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ACKNOWLEDGMENTS I would like to thank all organisations who made this research traineeship possible: The Conseil Regional d’Alsace and the European Community programme ERASMUS for the financial support, The ESSAIM, and particularly Professor Gerard Gissinger, who gave me this opportunity, with the help of Professor Mogens Blanke, The Universite de Haute Alsace for the invaluable help they offered me, I would like to express my gratitude to the following members of the Control Engineering Department at Aalborg University: Rafal Wisniewski & Roozbeh Izadi-Zamanabadi for their guidance, faithful supervision and enthusiasm, I would also like to thank the other teachers, researchers, students, secretaries and technicians, who have welcomed me, and shared their experience and expertise with me. They helped to create an enthusiastic working environment which has been invaluable to me. Eventually, I would like to express deep gratitude to my family and friends for their support.
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FOREWORD This report is the result of a six months training period at the Control Engineering Department of Aalborg University. The project was carried out during the period of March 3rd, 1999 to August 17th, 1999. The whole project concerns a guidance concept using potential functions based on the Hamiltonian formalism. The report provides therefore a software simulation test facility for spacecraft formation flying, featuring non-linear guidance functions and collision avoidance. The matlab files and some results are temporary1 available on the web page: http://www.geocities.com/Athens/Thebes/8039/index.html
Aalborg, August 1999
Eric Hueber
1
They’ll probably disappear while September because of my departure.
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DEPARTMENT OF CONTROL ENGINEERING RESEARCH ORGANIZATION The scope of the Department’s activities is control engineering, which includes all aspects of automation in relation to industrial processes and production plants. Processes, in this context, should be interpreted widely, as they include thermal as well as electro-mechanical processes. The area spans such tasks as decentralized PLC-controls and single-circuit regulations to master-control systems for large industrial processes. The Department focuses on automation, which besides traditional control engineering with control loops also include multidisciplinary aspects with complex systems combining mechanical, electronic and software engineering. In this broad area Information Technology (IT) plays an important role and IT is integrated in most of the projects carried out in the department. General control engineering includes: • • • • • • •
Feed back control of electro-mechanical objects (servo systems) or industrial processes Open-loop process control of continuous processes as well as batch processes. Also automatic planning and control of a production sequence according to given specifications Supervision, incl. automatic system monitoring Operator communications and computer based communication systems, incl. man-machine interface Safeguards, incl. alarm systems, fault detection, and fault diagnosis Distributed computer- and data systems and networks Real-time systems
The application of computers in process control implies a number of traditionally separated disciplines, the integration of which is a main feature of the Department’s activities. These activities are partly based on specific technical control problems, partly on new theories and methods, in which the applications are regarded as exemplifications. Among others, large industrial processes are investigated in relation to power stations, heating and ventilation automation control (HVAC), where new theories of multi-variable and nonlinear systems are tested. The Department is engaged in development and design of open sensor-based systems, in which advanced regulation and control strategies can be implemented in connection with co-operating robots. To increase reliability and durability in industrial process control systems, the Department carries out research in the development of systematic methods, both on superior control systems, distributed systems, and at an algorithmic level. The determination of parameters for new nonlinear loudspeaker models is an example of the application of advanced digital signal processing and system identification. Apart from solving specific problems, the purpose of the research is to extend the knowledge of new theories and methods. The work also involves development of new theories and methods.
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For example, the promising area within neural network has proven to be very suitable for system identification, simulation and regulation. In order to master all professional fields within control engineering the research in the Department of Control Engineering is organized in four main research areas: 1. 2. 3. 4.
