# Modeling and Control of Satellite Formations: A Survey

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## Abstract

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## 1. Introduction

## 2. Multiple-Input–Multiple-Output Formation Architecture

## 3. Circle Formation Architecture

## 4. Leader–Follower Formation Architecture

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ATI | Along-Track Interferometry |

AMDS | Autonomous Microsatellite Docking System |

ATO | Along Track Orbit |

CCN | Cooperative Circumnavigation |

CMKF | Converted Measurement Kalman Filter |

DGPS | Differential GPS |

EKF | Extended Kalman Filter |

EMFF | Electromagnetic Formation Flight |

ESA | European Space Agency |

FFG | Formation Flying Guidance |

GCO | General Circular Orbit |

GNSS | Global Navigation Satellite System |

GPS | Global Positioning System |

HCW | Hill–Clohessy–Wiltshire |

HEO | High Earth Orbit |

LEO | Low Earth Orbits |

LF | Leader – Follower |

LMI | Linear Matrix Inequality |

LISA | Laser Interferometer Space Antenna |

LVLH | Local-Vertical – Local-Horizontal |

LPR | Likins-Pringle Relation |

MAC | Medium Access Control |

MEMS | Microelectromechanical System |

MPC | Model Predictive Control |

MRP | Modified Rodrigues Parameter |

MIMO | Multiple-Input – Multiple-Output |

MVSS | Multiple Virtual Sub-Structures |

NRL | Naval Research Laboratory |

NTG | Nonlinear Trajectory Generation |

PCO | Projected Circular Orbit |

PD | Proportional-Differential |

PDC | Proportional-Differential-Consensus |

SAR | Synthetic Aperture Radar |

SDRE | System Dependent Riccati Equation Regulator |

SFF | Satellite Formation Flying |

SFFC | Spacecraft Formation Flight Control |

TVSMC | Time-Varying Sliding Mode Control |

UTKM | Unbiased Transformed Kalman Measurement |

## References

- Scharf, D.; Hadaegh, F.; Ploen, S. A survey of spacecraft formation flying guidance and control. Part 1: Guidance. In Proceedings of the 2003 American Control Conference (ACC 2003), Denver, CO, USA, 4–6 June 2003; Volume 2, pp. 1733–1739. [Google Scholar] [CrossRef]
- Alfriend, K.T.; Vadali, S.R.; Gurfil, P.; How, J.P.; Breger, L.S. Spacecraft Formation Flying: Dynamics, Control and Navigation. In Spacecraft Formation Flying; Alfriend, K.T., Vadali, S.R., Gurfil, P., How, J.P., Breger, L.S., Eds.; Butterworth-Heinemann: Oxford, UK, 2010; p. iv. [Google Scholar] [CrossRef]
- D’Errico, M. Relative Trajectory Design. In Distributed Space Missions for Earth System Monitoring; D’Errico, M., Ed.; Springer: New York, NY, USA, 2013. [Google Scholar] [CrossRef]
- Schilling, K.; Loureiro, G.; Zhang, Y.; Nchter, A.; Scharnagl, J.; Motroniuk, I.; Aumann, A. TIM: An Int. nano-satellite formation for photogrammetric earth observation. In Proceedings of the Int. Astronautical Congress, IAC, Washington, DC, USA, 21–25 October 2019; Volume 2019. [Google Scholar]
- Schilling, K.; Schechner, Y.; Koren, I. CloudCT—A formation of cooperating nano-satellites for cloud characterisation by computed tomography. In Proceedings of the Int.Astronautical Congress, IAC, Washington, DC, USA, 21–25 October 2019; Volume 2019. [Google Scholar]
- Freimann, A.; Petermann, T.; Schilling, K. Interference-Free Contact Plan Design for Wireless Communication in Space-Terrestrial Networks. In Proceedings of the 2019 IEEE International Conference on Space Mission Challenges for Information Technology (SMC-IT), Pasadena, CA, USA, 30 July–1 August 2019; pp. 55–61. [Google Scholar] [CrossRef]
- Busch, S.; Bangert, P.; Dombrovski, S.; Schilling, K. UWE-3, in-orbit performance and lessons learned of a modular and flexible satellite bus for future pico-satellite formations. Acta Astronaut.
