formation flying in earth, libration, and distant retrograde orbits david folta
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Formation Flying in Earth, Libration,
and Distant Retrograde Orbits
David FoltaNASA - Goddard Space Flight Center
Advanced Topics in AstrodynamicsBarcelona, SpainJuly 5-10, 2004
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Agenda
I. Formation flying – current and futureII. LEO Formations
Background on perturbation theory / accelerations - Two body motion - Perturbations and accelerations
LEO formation flying - Rotating frames - Review of CW equations, Shuttle - Lambert problems, - The EO-1 formation flying mission
III. Control strategies for formation flight in the vicinity of the libration points• Libration missions• Natural and controlled libration orbit formations
- Natural motion - Forced motion
IV. Distant Retrograde Orbit FormationsV. References
• All references are textbooks and published papers• Reference(s) used listed on each slide, lower left, as ref#
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NASA Enterprises of Space Sciences (SSE) and Earth Sciences (ESE) are a combination of several
programs and themes
NASA Themes and Libration Orbits
• Recent SEC missions include ACE, SOHO, and the L1/L2 WIND mission. The Living With a Star (LWS) portion of SEC may require libration orbits at the L1 and L3 Sun-Earth libration points.
• Structure and Evolution of the Universe (SEU) currently has MAP and the future Micro Arc-second X-ray Imaging Mission (MAXIM) and Constellation-X missions.
• Space Sciences’ Origins libration missions are the James Webb Space Telescope (JWST) and The Terrestrial Planet Finder (TPF).
• The Triana mission is the lone ESE mission not orbiting the Earth.
• A major challenge is formation flying components of Constellation-X, MAXIM, TPF, and Stellar Imager.
ESE
SEC Origins
SEU
SSE
Ref #1
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Earth ScienceLow Earth Orbit Formations
The ‘a.m.’ train • ~ 705km, 980 inclination, •10:30 .pm. Descending node sun-sync
-Terra (99): Earth Observatory- Landsat-7(99): Advanced land imager-SAC-C(00): Argentina s/c -EO-1(00) : Hyperspectral inst.
The ‘p.m.’ train•~ 705km, 980 inclination, •1:30 .pm. Ascending node sun-sync
- Aqua (02)- Aura (04)- Calipso (05)- CloudSat (05)- Parasol (04)- OCO (tbd)
Ref # 1, 2
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Space Science Launches Possible Libration Orbit missions
• FKSI (Fourier Kelvin Stellar Interferometer): near IR interferometer
• JWST (James Webb Space Telescope): deployable, ~6.6 m, L2
• Constellation – X: formation flying in librations orbit
• SAFIR (Single Aperture Far IR): 10 m deployable at L2,
• Deep space robotic or human-assisted servicing
• Membrane telescopes
• Very Large Space Telescope (UV-OIR): 10 m deployable or assembled in LEO, GEO or libration orbit
• MAXIM: Multiple X ray s/c
• Stellar Imager: multiple s/c form a fizeau interferometer
• TPF (Terrestrial Planet Finder): Interferometer at L2
• 30 m single dish telescopes
• SPECS (Submillimeter Probe of the Evolution of Cosmic Structure): Interferometer 1 km at L2
Ref #1
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Orbit Challenges Biased orbits when using large sun shades Shadow restrictions Very small amplitudes Reorientation and Lissajous classes Rendezvous and formation flying Low thrust transfers Quasi-stationary orbits Earth-moon libration orbits Equilateral libration orbits: L4 & L5
Future Mission Challenges Considering science and operations
Operational Challenges Servicing of resources in libration orbits Minimal fuel Constrained communications Limited V directions Solar sail applications Continuous control to reference trajectories Tethered missions Human exploration
Science Challenges Interferometers Environment Data Rates Limited Maneuvers
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Background on perturbation theory / accelerations
• Two Body Motion
• Atmospheric Drag
• Potential Models Forces
• Solar Radiation Pressure
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•Newton’s law of gravity of force is inversely proportional to distance
•Vector direction from r2 to r1 and r1 to r2
•Subtract one from other and define vector r, and gravitational constants
Fundamental Equation of Motion
Two – Body Motion
Ref # 3-7
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FORCES ONPROPAGATED ORBIT
• Equation Of Motion Propagated.