Control Theory Modeling and Estimation Fault-tolerant control systems Distributed Real-time systems
Control Theory Analysis and synthesis of feedback mechanisms in dynamic systems are the main subjects. The field is in essence interdisciplinary since on one hand it treats dynamic systems as abstract mathematical models that are related to any specific technology, and on the other hand it has applications in practically any industrial sector. The main research areas of the research group are: • • • • • •
Self tuning and adaptive control Control of nonlinear systems Robust and optimal control Control of large-scale systems Neural net for control Control engineering for new applications
Modeling and Estimation Modeling is a major part in the process of designing a control system. The Department wants to accumulate expertise in this area to derive nominal models and possible uncertainty models for the process. At the same time it is often required to estimate states if no or very noisy measurements are available. The main research areas cover: • • • •
Grey Box Modeling System Identification State estimation Neural net for system identification
Fault-tolerant Control Systems Fault tolerance in control is the ability to cope with faults in the components of a controlled plant, and ensure that faults do not develop into failure or emergencies. With an ever-increasing complexity in automation, the vulnerability to faults has become quite high, and experience has shown an increasing demand for better plant availability. The area of Fault tolerant control deals with: • • • • •
Methods for fault detection in industrial processes Methods for systematic design of fault tolerant control systems and components Software tools for support of design Analysis of discrete event systems for support of design of supervisory control Industrial tests of results to achieve experience for research and teaching activities
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Fail-safe systems with hardware redundancy
Distributed Real-time Control Systems Control systems are viewed as the total control of objects, low-level controllers, interconnecting networks, supervisory controllers and Human Computing Interface (HCI). Controller design relies heavily on a well-defined behavior of the underlying infrastructure, which is frequently taken for granted. However, since independently designed controllers exist in an environment of shared resources, independence is no longer obvious. Major issues within this area therefore include: • • • • •
Methods for analyzing temporal co-existence of task on multiprogramming platforms Capacity analysis of communication networks in control systems Distributions as a mean in fail resilient systems Real-time information exchange in distributed systems Object Oriented analysis, design and implementation of control systems
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ENGLISH ABSTRACT Autonomous onboard navigation, attitude, and orbit control are key capabilities vital to the realization of formation flying on future spacecraft and are necessary to operate these complex systems at reasonable costs while reducing the operational burden on the flight team. Moreover such autonomous systems reduce operation costs and increase longevity of the mission by minimizing thruster firings. This report describes a guidance concept using potential functions based on the Hamiltonian formalism. The first part of the work focuses on a simulator test facility concept for satellite formation flying. The simulations feature two different methods, Kepler’s laws and Newton’s equations. The second part reformulates the equations of motion from the Hamiltonian principle. This energetic viewpoint provides new ways of approach and opportunities for the control and guidance. The chapter about guidance presents a solution based on Hamiltonian equations of motion to arrange the spacecraft positions in the formation. This method for autonomous spacecraft guidance provides collisions avoidance and low propellant requirements. The project features therefore three different orbit simulators increasing in complexity and possibilities, and finally a solution of positions control for formation flying spacecraft with Hamiltonian potential function guidance.
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FRENCH ABSTRACT La navigation autonome et le contrôle d’orbite et d’orientation sont les qualites indispensables pour les vols en formation des futures missions spatiales. Elles sont essentielles aussi à la réalisation de ces projets complexes à un coût raisonnable, par la diminution de l’assistance des équipes d’analystes au sol. De plus, en économisant l’utilisation des propulseurs, de tels systèmes autonomes permettent de réduire les coûts d’opération et d’augmenter la durée de la mission Ce rapport décrit la conception d’un simulateur de satellites guidés par des fonctions de potentiel fondées sur une formulation hamiltonienne du problème. La première partie du travail porte sur la création de simulateurs d’orbites. Les simulations font appel à deux methodes: l’une basée sur les lois de Kepler et l’autre sur les équations de Newton. Dans la seconde partie, les équations du mouvement sont reformulées à partir du principe d’Hamilton. Leur aspect énergétique offre de nouvelles approches et perspectives pour le contrôle et la guidance. Le chapitre sur la guidance présente une solution, utilisant les équations hamiltonienne du mouvement, pour arranger la configuration spatiale d’une formation de satellites. Cette méthode doit s’appliquer a des satellites guidés de façon autonome. Elle garantit donc le contournement d’obstacle et l’économie de combustible. Le projet met donc en avant trois simulateurs d’orbite différents, de complexités et possibilités croissantes. Finalement, une solution mettant en jeu des fonctions de guidance est proposée, pour le contrôle de position de satellites volant en formation.