**2015**, 117, 73–89. [Google Scholar] [CrossRef] - Folta, D.; Bordi, F.; Scolese, C. Considerations on formation flying separations for earth observing satellite missions. Adv. Astronaut. Sci.
**1992**, 79, 803–822. [Google Scholar] - Bik, J.; Visser, P.; Jennrich, O. LISA satellite formation control. Adv. Space Res.
**2007**, 40, 25–34. [Google Scholar] [CrossRef] - Kim, D.Y.; Woo, B.; Park, S.Y.; Choi, K.H. Hybrid optimization for multiple-impulse reconfiguration trajectories of satellite formation flying. Adv. Space Res.
**2009**, 44, 1257–1269. [Google Scholar] [CrossRef] - Wu, S.F.; Fertin, D. Spacecraft drag-free attitude control system design with Quantitative Feedback Theory. Acta Astronaut.
**2008**, 62, 668–682. [Google Scholar] [CrossRef] - Schilling, K. Small Satellite Formations: Challenges in Navigation and its Application Potential. In Proceedings of the 28th Saint Petersburg International Conference on Integrated Navigation Systems (ICINS 2021), Saint Petersburg, Russia, 31 May–2 June 2021. [Google Scholar]
- Kramer, A.; Schilling, K. First demonstration of collision avoidance and orbit control for pico-satellites—UWE-4. Acta Astronaut.
**2021**, 185, 244–256. [Google Scholar] [CrossRef] - Mathavaraj, S.; Padhi, R. Satellite Formation Flying-High Precision Guidance Using Optimal and Adaptive Control Techniques; Springer: Singapore, 2021. [Google Scholar]
- Alfriend, K.T.; Vadali, S.R.; Gurfil, P.; How, J.P.; Breger, L. Spacecraft Formation Flying: Dynamics, Control and Navigation; Elsevier Ltd.: Oxford, UK, 2010. [Google Scholar]
- Schaub, H.; Vadali, S.; Junkins, J.; Alfriend, K. Spacecraft formation flying control using mean orbit elements. J. Astronaut. Sci.
**2001**, 48, 69–87. [Google Scholar] [CrossRef] - Vaddi, S.; Alfriend, K.; Vadali, S.; Sengupta, P. Formation establishment and reconfiguration using impulsive control. J. Guid. Control Dyn.
**2005**, 28, 262–268. [Google Scholar] [CrossRef] - Vassar, R.; Sherwoodt, R. Formation keeping for a pair of satellites in a circular orbit. J. Guid. Control Dyn.
**1985**, 8, 235–242. [Google Scholar] [CrossRef] - Ulybyshev, Y. Long-term formation keeping of satellite constellation using linear-quadratic controller. J. Guid. Control Dyn.
**1998**, 21, 109–115. [Google Scholar] [CrossRef] - Inalhan, G.; Tillerson, M.; How, J.P. Relative Dynamics and Control of Spacecraft Formations in Eccentric Orbits. J. Guid. Control Dyn.
**2002**, 25, 48–59. [Google Scholar] [CrossRef] - Hill, G.W. Researches in the Lunar Theory. Am. J. Math.
**1878**, 1, 5–26. [Google Scholar] [CrossRef] - Sedwick, R.; Miller, D.; Kong, E. Mitigation of Differential Perturbations. J. Astronaut. Sci.
**1999**, 47, 309–331. [Google Scholar] [CrossRef] - Clohessy, W.; Wiltshire, R. Terminal guidance system for satellite rendezvous. J. Aerosp. Sci.
**1960**, 27, 653–658. [Google Scholar] [CrossRef] - Lawden, D.F. Optimal Trajectories for Space Navigation; Butterworths: London, UK, 1963. [Google Scholar]
- Carter, T.E.; Humi, M. Fuel-Optimal Rendezvous near a Point in General Keplerian Orbit. J. Guid. Control Dyn.
**1987**, 10, 567–573. [Google Scholar] [CrossRef] - Carter, T.E. New Form for the Optimal Rendezvous Equations near a Keplerian Orbit. J. Guid. Control Dyn.
**1990**, 13, 183–186. [Google Scholar] [CrossRef] - Tillerson, M.; How, J. Formation flying control in eccentric orbits. In Proceedings of the AIAA Guidance, Navigation, and Control Conference and Exhibit, Montreal, QC, Canada, 6–9 August 2001. [Google Scholar]
- Schaub, H.; Alfriend, K.T. J
_{2}Invariant Reference Orbits for Spacecraft Formations. In Proceedings of the Flight Mechanics Symposium, Goddard Space Flight Center, Greenbelt, MD, USA, 18–20 May 1999. [Google Scholar] - Brouwer, D. Solution of the Problem of Artificial Satellite Theory Without Drag. Astronaut. J.
**1959**, 64, 378–397. [Google Scholar] [CrossRef] - Battin, R. An Introduction to the Mathematics and Methods of Astrodynamcis; AIAA Education Series: Reston, Virginia, 1987. [Google Scholar]
- Schaub, H.; Alfriend, K. Hybrid Cartesian and orbit element feedback law for formation flying spacecraft. J. Guid. Control Dyn.
