ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

Algorithms for Aerodynamic Control of Relative Motion Two Satellites in a Near Circular Orbit


Boris Rostislavich Andrievsky

Doctor of Technical Sciences, Leading Researcher, Laboratory for Control
of Complex Systems, Institute for Problems in Mechanical Engineering, RAS (IPME RAS),
Principal Researcher,
Department of Applied Cybernetics, Faculty of Mathematics and Mechanics,
St. Petersburg State University (St. Petersburg State University),
Leading Researcher of the Baltic State Technical University "VOENMEKH" D.F. Ustinova"

Nikolay Vladimirovich Kuznetsov

Doctor of Physical and Mathematical Sciences, Head of the Department
of Applied Cybernetics, St. Petersburg State University,
Head of the Laboratory of Information and Control Systems of the Institute
for Problems of Mechanical Engineering of the Russian Academy of Sciences (IPMash RAS)

Alexander Mikhailovich Popov

Ph.D. Associate Professor of the Department of Control Systems and Computer
Technologies of the Baltic State Technical University "VOENMEKH" n.a. D.F. Ustinov


In the recent years, there has been a growing interest in using aerodynamics drag force to eliminate relative drift between satellites moving in a group. This article is devoted to the development and study of some algorithms for decentalize stabilizing the relative position of two satellites moving in a near-circular Earth orbit. A brief review of approaches and results on methods of controlling a satellite group by changing the aerodynamic drag force is given. Control algorithms based on the modal (pole placement) approach, the passification-based and variable-structure control methods, linear and time-optimal partial stabilization are developed and studied. The robustness of control systems with respect to plant model parameters is examined. The possibility of the appearance of sustained oscillations as a consequence of the control action (the aerodynamic drag) boundedness is demonstrated.



