J.L. Fundamentals of Spacecraft Attitude Determination and Control. New York: Springer Science+Business Media, 2014.
https://doi.org/10.1007/978-1-4939-0802-8
36. Aghili F. A Prediction and Motion-Planning Scheme for Visually Guided Robotic Capturing of Free-Floating Tumbling Objects With Uncertain Dynamics. IEEE Transactions on Robotics. (2012). vol. 28, no. 3, pp. 634-649.
https://doi.org/10.1109/TRO.2011.2179581
37. Aghili F., Parsa K. Motion and Parameter Estimation of Space Objects Using Laser-Vision Data. Journal of Guidance, Control, and Dynamics. (2009). Vol. 32, No. 2, pp. 537-549,
https://doi.org/10.2514/1.37129
38. Stainfeld D., Rock S.M. Rigid Body Inertia Estimation with Applications to the Capture of a Tumbling Satellite. Proceedings of 19th AAS/AIAA Spaceflight Mechanics Meeting. (2009). Savannah, GA, pp. 343-356.
39. Kalman Filtering and Neural Networks (edited by S. Haykin). New York, Toronto: John Wiley&Sons, Inc., 2001. 284 p.
40. Arulampalam M. S., Maskell S., Gordon N., Clapp T. A tutorial on particle filters for online nonlinear/non-Gaussian bayesian tracking. IEEE Transfctions on Signal processing. (2002). vol. 50. No.2. pp. 174-188.
https://doi.org/10.1109/78.978374
41. Cheng Y., Crassidis J.L. Particle Filtering for Attitude Estimation Using a Minimal Local-Error Representation. Journal of Guidance, Control, and Dynamics. (2010). Vol. 33, No. 4, pp. 1305-1310.
https://doi.org/10.2514/1.47236
42. Schweppe F.C. Uncertain dynamic systems. Englewood Cliffs, N.J., Prentice-Hall, 1973. - 563 p.
43. Kuntzevich V.M., Lychak M. Guaranteed Estimates, Adaptation and Robustness in Control Systems. Berlin, New York: Springer-Verlag, 1981.
44. Chernousko, F. L. State estimation for dynamic systems. Boca Raton: CRC Press, 1994.
45. Kurzhanski A.B., Valyi I. Ellipsoidal Calculus for Estimation and Control. Boston: Birkhauser, 1997.
https://doi.org/10.1007/978-1-4612-0277-6
46. Chabane S.B., Maniu C. S., Alamo T., Camacho E.F., Dumur D. A New Approach for Guaranteed Ellipsoidal State Estimation. Preprints of the 19th World Congress. The International Federation of Automatic Control. (2014). Cape Town, South Africa. August 24-29. pp. 6533-6538.
https://doi.org/10.3182/20140824-6-ZA-1003.01629
47. Blanchini F., Miani S. Set-Theoretic Methods in Control. Switzerland: Springer International Publishing, 2015.
https://doi.org/10.1007/978-3-319-17933-9
48. Poznyak A., Polyakov A., Azhmyakov V. Attractive Ellipsoids in Robust Control. Switzerland: Springer International Publishing, 2014.
https://doi.org/10.1007/978-3-319-09210-2
49. Volosov V.V., Tyutyunnik L.I. Development and Analysis of Robust Algorithms for Guaranteed Ellipsoidal Estimation of the State of Multidimensional Linear Discrete Dynamic Systems. Part 1. Journal of Automation and Information Sciences. (2000). Vol.32, No. 3, pp. 37-46.
https://doi.org/10.1615/JAutomatInfScien.v32.i3.50
50. Volosov V.V., Tyutyunnik L.I. Development and Analysis of Robust Algorithms for Guaranteed Ellipsoidal Estimation of the State of Multidimensional Linear Discrete Dynamic Systems. Part 2. Journal of Automation and Information Sciences. (2000). vol.32, No. 11, pp. 13-23.
https://doi.org/10.1615/JAutomatInfScien.v32.i11.20
51. Salnikov N.N. On One Modification of Linear Regression Estimation Algorithm Using Ellipsoids. Journal of Automation and Information Sciences. (2012). Vol. 44, No. 3, pp. 15-32.