**2002**, 25, 387–393. [Google Scholar] [CrossRef] - Williams, T.; Wang, Z.S. Uses of solar radiation pressure for satellite formation flight. Int. J. Robust Nonlinear Control
**2002**, 12, 163–183. [Google Scholar] [CrossRef] - Hussein, I.; Scheeres, D.; Hyland, D. Control of a Satellite Formation For Imaging Applications. In Proceedings of the American Control Conference (ACC’2003), Denver, CO, USA, 4–6 June 2003; Volume 1, pp. 308–313. [Google Scholar]
- Gill, E.; Runge, H. Tight formation flying for an along-track SAR interferometer. Acta Astronaut.
**2004**, 55, 473–485. [Google Scholar] [CrossRef] - Bamler, R.; Hartl, P. Synthetic aperture radar interferometry. Inverse Probl.
**1998**, 14, R1–R54. [Google Scholar] [CrossRef] - Romeiser, R.; Breit, H.; Eineder, M.; Runge, H.; Flament, P.; de Jong, K.; Vogelzang, J. Current measurements by SAR along-track interferometry from a Space Shuttle. IEEE Trans. Geosci. Remote Sens.
**2005**, 43, 2315–2324. [Google Scholar] [CrossRef] - Romeiser, R.; Johannessen, J.; Chapron, B.; Collard, F.; Kudryavtsev, V.; Runge, H.; Suchandt, S. Direct Surface Current Field Imaging from Space by Along-Track InSAR and Conventional SAR. In Oceanography from Space; Barale, V., Gower, J., Alberotanza, L., Eds.; Springer: Dordrecht, The Netherlands, 2010. [Google Scholar] [CrossRef]
- Montenbruck, O.; Gill, E. Satellite Orbits—Models, Methods and Applications; Springer: Berlin/Heidelberg, Germany, 2001. [Google Scholar]
- Hussein, I.; Schaub, H. Stability and control of relative equilibria for the three-spacecraft Coulomb tether problem. Acta Astronaut.
**2009**, 65, 738–754. [Google Scholar] [CrossRef] - King, L.; Parker, G.; Deshmukh, S.; Chong, J.H. Spacecraft Formation-Flying Using Inter-Vehicle Coulomb Forces; Technical Report; NASA/NIAC; Michigan Technological University, Department of Mechanical Engineering–Engineering Mechanics: Houghton, MI, USA, 2002. [Google Scholar]
- King, L.B.; Parker, G.G.; Deshmukh, S.; Chong, J.H. Study of interspacecraft Coulomb forces and implications for formation flying. J. Propuls. Power
**2003**, 19, 497–505. [Google Scholar] [CrossRef] - Schaub, H.; Parker, G.; King, L.B. Challenges and prospects of Coulomb spacecraft formation control. J. Astronaut. Sci.
**2004**, 52, 169–193. [Google Scholar] [CrossRef] - Gombosi, T. Physics of the Space Environment; Cambridge University Press: Cambridge, UK, 1998. [Google Scholar] [CrossRef]
- Lappas, V.; Saaj, C.; Richie, D.; Peck, M.; Streetman, B.; Schaub, H. Spacecraft formation flying and reconfiguration with electrostatic forces. Adv. Astronaut. Sci.
**2007**, 127, 217–225. [Google Scholar] - Mullen, E.G.; Gussenhoven, M.S.; Hardy, D.A.; Aggson, T.A.; Ledley, B.G.; Whipple, E. SCATHA survey of high-level spacecraft charging in sunlight. J. Geophys. Res. Space Phys.
**1986**, 91, 1474–1490. [Google Scholar] [CrossRef] - Olsen, R.C.; Purvis, C.K. Observations of charging dynamics. J. Geophys. Res.
**1983**, 88, 5657–5667. [Google Scholar] [CrossRef] - Escoubet, C.P.; Fehringer, M.; Goldstein, M. The Cluster mission. Ann. Geophys.
**2001**, 19, 1197–1200. [Google Scholar] [CrossRef] - Torkar, K.; Riedler, W.; Escoubet, C.P.; Fehringer, M.; Schmidt, R.; Grard, R.J.L.; Arends, H.; Rudenauer, F.; Narheim, W.S.B.T.; Svenes, K.; et al. Active spacecraft potential control for Cluster—Implementation and first results. Ann. Geophys.
**2001**, 19, 1289–1302. [Google Scholar] [CrossRef] - Berryman, J.; Schaub, H. Static equilibrium configurations in GEO Coulomb spacecraft formations. Adv. Astronaut. Sci.
**2005**, 120, 51–68. [Google Scholar] - Berryman, J.; Schaub, H. Analytical Charge Analysis for 2- and 3-Craft Coulomb Formations. AIAA J. Guid. Control Dyn.
**2007**, 30, 1701–1710. [Google Scholar] [CrossRef] - Schaub, H.; Hussein, I.I. Stability and Reconfiguration Analysis of a Circulary Spinning 2-Craft Coulomb Tether. IEEE Trans. Aerosp. Electron. Syst.
**2010**, 46, 1675–1686. [Google Scholar] [CrossRef] - Azizi, S.; Khorasani, K. A distributed Kalman filter for actuator fault estimation of deep space formation flying satellites. In Proceedings of the 3rd Annual IEEE Systems Conference, Vancouver, BC, Canada, 23–26 March 2009; pp. 354–359. [Google Scholar] [CrossRef]
- Smith, R.; Hadaegh, F. Control topologies for deep space formation flying spacecraft. Proc. Am. Control Conf.
