IAF Astrodynamics CommitteeTECHNICAL KEYNOTES IAC2017
23rd John V. Breakwell Memorial Lecture
APPLIED ASTRODYNAMICS:FROM DYADICS TO UNIVERSITY SATELLITES
Filippo Graziani
Adelaide, 27 September 2017
Professor John V. Breakwell and myself30th IAC, Munich, Germany (1979)
Professor John V. BreakwellDurand Building, Stanford (1975)
Memories
223rd John V. Breakwell Memorial Lecture
• A tool I learnt to use under his direction is the dyadic.
• Dyadics allow to simplify our approach to complex problems, as I alwaysattempted to do.
• I would start this talk by discussing their relevance in Astrodynamics bymeans of some examples I used during my lectures.
❖ A special teacher
❖ A special scientist
❖ A special man
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John V. Breakwell Teaching
”An empty desk is a sign of an empty mind. JVB”
23rd John V. Breakwell Memorial Lecture
A dyad is a couple of vectors side by side.In mathematical words, it can be represented as a matrix.
A dyadic is a linear combination of dyads.
A unity dyadic is an identity operator
ˆ ˆˆ̂ ˆ̂u ii jj kk
ˆ ˆ ˆ ˆ ˆˆ̂ ˆ̂ ˆ ˆ ˆ ˆ ˆ ˆ x y zu a ii jj kk a i a i j a j k a k a i a j a k a
that is
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Dyads and Dyadics: a reminder
23rd John V. Breakwell Memorial Lecture
• As it was pointed out by D. DeBra, dyadics help to separate a probleminto two phases:
1) The thinking phase;2) The numerical evaluation phase.
• During the thinking phase, the notation is compact and helps to retain aclearer interpretation.
• These methods also prepare the problem for the numerical evaluationphase since they easily permit to transform the dyadic and polyadicnotation into coordinates convenient for digital programming.
• Perhaps the most remarkable simplification is achieved when the effectof a perturbation is considered. A consistent work saving is achievedthrough cancellation of the unperturbed solution in vector form.
3) a third phase, i.e. the experimental one, will be added later.
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Usefulness of Dyadics
23rd John V. Breakwell Memorial Lecture
Some Astrodynamics applications where dyadics play an important role:
ORBITAL DYADIC
GRAVITY GRADIENT DYADIC
INERTIA DYADIC Gravity Gradient StabilizationTethered Systems
Third body perturbationWobble due to Sun and MoonTidal effectsGravity Gradient StabilizationMagnetic Torque FormulationProximity OperationsRestricted three bodies problem formulationSphere of Influence ComputationExtra Vehicular ActivitiesLauncher Optimal GuidanceWeak Stability Boundary TransfersStructural Stability under GG compression
Wobble due to J2
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Dyadics in Astrodynamics
23rd John V. Breakwell Memorial Lecture
* *ˆ ˆ ˆcos sinr e p
2 2
* * * *ˆˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆcos sin ( )cos sinrr ee pp ep pe
1 1 ˆ ˆˆˆ ˆ ˆ ˆ ˆ2 2
rr ee pp u hh
e
h
ϑ*p
The average in ϑ* will be
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Orbital Average of in the Apsidal Frame
23rd John V. Breakwell Memorial Lecture
Orbital - Gyro Equivalence (W. Kaula)
A satellite orbiting a massive body may be regarded as a gyroscopewhose spin axis is the orbit angular momentum and whose mass isuniformly distributed along the orbit as a ring.