https://doi.org/10.1615/JAutomatInfScien.v44.i3.20
52. Salnikov N.N. Estimation of State and Parameters of Dynamic System with the Use of Ellipsoids at the Lack of a Priori Information on Estimated Quantities. Journal of Automation and Information Sciences. (2014). Vol. 46, No. 4, pp.60-75.
https://doi.org/10.1615/JAutomatInfScien.v46.i4.50
53. Leffens E.J., Markley F.L., Shuster M.D. Kalman Filtering for Spacecraft Attitude Estimation. Journal of Guidance. (1982). vol. 5, No. 5, pp. 417-429.
https://doi.org/10.2514/3.56190
54. Markley F.L. Multiplicative Versus Additive Filtering for Spacecraft Attitude Determination, 2003. https://ntrs.nasa.gov/citations/20040037784.
55. Crassidis J.L., Markley F.L., Cheng Y. Survey of Nonlinear Attitude Estimation Methods. Journal of Guidance, Control and Dynamics. (2007). vol.30, No. 1, pp.12-28
https://doi.org/10.2514/1.22452
56. Amel'kin N.I. Dinamika tverdogo tela[Rigid body dynamics]. Moscow: MIPT, 2012, 80 p.
https://studylib.ru/doc/1678659/n.i.-amel._kin-dinamika-tverdogo-tela. [in Russian].
57. Pontrjagin L.S. Obyknovennye differencial'nye uravnenija. [Ordinary differential equations]. Moscow: Nauka, 1982. 332 p. [in Russian].
58. Shor N.3., Stecenko S.I. Quadratic extremal problems and non-differential optimization. Kyiv: Naukova dumka, 1989. 208 p. [in Russian].
https://doi.org/10.1016/j.actaastro.2021.03.029
2. Savchuk A.P., Fokov A.A. Determination of non-cooperative object parameters in orbital service tasks. Technical Mechanics, № 4, pp. 30-45. (2018).[in Russian].
https://doi.org/10.15407/itm2018.04.030
3. Opromolla R. et. al. A review of cooperative and uncooperative spacecraft pose determination techniques for close proximity operations. Progress in Aerospace Sciences, vol. 93, pp.53-72. (2017).
https://doi.org/10.1016/j.paerosci.2017.07.001
4. Balahoncev V.G., Ivanov V.A., Shabanov V.I. Sblizhenie v kosmose[Approaching in space]. Moscow: Voenizdat, 1973, 240 p. [in Russian].
5. Bragazin A.F. Control of spacecraft approaching (navigation, guidance, motion correction. Koroljov: RKK "Jenergija", 2018. 470 p. [in Russian].
6. Farrell J.A. Aided navigation. GPS with High Rate Sensors. New York: The McGraw-Hill Companies, 2008. 553 p.
7. Fehse W. Automated Rendezvous and Docking of Spacecraft. Cambridge: Cambridge University Press, 2003. 517 p.
https://doi.org/10.1017/CBO9780511543388
8. Aghili F. Automated Rendezvous & Docking without Impact Using a Reliable 3D Vision System. Guidance, Navigation, and Control Conference, 2-5 August (2010), Toronto, Ontario Canada.
https://doi.org/10.2514/6.2010-7602
9. Opromolla R., Nocerino A. Uncooperative Spacecraft Relative Navigation With LIDAR-Based Unscented Kalman Filter. IEEE Access, vol. 7, pp. 180012-180026. (2019).
https://doi.org/10.1109/ACCESS.2019.2959438
10. Nocerino A., Opromolla R., Fasano G., Grassi M. LIDAR-based multi-step approach for relative state and inertia parameters determination of an uncooperative target. Acta Astronautica, vol. 181, pp. 662-678, (2021).
https://doi.org/10.1016/j.actaastro.2021.02.019
11. Aghili F., Kuryllo M., Okouneva G., English Ch. Fault-Tolerant Position/Attitude Estimation of Free-Floating Space Objects Using a Laser Range Sensor. IEEE Sensors Journal. (2011). vol. 11, No. 1, pp. 176-185.
https://doi.org/10.1109/JSEN.2010.2056365
12. Aghili F., Su C. Robust Relative Navigation by Integration of ICP and Adaptive Kalman Filter Using Laser Scanner and IMU. IEEE/ASME Transactions on Mechatronics. (2016). vol. 21. no. 4, pp. 2015-2026.
https://doi.org/10.1109/TMECH.2016.2547905
13. Kelsey J. M., Byrne J., Cosgrove M., Seereeram S., Mehra R. K. Vision-based relative pose estimation for autonomous rendezvous and docking. 2006 IEEE Aerospace Conference. (2006), pp. 20-39.
https://doi.org/10.1109/AERO.2006.1655916.