**2002**, 4, 2836–2841. [Google Scholar] [CrossRef] - Smith, R.S.; Hadaegh, F.Y. Closed-loop dynamics of cooperative vehicle formations with parallel estimators and communication. IEEE Trans. Automat. Contr.
**2007**, 52, 1404–1414. [Google Scholar] [CrossRef] - Smith, R.S.; Hadaegh, F.Y. A distributed parallel estimation architecture for cooperative vehicle formation control. In Proceedings of the IEEE American Control Conference, Minneapolis, MN, USA, 14–16 June 2006; pp. 4219–4224. [Google Scholar]
- Fax, J.A.; Murray, R.M. Information Flow and Cooperative Control of Vehicle Formations. IEEE Trans. Automat. Control
**2004**, 49, 1465–1476. [Google Scholar] [CrossRef][Green Version] - Proskurnikov, A.; Fradkov, A. Problems and methods of network control. Autom. Remote Control
**2016**, 77, 1711–1740. [Google Scholar] [CrossRef] - Andrievsky, B.; Fradkov, A.L.; Kudryashova, E.V. Control of Two Satellites Relative Motion over the Packet Erasure Communication Channel with Limited Transmission Rate Based on Adaptive Coder. Electronics
**2020**, 9, 2032. [Google Scholar] [CrossRef] - Prussing, J. Primer vector theory and applications. In Spacecraft Trajectory Optimization; Conway, B., Ed.; Cambridge Aerospace Series; Cambridge University Press: Cambridge, UK, 2010; pp. 16–36. [Google Scholar] [CrossRef]
- Lion, P.; Handelsman, M. Primer vector on fixed-time impulsive trajectories. AIAA J.
**1968**, 6, 127–132. [Google Scholar] [CrossRef] - Jezewski, D.J.; Rozendaal, H.L. An efficient method for calculating optimal free-space n-impulse trajectories. AIAA J.
**1968**, 6, 2160–2165. [Google Scholar] [CrossRef] - Gross, L.; Prussing, J. Optimal multiple-impulse direct ascent fixed-time rendezvous. AIAA J.
**1974**, 12, 885–889. [Google Scholar] [CrossRef] - Prussing, J.; Chiu, J. Optimal multiple-impulse time-fixed rendezvous between circular orbits. J. Guid. Control Dyn.
**1986**, 9, 17–22. [Google Scholar] [CrossRef] - Taur, D.R. Optimal, Impulsive, Time-Fixed Orbital Rendezvous and Interception with Path Constraints. Ph.D. Thesis, University of Illinois at Urbana-Champaign, Urbana-Champaign, IL, USA, 1989. [Google Scholar]
- Prussing, J.; Clifton, R. Optimal multiple-impulse satellite evasive maneuvers. J. Guid. Control Dyn.
**1994**, 17, 599–606. [Google Scholar] [CrossRef] - Taur, D.R.; Coverstone-Carroll, V.; Prussing, J.E. Optimal Impulsive Time-Fixed Orbital Rendezvous and Interception with Path Constraints. J. Guid. Control Dyn.
**1995**, 18, 54–60. [Google Scholar] [CrossRef] - Vinh, N. Integration of the primer vector in a central force field. J. Optim. Theory Appl.
**1972**, 9, 51–58. [Google Scholar] [CrossRef] - Massey, T.; Shtessel, Y. Continuous traditional and high-order sliding modes for satellite formation control. J. Guid. Control Dyn.
**2005**, 28, 826–831. [Google Scholar] [CrossRef] - Andrievsky, B.; Furtat, I. Disturbance Observers: Methods and Applications. II. Applications. Autom. Remote Control
**2020**, 81, 1775–1818. [Google Scholar] [CrossRef] - Levant, A. Universal SISO sliding-mode controllers with finite-time convergence. IEEE Trans. Automat. Control
**2001**, 49, 1447–1451. [Google Scholar] [CrossRef] - Utkin, V.; Guldner, J.; Shi, J. Sliding Mode Control in Electromechanical Systems; Taylor & Francis: Abingdon, UK, 1999. [Google Scholar]
- Brown, M.; Shtessel, Y.B. Disturbance Rejection Techniques for Finite Reaching Time Continuous Sliding Mode Control. In Proceedings of the American Control Conference (ACC 2001), Arlington, VA, USA, 25–27 June 2001; IEEE Publications: Piscataway, NJ, USA, 2001; Volume 6, pp. 4998–5003. [Google Scholar]
