Corso di Laurea Specialistica in Ingegneria Spaziale
COUPLED ORBITAL AND ATTITUDE
SDRE CONTROL SYSTEM FOR
SPACECRAFT FORMATION FLYING
Relatore: Prof. Mauro Massari
Tesi di Laurea di:
My first commitment is for my parents and relatives, which have always come along with me, patiently and carefully, in this path.
Most of all, I would like to thank anyone who egg on me to take out the best of myself, from professors in the school to coaches in sports, nevertheless clearly all my friends, school and university mates, who have always remembered me to trust in my possibilities and capabilities, and have actively supported me in my studies.
Arriving to current topics, which concern my thesis work, I would like to thank my advisor, Prof. Massari, which has been always interested and willing to listen to me during this period of research.
Sommario XXIII
1. Introduction 1
1.1 Spacecraft Formation Flying... 1
1.2 Control of Spacecraft Formations... 5
1.3 SDRE Technique for Control Design of Nonlinear Systems... 8
1.4 Thesis Breakdown Structure... 9
1.5 The Simulator... 10
2. Spacecraft Formation Flying Dynamics 13 2.1 Orbital Mechanics... 13
2.1.1 Orbital Dynamics... 14
2.1.2 Orbital Kinematics... 16
2.1.3 Relative Orbital Mechanics... 19
2.1.3.1 Simplified Relative Orbital Analysis... 23
2.2 Attitude Mechanics... 26 2.2.1 Attitude Dynamics... 28 2.2.2 Attitude Kinematics... 29 2.2.2.1 Quaternions... 30 2.2.2.2 Euler Angles... 31 2.2.2.3 Rodrigues Parameters... 33
2.2.3 Relative Attitude Mechanics... 37
2.3 Space Environment... 40
2.3.1 Gravitational Field... 42
2.3.1.1 Gravity Gradient Torque... 46
2.3.2 Third Body Attraction... 47
2.3.3 Magnetic Field... 48
2.3.4 Electromagnetic Radiation... 50
2.3.5 Atmosphere... 53
2.3.6 Small Forces... 54
2.3.6.1 Ocean and Solid Tides... 55
2.3.6.2 Relativistic Effects... 55
2.3.6.3 Yarkovsky Effect... 56
2.4 Simulator Implementation... 56
2.4.1 Numerical Integration... 58
2.4.2 User’s Input and Output... 59
2.4.4 Chief... 61
2.4.5 Deputy... 62
2.4.6 Deputy Target... 64
3. SDRE Control Technique 69 3.1 Linear Quadratic Optimal Control Theory... 69
3.1.1 Optimal Controller... 70
3.1.2 Optimal Observer... 72
3.1.3 ARE Solution... 74
3.2 SDRE Methodology for Control of Nonlinear Systems... 75
3.2.1 SDC Parameterization... 75
3.2.1.1 Multiplicity of SDC Parameterizations... 77
3.2.1.2 SDC Particular Cases... 77
3.2.2 SDRE-based Nonlinear Control Problem... 79
3.2.3 SDREF Nonlinear Problem... 80
3.3 Features of SDRE Technique... 81
3.3.1 Controllability and Observability... 81
3.3.2 Stability... 82
3.3.3 Sub-optimality... 83
3.3.4 Additional Degrees of Freedom... 83
3.3.5 Implementation of SDRE Solution... 84
3.4 Application of SDRE Technique for Formation Control... 85
3.4.1 Relative Orbital Dynamics... 85
3.4.2 Relative Attitude Dynamics... 87
3.4.3 Coupled Relative Orbital and Attitude Dynamics... 89
4. SDRE Control System Design 91 4.1 Actuators... 91
4.1.1 SDC Model of OACS... 91
4.1.2 Controllability... 96
4.1.3 Actuators Choice... 97
4.1.4 Actuators Modelization... 99
4.1.4.1 Electric Saturation of Commanded Action... 100
4.1.4.2 Actuators Dynamics... 101
4.1.4.3 Actuators Noise... 102
4.1.4.4 Introduction of Duty-Cycle... 103
4.1.4.5 Discrete Control for Chemical Actuators... 104
4.2 SDRE-based Controller... 105
4.2.1 Control Weight Matrices... 105
4.2.2 Coupling Control Matrix... 106
4.2.3 SDRE Timing... 106
4.3.1 Sensors Modelization... 109
4.3.2 Sensors Choice... 111
4.3.3 Observability... 111
4.4 SDREF... 112
4.4.1 Role of Integral States in the SDREF... 112
4.4.2 Measurement Weight Matrices... 113
4.4.3 SDREF Timing... 114
4.5 Regulator of the Control System... 115
4.5.1 Join of Controller and Observer... 115
4.5.2 User’s Input and Output of Control System... 116
5. Simulation Results 117 5.1 Sample Formation–Keeping Mission... 117
5.1.1 SDRE Control Law... 118
5.1.1.1 Hall Thrusters... 122
5.1.1.2 Discrete Control Implementation due to High and Pulsed Thrust... 123
5.1.2 SDRE Kalman Filter... 125
5.1.2.1 All-States Measurement: 21-order SDC state-model... 127
5.1.2.2 All-States Measurement: 15-order SDC state-model... 128
5.1.2.3 Position & Angles Measurement... 129
5.1.2.4 Orbital & Angular Velocity Measurement... 129
5.1.3 SDRE Control System... 130
5.2 Reconfiguration Maneuvers... 134
5.2.1 Attitude RC... 135
5.2.2 Orbital RC... 137
5.3 Trailing Formation–Keeping Mission... 141
5.4 Up-to-date Simulator Application: Proba-3... 146
5.4.1 Mission Analysis... 148
5.4.2 SDRE-based Controller applied to Proba-3 Mission... 149
5.4.3 Complete SDRE Control System for Proba-3 Mission... 153
6. Conclusions and Future Development 157 6.1 Final Remarks... 157
6.2 Implementations... 159
Appendices A. Fundamentals of Orbital Dynamics... i
B. Simulator Architecture... v
C. SDC State-Model Matrices... vii
C.1 State Matrix... vii
C.2 Control Matrix... x
1.1,3 The A-Train Formation architecture... 3
1.1,4 Spacecraft Formation Flying: missions... 4
2.1,1 Keplerian classical orbital elements... 16
2.1,2 Astrodynamics reference frames... 17
2.1,3 Hill reference frame... 20
2.1,4 Relative orbit solution of Hill’s LTI system... 25
2.2,1 Definition of CRP and MRP as stereographic projection of quaternions... 35
2.2,2 Rotational motion description on MRP space... 36
2.2,3 LISA heliocentric orbit and attitude motion... 36
2.3,1 Geoid... 43
2.3,2 Spherical harmonics of Earth gravity field... 44
2.3,3 Gravity gradient torque references... 46
2.3,4 Third body attraction references... 46
2.3,5 Earth magnetic field... 49
2.3,6 Eclipse models... 52
2.3,7 Solar cycle activity... 54
2.4,1 Formation Flying Simulator block diagram for the relative orbital and attitude control... 57
4.1,1 Spacecraft configuration and OACS arrangement... 92
4.1,2 Alternative OACS configurations... 93
4.1,3 FEEP concepts... 99
5.1,1 Inputs of sample Formation mission... 118
5.1,2 SDRE-based controller weights for sample Formation mission: FEEP OACS... 120
5.1,3 Outputs of sample Formation mission: FEEP OACS... 120
5.1,4 SDRE-based controller weights for sample Formation mission: high-thrust OACS... 124
5.1,5 Outputs of sample Formation mission: high-thrust OACS... 124
5.1,6 SDREF for sample Formation mission: weights... 125
5.1,7 Summary of SDREF outputs with various sensors sets... 125
5.1,8 Outputs summary of 21-order SDRE-based controller for complete Control System... 130
5.1,9 Outputs summary of 15-order SDRE-based controller for complete Control System... 130
5.1,10 Summary of SDREF outputs for complete Control System... 130
5.2,1 SDRE-based controller weights for orbital reconfiguration maneuver... 134
5.3,1 Inputs of sample Trailing Formation mission... 141
5.3,2 SDRE-based controller weights for sample Trailing Formation-Keeping... 143
5.3,3 Outputs of sample Trailing Formation mission: Hall thrusters OACS... 143
5.4,1 Inputs of Proba-3 tested mission... 147
5.4,2 Outputs of controller of Proba-3 tested mission... 151
5.4,3 SDRE-based controller weights for Proba-3 tested mission... 152
5.4,4 SDREF outputs: Control System of Proba-3 tested mission... 154
5.4,5 SDREF for Proba-3 tested mission: weights... 155
Note 2: control forces linecolor is set blue for coded 1 thrusters, cyan for coded 2 thrusters (see Fig. 4.1,1 for references).