14. Segal S., Carmi A., Gurfil P. Vision-Based Relative State Estimation of Non-Cooperative Spacecraft Under Modeling Uncertainty. Aerospace Conference. Piscataway(NJ): IEEE Publ. (2011). pp. 1-8.
https://doi.org/10.1109/AERO.2011.5747479
15. Oumer N.W., Panin G. Tracking and pose estimation of non-cooperative satellite for on-orbit servicing. Proseedings of the conference i-SAIRAS 2012. European Space Agency (ESA). i-SAIRAS. (2012), 4-6 Sep 2012, Turin, Italy.
16. D'Amico S., Benn M., Jørgensen J.L. Pose estimation of an uncooperative spacecraft from actual space imagery. International Journal of Space Science and Engineering. (2014). Vol.2. No.2, pp.171 - 189.
https://doi.org/10.1504/IJSPACESE.2014.060600
17. Yu X., Yu F., He Z. Stereo vision based relative state estimation for non-cooperative spacecraft with outliers. Proceedings of the 33rd Chinese Control Conference. (2014). pp. 763-769,
https://doi.org/10.1109/ChiCC.2014.6896723
18. Volpe R., Sabatini M., Palmerini G.B. Pose and Shape Reconstruction of a Noncooperative Spacecraft Using Camera and Range Measurements. International Journal of Aerospace Engineering. (2017). vol. 2017, Article ID 4535316, 13 p.
https://doi.org/10.1155/2017/4535316
19. Dementhon D.F., Davis L.S. Model-based object pose in 25 lines of code. International Journal of Computer Vision. (1995). vol. 15, pp.123-141.
https://doi.org/10.1007/BF01450852
20. Masutani Y., Iwatsu T., Miyazaki F. Motion estimation of unknown rigid body under no external forces and moments. Proceedings of the 1994 IEEE International Conference on Robotics and Automation (1994). vol.2, pp. 1066-1072.
https://doi.org/10.1109/ROBOT.1994.351227
21. Shijie et.al. Monocular Vision-based Two-stage Iterative Algorithm for Relative Position and Attitude Estimation of Docking Spacecraft. Chinese Journal of Aeronautics. (2010). vol. 23, issue 2, pp. 204-210.
https://doi.org/10.1016/S1000-9361(09)60206-5
22. Capuano V., Kim K., Hu J., Harvard A., Chung S. Monocular-Based Pose Determination of Uncooperative Known and Unknown Space Objects. Proceedings of the 69th International Astronautical Congress (IAC).(2018). Bremen, Germany, 1-5 October 2018.
23. Espinoza A.T., Setterfield T.P. Point-to-CAD 3D Registration Algorithm for Relative Navigation Using Depth-Based Maps.(2019). 2019 IEEE Aerospace Conference, pp. 1-7,
https://doi.org/10.1109/AERO.2019.8742148
24. Liang H., Wang J., Wang Y., Huo W. Monocular-vision-based spacecraft relative state estimation under dual number algebra. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering. (2020). vol. 234, pp. 221 - 235.
https://doi.org/10.1177/0954410019864754
25. Ivanov D. S., Karpenko S. O., Ovchinnikov M. Ju., Sakovich M. A. Relative motion determination of satellites at their separation on results of videoimages processing, Preprinty IPM im. M. V. Keldysha[Preprints of Keldysh IAM]. (2012), No. 057, 24 p.
26. Salnikov N.N., Melnychuk S.V., Gubarev V.F. Ellipsoidal Pose Estimation of an Uncooperative Spacecraft from Video Image Data. in book "Control Systems: Theory and Applications". River Publishers Series in Automation, Control and Robotics, 2018. pp. 169-195.