- Bailey, C.; McLain, T.; Beard, R. Fuel saving strategies for dual spacecraft interferometry missions. J. Astronaut. Sci.
**2001**, 49, 469–488. [Google Scholar] [CrossRef] - Udwadia, F.E.; Kalaba, R.E. A new perspective on constrained motion. Proc. R. Soc. Lond. Ser. A
**1992**, 439, 407–410. [Google Scholar] [CrossRef] - Udwadia, F.E.; Kalaba, R.E. A unified approach for the recursive determination of generalized inverses. Comput. Math. Appl.
**1999**, 371, 125–130. [Google Scholar] [CrossRef] - Udwadia, F.E. Nonideal constraints and Lagrangian dynamics. J. Aerosp. Eng.
**2000**, 13, 17–22. [Google Scholar] [CrossRef] - Udwadia, F.E.; Kalaba, R.E. What is the general form of the explicit equations of motion for constrained mechanical systems. J. Appl. Mech.
**2002**, 69, 335–339. [Google Scholar] [CrossRef] - Udwadia, F.E. Equations of motion for mechanical systems: A unified approach. Int. J. Nonlinear Mech.
**2002**, 69, 951–958. [Google Scholar] [CrossRef] - Udwadia, F.E. A new perspective on the tracking control of nonlinear structural and mechanical systems. Proc. R. Soc. Lond. Ser. A
**2003**, 459, 1783–1800. [Google Scholar] [CrossRef][Green Version] - Udwadia, F.E. Equations of motion for constrained multibody systems and their control. J. Optim. Theory Appl.
**2005**, 1273, 627–638. [Google Scholar] [CrossRef] - Udwadia, F.E. Optimal tracking control of nonlinear dynamical systems. Proc. R. Soc. Lond. Ser. A
**2008**, 464, 2341–2363. [Google Scholar] [CrossRef] - Udwadia, F.E.; Kalaba, R.E. Analytical Dynamics: A New Approach; Cambridge University Press: New York, NY, USA, 2008. [Google Scholar]
- Zhang, H.; Gurfil, P. Cooperative orbital control of multiple satellites via consensus. IEEE Trans. Aerosp. Electron. Syst.
**2018**, 54, 2171–2188. [Google Scholar] [CrossRef] - Monakhova, U.; Ivanov, D. Formation of a swarm of nanosatellites using decentralized aerodynamic control, taking into account communication constraints. Keldysh Inst. Prepr.
**2018**. [Google Scholar] [CrossRef] - Andrievsky, B.; Kuznetsov, N.; Popov, A. Algorithms for aerodynamic control of relative motion two satellites in a near circular orbit. Differ. Uravn. Protsesy Upr.
**2020**, 28–58. (In Russian) [Google Scholar] - Ivanov, D.; Monakhova, U.; Ovchinnikov, M. Nanosatellites swarm deployment using decentralized differential drag-based control with communicational constraints. Acta Astronaut.
**2019**, 159, 646–657. [Google Scholar] [CrossRef] - Zhou, B.; Li, Y. Autonomous orbit determination for two spacecraft with measurement conversion. In Proceedings of the Chinese Control Conference, CCC, Guangzhou, China, 27–30 July 2019; Volume 2019, pp. 4055–4059. [Google Scholar] [CrossRef]
- Lerro, D.; Bar-Shalom, Y. Tracking with debiased consistent converted measurments versus EKF. IEEE Trans. Aerosp. Electron. Syst.
**1993**, 29, 1015–1022. [Google Scholar] [CrossRef] - Mei, W.; Bar-Shalom, Y. Unbiased Kalman filter using converted measurements: Revisit. Proc. SPIE Signal Data Process. Small Targets
**2009**, 7445, 74450U. [Google Scholar] [CrossRef] - Shouman, M.; Bando, M.; Hokamoto, S. Output regulation control for satellite formation flying using differential drag. Adv. Astronaut. Sci.
**2019**, 168, 3433–3452. [Google Scholar] [CrossRef] - Schweighart, S.; Sedwick, R. High-Fidelity Linearized J
_{2}Model for Satellite Formation Flight. J. Guid. Control. Dyn.**2002**, 25, 1073–1080. [Google Scholar] [CrossRef] - Schweighart, S.; Sedwick, R. Cross-Track Motion of Satellite Formations in the Presence of J2 Disturbances. J. Guid. Control. Dyn.