  1. Kumar B., Ng A., Yoshihara K., De Ruiter A. Differential drag as a means of spacecraft formation control. IEEE Transactions on Aerospace and Electronic Systems. 2011. Vol. 47, no. 2. P. 1125-1135
  2. Pé rez D., Bevilacqua R. Lyapunov-based Spacecraft Rendezvous Maneuvers using Differential Drag. Proc. AIAA Guidance, Navigation, and Control Conference, Portland, Oregon. 2011. 08 - 11 Aug. P. AIAA 2011-6630
  3. Varma S., Kumar K. Multiple satellite formation flying using differential aerodynamic drag. J. Spacecraft and Rockets. 2012. Vol. 49, no. 2. P. 325-336
  4. Horsley M., Nikolaev S., Pertica A. Small satellite rendezvous using differential lift and drag. J. Guidance Control Dyn. 2013. Vol. 36, no. 2. P. 445-453
  5. Kumar K., Misra A., Varma S. et al. Maintenance of satellite formations using environmental forces. Acta Astronautica. 2014. Vol. 102. P. 341-354
  6. Dellelce L., Kerschen G. Optimal propellantless rendez-vous using differential drag. Acta Astronaut. 2015. Vol. 109. P. 112-123
  7. Ivanov D., Monakhova U., Ovchinnikov M. Nanosatellites swarm deployment using decentralized differential drag-based control with communicational constraints. Acta Astronautica. 2019. Vol. 159. P. 646-657
  8. Ivanov D., Biktimirov S., Chernov K. et al. Writing with Sunlight: Cubesat formation control using aerodynamic forces. Proc. Int. Astronautical Congress, IAC. Vol. 2019-October. 2019
  9. Tang A., Wu X. LEO satellite formation flying via differential atmospheric drag. Int. Journal of Space Science and Engineering. 2019. Vol. 5, no. 4. P. 289-320
  10. Shouman M., Bando M., Hokamoto S. Output regulation control for satellite formation flying using differential drag. J. Guidance Control Dyn. 2019. Vol. 42, no. 10. P. 2220-2232
  11. Smith B., Capon C., Brown M. Ionospheric drag for satellite formation control. J. Guidance Control Dyn. 2019. Vol. 42, no. 12. P. 2590-2599
  12. Traub C., Herdrich G., Fasoulas S. Influence of energy accommodation on a robust spacecraft rendezvous maneuver using differential aerodynamic forces. CEAS Space Journal. 2020. Vol. 12, no. 1. P. 43-63
  13. Monakhova U., Ivanov D., Roldugin D. Magnetorquers attitude control for differential aerodynamic force application to nanosatellite formation flying construction and maintenance. Advances in the Astronautical Sciences. 2020. Vol. 170. P. 385-397
  14. Traub C., Romano F., Binder T. et al. On the exploitation of differential aerodynamic lift and drag as a means to control satellite formation flight. CEAS Space Journal. 2020. Vol. 12, no. 1. P. 15-32
  15. Leonard C. Formationkeeping of Spacecraft via Differential Drag. Master’s thesis, Massachusetts Inst. Technol., Cambridge, MA, USA, 1986
  16. Hill G. W. Researches in the Lunar Theory. American J. Mathematics. 1878. Vol. 1, no. 1. P. 5-26
  17. Clohessy W., Wiltshire R. Terminal guidance system for satellite rendezvous. J. Aerospace Sciences. 1960. P. 653-658
  18. Sedwick R., Miller D., Kong E. Mitigation of Differential Perturbations. J. Astronautical Sciences. 1999. Vol. 47, no. 3-4. P. 309-331
  19. Bolotin S. V., Treshchev D. V. [Hill's formula] Formula Khilla. Uspekhi matem. nauk, 2010; 65(2): 3-70.. (In Russ. )
  20. Schweighart S., Sedwick R. High-Fidelity Linearized J2 Model for Satellite Formation Flight. J. Guid. Control. Dyn. 2002. Vol. 25, no. 6. P. 1073-1080
  21. Schlanbusch R., Kristiansen R., Nicklasson P. Spacecraft formation reconfiguration with collision avoidance. Automatica. 2011. Vol. 47, no. 7. P. 1443-1449
  22. Monakhova U. V., Ivanov D. S. [Formation of a swarm of nanosatellites using decentralized aerodynamic control, taking into account communication constraints] Formirovaniye roya nanosputnikov s pomoshch'yu detsentralizovannogo aerodinamicheskogo upravleniya s uchetom kommunikatsionnykh ogranicheniy. Preprinty IPM im. M. V. Keldysha. 2018; (151), 1:32. (In Russ. ) Available at:
  23. Kwakernaak H., Sivan R. Linear Optimal Control Systems. New York, Wiley-Interscience, 1972
  24. Andriyevskiy B. R., Fradkov A. L. Izbrannyye glavy teorii avtomaticheskogo upravleniya s primerami na yazyke MATLAB [Selected chapters of the theory of automatic control with examples in MATLAB]. Sankt-Peterburg, Nauka Publ, 1999. (In Russ. )
  25. Pavlov A. A. Sintez releynykh sistem, optimal'nykh po bystrodeystviyu (metod fazovogo prostranstva) [Synthesis of relay systems, optimal in transient time (phase space method)]. Moscow, Nauka Publ, 1966. (In Russ. )
  26. Omar S., Bevilacqua R. Guidance, Navigation, and Control Solutions for Spacecraft Re-Entry Point Targeting Using Aerodynamic Drag. Acta Astronautica. 2019. Feb. Vol. 155. P. 389-405
  27. Schweighart S., Sedwick R. Cross-Track Motion of Satellite Formations in the Presence of J2 Disturbances. J. Guid. Control. Dyn. 2005. Jul. -Aug. Vol. 28, no. 4. P. 824-826
  28. Wang D., Wu B., Poh E. K. Satellite Formation Flying Relative Dynamics, Formation Design, Fuel Optimal Maneuvers and Formation Maintenance. Ed. by S. Tzafestas. Springer, 2017. Vol. 87 of Intelligent Systems, Control and Automation: Science and Engineering
  29. Aksenov Ye. P. Teoriya dvizheniya iskusstvennykh sputnikov Zemli [The theory of motion of artificial earth satellites]. Moscow, Nauka Publ, 1977. (In Russ. )
  30. Bordovitsyna T. V., Avdyushev V. A. Teoriya dvizheniya iskusstvennykh sputnikov Zemli. Analiticheskiye i chislennyye metody: Ucheb. posobiye [The theory of motion of artificial earth satellites. Analytical and numerical methods: Textbook]. Tomsk Univ, 2007. (In Russ. )
  31. Bevilacqua R., Hall S., J., Romano M. Multiple Spacecraft Assembly Maneuvers by Differential Drag and Low Thrust Engines. Celestial Mechanics and Dynamical Astronomy. 2010. Vol. 106. P. 69-88
  32. Bevilacqua R., Romano M. Rendezvous Maneuvers of Multiple Spacecraft by Differential Drag under J2 Perturbation. J. Guidance, Control and Dynamics. 2008. Vol. 31, no. 6. P. 1595-1607
  33. Kim D. -Y., Woo B., Park S. -Y., Choi K. -H. Hybrid optimization for multiple-impulse reconfiguration trajectories of satellite formation flying. Advances in Space Research. 2009. Vol. 44, no. 11. P. 1257-1269
  34. Vaddi S., Alfriend K., Vadali S., Sengupta P. Formation establishment and reconfiguration using impulsive control. J. Guidance Control Dyn. 2005. Vol. 28, no. 2. P. 262-268
  35. Vaddi S. Modeling and Control of Satellite Formations: Ph. D. thesis. Department of Aerospace Engineering, Texas A& M University. Texas A& M University, 2003
  36. Ovchinnikov M. Yu., Tkachev S. S. [Determination of the parameters of the relative motion of two satellites using trajectory measurements. Space exploration]. Opredeleniye parametrov otnositel'nogo dvizheniya dvukh sputnikov s pomoshch'yu trayektornykh izmereniy. Kosmicheskiye issledovaniya. 2008; (6): 553-558. (In. Russ. )
  37. Ivanov D., Karpenko S., Ovchinnikov M., Sakovich M. Satellite relative motion determination during separation using image processing. Int. J. Sp. Sci. Eng. 2014. Vol. 2, no. 4. P. 365-379
  38. Fehse W. Automated rendezvous and docking of spacecraft. New York: Cambridge University Press, 2003
  39. Persson S., Jacobsson B., Gill E. PRISMA - Demonstration mission for advanced rendezvous and formation flying technologies and sensors. Int. Astronautical Federation - 56th Int. Astronautical Congress 2005. Vol. 4. 2005. P. 2403-2412
  40. Persson S., Bodin P., Gill E. et al. PRISMA - An autonomous formation flying mission. Europ. Space Agency, (Special Publication) ESA SP. 2006. Vol. 625 SP
  41. Persson S., Veldman S., Bodin P. PRISMA - A formation flying project in implementation phase. Acta Astronautica. 2009. Vol. 65, no. 9-10. P. 1360-1374
  42. Nemirovsky A. S., Danilovich O. S., Marimont Yu. I., i dr. Radioreleynyye i sputnikovyye sistemy peredachi: Uchebnik dlya vuzov. Pod red. A. S. Nemirovskogo [Radio-relay and satellite transmission systems: Textbook for universities. A. S. Nemirovsky, Ed. ] Moscow, Radio i svyaz Publ, 1986. (In Russ. )