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STANDARD NOTATION Coordinate systems BCS FCS GCS OCS
Body spacecraft Coordinate System Formation Coordinate System Global Coordinate System Orbit Coordinate System
List of symbols a b e i m n r E M V
φ θ σ ν ω Ψ Ω F
r r& &r&
∆v
Semi-major axis Semi-minor axis Eccentricity Inclination Mass of the spacecraft Mean motion Distance between Earth and satellite centers. Eccentric anomaly Mean anomaly Potential Elevation Right ascension Obstruction potential True anomaly Argument of perigee Vernal Equinox Right ascension of the ascending mode Velocity kick force vector Position vector Velocity vector Acceleration vector Velocity kick vector
Acronyms and abbreviations ESA KOE NASA
European Space Agency Keplerian Orbital Elements National Aeronautics and Space Administration
Parameters of equations and variables used in this report are in italic font. As well, an italic font identifies keywords, to remind the reader they are explained in the glossary at the end of this report.
TABLE OF CONTENTS
ix
TABLE OF CONTENTS Acknowledgments ....................................................................................................................................... i Foreword .................................................................................................................................................... ii Department of Control Engineering ....................................................................................................... iii English Abstract ....................................................................................................................................... vi French Abstract ....................................................................................................................................... vii Standard Notation .................................................................................................................................. viii
1.
INTRODUCTION ......................................................................................... 1
1.1
Problem Formulation ................................................................................................................... 2
1.2
Outline ........................................................................................................................................... 3
2. DEFINITIONS.............................................................................................. 4 2.1 Coordinate Systems ...................................................................................................................... 4 2.1.1 Global Coordinate System (GCS).......................................................................................... 4 2.1.2 Orbit Coordinate System (OCS) [er eθ eφ].............................................................................. 5 2.1.3 Spacecraft Body (BCS) Coordinate System [xBCS yBCS zBCS] ................................................ 6 2.1.4 Formation (FCS) Coordinate System [xFCS yFCS zFCS]............................................................ 6 2.2
Spacecraft and its Functionality.................................................................................................. 6
2.3
Formation Flying .......................................................................................................................... 7
2.4
Chapter Conclusion...................................................................................................................... 8
3.
DESCRIPTION OF THE ORBIT.................................................................. 9
3.1 Gravitation and Mechanics.......................................................................................................... 9 3.1.1 Principles and Laws ............................................................................................................... 9 3.1.2 Ellipses................................................................................................................................. 10 3.1.3 Orbit Perturbations............................................................................................................... 13 3.2 Implementation........................................................................................................................... 15 3.2.1 With Kepler’s equation........................................................................................................ 15 3.2.2 With Newton’s equations..................................................................................................... 17 3.3
Chapter Conclusion.................................................................................................................... 23
4. LAGRANGIAN AND HAMILTONIAN EQUATIONS ................................. 24
x
TABLE OF CONTENTS
4.1
Introduction................................................................................................................................. 24
4.2
Equations of Motion ................................................................................................................... 27
4.3
Chapter Conclusion .................................................................................................................... 30
5. POTENTIAL FUNCTION GUIDANCE ....................................................... 31 5.1
Autonomous Guidance ............................................................................................................... 31
5.2
The PFG Principle ...................................................................................................................... 31
5.3
Hamiltonian Potential Function Guidance ............................................................................... 33
5.4 Potential Functions ..................................................................................................................... 35 5.4.1 Goal Function....................................................................................................................... 35 5.4.2 Obstacle Function ................................................................................................................ 35 5.4.3 Conclusion ........................................................................................................................... 36 5.5
Artificial Friction ........................................................................................................................ 36
5.6
Chapter Conclusion .................................................................................................................... 38