• External Accelerations Caused By Perturbations
a = anonspherical+adrag+a3body+asrp+atides+aother
+ accelerations
Ref #3-7
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Gaussian Lagrange Planetary Equations
• Changes in Keplerian motion due to perturbations in terms of the applied force. These are a set of differential equations in orbital elements that provide analytic solutions to problems involving perturbations from Keplerian orbits. For a given disturbing function, R, they are given by
Ref # 3-7
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Geopotential
• Spherical Harmonics break down into three types of terms Zonal – symmetrical about the polar axis Sectorial – longitude variations Tesseral – combinations of the two to model specific regions• J2 accounts for most of non-spherical mass• Shading in figures indicates additional mass
Ref # 3-7
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Potential Accelerations
2 11
1
sincossin1
sin1l
l
lllll
l
l
lm
mmm mSmCPr
Pr
RJ
r
The coordinates of P are now expressed in spherical coordinates , where is the geocentric latitude and is the longitude. R is the equatorial radius of the primary body and is the Legendre’s Associated Functions of degree and order m. The coefficients and are referred to as spherical harmonic coefficients. If m = 0 the coefficients are referred to as zonal harmonics. If they are referred to as tesseral harmonics, and if , they are called sectoral harmonics.
,,r sinmPl
0ml 0ml
Simplified J2 acceleration model for analysis with acceleration in inertial coordinates
-3J2Re2 ri
2r51-
5rk2
r2ai =
-3J2Re2 rj
2r51-
5rk2
r2aj =
-3J2Re2 rj
2r53-
5rk2
r2ak =
+ + x + y + z
a = = i j k
Ref # 3-7
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Atmospheric Drag
• Atmospheric Drag Force On The Spacecraft Is A Result Of Solar Effects On The Earth’s Atmosphere
• The Two Solar Effects:– Direct Heating of the Atmosphere– Interaction of Solar Particles (Solar Wind) with the Earth’s Magnetic
Field• NASA / GSFC Flight Dynamics Analysis Branch Uses Several models:
– Harris-Priester • Models direct heating only• Converts flux value to density
– Jacchia-Roberts or MSIS• Models both effects• Converts to exospheric temp. And then to atmospheric density• Contains lag heating terms
Ref # 3-7
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F10.7 RADIO FLUX OBSERVED AND PREDICTED
0
50
100
150
200
250
300
350
1940 1950 1960 1970 1980 1990 2000 2010
YEAR
F10
.7 R
AD
IO F
LU
X
F10.7 - observed
F10.7 - Predicted
Solar Flux Prediction
Historical Solar Flux, F10.7cm values Observed and Predicted (+2s) 1945-2002
Ref # 3-7
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Drag Acceleration
• Acceleration defined as
Cd A va2 v
m12
a =
A = Spacecraft cross sectional area, (m2)Cd = Spacecraft Coefficient of Drag, unitless m = mass, (kg) = atmospheric density, (kg/m3)va = s/c velocity wrt to atmosphere, (km/s)v = inertial spacecraft velocity unit vectorCd A = Spacecraft ballistic propertym
• Planetary Equation for semi-major axis decay rate of circular orbit (Wertz/Vallado p629), small effect in e
a = - 2Cd A a2
m
^
^
Ref # 3-7
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s / c
26 3
8
6 2SR
SR s / c
1ˆF GA s, but
c
G 1350 watts / m4.5x10 watt sec/ m
c 3x10 m / s
4.5x10 N / m P
Therefore,
ˆF P A s 3
Where G is the incident solar radiation per unit area striking the surface, As/c. G at 1 AU
= 1350 watts/m2 and As/c = area of the
spacecraft normal to the sun direction. In general we break the solar pressure force into the component due to absorption and the component due to reflection
s n
s n
F F F
Where F is the force in the solar direction and F is the force normal to the surface
SP s / c
s / c
s / c
ˆ ˆF P A [ s 2 n]
where
absorptivity coefficient, 0 1, 1
reflectivity coefficient of specular reflection, 0 1
n is a unit vector normal to the surface, A
A is the area normal to the sun direction
Solar Radiation Pressure Acceleration
2
R I2
R I
From Lagrange 's planetary equations
da 2 a(1 e ){esin F F }
dt rn 1 e
Where F and F are the radial and in track solar pressure forces.