2.4,1 Numerical integration trade-off... 59
2.4,2 Sample chief mission... 62
2.4,3 Sample Formation mission... 63
2.4,4 Gravitation field attraction... 63
2.4,5 Disturbance forces and torques... 63
2.4,6 Disturbance forces particularities... 64
2.4,7 J2-free sample Formation... 68
4.1,1 Alternative OACS configurations trade-off... 96
4.1,2 Actuators modelization: saturation... 100
4.1,3 Actuators modelization: dynamics... 101
4.1,4 Actuators modelization: noise... 103
4.1,5 Actuators modelization: pulsed and discrete control... 105
4.2,1 Coupling control matrix trade-off... 107
4.2,2 SDRE timing... 108
4.4,1 SDREF timing... 114
5.1,1 Sample Formation mission: alternative descriptions... 118
5.1,2 Sample Formation mission: performances... 119
5.1,3 Sample Formation mission: consumptions... 120
5.1,4 Sample Formation mission: controller actions... 121
5.1,5 Sample Formation mission: long-term analysis... 122
5.1,6 Hall thrusters: performances... 122
5.1,7 High-thrust: performances... 123
5.1,8 High-thrust: control actions... 124
5.1,9 SDREF: observer actions... 126
5.1,10 SDREF: all-states measurement, 21-order SDC state-model... 127
5.1,11 SDREF: all-states measurement, 15-order SDC state-model... 128
5.1,12 SDREF: position & angles measurement... 129
5.1,13 SDREF: orbital & angular velocity measurement... 129
5.1,14 Complete Control System for sample Formation mission (FEEP OACS, all-states measurement, 21-order SDC state-model): performances... 131 5.1,15 Complete Control System for sample Formation mission
(FEEP OACS, all-states measurement):
Control System actions... 132
5.1,16 Complete Control System for sample Formation mission (FEEP OACS, all-states measurement, 15-order SDC state-model): performances... 133
5.1,17 Complete Control System for sample Formation mission (FEEP OACS, all-states measurement): Control System performances... 134
5.2,1 Attitude reconfiguration maneuver: performances... 136
5.2,2 Attitude reconfiguration maneuver: control actions... 137
5.2,3 Orbital reconfiguration maneuver: non-optimal trajectory... 138
5.2,4 Orbital reconfiguration maneuver: SDRE solution trajectory... 139
5.2,5 Orbital reconfiguration maneuver: control actions... 139
5.2,6 Orbital reconfiguration maneuver: performances... 140
5.3,1 Sample Trailing Formation... 142
5.3,2 Trailing Formation Flying: control actions... 143
5.3,3 Trailing Formation Flying: performances... 144
5.3,4 Trailing Formation Flying: control actions analysis... 145
5.4,1 Proba-3 mission analysis... 147
5.4,2 Proba-3 Formation Flying... 148
5.4,3 Proba-3 Occulter disturbances... 148
5.4,4 Deputy Target relative orbital maneuvers... 149
5.4,5 Controller of Proba-3 tested mission: performances... 151
5.4,6 Controller of Proba-3 tested mission: control actions... 152
5.4,7 Control System of Proba-3 tested mission: performances... 153
5.4,8 Observer of Proba-3 tested mission: performances... 154
a acceleration semi-major axis
ap index of geomagnetic activity
b bias term ecliptic latitude moment arm
nonlinear control-affine equation matrix semi-minor axis
c radiative coefficient: specular reflectance (sr), diffusive reflectance (dr), transmittance (t), absorbance (a), emission (e)
speed of light
e eccentricity
error: target (t), observation (o), measurement (y) Euler axis
d process disturbance
f focal distance
nonlinear differential equation vector
g CRP vector
intensity of gravitational field
h altitude
declination of RSE
specific angular momentum
i inclination
k Gauss gravitational constant
l ecliptic longitude generic length
m mass
mm magnetic dipole momentum
mp propellant mass consumption
n control noise mean motion state dimension
o observed state vector
p orbital parameter or semi-latus rectum pressure
q momentum
intensity of electromagnetic flux quaternions vector
r measurement noise position
rP2P1 distance of P2 from P1 points
s stable state
t specific kinetic energy time u argument of latitude control action v velocity x state vector y measurement output z performance output A state matrix
AF1F2 attitude matrix, related to rotation from F1 to F2 frames
Az azimuth of RSE
AU Astrodynamics Unit, Earth-Sun mean distance: . 11 1 4962 10 mi B induction of magnetic field
Bd disturbance matrix
Bu control matrix
Cn,m EGM first coefficient
Cy state-measurement matrix
Cz state-performance matrix k
continuous functions set, k derivative order
D output-input matrix: measurements: control (yu), disturbance (yd), noise(yr)
performances: control (zu), disturbance (zd)
E eccentric anomaly
EA321 Euler angles, combination 321
F force
generic frame
F10.7 index of solar activity
G barycenter
Cavendish gravitational constant
H angular momentum Hamiltonian matrix hyperbolic anomaly
H
Hamiltonian functionalI generic inertial frame
Ie intensity of electric current
Is specific impulse
Itot total impulse
J cost functional symplectic matrix
J2 Principal Earth gravity field zonal harmonic magnitude
Jn.m EGM coefficient magnitude
K controller gain matrix
L observer gain matrix
L
Lagrangian functionalLST Local Sidereal Time
M generic matrix mean anomaly
MO torque around pole O
N state weight matrix for optimal observer normal of surface
direction of line of nodes
O generic pole of rotation
P parabolic anomaly generic point power
sensitivity matrix
Pn Legendre polynomial
Pnm associated Legendre polynomial
PSD Power Spectral Density function
PSDWN White Noise approximation of PSD Q state weight matrix for optimal controller
quaternions kinematics matrix
R control weight matrix for optimal controller range of RSE
reference length real numbers set
RU length Unit, Earth semi-major axis: 6.378,15km
S coupled state-control weight matrix for optimal observer static moment
Schur form matrix
Sn,m EGM second coefficient
T period thrust
Tm time of maneuver
U eigenvector matrix unitary matrix
V control weight matrix for optimal observer
W weight matrix α generic angle
right-ascension SDC parameter
αA coefficient of attitude matrix direct transformation with MRP β Solar angle
δ declination
ε specific total energy φ latitude
ϕ ϕe
first Euler angle Euler angle
γ Ares Constellation time-dilation coefficient θ polar anomaly
second Euler angle ϑ co-latitude
λ eigenvalue
λn,m EGM coefficient anomaly
μ P 2B-R gravitational constant of planet P ν true anomaly
η damping parameter: 2
1−ξ
ψ third Euler angle longitude ρ relative position σ MRP vector standard deviation τ time constant ω0 bandwidth critical frequency
ωF1F2 angular velocity, related to rotation from F1 to F2 frames ϖ argument of pericenter
ξ damping factor
υ specific potential energy of gravitational field Δ accessibility distribution
difference variation
Δv impulse consumption Λ dual sensitivity matrix