  43. Genike A. A., Pobedinskiy G. G. Global'nyye sputnikovyye sistemy opredeleniya mestopolozheniya i ikh primeneniye v geodezii. Izd. 2-ye, pererab. i dop. [Global satellite positioning systems and their application in geodesy. 2nd Ed. ] Moscow, Kartgeotsentr Publ, 2004. (In Russ. )
  44. Renga A., Grassi M., Tancredi U. Relative navigation in LEO by carrierphase differential GPS with intersatellite ranging augmentation. Int. J. Aerosp. Eng. 2013. Vol. 2013. 11 p.
  45. Bragin V., Vagaitsev V., Kuznetsov N., Leonov G. Algorithms for finding hidden oscillations in nonlinear systems. The Aizerman and Kalman conjectures and Chua’s circuits, J. Computer and Systems Sciences Int. 2011; 50(4): 511-543
  46. Dudkowski D., Jafari S., Kapitaniak T. et al. Hidden attractors in dynamical systems. Physics Reports. 2016. Vol. 637. P. 1-50
  47. Andriyevskiy B. R., Kuznetsov N. V., Kuznetsova O. A., Leonov G. A., Mokayev T. N. [Localization of hidden oscillations in flight control systems] Lokalizatsiya skrytykh kolebaniy v sistemakh upravleniya poletom. Trudy SPIIRAN. 2016; 6(49): 5-31. (In Russ. )
  48. Andrievsky B., Kuznetsov N., Leonov G. Methods for suppressing nonlinear oscillations in astatic auto-piloted aircraft control systems. J. Computer and Systems Sciences International. 2017; 56(3): 455-470
  49. Kuznetsov N. Theory of hidden oscillations and stability of control systems. J. Computer and Systems Sciences International. 2020. Vol. 59, no. 5. P. 647-668
  50. Fradkov A. L. [Quadratic Lyapunov functions in the problem of adaptive stabilization of a linear dynamic plant] Kvadratichnyye funktsii Lyapunova v zadache adaptivnoy stabilizatsii lineynogo dinamicheskogo ob" yekta. Sibirskiy matematicheskiy zhurnal. 1976; 17(2): 436-445. (In Russ. )
  51. Fomin V. N., Fradkov A. L., Yakubovich V. A. Adaptivnoye upravleniye dinamicheskimi ob" yektami [Adaptive control of dynamical plants]. Moscow, Nauka Publ, 1981. (In Russ. )
  52. Fradkov A. L. [Adaptive control in complex systems: Search-free methods] Adaptivnoye upravleniye v slozhnykh sistemakh: Bespoiskovyye metody. Moscow, Nauka Publ, 1990. (In Russ. )
  53. Fradkov A. L., Miroshnik I. V., Nikiforov V. O. Nonlinear and Adaptive Control of Complex Systems. 1999. Springer Science & Business Media
  54. Fradkov A. L. Passification of Non-square Linear Systems and Feedback Yakubovich-Kalman-Popov Lemma. Europ. J. of Control. 2003. no. 6. P. 573-582
  55. Andrievskii B., Selivanov A. New Results on the Application of the Passification Method. A Survey. Autom. Remote Control. 2018; 79(6): 957-995
  56. Fradkov A. L. Synthesis of an adaptive system for linear plant stabilization. Autom. Remote Control, 1974. 35(12): 1960-1966
  57. Andrievsky B. R., Churilov A. N., Fradkov A. L. Feedback Kalman-Yakubovich lemma and its applications to adaptive control. Proc. 35th Conference on Decision and Control (CDC’96). Kobe, Japan: IEEE, 1996. December. P. 4537-4542
  58. Gusev S. V., Likhtarnikov A. L., Kalman-Popov-Yakubovich lemma and the S-procedure: A historical essay. Autom. Remote Control, 2006. 67(11): 1768-1810
  59. Utkin V. I. Skol'zyashchiye rezhimy i ikh primeneniya v sistemakh s peremennoy strukturoy [Sliding modes and their applications in systems with variable structure]. Moscow, Nauka Publ, 1974. (In Russ. )
  60. Massey T., Shtessel Y. Continuous traditional and high-order sliding modes for satellite formation control. J. Guidance Control Dyn. 2005. Vol. 28, no. 4. P. 826-831
  61. Grishin, A. A., Lenskii, A. V., Okhotsimsky, D. E., Panin, D. A., Formal'skii, A. M. A control synthesis for an unstable object. An inverted pendulum. 2002, J. Computer and Systems Sciences International, 2002, 41(5): 685-694

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