Ref # 3-7
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Other Perturbations
• Third Bodya3b = -/r3 r = (rj/rj
3 – rk/rk3)
– Where rj is distance from s/c to body and
rk is distance from body to Earth
• Thrust – from maneuvers and out gassing from instrumentation and materials
Inertial acceleration: x = Tx/m, y=Ty/m, z=Tz/m
• Tides, others
.. .. ..
- - -
Ref # 3-7
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Ballistic Coefficient
• Area (A) is calculated based on spacecraft model.– Typically held constant over the entire orbit– Variable is possible, but more complicated to model– Effects of fixed vs. articulated solar array
• Coefficient of Drag (Cd)is defined based on the shape of an object.– The spacecraft is typically made up of many objects of different shapes.– We typically use 2.0 to 2.2 (Cd for A sphere or flat plate) held constant
over the entire orbit because it represents an average
• For 3 axis, 1 rev per orbit, earth pointing s/c: A and Cd do not change drastically over an orbit wrt velocity vector– Geometry of solar panel, antenna pointing, rotating instruments
• Inertial pointing spacecraft could have drastic changes in Bc over an orbit
Ref # 3-7
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Ballistic Effects
Varying the mass to area yields different decay ratesSample: 100kg with area of 1, 10, 25, and 50m2, Cd=2.2
5250
5500
5750
6000
6250
6500
6750
0 5 10 15 20 25 30
FreeFlyer Plot Window2/11/2003
Km
sat100to1.ElapsedDays (Days)
sat100to1.BL_A (Km) Sat100to10.BL_A (Km)
Sat100to25.BL_A (Km) Sat100to50.BL_A (Km)
50m2 25m2
10m2
Ref # 3-7
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Numerical Integration
• Solutions to ordinary differential equations (ODEs) to solve the equations of motion.
• Includes a numerical integration of all accelerations to solve the equations of motion
• Typical integrators are based on– Runge-Kutta
formula for yn+1, namely:
yn+1 = yn + (1/6)(k1 + 2k2 + 2k3 + k4) is simply the y-value of the current point plus a weighted average of four different y-
jump estimates for the interval, with the estimates based on the slope at the midpoint being weighted twice as heavily as the those using the slope at the end-points.
– Cowell-Moulton– Multi-Conic (patched)– Matlab ODE 4/5 is a variable step RK
Ref # 3-7
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Coordinate Systems
Geocentric Inertial (GCI)
Greenwich Rotating Coordinates (GRC)
• Origin of reference frames:PlanetBarycenterTopographic
• Reference planes:Equator – equinoxEcliptic – equinoxEquator – local meridianHorizon – local meridian
Most used systems GCI – Integration of EOM ECEF – Navigation Topographic – Ground station
Ref # 3-7
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Describing Motion Near a Known Orbit
)(
)(
)(
)(**
**
tvv
trr
tvv
trr
Chief (reference Orbit)
r*_
r_
v_v*
_
)()()(
)()()(*
*
tvtvtv
trtrtr
Relative Motion
),(),(/1)(2
2
vrfvrFmtrdt
d
What equations of motion does the relative motion follow?