Σ MRP kinematics matrix sum
Ω right-ascension of ascent node Υ potential of magnetic field
Superscripts
• time-derivative
∧ versor
∼ dual
c control action
d disturbance (orbital perturbation: p) u commanded action
Subscripts
In order of writing:
integral state I
body c chief spacecraft
d deputy spacecraft t deputy target E, ⊕ Earth M Moon S, Sun initial condition 0
reference system frame ID
component due to related frame
Coordinate Reference Systems
ECEF: x,y,z Earth Centered Earth Fixed frame
ECI: x,y,z Earth Center Inertial frame
GE: x,y,z Geocentric Equatorial frame
Ge: x,y,z Geocentric Ecliptic frame
Hb: r,θ (or t),h Hill’s frame of body b
PIb: x,y,z Principal of Inertia frame of body b
SE: x,y,z Sun Equatorial frame
2B-R Two Body Restricted
3B Three Body
ACS Attitude Control System ARE Algebraic Riccati Equation app apparent
ATT Attitude
CW Clohessy-Wiltshire CP Chemical Propulsion CRP Classical Rodrigues Parameters CS Coronagraph Spacecraft (Proba-3) DEM Digital Elevation Model
DC Duty-Cycle EGM Earth Gravity Model EP Electrical Propulsion ESA European Space Agency EWSK East-West Station-Keeping FEEP Field Emission Electric Propulsion
FF Formation Flying
FH Finite-Horizon FK Formation-Keeping GEO Geostationary Earth Orbit GMT Greenwich Meridian Time
GNC Guidance, Navigation and Control system GPS Global Position System
GSO Geosynchronous Earth Orbit GT Ground-Track
H2 H2 Norm Control HEO Highly Elliptical Orbit
HJB Hamilton-Jacobi-Bellman H∞ H∞ Norm Control
IGRF International Geomagnetic Reference Field ID Identifier
IH Infinite-Horizon IR Infrared
JD Julian Day
LEO Low Earth Orbit LQ Linear Quadratic LQG Linear Quadratic Regulator LQR Linear Quadratic Gaussian LTI Linear Time-Invariant MEO Medium Earth Orbit
MJD Modified Julian Day
MRP Modified Rodrigues Parameters
MSISE Mass Spectrometer and Incoherent Scatter radar, from ground to Exosphere
NASA National Aeronautics and Space Administration NL Nonlinear
NRL National Research Laboratory NSSK North-South Station-Keeping OACS Orbital and Attitude Control System OCS Orbital Control System
ODE Ordinary Differential Equation ORB Orbital
OS Occulter Spacecraft (Proba-3) PPU Power Processing Unit RC Reconfiguration RH Receding-Horizon RMS Root Mean Square ROA Region of Attraction
RSE Relative Spherical Elements RP Rodrigues Parameters SAR Synthetic-Aperture Radar SDC State-Dependent Coefficients SDRE State-Dependent Riccati Equation sec secular derivative
SK Station-Keeping SRP Solar Radiation Pressure SSO Sun-Synchronous Orbit SOI Sphere Of Influence
TI Time-Invariant TV Time-Variant
UT Universal Time
exploit the huge possibilities of this kind of spacecraft configuration, which consist in lowering inter-satellite range to allow new possible spacecraft operations. Hence, autonomous control is the most important field that must be developed, in order to provide suitable safety and performances at these small distances. The aim of this work is to apply one of the current major approaches to solve nonlinear control problems, the State-Dependent Riccati Equation technique, for the relative orbital and attitude dynamics of Formation sub-satellites, making use of a coupling actuator device.
The approach has consisted in designing a proper thruster set as actuator system and building a true-life Simulator for the relative deputy satellite motion, that can be used for every kind of Earth orbit Formation, in order to implement a nonlinear SDRE control system algorithm. All the major aspects of actuators modelization and sensors set choice have been considered, and the design of controller and Kalman Filter takes into consideration the important practical issue of SDRE timing. In particular, continuous electrical low-thrust and pulsed chemical high-thrust have both been considered for the actuator implementation. Simulations have been conducted choosing some typical sample mission requirements, and using a new kind of J2-free Formation long-period maintenance strategy: in particular, ESA Proba-3 requirements have been used as up-to-date test-bed for the designed SDRE coupled relative control system. Results obtained have proved that this coupled control system is a feasible actuator solution for Formation-Keeping and attitude pointing: low thrust propulsion has resulted the best solution, and has the advantage of not only do not compromise pointing performance, but also to allow to intrinsically moderate high frequency aspects of attitude control.
In conclusion, SDRE methodology provides good performances and advanced possibilities to deal with relative control of future complex Formation Flying missions.
Keywords: Formation Flying, nonlinear control, State-Dependent Riccati Equation, SDRE, coupled relative control, Orbital and Attitude Control System, SDRE timing, J2-free Formation.
sfrutteranno ancora di più le enormi possibilità offerte da questa architettura, mantenendo i satelliti sempre più vicini per compiere complesse operazioni nello spazio. Quindi, il controllo automatico è la maggiore area di sviluppo, in quanto deve garantire adeguate sicurezza e prestazioni a tali ristrette distanze. L’obiettivo del mio lavoro è di applicare uno dei moderni metodi per il controllo non lineare, la soluzione dell’equazione di Riccati in funzione dello stato (SDRE), alla dinamica relativa orbitale e d’assetto dei sotto-satelliti nelle Formazioni, utilizzando degli attuatori accoppiati.
Il metodo è consistito nel progetto di un sistema di razzetti come attuatori e nella costruzione di un simulatore del moto relativo per satelliti in Formazione, utilizzabile per tutte le orbite terrestri, in cui viene implementato l’algoritmo di controllo non lineare SDRE. Tutti gli aspetti per la modellizzazione degli attuatori e la scelta dei sensori sono stati considerati, ed il progetto di controllore e filtro di Kalman considera l’importante problematica della temporizzazione dell’algoritmo SDRE; inoltre, sia la propulsione elettrica che quella chimica ad alta spinta pulsata vengono testate come sistema di attuazione. Le simulazioni sono state condotte con tipici requisiti di missione, ed usando una nuova strategia per il mantenimento di Formazioni senza controllo di J2: in particolare, la missione Proba-3 dell’ESA è stata usata come banco di prova per testare il sistema di controllo progettato. I risultati ottenuti hanno dimostrato che il controllo accoppiato è un soluzione fattibile per il mantenimento orbitale e di puntamento: la propulsione elettrica è risultata la soluzione migliore, avendo il grande vantaggio di non pregiudicare le prestazioni di puntamento, ma anche di moderare intrinsecamente in alta frequenza il controllo d’assetto.