27. Grishin V.A., Zhukov B.S. Peculiarities of image recognition at its application to relative navigation tasks at spacecraft docking. Modern problems of Earth Remote Control. (2020). Vol. 17, No 7. pp. 58-66.
https://doi.org/10.21046/2070-7401-2020-17-7-58-66
28. Molina Saqui J.C., Tkachev S.S. Kalman filter application for the angular motion estimation by video processing. Keldysh IAM Preprints. (2021). No. 27. 27 p.
https://doi.org/10.20948/prepr-2021-27-e
29. J.-F. Shi, S. Ulrich, S. Ruel, Spacecraft pose estimation using principal component analysis and a monocular camera. AIAA Guidance, Navigation, and Control Conference. (2017). Grapevine, Texas, 9 - 13 January 2017, p. 1034.
30. Gubarev V., Salnikov N., Melnychuk S., Shevchenko V., Maksymyuk L. Special Cases in Determining the Spacecraft Position and Attitude Using Computer Vision System. Chapter 10 in the book "Advanced Control Systems: Theory and Applications". (2021). River Publishers Series in Automation, Control and Robotics. pp. 289-316.
https://doi.org/10.1201/9781003337010-12
31. Volosov V., Salnikov N., Melnychuk S., Shevchenko V. Control Synthesis of Rotational and Spatial Spacecraft Motion at Approaching Stage of Docking. Chapter 12 in the book "Advanced Control Systems: Theory and Applications". (2021). River Publishers Series in Automation, Control and Robotics. pp.331-364.
https://doi.org/10.1201/9781003337010-14
32. Koshkin N., Melikyants S., Korobeinikova E., Shakun L., Strakhova S., Kashuba V., Romanyuk Ya., Terpan S. Simulation of the orbiting spacecraft to analysis and understand their rotation based on photometry. Odessa Astronomical Publications. (2019). vol. 32, pp.158-161.
https://doi.org/10.18524/1810-4215.2019.32.183899
33. Koshkin N., Shakun L., Kozhukhov O., Kozhukhov D., Mamarev V., Prysiaznyi V., Ozeryan A., Bilinsky A., Kudak V., Neubauer I. Simultaneous multi-site photometry of LEO satellites for rotation characterization. Proceedings of the 8th European Conference on Space Debris. (2021). Darmstadt, Germany, 20-23 April 2021.
http://conference.sdo.esoc.esa.int.
34. Sarychev V.A., Paglione P., Guerman A.D. Stability of Equilibria for a Satellite Subject to Gravitational and Constant Torques. Journal of Guidance Control and Dynamics. (2008). vol. 31. № 2. P. 386-394.
https://doi.org/10.2514/1.28753
35. Markley, F.L., Crassidis, J.L. Fundamentals of Spacecraft Attitude Determination and Control. New York: Springer Science+Business Media, 2014.
https://doi.org/10.1007/978-1-4939-0802-8
36. Aghili F. A Prediction and Motion-Planning Scheme for Visually Guided Robotic Capturing of Free-Floating Tumbling Objects With Uncertain Dynamics. IEEE Transactions on Robotics. (2012). vol. 28, no. 3, pp. 634-649.
https://doi.org/10.1109/TRO.2011.2179581
37. Aghili F., Parsa K. Motion and Parameter Estimation of Space Objects Using Laser-Vision Data. Journal of Guidance, Control, and Dynamics. (2009). Vol. 32, No. 2, pp. 537-549,
https://doi.org/10.2514/1.37129
38. Stainfeld D., Rock S.M. Rigid Body Inertia Estimation with Applications to the Capture of a Tumbling Satellite. Proceedings of 19th AAS/AIAA Spaceflight Mechanics Meeting. (2009). Savannah, GA, pp. 343-356.
39. Kalman Filtering and Neural Networks (edited by S. Haykin). New York, Toronto: John Wiley&Sons, Inc., 2001. 284 p.
40. Arulampalam M. S., Maskell S., Gordon N., Clapp T. A tutorial on particle filters for online nonlinear/non-Gaussian bayesian tracking. IEEE Transfctions on Signal processing. (2002). vol. 50. No.2. pp. 174-188.
https://doi.org/10.1109/78.978374
41. Cheng Y., Crassidis J.L. Particle Filtering for Attitude Estimation Using a Minimal Local-Error Representation. Journal of Guidance, Control, and Dynamics. (2010). Vol. 33, No. 4, pp. 1305-1310.
https://doi.org/10.2514/1.47236
42. Schweppe F.C. Uncertain dynamic systems. Englewood Cliffs, N.J., Prentice-Hall, 1973. - 563 p.
43. Kuntzevich V.M., Lychak M. Guaranteed Estimates, Adaptation and Robustness in Control Systems. Berlin, New York: Springer-Verlag, 1981.