CONCLUSION ................................................................................................. 41 References and Bibliography .................................................................................................................. 43 Glossary .................................................................................................................................................... 44 Appendices................................................................................................................................................ 47
INTRODUCTION
1
1. INTRODUCTION This chapter explains the goal and interests of this subject. At first, it presents the expectations of the ESA and NASA scientists. Secondly it describes our problem with our goals. Eventually an outline closes this chapter. “Enhanced formation flying and space vehicle autonomy will revolutionize space and Earth science missions and enable many small, inexpensive spacecraft to fly in formation and gather concurrent science data.” Nowadays space formation flying and autonomous vehicles have become important research topics due to their actuality and are subject to intensive research activities. We are establishing the bases of the next ten years space missions (see Figure 1-1) and obviously for an important step in the space exploration. Autonomous Navigation
1996 Real Time On Board Orbit Determination
On Board Maneuver Planning
Autonomous Constellation Control
2001
1999 Real Time On Board ∆v Determination
1-Way 2-Way Formation Formation Flying Flying
Real Time On Board ∆v Execution
Chase S/C reacts to Data from Target S/C
True Formation Flying
2006 Chase & Target S/C react in concert
Virtual Platform
2010 Multiple S/C Navigation collectively & autonomously
Navigation & Attitude reacting collectively
Figure 1-1: The NASA Guidance, Navigation & Control Technology Office roadmap
Up until now, no space missions have used formation flight of spacecraft, but in the future both NASA and ESA have planned missions. The goal of the NASA Orion Project, scheduled to launch in 2000, is to demonstrate formation flying with 3 - 6 satellites. For this project formation flying, the satellites stay within a couple of hundred meters of each other, and they are under autonomous, closed-loop control. This means that ground controllers can ask the satellites to arrange themselves in a specific formation, but the satellites themselves figure out how to achieve that goal. Space Technology 3, scheduled to launch in 2003, will test technologies and flying concepts that will benefit NASA's Origins Program, which seeks answers to the origins of our universe by studying distant stars and their planets. By sending interferometers into space, NASA's goal is to image extremely distant stars, and ultimately even find and image planets like Earth around other stars. Multiple spacecraft containing telescopes and flying in precise formation at great distances from each other (acting as one enormous telescope system) will make these goals possible. ESA has planned two missions namely the Laser Interferometry Space Antenna (LISA) and the DARWIN mission. The LISA project is to detect and study low frequency astrophysical gravitational radiation, for the use of application to astrophysics, cosmology, and fundamental physics. The DARWIN mission is a space interferometer. These two missions are expected as landmarks in the history of space investigations. All these challenging space science missions are based on formation flying spacecraft. The control of the formation from the ground would require a numerous and expensive flight team. Moreover, the complexity of any ground-based control for relative spacecraft positioning would probably not provide sufficiently rapid corrective commands. Thus the onboard autonomy for
INTRODUCTION
2
formation acquisition needs further researches. This report is an investigation of this new field for space missions.
1.1 PROBLEM FORMULATION The aim of this project is to conceive a guidance control based on the Hamiltonian formalism. Therefore, it should provide a software simulation test facility for spacecraft formation flying, featuring guidance functions and collision avoidance. At first we want to conceive a simulation of spacecraft in orbit, while respecting the three Kepler’s fundamental laws and Newton’s principles of mechanics. Moreover, we wish to include the space perturbations encountered by satellites. Actually small forces lead to slight modifications of orbits. This is caused by: additional gravitational fields, the solar wind, attitude management thruster firings, atmospheric friction, the planets’ flatness... For a simulation of spacecraft formation flying, the software should be able to incorporate other spacecraft. Furthermore, the whole realization should allow testing several formation behaviors, and finally point out some methods and solutions to navigation, collision avoidance, orbit transfers, obviously all in agreement to celestial mechanics. Concerning the orbit simulation, a lot of software used by radio amateurs can be found on the market. But all these software are only aimed at tracking satellites. They don’t feature thruster firings or perturbation effects on the orbit shape. The autonomous guidance research is based on a work made by [McInnes], which presents a potential solution for guidance of spacecraft. The publication concerns artificial potential functions and describes the autonomous guidance of a spacecraft in a space containing other bodies. However, according to [Carter], most of the terminal rendezvous studies use linearized equations of motion (e.g. Clohessy-Wiltshire equations). In this project, equations of motion used for the guidance should be built on the Hamiltonian principle. This formulation is nonlinear and provides the true equations of motion. For this project, the thrust force has been chosen ideal. This means that this force has no restriction like limit, dead zone or noise. As well the sensors give the exact positions. The purpose will only focus on position control. Indeed attitude control will be considered as perfect. The work realized doesn’t take in account any time constraint. According to works done on this subject, the time is not an essential parameter compared to the other requirements, like propellant. The mission chosen for the implementation is around the Earth. The orbit of the formation is the same as the Moon. The Spacecraft used weight 500 kg. The formation baseline (i.e. the distance to keep between the bodies) will be within 10 meters up to 1 km. Concerning the number of spacecraft, 2 to 20 vehicles are expected for interferometer missions. But our goal is to observe behaviors of satellites in orbit operating motions in their formation using guidance functions. Therefore a 1 - 3 spacecraft mission matches our goal. Eventually the requested position accuracy is ±1 cm.