In conclusione, il metodo SDRE fornisce buone prestazioni e possibilità avanzate per il controllo relativo delle future e complesse missioni di volo in Formazione.
Parole chiave: volo in Formazione, controllo non lineare, State-Dependent Riccati Equation, SDRE, controllo relativo accoppiato, sistema di controllo orbitale e di assetto, temporizzazione SDRE, Formazioni libere in J2.
1.1 Spacecraft Formation Flying
Spacecraft Formations have become, in the last decade, a topic of great interest in the space mission design both for NASA and ESA. Basically, we talk of Formation Flying when two or more spacecraft move in close proximity, relatively to orbit size: for common Earth orbits, that means about few kilometers. So the main difference from Spacecraft Constellation is the objective of the mission: when the purpose of a space mission cannot be accomplished by a single spacecraft due to typical technical difficulties and size constraints, we achieved it with a Formation of satellites, performing an unified space mission together; instead, Constellation objective is, since the beginning, to be
constituted from different satellite missions.
Keeping in mind the design philosophy that all the spacecraft in a Formation want to constitute a single bigger one, it’s important to say that every Formation Flying mission is defined by the absolute dynamics of a reference object, which can be the chief satellite of the Formation, or an asteroid, or even a non-physical
point of interest for the mission, and the relative movement around this reference of the real deputies satellites. Other terms in use for the reference
satellite are master or leader, while the others spacecraft are also referred as slave or follower.
The aim of this thesis work is to study one of the most important issue of spacecraft Formation Flying, which is the relative control of deputy satellite movement, using a modern nonlinear technique, and considering the coupling of the two actuators systems, which is using an unique set of thrusters for controlling both orbital and attitude motion: in addition, also practical aspects of physical implementation of the control system will be investigated.
Formation missions so far have been characterized by one kind of simple position of two spacecraft, called Trailing configuration, where the second satellite follows the leader around its same orbit, maintaining a quasi-fixed relative position, which is a difference in phase angle with respect to the absolute central body. In fact, that configuration has been of large interest for Earth observation missions, because targets can be viewed from multiple angles at multiple times: that allows to realize a differential interferometric synthetic-
Fig. 1.1,1. Spacecraft Formation Flying: concepts. At left, example of Trailing Formation: Landsat-7
being followed by EO-1, covering the same area at different times. At right, example of Cluster Formation, for the proposed TechSat-21 mission.
aperture radar (D-InSAR)1. Historical example of Trailing Formation is the latest satellite of the NASA Landsat program, Landsat 7, launched on 1999 to provide up-to-date and cloud-free Earth images and maps, which constitutes a Formation pair with Earth Observing 1, that trails the leader one minute after, on the same orbit. A-Train is a NASA-CNES upgradable Formation for surface and atmosphere study, constituted by a train of seven satellites: starting from the head we have OCO (now failed), Aqua (first one to be launched, in 2002),
Fig. 1.1,2. SAR applications. At left, SAR polarimetry image of Death Valley, that allows materials
identification. At center, SAR interferometry image of Kilauea (Hawaiian volcano), showing topographic fringes. At right, 3D rendering of a DEM of Tithonium Chasma (Martian canyon).
1
SAR is a technique which provides both high field of view and resolution, using the relative motion between a microwave antenna and the target region, widely used for remote sensing and mapping of the surfaces of both the Earth and other planets. But with two samples, interferometry analysis of the phase difference provides also information about the angle from which the radar echo returned: if the two samples are simultaneously, that allows to determine the position in three dimensions, so with altitude, of the image pixel (digital elevation model, DEM); if the samples are taken at different times, that allows to determine the terrain motion.
Fig. 1.1,3. The A-Train Formation architecture.
CloudSat, CALIPSO, PARASOL (now removed), Glory (scheduled to launch in 2011) and Aura. But the more successful mission of that kind has been GRACE (Gravity Recovery And Climate Experiment): placed in polar LEO, with the use of a K-band microwave tracking system for the accurate measurements of the inter-spacecraft range change, and of course an high sensitive gradiometer, it described the high order EGM2008 spherical harmonics of Earth gravity field, considering also the temporal effects of ocean and solid tides.
In recent years, in-deep studies of Formation Flying concepts have made possible the realization of more complicated missions. Formations in Cluster configuration are constituted by various satellites tightly spaced. ESA Cluster mission has been the forerunner (in collaboration with China National Space Agency): in 2000, Rumba, Tango, Salsa and Samba constituted the angles of a tetrahedron Formation which still provides important results on the high Magnetosphere dynamics. ESA has also in-plan two important missions. Proba-3 is specifically designed to develop technologies enabling the future Formation Flying missions, using two spacecraft in HEO to test new space components in order to perform relative orbital and attitude maneuvers never done before: however, the main objective is to study, during apogee passage, the coronal emissions of the Sun, placing the chief satellite in-shadow of the deputy. With these ambitious objectives, Proba-3 can be the milestone for new horizon developments in exploitation of Formation Flying. LISA (Laser Interferometer Space Antenna, in collaboration with NASA) is, maybe, the most popular and anticipated mission in the science world: its purpose is to prove the existence of gravitational waves, which are theorized to be responsible of the time delay near a gravity field. That will be proved using a Cluster configuration of three spacecraft in heliocentric orbit, which will maintain a triangular position while chief center position changes. NASA has also planned some ambitious Cluster Formations so far, as Terrestrial Planet Finder, New Millennium Interferometer (actually shifted in various program names, precisely Deep Space and Space Technology) and TechSat-21, but all of them have been postponed or cancelled due to funding lack: hope is now posed in Magnetospheric Multiscale Mission.
Fig. 1.1,4. Spacecraft Formation Flying: missions. At left, concept of GRACE satellite pair. At right,
operational architecture of LISA Pathfinders.
Satellite Formation Flying is a configuration that has numerous potential benefits to solve common problems in space mission design. Basically, weight
and cost are the clearest issues that Formations are able to solve, because
smaller and cheaper satellites lead indeed to reduce launch and life cycle cost, simplify the design, and have fast build times. But this kind of missions are intrinsically characterized by two other important items: flexibility and reliability. The fact that a Formation is constituted by different satellites doesn’t
make it static: so reconfiguration maneuvers can be executed to produce a lot of very different applications and studies from one single space mission, even in order to meet changes in operational needs; that also means that a Formation has the ability to upgrade with time. The second property is a great deal for satellite clusters: possible problems of one single spacecraft don’t affect mission failure, because spare satellites can replace it; that’s an important point for maintenance of space missions. These unique advantages are not only used in the civilian sector, for purposes like science observations and improved communications as mentioned before: from a military standpoint, Formation has a great
survivability, as multiple satellites are obviously harder to destroy than a single
one; the U.S. Air Force is exploring Formations for use in surveillance, passive radiometry, terrain mapping, navigation, communications, and ground target identification.
