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# Orbital Mechanics and Formation Flying: A Digital Control Perspective

• Published: October 2011
• Region: Global
• 438 Pages

Aimed at students, faculty and professionals in the aerospace field, this book provides practical information on the development, analysis, and control of a single and/or multiple spacecraft in space. This book is divided into two major sections: single and multiple satellite motion. The first section analyses the orbital mechanics, orbital perturbations, and attitude dynamics of a single satellite around the Earth. Using the knowledge of a single satellite motion, the translation of a group of satellites called formation flying or constellation is explained. Formation flying has been one of the main research topics over the last few years and this book explains different control approaches to control the satellite attitude motion and/or to maintain the constellation together.

The control schemes are explained in the discrete domain such that it can be easily implemented on the computer onboard the satellite. The key objective of the book is to show the reader the practical and the implementation process in the discrete domain.

Key features:

- explains the orbital motion and principal perturbations affecting the satellite
- uses the Ares V rocket as an example

Introduction
- Introduction to the book
- Book division

Two body orbital motion
- Introduction to orbital motion
- Constraints and generalized coordinates
- Lagrange’s equation
- System of particles
- Two body orbital motion problem
- Orbital equations of motion
- Energy and velocity of orbiting bodies
- Escape velocity
- Earth coordinate inertial (ECI) system
- Period of an orbit
- Development of Kepler’s equation
- Suggested problems

Orbital perturbations in the two body motion
- Introduction to disturbance effects
- Lagrange planetary equations
- Perturbation due to the earth oblateness
- The near-Earth atmosphere effects
- Other disturbance effects
- Suggested problems

Frame rotations and quaternions
- Introduction to rotations and quaternions
- Two-dimensional frame rotations
- Three-dimensional frame rotations
- Example of frame rotations
- Quaternion defi nition and rotations
- Quaternion to Euler angle relations
- Suggested problems

Rigid body motion
- Introduction to attitude dynamics
- Rate of change of a vector
- Moment of inertia
- Principal moments of inertia
- Energy formulation
- Rate of change of a quaternion
- Ares V equations of motion
- Suggested problems

Environmental and actuator torques
- Introduction to torque formulation
- Environmental torques
- Actuator (or control) torques
- Suggested problems

Continuous and digital control systems
- Introduction to methods of designing continuous and discrete control systems
- Ares V equations of motion for first stage flight
- Continuous control formulation
- Discrete control formulation
- Suggested problems

Example
- Introduction to examples in spacecraft attitude dynamics and control
- Nanosatellite problem defi nition
- B-dot controller for fast corrections
- Linear quadratic regulator for attitude correction
- Linear quadratic regulator control weight design
- Suggested problems

Formation flying
- Introduction to formation flying
- Tschauner–Hempel formulation
- Clohessy–Wiltshire formulation
- Earth oblateness and solar effects in formation flying
- Lawden solution
- Discrete optimal control problem for formation flying
- Formation flying controller implementation
- Suggested problems

Deployment procedure for a constellation
- Desired conditions of the satellites in the proposed tetrahedron constellation
- Transfer from a circular orbit to the elliptical orbit (stage 1)
- Station-keeping procedure (stage 2)
- Deployment procedure for the tetrahedron constellation
- Remarks
- Suggested problems

Reconfiguration procedure for a constellation
- Introduction to the reconfi guration process of a constellation
- Data mining process of the Lagrange Planetary equations
- Fuzzy logic controller
- Phase I to II in-plane motion fuzzy logic control system
- Phase II to III in-plane motion fuzzy logic controller
- Out-of-plane motion correction
- Some solutions for the reconfiguration procedures
- Implementation of the fuzzy logic controller
- Adaptive control scheme for reconfiguration procedure
- Remarks
- Suggested problems

Dr. Pedro A. Capó-Lugo works as an Aerospace Engineer in the guidance, navigation, and control analysis and design group at NASA George C. Marshall Space Flight Center in Huntsville, Alabama. He has worked in the analysis of control systems of the Ares rockets in the Constellation program, and has analyzed and developed control systems for different satellite missions which include nano and cube satellites. One of his main research interests is formation flying.

Dr. Peter M. Bainum has 50 years industrial and academic experience. His research in aerospace systems dynamics and control resulted in 220 authored/co-authored publications. His current research interests include: formation flying and dynamics; and control of large flexible space structures. Honours include Fellow AIAA, AAS, AAAS, BIS; Honorary Member Japanese Rocket Society (JRS); IAA Member, and recipient of AIAA International Cooperation, IAF Malina Education, AAS Dirk Brouwer and Sen. Spark Matsunaga International Cooperation Awards. His experience includes participation in the Gemini mission, Apollo program proposal, Department of Defense Gravity Experiment Satellite, Small Astronomy Dual Spin Spacecraft, the Shuttle-Tethered-Subsatellite program, and the NASA proposed Cluster Formation Flying program. Dr. Bainum holds the position of Distinguished Professor of Aerospace Engineering, Emeritus, Howard University, Washington, DC, USA.

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