dhh h
dt
An original sketch by J.V. Breakwell
h
z
2
2 2
3 ˆˆ ˆ 2
RJ z h z
p
Analogy to Poisson rule:
4.8 / year
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Wobble due to Earth Oblateness
23rd John V. Breakwell Memorial Lecture
(for GEO)
From Earth quadrupole moment
2
22
2 3
ˆ ˆ1 3
2J
r zRU J
r
22
2
2 4
3ˆ ˆ ˆˆ ˆ ˆ1 5 2
2J
Rf J r z r r z z
r
2
2
2 4ˆ ˆ ˆ3 J
Rdhr f J r z r z
dt r
22 2
2 2 *4ˆ ˆ ˆˆ ˆ ˆ ˆ3 3 (1 cos )
R Rdh rJ r z r z J h e z r r z
d r h p
2 2
2 22 2
3 3ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ2 2
R RdhJ hz u hh z J hz hh z
d p p
2
2 2
3 ˆˆ ˆ 2
RJ z h z
p
The J2 perturbation in vectorialform is
The effect is
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Wobble due to Earth Oblateness
23rd John V. Breakwell Memorial Lecture
The Gravity Gradient represents the linearized difference ofthe gravity acceleration in two different points
))(()(|)()( oooo rrorrgrgrg
rr
rrr
rr
rr
rg
3432
ˆ3)()ˆ()(
ukkjjiikzjyixkz
j
y
i
x
kzjyixr
ˆˆˆ̂ˆˆ)ˆˆˆ)(ˆ
)(ˆ)(ˆ)(()ˆˆˆ(
3ˆˆ( ) (3 )g r rr u G
r
Its evaluation can be conveniently achieved with dyadics
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Gravity-Gradient Dyadic
23rd John V. Breakwell Memorial Lecture
Tethered Systems
Third body perturbationWobble due to Sun and MoonTidal effectsWeak Stability Boundary TransfersLauncher Optimal Guidance
Wobble due to J2
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Dyadics in Astrodynamics
23rd John V. Breakwell Memorial Lecture
ORBITAL DYADIC
GRAVITY GRADIENT DYADIC
INERTIA DYADIC
The third body perturbation effect is a secular one, due to Solar and Lunargravity gradients, with the orbital axis preceding around the ecliptic axis orthe Moon orbit plane axis.
The effect can be averaged over 1 year (Sun) or 1 month (Moon).
This is a direct gravity gradient effect, easy to evaluate through dyadicformulation.
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Wobble Due To Third Body (Sun And Moon)
23rd John V. Breakwell Memorial Lecture
An original sketch by J.V. Breakwell
* ** * * * * *3 3* *
ˆ ˆ ˆ ˆ 3 3 dh
r f r G r r r r u r r r r rdt R R
Average along S/C orbit
Average in third body period
2 2
* ** * ** *3 3
* *
3 3ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ2 2
r rdhR u hh R R hh R
dt R R
2
** *3
*
3 ˆ ˆ ˆ ˆ4
rdhh h h h
dt R
2
** * *3
*
3 ˆ ˆ ˆ 4
rh h h
R h
dhh h
dt Analogy to Poisson
R*
r*r
h
μ*
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Third Body (Sun and Moon) Direct Effects
23rd John V. Breakwell Memorial Lecture
Combined J2, Solar and Lunar perturbations affect GEO platforms, dictating aNorth-South correction maneuver at node to limit inclination.
N
Δimax≈15°
ecliptic
pole
T ≈ 54 years
E.M. Soop “Handbook of Geostationary Orbits”14
Station Keeping Inclination Maneuvers for GEO
23rd John V. Breakwell Memorial Lecture
InclinationLimit (deg)
𝛥V permaneuver (m/s)
Average Time between maneuvers (days)
0.01 1 10Δ v=2v sinΔi
B. N. Agrawal “Design of Geosynchronous Spacecraft”
A different view about periodicinclination corrections:
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Station Keeping Inclination Maneuvers for GEO
23rd John V. Breakwell Memorial Lecture
Tethered Systems
Third body perturbationWobble due to Sun and Moon
Tidal effectsWeak Stability Boundary TransfersLauncher Optimal Guidance
Wobble due to J2
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Dyadics in Astrodynamics
23rd John V. Breakwell Memorial Lecture
ORBITAL DYADIC
GRAVITY GRADIENT DYADIC
INERTIA DYADIC
*
3
o
ar
*
3
2
o
ar
*
3
2
o
ar
*
3
o
ar
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Gravity Dyadic as Tide Tensor
23rd John V. Breakwell Memorial Lecture
** * 3
ˆ ˆ( ) ( ) | ( ) (3 )o o o o oo
a r a r g r r r r ur
=
SUN AND MOON
DEFORMATION ON THE EARTH
change in the distributionof the EARTH mass
variations in the gravitational potential (TIDAL POTENTIAL)
proportional to the elastic propertiesof the Earth
Modeled by Love Numbers
Tidal orbital perturbations
2
** 2
* *
rU r S
r r
2
** 2
* *
RU R S
r r
2
*2 2
* *
RT R K S
r r
2 3 5
*2 2 2 * 23 3
* * *
R R RT r K S K S
r r r r r
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Sun and Moon Indirect (Tidal) Effects
23rd John V. Breakwell Memorial Lecture
Small variations are observable through high accuracy measurements, like the onesproposed by:• Two counter-orbiting polar satellites (Stanford, 1975, Breakwell-Van Patten-Everitt)• LARES Project (Scuola di Ingegneria Aerospaziale, 2012, Paolozzi-Ciufolini)
UniCubeSat-GG inside the deployer
LARES
The 1975 paper proposing thedual-use to investigate tidaleffects, co-authored by JVB.