The basic concepts behind relative satellite motion were developed in the 1960’s by W.H. Clohessy and R.S. Wiltshire, who were working on methods of vous for on-orbit operations [1]. Although developed specifically for rendez-vous, these relative equations of motion work well in the design of useful relative orbits. CW equations assume that the leader satellite exists in a circular orbit, and are nonlinear. A linear form of the CW equations can be derived with the additional assumption that the distance between the leader and follower satellites is small in comparison to their orbital radii; these are the well-known Hill’s equations for rendez-vous and docking maneuvers. Although all these
limitations, Hill’s solution (a pseudo-ellipse, expressed in the moving Orbital frame of the master satellite, from that known as Hill’s frame) characterizes the behavior of deputy motion for all of the most common Formation designs. An useful parameterization of this special case of relative orbit is presented in Irvin’s thesis [2]. The big difference is the fact that rendez-vous and docking are fast operations, relating to spacecraft lifetime, so accurate Formation mission analysis should be avoided to use linearized approximations.
Attitude dynamics of spacecraft in Formation Flight is obviously as important as orbital motion: in fact, all the Formation mission objectives can’t be accomplished if the deputies are not good oriented to achieve chief mission purposes. The main concept is that relative attitude dynamics problem has nothing of special for space mission field, but it’s just the angular motion of two rotating frames, resulting from the solution of the nonlinear Euler equations. Issues are instead related to introduction of attitude equations to complete the six degrees of freedom characterization of a rigid body: forces related to configurations and torques related to position bring on coupling of the two dynamic behaviors.
1.2 Control of Spacecraft Formations
The most important issue that involves Formation missions is, without of doubt, the autonomous control of its sub-satellites, which is necessary to provide fast
and accurate response times that would not be achieved with ground control. Control of spacecraft Formations is so difficult, and so important, because it is characterized by a lot of peculiarities.
First of all, it is a centralized control; actually difference between centralized
and decentralized terms depends on the reference topic: if we think about Formation architecture as a system network, centralized control means that sub-satellites motion is described relatively to the chief position, while decentralized control indicates that sub-satellites position is treated independently from chief one; whereas, if we put the accent on single spacecraft system, the two terms are shifted in meaning. In this thesis, first description is the reference, so only the chief satellite (if it exists) is designed with a classical absolute controller, whereas deputies satellites are controlled in a relative way, and very accurate
positioning is request just because these spacecraft are separated by only few kilometers, while they fly at absolute speeds of km/s! This means that target deputies trajectories and attitudes must be always designed relatively to chief position; then, control must keep into account collision avoidance strategies for
maneuvers or configurations where distances between satellites are set to be very close; in addition, control system requires also accurate measurement of relative state: typically, Formations use on-board high precision differential
interferometry sensors (millimeter resolution) to provide the relative position signal. Recent analytical studies (Chang, Park, Choi [3]) are considering the hypothesis to provide the opportunity to switch attitude control of deputies satellites into a decentralized algorithm, in order to build a selective control strategy for big Cluster Formations that will allow to choose the optimal
solution in case of long-period reconfiguration maneuvers or control problems coming out from one or some sub-satellites; this solution derives from multi-agents theory, which can be applied to every cluster of autonomous controllers.
In addition, control of spacecraft in a Formation involves close relation of both translational and rotational motion, an aspect that is not widely been considered
so far in literature: in fact, satellite target configuration is indeed dependent of its position in space, but for Formation Flying that means that deputies mission operations constrain relative attitude not only respect to the chief one, but also to the relative orbit position in respect to the chief, and viceversa. But the rigid body six degrees of freedom are also naturally coupled in dynamics: forces like
aerodynamic drag and radiation pressure are dependent on configuration, whereas torques produced by gravity gradient and geomagnetic induction are dependent on position. Last, but not least, coupling of the two dynamics also spills in the control system, at two levels: controller algorithm must consider it
for accurate determination of the feedback law; but also actuators give the last
source of dynamics coupling: first of all, because thrust resultant may not be perfectly aligned with the instantaneous center of mass of the spacecraft, and secondly, because thrusters may be used for both orbital (simply named OCS) and attitude control (simply named ACS). Coupling of orbital and attitude control has indeed the advantage of lighten propulsion system design (easily named OACS), but the problem is that the two dynamics have very different time response: attitude pointing requires fast and fine control, while orbital
positioning is intrinsically slower and powerful. This thesis will so focus on coupled control of orbital and attitude dynamics of relative deputy motion, using OACS as actuator system.
Another important peculiarity of Formation control is indeed the fact that it is intrinsically nonlinear in both orbital and attitude frame. That entails the use of
two types of control techniques: take out the nonlinearities from the model, and so use classical control theory, or use appropriate nonlinear techniques. Last solution has the advantage of not making approximations, which will give a loss of performance of the control system, and that is a necessary requirement for Formation control, because model errors, propagating for the appropriate mission analysis time, will determine conspicuous control action to be compensated: so difference between use of rendez-vous/docking simplified
equations and exact nonlinear model for Formations is now well understood. The problem of nonlinear control techniques is that an unified analytical theory cannot be done: so, if analytical approach is too hard or impossible, control laws
must be verified via a computer simulator, in order to analyze response
performance and test the stability of the system.
First analysis of Formation orbital control, throughout literature, have been conducted via linearization approach: that made possible the use of classical linear control techniques as LQR, LQG, H2, H∞. Vassar and Sherwood [4] made in 1985 the very first investigation of the formation-keeping control problem, but Redding et al. [5] were the firsts to design an LQR controller to contrast the applied disturbance actions, using an appropriate Kalman Filter to recuperate the entire relative state in Hill’s frame (a similar work then has been done by Kapila [6]). In 2000, Yedavalli and Sparks [7] realized an hybrid control system analysis. Inhalan et al. [8] used instead a linearization based on small curvature assumption. Finally, elliptic orbit case for the reference satellite started to be taken into consideration: first major works are due to Lawden [9], McInnes [10], Manikonda at al. [11], Kapila et al. [12], and Chao [13].
Formation attitude control has recently been investigated by Chang et al. [3] and Xing et al. [14], both using quaternions for kinematics description: in this thesis, a modern attitude set, that is considered the best evolution of quaternions, is used for the parameterization of deputy relative attitude.
Very few researchers have considered orbital and attitude coupled dynamics in satellite Formation Flying control design: major results were obtained by Wang at al. [15], Kapila et al. [16] and Naasz et al. [17].
Nonlinear control strategies start to be investigated only in recent years: Alfriend, Schaub and Gim in [18],[19],[20] studied the influence of simplified assumptions of orbit circularity, differential perturbations (Earth zonal harmonics overall) on the variation of orbital elements and fuel consumption for a Formation. Schaub and Junkins [21] formulated then an alternative, in respect to Cartesian, description of relative motion using orbital elements difference, with Gauss planetary equations approach; that work was upgraded by Breger and How [22] including J2 effect.
The major nonlinear control techniques for state-model systems, very sensitive to the case in study, can so be summarized:
• Control Lyapunov Function (CLF): the method consists to find an analytical Lyapunov function for the nonlinear system, and then apply linear control laws to it.
• Satisficing: it is an application of CLF with parameterized control laws, based to a concept of decision-making theory, which is to meet criteria for adequacy, rather than to identify the optimal solution.
• Dynamic Inversion: this method wants to build a feedforward law, but it’s very difficult to successfully apply.
• Nonlinear Optimal Control: it consists in solving Hamilton-Jacobi-Bellman equation (HJB) for particular cases of nonlinear systems.
• Sliding Modes: this technique defines a Lyapunov function and its surface on the state domain, and search the discontinuous feedback law to carry the state on that stability surface.