INTRODUCTION
3
1.2 OUTLINE This report is split into four parts: •
The first part (Definitions) gives the bases for the orbit description and the path planning. It describes the different coordinate systems used in this report, the different parts of a spacecraft, and finally presents the meaning of a formation flight.
•
The second part (Description of the Orbit) describes the celestial mechanics from the basic laws of gravitation and mechanics until the perturbations of the space environment. Then it describes the implementation and the simulation of the theory
•
Part three (Lagrangian and Hamiltonian Equations) deals with a new way of describing equations of motion: the Hamiltonian formalism. This methodology provides a more energetic point of view than the two previous methods.
•
The last part (Potential Function Guidance) defines the PFG control. It also explains the inclusion of the control action in the Hamiltonian equations of motion. Finally, after the function descriptions, some artificial drag is added in the system to make it stable.
At last, an overall conclusion of the project is given with a resume of the work done and different ideas for future directions to explore.
DEFINITIONS
4
2. DEFINITIONS The objective of this chapter is to give the bases for the orbit description and the guidance. It describes the different coordinate systems used in this report, the different parts of a spacecraft and presents the meaning of a formation flight.
2.1 COORDINATE SYSTEMS This section describes the various coordinate systems used later. In order to explore the space environment, coordinates must be developed to consistently identify the observer’s reference, an intermediary reference, and the formation’s reference. 2.1.1 GLOBAL COORDINATE SYSTEM (GCS) Because space is observed from Earth, Earth's coordinate systems must be established before space can be mapped. This system is also called World Coordinate System or Earth Centered Coordinate System. It’s a right-hand coordinate system, with its origin in the Earth’s center. The XY-plane is parallel with the equatorial plane. The +X-axis is fixed in the direction of vernal equinox. The +Z is normal to the XY-plane, oriented toward the North Pole. The Y-axis is formed by the cross product of Z- and X- axes. All axes are defined according to one epoch. This system is usually associated with the Global Spherical Coordinates (see Figure 2-1). The elevation or latitude component of the coordinates, φ, is denoted declination. Similarly, the azimuth or longitude component, θ, is known as right ascension. z
r φ y θ x Figure 2-1: The GCS, with X pointing toward the vernal equinox, Z toward the North Pole and Y = Z × X, and the Global Spherical Coordinates r, θ and φ.
In Figure 2-1 r is the radius (distance between the Earth center and the satellite center), φ is the declination (such as φ& has the same direction as +x-axis), and θ is the right ascension (such as θ& has the same direction as +z-axis). The Keplerian Orbital Elements (see next chapter), are defined in this system.
DEFINITIONS
5
2.1.2 ORBIT COORDINATE SYSTEM (OCS) [er eθ eφ] This local tangent coordinate system2 is handy to determine position, velocity and acceleration of an orbiting body observed from the Earth. This is the intermediary coordinate system, which does the link between the GCS and the BCS (see next subsection). The three axes of this system are: er unit vector in radial direction eθ unit vector in transverse direction eφ unit vector in normal direction eφ
er
• r
Ψ
θ
eθ S
φ
Figure 2-2: Orbit Coordinate System [er eθ eφ] with the Global Spherical Coordinates
In Figure 2-2, S is the satellite’s center of gravity which coincides with the origin of the OCS. This system allows a handy formulation of the celestial mechanics (see subsection 3.2.2). Since the components of the Global Spherical Coordinates are r, θ and φ, er is in the direction of increasing r, eθ in the direction of increasing3 θ, and eφ in the direction of increasing2 φ.