44. Chernousko, F. L. State estimation for dynamic systems. Boca Raton: CRC Press, 1994.
45. Kurzhanski A.B., Valyi I. Ellipsoidal Calculus for Estimation and Control. Boston: Birkhauser, 1997.
https://doi.org/10.1007/978-1-4612-0277-6
46. Chabane S.B., Maniu C. S., Alamo T., Camacho E.F., Dumur D. A New Approach for Guaranteed Ellipsoidal State Estimation. Preprints of the 19th World Congress. The International Federation of Automatic Control. (2014). Cape Town, South Africa. August 24-29. pp. 6533-6538.
https://doi.org/10.3182/20140824-6-ZA-1003.01629
47. Blanchini F., Miani S. Set-Theoretic Methods in Control. Switzerland: Springer International Publishing, 2015.
https://doi.org/10.1007/978-3-319-17933-9
48. Poznyak A., Polyakov A., Azhmyakov V. Attractive Ellipsoids in Robust Control. Switzerland: Springer International Publishing, 2014.
https://doi.org/10.1007/978-3-319-09210-2
49. Volosov V.V., Tyutyunnik L.I. Development and Analysis of Robust Algorithms for Guaranteed Ellipsoidal Estimation of the State of Multidimensional Linear Discrete Dynamic Systems. Part 1. Journal of Automation and Information Sciences. (2000). Vol.32, No. 3, pp. 37-46.
https://doi.org/10.1615/JAutomatInfScien.v32.i3.50
50. Volosov V.V., Tyutyunnik L.I. Development and Analysis of Robust Algorithms for Guaranteed Ellipsoidal Estimation of the State of Multidimensional Linear Discrete Dynamic Systems. Part 2. Journal of Automation and Information Sciences. (2000). vol.32, No. 11, pp. 13-23.
https://doi.org/10.1615/JAutomatInfScien.v32.i11.20
51. Salnikov N.N. On One Modification of Linear Regression Estimation Algorithm Using Ellipsoids. Journal of Automation and Information Sciences. (2012). Vol. 44, No. 3, pp. 15-32.
https://doi.org/10.1615/JAutomatInfScien.v44.i3.20
52. Salnikov N.N. Estimation of State and Parameters of Dynamic System with the Use of Ellipsoids at the Lack of a Priori Information on Estimated Quantities. Journal of Automation and Information Sciences. (2014). Vol. 46, No. 4, pp.60-75.
https://doi.org/10.1615/JAutomatInfScien.v46.i4.50
53. Leffens E.J., Markley F.L., Shuster M.D. Kalman Filtering for Spacecraft Attitude Estimation. Journal of Guidance. (1982). vol. 5, No. 5, pp. 417-429.
https://doi.org/10.2514/3.56190
54. Markley F.L. Multiplicative Versus Additive Filtering for Spacecraft Attitude Determination, 2003. https://ntrs.nasa.gov/citations/20040037784.
55. Crassidis J.L., Markley F.L., Cheng Y. Survey of Nonlinear Attitude Estimation Methods. Journal of Guidance, Control and Dynamics. (2007). vol.30, No. 1, pp.12-28
https://doi.org/10.2514/1.22452
56. Amel'kin N.I. Dinamika tverdogo tela[Rigid body dynamics]. Moscow: MIPT, 2012, 80 p.
https://studylib.ru/doc/1678659/n.i.-amel._kin-dinamika-tverdogo-tela. [in Russian].
57. Pontrjagin L.S. Obyknovennye differencial'nye uravnenija. [Ordinary differential equations]. Moscow: Nauka, 1982. 332 p. [in Russian].
58. Shor N.3., Stecenko S.I. Quadratic extremal problems and non-differential optimization. Kyiv: Naukova dumka, 1989. 208 p. [in Russian].