• Recursive Backstepping: this control strategy is applicable for a special class of nonlinear systems, that are constituted by cascaded subsystems, and the aim is so to start the design process at the known-stable subsystems and back out new controllers that progressively stabilize the irreducible ones.
Optimization was then applied to nonlinear techniques: Folta and Quinn presented the 3D Method [23], a suitable approach for Formation orbital control even at great relative distance; Massari paved the way to collision avoidance issues, englobing them in the optimal control formulation [24].
1.3 SDRE Technique for Control Design of Nonlinear Systems
In the second half of the 1990s, Cloutier, Mracek and D’Souza [25],[26] presented a new nonlinear control methodology, based on a systematic procedure and so applicable to all nonlinear systems.
This technique is known as SDRE, because the control gain matrix is obtained from the solution of the State-Dependent Riccati Equation. This equation has
the same exact structure obtained from Linear Quadratic Optimal Control theory, but the peculiarity is that all the matrices are state dependent. The procedure so consists on transforming the nonlinear system dynamics in an identically pseudo-linear form, where the classical A-B-C-D matrices are not constant, but their coefficients are function of the state itself: this parameterization of a linear system is called State-Dependent Coefficient (SDC). So SDC system, at a given state, can be controlled with an Algebraic Riccati Equation (ARE) solution, to give the gain matrix K, in order to optimize the associate state-control energetic cost function, where the weight matrices Q-R can also be expressed in an SDC form.
Obviously, identical considerations can be made to design a Kalman Filter for the nonlinear system (SDREF [27]).
Advantages of SDRE technique are evident. First of all, it is a systematic
methodology, which allows to describe systems in the well-known linear-state model. Second, it allows to apply every linear technique, with the state
dependency assumption. Then, SDC form provides new opportunities to improve control system performances: for example, weight matrices can be
changed to take consideration of variable trade-offs during a physical simulation, but also controllability and observability can be modified with appropriate choice of the SDC parameterization.
Although all this benefits, SDRE isn’t fault-free. First thing to be underlined is that this formulation gives a sub-optimal solution of the control system. Then, it doesn’t provide an analytical expression of the closed-loop system, so
simulations typically involve a lot of calculations at every step. Last but not
least, state dependency of the SDC system doesn’t provide warranty of global asymptotic stability: in fact, it has been proved [28] that stability is provided
only in a region close to the origin of the state domain; analytical demonstration of stability of the system in all the space domain requires to find a Lyapunov function, with star-convex property [29] or ability to be used in the Satisficing technique [30], and so, as said before, can be derived only for some simple cases.
In fact, after the presentation of this new methodology, some in-depth studies on estimation of the region of stability have been done. Erdem and Alleyne [31] found an analytical estimation for the case of three-order system; successive work of the same two researchers [32], Bracci et al. [33], produced a technique based on vector norms, but which presents high computational load for medium-high order systems.
So the only way to assure that the control system is stabilizing is always to test him on a computer simulator, under various state initial values.
SDRE methodology has immediately found application on Formation Flying control: Irvin and Jacques demonstrated that an SDRE controller has better performance on an LQR/Hill’s equation-based controller for a Formation Flying mission where the chief orbit is circular and the deputy moves in close proximity [2]. Cloutier and Stansbery provided the first SDRE controller for orbital and attitude control of spacecraft in Formation, which follows tracking signals using integral servomechanism implementation [34].
Nonlinear SDRE-based control, in fact, has been successfully used in a various number of engineering fields so far: spacecraft control, autonomous pilots for missiles and aircrafts, process control, robots, biomedical devices (artificial pancreas), Maglev trains.
1.4 Thesis Breakdown Structure
This thesis follows the work done so far by two previous graduating students on Formation Flying control issues, that produced an SDRE-based orbital controller and observer algorithms [35] and an SDRE-based orbital and attitude controller [36]. The aim is now to implement a Simulator for Formation Flying missions to test the performances for station-keeping and reconfiguration maneuvers of an
SDRE-based complete orbital and attitude control system, considering now the
case of dynamics coupling also due to control actuation, that is an OACS.
Actuators modelization will be accurately considered for the analysis of the trade-offs that will bring to the design of the control system: the background aim is so to implement SDRE methodology without forgetting also practical issues,
control system will be simulated on real and up-to-date Formation Flying missions.
My work starts with the building of the Formation Flying dynamics in the Simulator, whose specifics are presented on the next section. Chapter 2 provides a basic analytical description of orbital and attitude mechanics for the propagation of the dynamics of spacecraft Formations; also, modelization of space environment is highlighted for the determination of forces and torques acting on satellites. The final part of this chapter is dedicated to the choice of target dynamics for the deputy satellite, which is the reference signal that control system employs to move the spacecraft, following desired requirements; during this work, an important result on target choice for J2-free Formations mechanics has been found: the innovative suggestion is applicable when relative orbit has to be maintained for long-term periods, providing both propellant consumption saving and stability of the inter-satellite range.
Chapter 3 presents an analytical background of the SDRE methodology, with its important results obtained so far in applied control theory. Second part of the chapter is dedicated to derivation of SDC form of the differential equations describing Formation Flying dynamics.
In Chapter 4, SDRE technique is applied to the design of a control system for relative orbital and attitude motion of Formation satellites. First of all, actuators modelization is analyzed, for the case of control coupling. Second item of the chapter is the implementation of the SDRE-based control in the Simulator. Then, a discussion is dedicated to sensors choice and modelization for Formation Flying applications. Consequently, SDREF is then implemented in the Simulator to derive a Kalman Filter. In conclusion, the two problems are matched: a complete control system is derived for the problem of relative orbital and attitude dynamics of Formation Flying satellites.
Chapter 5 collects all the results obtained from the Simulator that have been used to validate the trade-offs analyzed in the previous chapter. Last part of the chapter is dedicated to apply the Simulator for up-to-date Formation Flying missions: Proba-3 requirements provide the test-bed for the designed control system.
In conclusion, Chapter 6 addresses so final remarks and future possible implementations.
1.5 The Simulator
As discussed before, nonlinear control algorithm performances require to be tested via computer simulation. In addition, accurate modelization of space environment is extremely important for the design of mission requirements of the spacecraft control system.
Analysis of Formation control issues handled during this thesis work are conducted on a Simulator of the spacecraft dynamics. This Simulator has been built step-by-step, according to the matters dealing with this thesis.
As explained before in previous section, the implementation of the Formation satellites dynamics has been the first of these steps: the analysis is conducted for
the basic case of a centralized control of a pair of Formation satellites, a chief and a deputy, in Earth orbit environment. Extension of control issues for
centralized Cluster Formations is, in fact, easily implementable multiplying dynamics and control system of the deputy satellite, and will not be analyzed in this thesis.
The following steps are dedicated to the control system design: controller and observer are first applied independently; coupling of the two systems is so the final stage.
A presentation of user’s input and output is provided in this thesis: precisely, Par. 2.4.2 is dedicated to orbital and attitude dynamics, whereas Par. 4.5.2 focuses on control system requirements. Clearly, a round-up of all the possible output graphs is spread all over the thesis.
Formation Flying mission involves maintenance and reconfiguration of the geometric position between each spacecraft member in order to follow mission requirements. Obviously, mission objectives constrain also the spacecraft orientation: each operation mode that characterize a Formation mission usually modifies spacecraft attitudes; by instance, inter-satellite communication requires adequate alignment between transmitter and receiver devices of the two satellites, while on-board science instruments need to coordinate actions between Formation members to acquire their function with adequate pointing. Again, it’s important to underline that, in a Formation, position and attitude of deputies satellites are meaningful if they are expressed relatively to chief position and attitude, rather than in an absolute reference.