2
Also called local horizontal coordinate system. Such systems have a reference plane at any point always tangent to the sphere centered on the origin of the orbited body and passing through the center of the coordinate system in question. 3 For a prograde orbit.
DEFINITIONS
6 2.1.3 SPACECRAFT BODY (BCS) COORDINATE SYSTEM [xBCS yBCS zBCS]
Each spacecraft has its own body coordinate system. The guidance functions need this intermediary system to process the velocity kick vector. Origin is defined in the spacecraft center of gravity. It is a right orthogonal coordinate system fixed in the spacecraft structure. The axes are aligned with the principal axes of the satellite body. Note that in this project it is assumed that attitude control is ideal thus BCS coincides with GCS. z BCS
⋅
Z GCS
y BCS
S
x BCS r
φ
XGCS
YGCS
θ
Figure 2-3: The BCS as a translation, by the vector r, of the GCS.
2.1.4 FORMATION (FCS) COORDINATE SYSTEM [xFCS yFCS zFCS] The motion of each spacecraft is studied in this system. This system allows observing the general formation behavior as if we were in the master spacecraft. Since here the attitude is not discussed the FCS is equal to the master spacecraft’s OCS. Thus the center of the FCS coincides with the master body center and the unit vectors composing the FCS are defined as
[x FCS
y FCS
z FCS ] = [e r
eθ
eφ ]
.
master
2.2 SPACECRAFT AND ITS FUNCTIONALITY A spacecraft system can be divided into two principal elements, the payload and the bus. The payload is the motivation for the mission itself (e.g. the separated spacecraft interferometry missions). In order that this may function it requires certain resources which will be provided by the bus. In particular it should be possible to identify the following functional requirements, for a formation-flying mission: 1.
The payload must be pointed in the correct direction,
2.
The payload must be operable,
DEFINITIONS 3.
The data from the payload must be communicated to the ground,
4.
The desired formation arrangement must be maintained,
5.
The payload must be operable and reliable over some specified period of time,
6.
An energy source must be provided to enable the above functions to be performed.
7
These requirements lead on to the breakdown into subsystems. The number and the variety of the subsystems depend on the payload requirements. For our purpose, the bus should feature these subsystems: (numbers in brackets refer to the above functions) -
Attitude control (1) Data handling (2) Communication ground-spacecraft (3) Propulsion (1 & 4) Sensors (1 & 4) Position control (4) Communication spacecraft-spacecraft (4) Thermal (5) Power (6)
We’ll admit, for this project, that the attitude control is perfect and instantaneous. The sensors and the actuators could be later studied, to give a more realistic simulation of the guidance control. This report doesn’t deal with the inter-spacecraft communication methods either. However the main subsystem studied is the control of position. Finally, here is the definition of a satellite, which is a particular spacecraft. A satellite is a small body, which orbits a larger one. Earth-orbiting spacecraft are called satellites. While deep-space vehicles are technically satellites of the sun or of another planet, or of the galactic center, they are generally called spacecraft instead of satellites.
2.3 FORMATION FLYING To give an overview of the formation behavior, different propositions and ideas are presented in this section and a description of how they detect their different positions and velocities. All the spacecraft have to be placed in precise positions in the formation and should be capable of changing or adjusting the formation geometry. Furthermore during formation acquisition the need of position information is required and position-sensors are therefore necessary. The formation is a master/slave system. Each spacecraft needs its position and the others’ position in the formation, to compute its proper guidance function. The master sends positions, velocities, and goal positions of each satellite to the different slaves. The master is also the only body tracked from Earth. Obviously, with its data of the formation, it can restitute all the formation configuration to a ground base. The formation flying is not necessarily composed with identical spacecraft. But if all the bodies are similar, worthwhile advantages appear. At first, it’s a security. If the master presents a failure, the formation will have the possibility to vote another “sane” master.
8
DEFINITIONS
Secondly the formation will be able to improve the accuracy of individual position measurement4. The master should in this case feature a collect and process of all the position measurements made by all the slave spacecraft, in order to do an average and consequently minimize errors. On top of that we’ll dismiss a problem of a body hiding another one, that might happen if only one spacecraft makes the detection. A third possibility is to elect as master the spacecraft with the less propellant. Like in the real life, the older we are, the more official we become. Obviously this case is efficient only if the master orders the formation configuration without firing its thrusters. After that the master or one spacecraft of the formation, sends the positions in the GCS to the Earth for an Earth tracking. This operation will be used in the Orion Project (see Chapter 1) to check the Orbital Elements of the formation.