**2005**, 28, 824–826. [Google Scholar] [CrossRef] - Leonov, A.G.; Lavrenov, A.N.; Prokhorchuk, Y.A.; Palkin, M.V. Method of group orbital motion of artificial satellites RU Patent 2011153048/11, 20 July 2016. (In Russian).
- Tragesser, S. Formation flying with tethered spacecraft. In Proceedings of the Astrodynamics Specialist Conference, Denver, CO, USA, 14–17 August 2000; pp. 1–9. [Google Scholar]
- Pringle, J.R. Bounds on the librations of a symmetrical satellite. AIAA J.
**1964**, 2, 908–912. [Google Scholar] [CrossRef] - Likins, P. Stability of a symmetrical satellite in attitudes fixed in an orbiting reference frame. J. Astronaut. Sci.
**1965**, 12, 18–24. [Google Scholar] - Natarajan, A.; Schaub, H. Linear dynamics and stability analysis of a two-craft coulomb tether formation. J. Guid. Control Dyn.
**2006**, 29, 831–838. [Google Scholar] [CrossRef] - Natarajan, A.; Schaub, H. Hybrid control of orbit normal and along-track two-craft Coulomb tethers. Aerosp. Sci. Technol.
**2009**, 13, 183–191. [Google Scholar] [CrossRef] - Natarajan, A.; Schaub, H.; Parker, G.G. Reconfiguration of a Nadir-Pointing 2-Craft Coulomb Tether. J. Br. Interplanet. Soc.
**2007**, 60, 209–218. [Google Scholar] - Hussein, I.I.; Schaub, H. Invariant Shape Solutions of the Spinning Three Craft Coulomb Tether Problem. Celest. Mech. Dyn. Astron.
**2006**, 96, 137–157. [Google Scholar] [CrossRef] - Ferguson, P.; How, J. Decentralized estimation algorithms for formation flying spacecraft. In Proceedings of the AIAA Guidance, Navigation Control Conference, Austin, TX, USA, 11–14 August 2003. AIAA paper number: 2003–5442. [Google Scholar]
- Chang, I.; Park, S.Y.; Choi, K.H. Nonlinear attitude control of a tether-connected multi-satellite in three-dimensional space. IEEE Trans. Aerosp. Electron. Syst.
**2010**, 46, 1950–1968. [Google Scholar] [CrossRef] - Bilal, M.; Vijayan, R.; Schilling, K. SDRE control with nonlinear J2 perturbations for nanosatellite formation flying. IFAC-PapersOnLine
**2019**, 52, 448–453. [Google Scholar] [CrossRef] - Zeng, G.; Hu, M. Finite-time control for electromagnetic satellite formations. Acta Astronaut.
**2012**, 74, 120–130. [Google Scholar] [CrossRef] - Kong, E.; Kwon, D.; Schweighart, S.; Elias, L.; Sedwick, R.; Miller, D. Electromagnetic formation flight for multisatellite arrays. J. Spacecr. Rocket.
**2004**, 41, 659–666. [Google Scholar] [CrossRef] - Inampudi, R.; Schaub, H. Optimal reconfigurations of two-craft Coulomb formation in circular orbits. J. Guid. Control Dyn.
**2012**, 35, 1805–1815. [Google Scholar] [CrossRef] - Zhou, J.; Hu, Q.; Friswell, M. Decentralized finite time attitude synchronization control of satellite formation flying. J. Guid. Control Dyn.
**2013**, 36, 185–195. [Google Scholar] [CrossRef] - Nair, R.; Behera, L.; Kumar, V.; Jamshidi, M. Multisatellite formation control for remote sensing applications using artificial potential field and adaptive fuzzy sliding mode control. IEEE Syst. J.
**2015**, 9, 508–518. [Google Scholar] [CrossRef] - AlandiHallaj, M.; Assadian, N. Multiple-horizon multiple-model predictive control of electromagnetic tethered satellite system. Acta Astronaut.
**2019**, 157, 250–262. [Google Scholar] [CrossRef] - Sparks, A. Satellite formationkeeping control in the presence of gravity perturbations. In Proceedings of the American Control Conference, Chicago, IL, USA, 28–30 June 2000; Volume 2, pp. 844–848. [Google Scholar]
- Kang, W.; Xi, N.; Sparks, A. Theory and applications of formation control in a perceptive referenced frame. In Proceedings of the IEEE Conference on Decision and Control, Sydney, NSW, Australia, 12–15 December 2000; Volume 1, pp. 352–357. [Google Scholar]
- McInnes, C. Autonomous ring formation for a planar constellation of satellites. J. Guid. Control Dyn.
**1995**, 18, 1215–1217. [Google Scholar] [CrossRef] - Milam, M.; Petit, N.; Murray, R. Constrained trajectory generation for micro-satellite formation flying. In Proceedings of the AIAA Guidance, Navigation, and Control Conference and Exhibit, Montreal, QC, Canada, 6–9 August 2001. [Google Scholar]
- Tabuada, P.; Pappas, G.; Lima, P. Feasible formations of multi-agent systems. Proc. Am. Control Conf.