Important references for delving into basic arguments of this chapter are definitely [21], [37], [38], [39], [40], [41].
2.1 Orbital Mechanics
Celestial mechanics deals with dynamics of a system of bodies subjected to one kind of force, which is gravitational attraction, as expressed by Newton’s law of Universal Gravitation (Eq. 2.1,1), that describes the force acting on body 2 by the mass of body 1, in an inertial reference system:
,1 1 1 2 1 2 2 2 2 2 ,1 1 2 2 ˆ ˆ G G m F G m g m r F r = = = − (2.1,1)
where the field character is highlighted by the existence of the gravitational conservative (because space dependent) field g, and the attraction character is
showed by the direction through the acting body; G is the Cavendish
gravitational constant, that can be also expressed by the use of Gaussian gravitational constant k and Sun mass m :
2
Describing the position during time of this system of bodies is the well-known N-body problem: analytical solution of this problem does not exist, except when the number of bodies is limited. The most important case is the 2-Body problem (2B), because motion of celestial bodies is often dominated by gravitational attraction of one massive star or planet, whose solution is obtained considering an inertial reference frame (so assuming its movement to be at constant speed) centered on the barycenter of the system, which is on the conjunction between the two bodies, and with the further hypothesis of neglecting the gravitational action of one of the two bodies, that brings to the restricted formulation of the problem (2B-R): dynamic equivalence is expressed by setting one of the two bodies fixed (let body 1 to be the primary, so assuming its position known), with the mass of the entire system M, and to study the motion of the other one,
having a so-called reduced mass mr, around it. This problem is now characterized by only three degrees of freedom to be determinate, neglecting to consider the six absolute constants of motion of the reference, which are its velocity and initial position. Curiously, the problem is overcome expressing so the solution in a relative way, as dynamics of Formation is defined. For most practical cases, restricted formulation is well applicable: for spacecraft or natural satellites dynamics around Earth, Sun or others planets (apart from motion around Sun of planets too) the formulas of the two new masses coincide with the two related physical values, due to the fact that mass ratio is very high (m1>>m2). 1 2 1 1 2 2 r M m m m m m m m M = + ≈ = ≈ (2.1,3) 2.1.1 Orbital Dynamics
Dynamic equations of the 2B-R problem are easily obtained by Newton’s second law of dynamics by expressing the gravitational force with Eq. 2.1,1 and using dynamic equivalence of masses from Eq. 2.1,3; due to mass proportionality, the law can be stated directly in the acceleration form:
1 ,1 1 1 1 1 2 2 2 2 1 2 2 1 3 2 2 2 ˆ G r GM m r F r g r r r r μ = − ⇒ = = − = − (2.1,4)
where μ is the primary gravitational constant, which for the Earth is equal to
. , , 14 3/ 2
3 986 004 415 10 mi s . Now let neglect subscripts and superscripts for expression of position.
As it can be seen, this is a nonlinear second order differential equation system, that can be expressed in six ordinary differential equations due to kinematic equivalence of velocity: 3 r v v r r μ = ⎧ ⎪ ⎨ = − ⎪ ⎩ (2.1,5)
The system represents the equations of a central motion, because acceleration is instantaneously directed toward the center body, and so trajectory evolves in a plane. Six first order equations are matched with as many integrals of motion, which are the specific angular momentum vector h, the eccentricity vector e and the specific total energy ε (see Appendix A for the derivation of integrals of motion).
Solution can be easily expressed on polar coordinates (see Appendix A for the complete derivation using Binet’s formula), in the plane of the motion:
( )
(
)
0 1 cos p r e θ θ θ = + − (2.1,6)with the introduction of the orbital parameter p=h2/μ and the magnitude and the direction, given an arbitrary reference direction, of eccentricity vector in orbit plane. Eq. 2.1,6 describes a conic trajectory, with the primary body in one focus position, and eccentricity is the discriminant value of the well-known possible cases of elliptic (e<1, with circular case for e=0) and open trajectory cases of hyperbola (e>1) and parabola (e=1). It’s possible to demonstrate that also total energy can be a discriminant for closed (ε<0, with a minimum for the circular
case) and open orbits (ε>0, with parabolic case for ε=0). It’s important to
underline that the orbital parameter has a physical meaning as the radial distance in quadrature position from the reference direction, and minimum distance (pericenter) occurs along positive direction of eccentricity, while, in case of close orbits, maximum distance (apocenter) occurs at opposite direction: for that reason, the direction of eccentricity defines the apsidal line.
So the solution found describes the kind of trajectory, whose position in space dependent also from the orbital plane attitude. From the six integrals of motion it’s possible to derive six orbital elements, which leads to more physical comprehension of orbits: many sets of parameters exist, but Keplerian classical orbital elements are the most used. From directions of angular momentum and eccentricity in the orbital plane, it’s possible to derive three angles: the right ascension of ascent node Ω and the inclination i define the first one and so the
Fig. 2.1,1. Keplerian classical orbital elements.
plane position, whereas the argument of pericenter ϖ defines the second; eccentricity magnitude e, as previously said, describes the orbit geometry; from
total energy, another parameter is derived, which is the semi-major axis a, that
really defines the size, and so the energy, of the orbit.
An orbit is now completely defined, so the sixth parameter outlines the time dependency of the 2B-R problem, which was included in the expression of the integrals of motion, and that is simply the tangential position of the second body, which is called the true anomaly ν, as the angle from eccentricity direction, so from pericenter position.
2.1.2 Orbital Kinematics
Kinematics requires first of all the definition of coordinates reference systems. Two kind of reference frames must be distinct: inertial and non-inertial.
Inertial frames are, in fact, considered fixed because their motion become important only in centenarian and millenarian analysis: far stars are optimum points for reference directions. For this reason, these frames need a reference time to be referred, that is called epoch: J2000 epoch is the actual frame in use and dates are calculated in days from midnight of Greenwich meridian (GMT) on January 1st 2000, which is the 2000 Modified Julian Date time system (MJD2000). The most practical way to define a celestial reference system, which is obviously formed by an origin point and three orthogonal axis, is to declare three elements: the central body, a plane that passed through it, and a direction on the plane. This reference direction constitutes the first axis, and is historically chosen as the direction of Ares Constellation, often called γ point: J2000 considers γ position at the start of this epoch. The central body depends on the analysis: obviously, for a 2B-R problem, it must be the center of mass of the primary body. Last, two kinds of reference planes are the most useful: the plane of orbital motion, or the equatorial plane of the central body.
Fig. 2.1,2. Astrodynamics reference systems. At left, relationship between Ge and GE (or ECI) frames.
At center, relationship between ECI and ECEF frames. At right, relationship between ECEF and SEZ/TSE frames. Reference direction for Vernal Equinox is J2000.
In this thesis, due to the fact that spacecraft on Earth orbits are analyzed, three inertial systems have been used: Geocentric Equatorial or indifferently called Earth-Centered Inertial (GE and ECI), Heliocentric Ecliptic (Se, where Earth ecliptic is the reference plane) and Geocentric Ecliptic (Ge). Directions in these systems are defined by two angles: declination for out-of-plane component (δ for Equatorial, b as Ecliptic latitude) and right ascension for in-plane
components (α for Equatorial, l as Ecliptic longitude).