The relativity experimentrequiring two counter-orbiting spacecraft (1975).
The real mission to study tidal effects with the involvement of
the School of Rome (2012).19
Sun and Moon Tidal Effects
23rd John V. Breakwell Memorial Lecture
Tethered Systems
Third body perturbationWobble due to Sun and MoonTidal effects
Weak Stability Boundary TransfersLauncher Optimal Guidance
Wobble due to J2
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Dyadics in Astrodynamics
23rd John V. Breakwell Memorial Lecture
ORBITAL DYADIC
GRAVITY GRADIENT DYADIC
INERTIA DYADIC
References:• J. Kawaguchi et al., “ON MAKING USE OF LUNAR AND SOLAR GRAVITY ASSISTS IN LUNAR-A, PLANET-B MISSIONS” Acta
Astronautica Vol. 35. No. 9-l I, pp. 633-642, 1995.• E.A. Belbruno et al., “CALCULATION OF WEAK STABILITY BOUNDARY BALLISTIC LUNAR TRANSFER TRAJECTORIES” AIAA
2000-4142, AIAA/AAS Astrodynamics Specialist Conference, 2000.• M. Bello Mora et al., “A SYSTEMATIC ANALYSIS ON WEAK STABILITY BOUNDARY TRANSFERS TO THE MOON” IAF-00-
A.6.03, 51st International Astronautical Congress, 2000.• P. Teofilatto et al., “ON THE DYNAMICS OF WEAK STABILITY BOUNDARY LUNAR TRANSFERS” Celestial Mechanics and
Dynamical Astronomy 79: 41–72, 2001.
o Mission analysis has been developed for decades on the basis of two bodyproblem models with perturbations (planetary missions) and patched twobody problem (interplanetary missions).
o More recently, the WSB method has been used to analyse interplanetarymissions in the context of restricted three body problems with thefundamental effect of a fourth attracting body to accomplish the mission(such as Earth-Moon-Sat plus Sun). The result is a bigger flexibility andextension of launch windows and propellant saving at the cost of longerduration missions.
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Weak Stability Boundary Orbits
23rd John V. Breakwell Memorial Lecture
- The spacecraft is injected into a circular parkingorbit.
- A perigee kick manoeuvre ΔV1 injects thespacecraft into a lunar transfer orbit.
- An unpowered Moon swingby injects thespacecraft into a high eccentric orbit with apogeeof about 1.4 million km in WSB-Earth.
- A second manoeuvre ΔV2 introduces a smallcorrection in order to inject the spacecraft into atrajectory which joins the apogee and the Moonorbit.
- When the spacecraft approaches the Moon, bothEarth (gravity gradient effect of the Earth) and Sun(gravity gradient effect of the Sun) contribute tothe lunar capture.
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An Example:
23rd John V. Breakwell Memorial Lecture
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Sun Gravity Gradient Effects
23rd John V. Breakwell Memorial Lecture
Gravity gradient acts at the orbit aphelionSemi-major axis (a) increasesperihelion height increases
Gravity gradient acts at the orbit perihelionSemi-major axis (a) decreases
Aphelion height decreases
90 days later
J. Kawaguchi et al., “ON MAKING USE OF LUNAR AND SOLAR GRAVITY ASSISTS IN LUNAR-A, PLANET-B MISSIONS”
Indeed, gravity gradientoffers a useful explanation ofthe advantage of the longtransfers.