2.4 CHAPTER CONCLUSION In this chapter, all the coordinates systems used in this report are described. They are obviously useful for orbit simulation. Then an overview of a spacecraft with its different parts and their functionality is defined. By this description, the reader is able to understand how interact the guidance functions with other subsystems working in a spacecraft. Finally, a formation flight mission is presented and ideas added about possible cooperative behaviors.
4
An accuracy increase can be obtained for the attitude determination as well (but it’s not our purpose).
DESCRIPTION OF THE ORBIT
9
3. DESCRIPTION OF THE ORBIT The motion of a spacecraft is specified by its position, velocity, attitude, and angular velocity. The first two quantities describe the translational motion of the center of mass of the spacecraft and are those which match our purpose. The position and the velocity are subject to celestial mechanics. This chapter develops the celestial mechanics from the basic laws of gravitation and mechanics until the perturbations of the space environment. Then the theory is validated by implementations and simulations.
3.1 GRAVITATION AND MECHANICS This paragraph describes the force of gravity, characteristics of ellipses, and the concepts of Newton's principles of mechanics. The theory of celestial mechanics is required in order to predict the motion of a satellite. With the intention of evaluating how the space effects act on maneuver, perturbations of the orbit will close this part. 3.1.1 PRINCIPLES AND LAWS 3.1.1.1 KEPLER’S LAWS OF ORBITAL MOTION Gravitation is the mutual attraction of all masses in the universe. The concepts associated with planetary motions developed by Johannes Kepler (1571-1630) describe the positions and motions of objects in our solar system. Isaac Newton (1643-1727) explained why Kepler's laws worked, in terms of gravitation. Since planetary motions are orbits, and all orbits are ellipses, a review of ellipses follows. The German mathematician Kepler solved the problem of non-circular orbits. He described planetary motion according to three laws. Each of these laws is illustrated by an applet. First law (1609): The orbit of each planet is an ellipse, with the sun at one focus. Second law (1609): The line joining the planet to the Sun sweeps out equal areas in equal intervals of time. Third law (1619): The square of a planet's orbital period is proportional to the cube of its mean distance from the sun. Kepler’s laws apply not just to planets orbiting the Sun, but to all cases in which one celestial body orbits another 3.1.1.2 NEWTON'S PRINCIPLES OF MECHANICS Whereas the Kepler’s laws summarize observations, the Newton’s principles brought the theory in 1685. Newton realized that the force that makes apples fall to the ground is the same force that makes the planets "fall" around the Sun. He established that a force of attraction toward the sun becomes weaker in proportion to the square of the distance from the sun. Also, the equation of the force for the magnitude due to gravity is:
F =−
µm ⎛ r ⎞ ⎜ ⎟ r2 ⎝ r ⎠
(3.1)
DESCRIPTION OF THE ORBIT
10
where µ (= G⋅M)is the gravitational constant5, m the mass of the body submitted to the attraction, and r the distance between the centers of the two bodies. F has the direction of the unit vector (r/r), where locates m from M. Now with the Newton’s second law of gravitation, the acceleration vector of the satellite is
&r& = −
µ r3
r
(3.2)
This equation is independent of the mass. A solution of this motion equation is [Prussing]
a (1 − e 2 ) r= 1 + e cos θ
(3.3)
where a is the semi-major axis, e the eccentricity and θ the true anomaly. 3.1.2 ELLIPSES Satellites launched from the Earth can be described by different kinds of orbit. These orbits can be hyperbolic or elliptical. Hyperbolic orbits are generally used to observe other planets. Whereas elliptical ones (ellipse or circular) are used for Earth observation and communications. Such trajectories are obviously most often used. An ellipse is a closed plane curve generated in such a way that the sum of its distances from two fixed points (called the foci) is constant. In Figure 3-2 below, Distance A + B is constant for any point on the curve.