Non-inertial frames are used when rotation of the reference, so its attitude, become important for analysis. Taking into consideration Earth rotation, Earth-Centered Earth-Fixed frame (ECEF) is the rotating GE, where the first axis is continuously aligned with Greenwich Meridian. If surface position on Earth is important, topocentric frames are used: these systems are obtained considering a spherical Earth, and are defined by multifaceted choices of the axis orientation. For example we have Top-East-Nord and North-East-Down (TEN and NED, defined by their latitude φ and longitude ψ), that differs by use of Zenith (often Z is used in place of T) or Nadir radial direction; Top-East and South-East-Zenith (TSE and SEZ, defined by its co-latitude ϑ and longitude). Considering spacecraft orbit motion, and neglecting 2B-R assumptions, so orbit plane isn’t fixed, the Perifocal frame (PF) has the instantaneous focus, orbit plane and eccentricity direction as the reference elements.
A spacecraft-centered system used in orbital mechanics is the Orbital or Hill’s frame (O, H or RCO): its axis are the instantaneously radial, cross-track (also said transverse) and out-of-plane directions, so, for a 2B-R problem, this frame rotates with the instantaneous orbital angular velocity and acceleration:
(
)
(
)
2 2 3 3 3 0 0 1 cos 0 , 0 2 1 cos sin 2 h H H r h h h h v h e r p r v e e p r θ μ ω ν ν ω ω μ ω ω ω ν ν ν ω ⎧ = = = + = ⎡ ⎤ ⎡ ⎤ ⎪ ⎪ ⇒ =⎢ ⎥ =⎢ ⎥ ⎨ ⎢ ⎥ ⎢ ⎥ ⎪ = = − + = − ⎢ ⎥⎣ ⎦ ⎢ ⎥⎣ ⎦ ⎪⎩ (2.1,7)Importance of this frame is well explained by the easy way to express the state vector, that can be reported in the inertial frame with a three consecutive matrix rotations of the four keplerian angles, where u=ϖ+ν is the argument of latitude, and N is the line of nodes direction, which is the line connecting the two point
(nodes) of the orbit that intersect the reference plane, considered positive toward the ascent node.
( )
( )
1 0 1 cos 0 sin 1 cos 0 r H T I I H H h T I r I H H H h r p r r e r A r r v e v A v v v e p v θ θ ν ν μ ν ⎧ ⎡ ⎤ ⎡ ⎤ ⎪ =⎢ ⎥= ⎢ ⎥ ⎪ ⎢ ⎥ + ⎢ ⎥ ⎧ = ⎪ ⎢ ⎥ ⎢ ⎥ ⎪ ⎣ ⎦ ⎣ ⎦ ⇒⎪ ⎨ ⎨ ⎡ ⎤ ⎡ ⎤ ⎪ ⎢ ⎥ ⎢ ⎥ ⎪ =⎩ ⎪ = = + ⎢ ⎥ ⎢ ⎥ ⎪ ⎢ ⎥ ⎢ ⎥ ⎪ ⎣ ⎦ ⎣ ⎦ ⎩ (2.1,8)( )
ˆ( )
ˆ( )
ˆ 3( ) ( ) ( )
1 3 I H A = A uh A iN A Ω =x A u A i A Ω (2.1,9)State of an orbiting point can so be described in a reference system in two ways: by the six dimension state vector of position and velocity, and by one set of orbital elements. Transformations among these sets or the state vector can be obtained with rotation matrices and parameter definitions. From all the possible sets of orbital elements, a mention must be done for Equinoctial elements,
{p,f,g,h,k,L}, which are the only ones that not meet singularities of angles and
formulas switch for elliptic/hyperbolic case.
Equation of motion 2.1,6 is expressed in polar coordinates, and it has been used to describe the trajectory of the motion. But the aim of kinematics is to express motion law of position during time. So that brings to derive the expression of true anomaly (as previously said, this is the only variable 2B-R element), as a function of time. Kepler equation of time sets the relationship between time of flight (TOF) from pericenter passage and angular position, which differs among cases of ellipse, parabola and hyperbola. Angular position is expressed by definition of different kinds of anomalies, each one starting from eccentricity direction: true anomaly ν is calculated from the first focus of the orbit, where the primary body is located; eccentric, parabolic and hyperbolic anomalies E, P
and H are calculated from the orbit center of the ellipse, parabola and hyperbola;
mean anomaly M is a projection of orbit motion on an equivalent circle with
1 tan tan 2 1 2 tan 2 1 tanh tan 2 1 2 E e e P H e e ν ν ν ⎧ − = ⎪ + ⎪ ⎪ = ⎨ ⎪ ⎪ − = ⎪ + ⎩ (2.1,10)
(
P)
M =n t−t =nTOF (2.1,11) 3 sin 1 1 2 6 sinh M E e E M P P M e H H = − ⎧ ⎪⎪ = + ⎨ ⎪ = − ⎪⎩ (2.1,12)In conclusion, whereas determination of time of flight is trivial from the angular position, solution of the inverse Kepler equation, as it is requested in order to obtain an explicit time dependency of the law of motion, can be done only for the banal circular orbit. This means that a direct time integration needs a step-by-step resolution of a nonlinear equation: in fact, due to regularity and simplicity of the relationships 2.1,12, and so their derivatives, application of Newton iterative method for the solution of nonlinear equations is successfully applicable, with adequate tolerance constraints.
2.1.3 Relative Orbital Mechanics
Formation Flying orbital mechanics of deputies satellites is formulated with their relative state vectors in regard to chief one. So it’s important to derive an equation of relative dynamics in a chief-centered reference system. Two possibilities come out: a chief-fixed frame, maybe along its principal inertia axis, which is useful for mission operations of the Formation, but not adequate for physical analysis; so the choice is an orbital frame, as the Hill’s frame, rotating instantaneously with the orbital angular velocity and acceleration: its axis are been defined in previous section, and can be seen in Fig. 2.1,3.
First thing to do is to highlight relative terms that appears in kinematics relationships, due to the fact that derivation is conducted in a non-inertial frame: from now, let indicate, when it is important, chief and deputy physical quantities with subscripts c and d respectively; r, v and a representing the absolute quantities, ρ ρ, andρtheir relative counterparts. All quantities are expressed in chief Hill’s frame Hc.
Fig. 2.1,3. Hill reference frame. d c d c app d c app r r v v v a a a ρ ρ ρ ⎧ = + ⎪ = + + ⎨ ⎪ = + + ⎩ (2.1,13)
This is an elegant way to deal with relative motion, which is enclosing all the additive terms, originated by derivation on a moving reference system, in a single apparent term in kinematics, in order to highlight the birth of additive actions in dynamics: these terms are called apparent because they aren’t originated by a material action, but only from the fact that the reference system is not inertial, but it is moving with a net acceleration. These apparent terms are:
(
)
2 app D app D C v v a a a ω ρ ω ρ ω ω ρ ω ρ = = ∧ ⎧⎪ ⎨ = + = ∧ + ∧ ∧ + ∧ ⎪⎩ (2.1,14)where the difference between dragging and Coriolis terms is emphasized.
It’s now possible to derive the equation of relative dynamics for orbital motion: considering the two spacecraft as independently subjected to 2B-R equations, obviously with the same central body, and considering also the effect of disturbance and control forces d and u, relative motion is obtained from the difference of the two equations in terms of acceleration, each one expressed in Hc frame.
