**1.**

**Introduction**

This course, given as a graduate-level course (#088900) in the Faculty of Aerospace Engineering, will expose the participants to the emerging technology of distributed space systems, a concept of distributing the functionality of a single spacecraft between several closely-flying satellites. The students will learn modeling techniques of relative spacecraft motion using various dynamical models, different control strategies that enable a myriad of cooperative tasks, and basic relative navigation methodologies. The students will also get acquainted with the fundamental system engineering tradeoffs associated with the design of multiple-spacecraft missions, and be exposed to a number of applications such as sparse-aperture imaging, geolocation and remote sensing.

**2.**

**Course Learning Objectives**

The course is aimed at extending the knowledge and understanding of space systems by presenting the challenges associated with multi-spacecraft systems. Compared to traditional courses, this course will expose the students to the possibility of designing more efficient space systems by distributing the functionality among several cooperating spacecraft. The students will be thus familiar with the forefront of space systems technology, devoted to the development, research and design of multiple spacecraft missions. In particular, the course learning objectives are as follows:

1) To understand the description of non-Keplerian motion in rotating coordinates;

2) To be able to formulate astrodynamic problems using analytical methods;

3) To present the forefront of current research in spacecraft formation flying;

4) To learn how to model relative motion using orbital elements;

5) To be able to design and simulate cooperative control systems;

6) To gain a systematic view of the distributed space systems engineering;

7) To understand the design and operation of precision electric propulsion devices;

8) To be familiar with future applications that require multiple spacecraft.

**3.**

**Short Syllabus**

Keplerian orbital mechanics. Orbital perturbations. The general relative motion problem. Impulsive stationkeeping. Linear formation flying dynamics and control. High-order relative motion equations. Formulation of relative motion using orbital elements. Canonical modeling of relative motion. Perturbation-invariant formations. Nonlinear formation control. Centralized and de-centralized formationkeeping. Low-thrust propulsion for formation flying. Relative navigation in space. Applications: Sparse-aperture imaging, geolocation, remote sensing.

**Hours Subject**

1-3 |
Keplerian orbital mechanics: Motion in a central field, conic sections, classical orbital elements, the time equation, coordinate system |

4-6 |
Orbital perturbations: Gauss’s variational equations, Lagrange’s planetary equations, influence of oblateness, drag, solar pressure, third body effects. |

7-9 |
The general relative motion problem: Nonlinear relative motion equations, the energy matching condition, impulsive formationkeeping, formulation using polar coordinates. |

10-12 |
Linear relative motion dynamics: Hill-Clohessy-Wiltshire (HCW) equations, linear formation flying control strategies, LQR formationkeeping. |

13-15 |
High-order relative motion equations: Lagrangian mechanics, Euler-Lagrange equations, Legendre polynomials. |

16-18 |
Relative motion modeling using classical osculating orbital elements: Motion relative to circular and elliptic reference orbits. |

19-21 |
Hamiltonian dynamics: Legendre transformation, canonical transformations, Hamilton’s equations. |

22-24 |
Canonical modeling of relative motion: epicyclic orbital elements. |

25-27 |
Perturbation-invariant relative motion: J2-invariant motion, frozen formations. |

28-30 |
Nonlinear formation flying control: Lyapunov-based methods, feedback linearization, intelligent control, precision formation flying, deep-space formation flying, fuel optimization. |

31-33 |
Systems engineering aspects: Centralized and de-centralized spacecraft formation flying control, survivability, adaptability, flexibility, safe-mode operation. |

34-36 |
Propulsion for stationkeeping and formationkeeping: FEEPs, PPTs, Hall thrusters, propulsion tradeoffs. |

37-40 |
Relative navigation; Applications: Future and existing projects, large-baseline interferometry, sparse-aperture imaging, geolocation, remote sensing. |

5.

5.

**Bibliography**

Alfriend, K.T., Vadali, S.R., Gurfil, P., How, J.P., Breger, L.,

*Spacecraft Formation Flying: Dynamics, Control and Navigation*, Elsevier, Oxford, UK, 2010.Battin, R. H.,

*An Introduction to the Mathematics and Methods of Astrodynamics*, AIAA Education Series, 5^{th}Ed., 1999.Bate, R. R., Mueller, D. D., and White, J. E.,

*Fundamentals of Astrodynamics*, Dover, 1971.Goldstein, H., Poole, C., and Safko, J.,

*Classical Mechanics,*Addison-Wesley , 3^{rd}Ed., 2002.Clohessy, W. H., and Wiltshire, R. S., “Terminal Guidance System for Satellite Rendezvous”,

*Journal of the Astronautical Sciences*, Vol. 27, No. 9, Sep. 1960, pp. 653-678.Inalhan, G., Tillerson, M., and How, J. P., “Relative Dynamics and Control of Spacecraft Formations in Eccentric Orbits”,

*Journal of Guidance, Control and Dynamics,*Vol. 25, No. 1, January 2002, pp. 48-60.Alfriend, K. T., and Schaub, H., “Dynamics and Control of Spacecraft Formations: Challenges and Some Solutions”,

*The Journal of the Astronautical Sciences,*Vol. 48, No. 2, April 2000, pp. 249-267.Schaub, H., Vadali, S. R., and Alfriend, K. T., “Spacecraft Formation Flying Control Using Mean Orbital Elements”,

*The Journal of the Astronautical Sciences*, Vol. 48, No.1, 2000, pp.69-87.Schaub, H., and Alfriend, K., “Hybrid Cartesian and Orbit Elements Feedback Law for Formation Flying Spacecraft”,

*Journal of Guidance, Control, and Dynamics,*Vol. 25, No. 2, March-April 2002, pp. 387-393.