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Gravity gradient exploited in WSB
23rd John V. Breakwell Memorial Lecture
Tethered Systems
Third body perturbationWobble due to Sun and MoonTidal effectsWeak Stability Boundary Transfers
Launcher Optimal Guidance
Wobble due to J2
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Dyadics in Astrodynamics
23rd John V. Breakwell Memorial Lecture
ORBITAL DYADIC
GRAVITY GRADIENT DYADIC
INERTIA DYADIC
Introducing Lagrangianmultipliers
0
, ,
T
J L x u t dt
( , , )x f x u t
0 0
, , ( ) , ,
T T
J L x u t dt t f x u t x dt
H L f
0
, ,
T
J H x u x dt
Functional
Constrained by dynamics
Defining the Hamiltonian
and integrating by part
Leading to Pontryagin conditions
H
x
Hx
0H
u
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Optimal Guidance of Launchers
23rd John V. Breakwell Memorial Lecture
=
ˆ ˆ 0v U uu
If T/m is constant, the Lawden primer vector behavior is convenientlydescribed by dyadics
In our case:Cost functionDynamicsHamiltonian
v
Hg
r
r
H
v
2vH T
m m
ˆ0v
H Tu
u u m
ˆ ˆv u
( )
o sp
r v
Tv g r
m
Tm
g I
finalJ m
Pontryagin conditions become:
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Optimal Guidance of Launchers (2)
23rd John V. Breakwell Memorial Lecture
( )m r vo sp
T T uH v g r
g I m u
H = -
within the simplifying assumption of flat Earth and constant gravity (applicable to students sounding rockets), it follows:
v vG
0
a bt ˆopta bt
ua bt
The classical Bilinear tangent law
Lawden primer vector
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An Application:
23rd John V. Breakwell Memorial Lecture
Tethered Systems
Third body perturbationWobble due to Sun and MoonTidal effectsWeak Stability Boundary TransfersLauncher Optimal Guidance
Wobble due to J2
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Dyadics in Astrodynamics
23rd John V. Breakwell Memorial Lecture
ORBITAL DYADIC
GRAVITY GRADIENT DYADIC
INERTIA DYADIC
The dyadic notation is particularly well suited to the attitude motion of a rigidbody (attitude dynamics) since the inertia is described most easily by a dyadic:
ˆ ˆˆ̂ ˆ̂I Aii Bjj Ckk
The gravity torque acting on a body is:
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Inertia Dyadic
23rd John V. Breakwell Memorial Lecture
A sketch from Beletskii“Essais sur le mouvement des corps cosmiques”
The classical application is offered by gravity gradient stabilization (one morechance to recall Stanford’s AeroAstro contribution!).
Gravity gradient – among many othercases of interest – may be also appliedto a proposed NASA-Goddard missioncalled BOLAS.
Two 12U CubeSats linked by a 112 mileslong tether flying in a very low altitudelunar orbit.
Thinking about such amission allows to move tosmall platforms…
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Dyadics (Attitude)
23rd John V. Breakwell Memorial Lecture
UNISAT 2000
UNISAT-2 2002
UNISAT-3 2004
UNISAT-4 2006
EDUSAT 2011
UNICUBE-GG 2012
UNISAT-5 2013
UNISAT-6 2014
TUPOD 2017
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Microsatellites launched by GAUSS
23rd John V. Breakwell Memorial Lecture
• Along the years of studying and teaching, the possibility of realapplications has been always stressed, together with the support toyoung scholars and the chance and the challenge of team work.
• Researches carried out at University of Rome through GAUSS (Gruppodi Astrodinamica Università degli Studi la Sapienza).
• 2012: the Group became the private company GAUSS (Group ofAstrodynamics for the Use of Space Systems).
• Along the years GAUSS grew up….
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The Experience of GAUSS
23rd John V. Breakwell Memorial Lecture
▪ Spacecraft mass: about 25 kg.
▪ Spacecraft may be launched into escapetrajectory to Moon as a piggy-back payload.
▪ As launch vehicle, a potential candidate maybe the Russian Soyuz/Fregat. The Fregatupper stage will provide acceleration fromparking orbit to the escape trajectory toMoon (ΔV = 3150 m/s), then the payload willbe separated and it will continue itsautonomous flight.
Fregat Upper Stage provides with accelerationto the Moon escape trajectory
Soyuz-2Launch Vehicle provides launching into the parking orbit
International Moon Mission
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More to come….
23rd John V. Breakwell Memorial Lecture
A microsatellite to the Moon (and beyond): A dream…
This is the mission we are trying to achieve.It is a dream… but sometimes dreams come true.
Credit: SpaceX
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