⋅
Foc i
Dista nc e A
⋅
Dista nc e B
Any p oint on the c urve Figure 3-1: ellipse foci
An ellipse also results from the intersection of a circular cone and a plane cutting completely through the cone. The maximum diameter is called the major axis. It determines the size of an ellipse. Half the maximum diameter, the distance from the center of the ellipse to one end, is called the semi-major axis.
µ stands for the universal constant of gravitation G (=6.67259⋅10-11 m3kg-1s-2) multiplied by the mass of the orbited body (µ = 398.6⋅103 km3s-2 for the Earth).
5
DESCRIPTION OF THE ORBIT
11
An Ellipse from a Conica l Section
Ma jor a nd Minor Axes
Ellip se
Circ ula r Ba se of Cone
Semi-ma jor a nd Semi-minor Axes
Figure 3-2: Properties of ellipses
The shape of an ellipse is determined by how close together the foci are, in relation to the major axis.
Eccentricity =
distance between the foci major axis
(3.4)
If the foci coincide, the ellipse is a circle. Therefore, a circle is an ellipse with an eccentricity of zero.
⋅⋅⋅⋅
⋅⋅⋅⋅ Foc i for va rying Ec c entric ities
Figure 3-3: Same major axis for various eccentricities
In elliptical orbits, the most distant point from the center of the orbited body is called the apofocus (apogee for the Earth), and for the closest point perifocus (perigee for the Earth). The straight line connecting apogee, perigee and the two foci is called the line of apsides. If rp, ra, and Re are the perigee height, apogee height, and radius of the Earth, respectively, then for an Earth satellite, the major axis is equal to 2⋅Re + rp + ra. However to specify an orbit, its size and shape aren’t sufficient. Orientation of the orbit plane in space (see Figure 3-4) is also necessary. The Keplerian Orbital Elements define a unique orbit in the space.
DESCRIPTION OF THE ORBIT
12
Sa tellite
• ν
First p oint of Aries
Pla ne of Ea rth’ s eq ua tor
Ψ
ω Ω
i Asc end ing no d e (RAAN) Direc tion of sa te llite m otion
Orb it
Figure 3-4: Keplerian Orbital Elements. Ψ marks the direction of the Vernal Equinox. Ω is measured in the plane of the Earth’s equator, ω and ν are measured in the orbit plane.
Moreover the position of a satellite in its orbit is necessary. The true anomaly, ν, gives the angle from the perigee point to the satellite in the orbital plane. The mean anomaly, M, can also give a similar angle at any time, but M is a trivial calculation of no physical interest. The real angle is given by the true anomaly, which is difficult to calculate. The eccentric anomaly, E, is introduced as an intermediate variable relating the other two. E is defined with ν in the following figure.
Loc a tion of sa te llite
•
E
ν
Ap og e e
Figure 3-5: Definition of True Anomaly, ν, and Eccentric Anomaly, E. The outer figure is a circle with radius equal to the semi-major axis.
The mean eccentric anomalies are related by Kepler’s equation:
DESCRIPTION OF THE ORBIT
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M = E − e sin E
(3.5)
where e is the eccentricity. E is then related to ν by Gauss’ Equation [Wertz]: 1
⎛E⎞ ⎛ν ⎞ ⎛ 1 + e ⎞ 2 tan⎜ ⎟ = ⎜ ⎟ tan⎜ ⎟ ⎝2⎠ ⎝ 2 ⎠ ⎝1− e ⎠
(3.6)
3.1.3 ORBIT PERTURBATIONS Various perturbing forces can be included in order to produce a better approximation to a satellite’s orbit. At low altitude (π/2 or φπ/2
In order to correct the problem, equations must be modified by inverting the action of the forces along eθ and eφ. Therefore, thrust forces, Fθ and Fφ, must be inverted if φ is not within -π/2 and π/2. However, the φ can take any value, but these values are hidden in the equations of motion. Actually, just before the output of the equations, a function constraints φ to [-π/2;π/2]. The output of the spacecraft’s dynamic bloc always provides a φ in accordance with the OCS definition. Note: The two problems described here might happen rarely. Such behavior would be exceptional. However a simulation could present this phenomenon with adequate initial conditions.
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This case happens when θ& = 0 .