**2001**, 1, 56–61. [Google Scholar] [CrossRef] - Pappas, G.J.; Lygeros, J.; Tilbury, D.; Sastry, S. Exterior differential systems in control and robotics. In Essays on Mathematical Robotics; Baillieul, J., Sastry, S., Sussmann, H., Eds.; IMA Volumes in Mathematics and its Applications; Springer: New York, NY, USA, 1998; Volume 104, pp. 271–372. [Google Scholar]
- Bicchi, A.; Christensen, H.; Prattichizzo, D. Control Problems in Robotics; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2002. [Google Scholar]
- Won, C.H.; Ahn, H.S. Nonlinear orbital dynamic equations and state-dependent riccati equation control of formation flying satellites. J. Astronaut. Sci.
**2003**, 51, 433–449. [Google Scholar] [CrossRef] - Palmer, P. Optimal relocation of satellites flying in near-circular-orbit formations. J. Guid. Control Dyn.
**2006**, 29, 519–526. [Google Scholar] [CrossRef] - Izzo, D.; Pettazzi, L. Autonomous and distributed motion planning for satellite swarm. J. Guid. Control Dyn.
**2007**, 30, 449–459. [Google Scholar] [CrossRef] - Kumar, K.; Bang, H.; Tahk, M. Satellite formation flying using along-track thrust. Acta Astronaut.
**2007**, 61, 553–564. [Google Scholar] [CrossRef] - Sengupta, P.; Vadali, S.; Alfriend, K. Averaged relative motion and applications to formation flight near perturbed orbits. J. Guid. Control Dyn.
**2008**, 31, 258–272. [Google Scholar] [CrossRef] - He, Q.; Han, C. Dynamics and control of satellite formation flying based on relative orbit elements. In Proceedings of the AIAA Guidance, Navigation and Control Conference and Exhibit, Honolulu, Hawaii, 18–21 August 2008. [Google Scholar]
- Cho, H.C.; Park, S.Y. Analytic solution for fuel-optimal reconfiguration in relative motion. J. Optim. Theory Appl.
**2009**, 141, 495–512. [Google Scholar] [CrossRef] - Baoyin, H.; Junfeng, L.; Yunfeng, G. Dynamical behaviors and relative trajectories of the spacecraft formation flying. Aerosp. Sci. Technol.
**2002**, 6, 295–301. [Google Scholar] [CrossRef] - Dar’in, A.; Kurzhanskii, A.; Seleznev, A. The Dynamic Programming Method in Impulsive Control Synthesis. Diff. Equat.
**2005**, 41, 1566–1576. [Google Scholar] [CrossRef] - Kurzhanski, A.; Daryin, A. Dynamic Programming for Impulse Feedback and Fast Controls; Lecture Notes in Control and Information Sciences; Springer: London, UK, 2020. [Google Scholar] [CrossRef]
- Hartmann, J.W.; Coverstone-Carroll, V.; Williams, S. Optimal interplanetary spacecraft trajectories via a Pareto genetic algorithm. Adv. Astronaut. Sci.
**1998**, 99, 1439–1454. [Google Scholar] [CrossRef] - Woo, B.; Coverstone, V.; Cupples, M. Low-Thrust Trajectory Optimization Procedure for Gravity-Assist, Outer-Planet Missions. J. Spacecr. Rocket.
**2006**, 43, 121–129. [Google Scholar] [CrossRef] - Vaddi, S. Modeling and Control of Satellite Formations. Ph.D. Thesis, Department of Aerospace Engineering, Texas A&M University, College Station, TX, USA, 2003. [Google Scholar]
- Cho, H.; Yu, A. New approach to satellite formation-keeping: Exact solution to the full nonlinear problem. J. Aerosp. Eng.
**2009**, 22, 445–455. [Google Scholar] [CrossRef] - Vaddi, S.S.; Vadali, S.R.; Alfriend, K.T. Formation flying: Accommodating nonlinearity and eccentricity perturbations. J. Guid. Control Dyn.
**2003**, 26, 214–223. [Google Scholar] [CrossRef] - Prussing, J.E.; Conway, B.A. Orbital Mechanics; Oxford University Press: New York, NY, USA, 1993. [Google Scholar]
- Cho, H.; Udwadia, F. Explicit solution to the full nonlinear problem for satellite formation-keeping. Acta Astronaut.
**2010**, 67, 369–387. [Google Scholar] [CrossRef] - Wu, B.; Wang, D.; Poh, E. Decentralized sliding-mode control for attitude synchronization in spacecraft formation. Int. J. Robust Nonlinear Control
**2013**, 23, 1183–1197. [Google Scholar] [CrossRef] - Ghadami, R.; Shafai, B. Decomposition-based distributed control for continuous-time multi-agent systems. IEEE Trans. Automat. Contr.
**2013**, 58, 258–264. [Google Scholar] [CrossRef] - Vetrella, A.; Fasano, G.; Accardo, D.; Moccia, A. Differential GNSS and vision-based tracking to improve navigation performance in cooperative multi-UAV systems. Sensors
**2016**, 16, 2164. [Google Scholar] [CrossRef] - Scharnagl, J.; Kempf, F.; Schilling, K. Combining distributed consensus with robust H
_{∞}-control for satellite formation flying. Electronics**2019**, 8, 319. [Google Scholar] [CrossRef] - Mashtakov, Y.; Ovchinnikov, M.; Petrova, T.; Tkachev, S. Two-satellite formation flying control by cell-structured solar sail. Acta Astronaut.
**2020**, 170, 592–600. [Google Scholar] [CrossRef] - Mashtakov, Y.; Petrova, T.; Tkachev, S. Relative motion control of two satellites by changing the reflective properties of the solar sails surface. Adv. Astronaut. Sci.
**2020**, 170, 399–416. [Google Scholar] - Monakhova, U.; Ivanov, D.; Roldugin, D. Magnetorquers attitude control for differential aerodynamic force application to nanosatellite formation flying construction and maintenance. Adv. Astronaut. Sci.
**2020**, 170, 385–397. [Google Scholar] - Liu, H.; Tian, Y.; Lewis, F.; Wan, Y.; Valavanis, K. Robust formation flying control for a team of satellites subject to nonlinearities and uncertainties. Aerosp. Sci. Technol.
**2019**, 95, 105455. [Google Scholar] [CrossRef] - Li, D.; Li, C.; Ma, G.; Lv, Y.; He, W.; Ge, S. Cooperative circumnavigation of a host spacecraft by multiple microsatellites. In Proceedings of the 2019 18th Europ. Control Conference, ECC 2019, Naples, Italy, 25–28 June 2019; pp. 1068–1072. [Google Scholar] [CrossRef]
- Vetrella, A.; Fasano, G.; Accardo, D. Attitude estimation for cooperating UAVs based on tight integration of GNSS and vision measurements. Aerosp. Sci. Technol.
**2019**, 84, 966–979. [Google Scholar] [CrossRef] - Mashtakov, Y.; Shestakov, S. Maintain tetrahedral satellite array configuration using uniaxial control. Keldysh Inst. Prepr. M.V. Keldysh
**2016**. (In Russian) [Google Scholar] [CrossRef] - Mashtakov, Y.; Shestakov, S. Controlling a tetrahedral satellite array using atmospheric drag. In Proceedings of the XLIII Academic Readings on Cosmonautics Dedicated to the Memory of Academician S.P. Korolev and Other Prominent Russian Scientists—Pioneers of Space Exploration, Moscow, Russia, 29 January–1 February 2019; pp. 307–308. (In Russian). [Google Scholar]
- Michael Ross, I.; King, J.; Fahroo, F. Designing optimal spacecraft formations. In Proceedings of the AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Monterey, CA, USA, 5–8 August 2002. [Google Scholar]

**Figure 1.**Illustration of a group of three Coulomb spacecraft orbiting in deep space around the center of mass of their cluster.

**Figure 2.**Geometry of the orbital motion of the lead and trailing satellites according to [120].

**Figure 3.**Coordinate systems in [130]. (Earth s.c. (X–Y–Z) and s.c. Hill (x–y–z). The distance between the leading and trailing satellites on the yz-plane of the local rotating coordinate system x–y–z must be equal to the specified $\rho $).

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**MDPI and ACS Style**

Andrievsky, B.; Popov, A.M.; Kostin, I.; Fadeeva, J.
Modeling and Control of Satellite Formations: A Survey. *Automation* **2022**, *3*, 511-544.
https://doi.org/10.3390/automation3030026

**AMA Style**

Andrievsky B, Popov AM, Kostin I, Fadeeva J.
Modeling and Control of Satellite Formations: A Survey. *Automation*. 2022; 3(3):511-544.
https://doi.org/10.3390/automation3030026

**Chicago/Turabian Style**

Andrievsky, Boris, Alexander M. Popov, Ilya Kostin, and Julia Fadeeva.
2022. "Modeling and Control of Satellite Formations: A Survey" *Automation* 3, no. 3: 511-544.
https://doi.org/10.3390/automation3030026