),(),()()())()(( ***2
2
2
2*
2
2
2
2
vrfvrftrdt
dtr
dt
dtrtr
dt
dr
dt
d
),(),()( **2
2
vrfvrftrdt
d
A local system can be established by selection of a central s/c or center point and using the Cartesian elements to construct the local system that rotates with respect to a fixed point (spacecraft)
Ref # 5,6,7
Deputy
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Substituting in yields a linear set of coupled ODEs
This is important since it will be our starting point for everything that follows
...|)()()( ***
rr
frfrrfrf
rr
)(|)()()()( ***2
2
* rfrr
frfrfrftr
dt
drr
rr
fr
dt
drr *|
2
2
As the last equation stands it is exact. However if r is sufficiently small, the term f(r) can be expanded via Taylor’s series …---
Describing Motion Near a Known Orbit
Ref # 5,6,7
Lets call this (1)
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*
*ˆ
r
rR
**
**
ˆvr
vrN
RNT ˆˆˆ
• If the motion takes place near a circular orbit, we can solve the linear system (1) exactly by converting to a natural coordinate frame that rotates with the circular orbit
RT
N^
^
^r*_
v*_
The frame described is known as - Hill’s - RTN - Clohessy-Wiltshire - RAC - LVLH - RIC
Describing Motion Near a Known Orbit
Ref # 5,6,7
Any vector relative to the chief is given by: NrTrRrrNTR
ˆˆˆˆˆˆ
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First we evaluate inarbitrary coordinates
)()(33 ii x
rfr
rf
ijjiijj
i
rxx
rx
rxx
f
r
f 353
3)(
)( 25 ijji rxx
rr
f
222
222
222
5
233
323
332
xyzyzxz
yzzxyxy
xzxyzyx
r
100
010
002
100
010
002
| 23*
* nrr
frr
Only in RTN , where the chief has a constant position =Rr*, the matrix takes a formxRTN = r* , yRTN = zRTN = 0
^
Describing Motion Near a Known Orbit
Ref # 5,6,7
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Transforming the Left Hand Side of (1)
Now convert the left hand side of (1) to RTN frame,
Newton’s law involves 2nd derivatives:
fixedmoving dt
d
dt
d
Dt
D
)(22
2
2
2
2
2
rrdt
dr
dt
dr
dt
dr
Dt
D
)(22
2
2
2
2
2
rrDt
Dr
dt
dr
dt
dr
Dt
D
n
0
0
z
y
x
r 0Dt
D
dt
d
z
y
x
rDt
D
0
xn
yn
rDt
D
0
)( 2
2
yn
xn
r
Ref # 5,6,7
In the RTN frame
00
2
22
2
2
2
2
2
2
2
2
yn
xn
xn
yn
zn
yn
xn
rDt
D
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Transforming the EOM yields Clohessy-Wiltshire Equations
122 knxyxny
12 2nkxnx
n
knt
n
vntxx 10
0
2)sin()cos(
010
0
2)cos(
2)sin(2 yt
n
knt
n
vntxy
)sin()cos( 00 nt
nntzz
A “balance” form will have no secular growth, k1=0
Note that the y-motion (associated with Tangential) has twice the amplitude of the x motion (Radial)Ref # 5,6,7
nz
xn
ynxn
z
y
x
rr
fr
dt
d RTN
rr
2
23
|
22
2
2
*
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0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
-10 -5 0 5 10 15 20 25 30 35
+1 m/s in Alongtrack Velocity
main
.Rad
ialSe
para
tion
(Km
)
main.AlongArcTrackSeparation (Km)
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5
+1m/s in Radial Velocity6/4/2004
main
.Rad
ialSe
para
tion
(Km
)
main.AlongArcTrackSeparation (Km)
-0.00300
-0.00275
-0.00250
-0.00225
-0.00200
-0.00175
-0.00150
-0.00125
-0.00100
-0.045 -0.040 -0.035 -0.030 -0.025 -0.020 -0.015 -0.010 -0.005 0.000
-1 m in Radial Position6/4/2004
mai
n.Ra
dial
Sepa
ratio
n (K
m)
main.AlongArcTrackSeparation (Km)
Initial Location 0,0,0
Initial Location 0,0,0
Initial Location 0,0,-1
A numerical simulation using RK8/9 and point mass
Effect of Velocity (1 m/s) or Position(1 m) Difference
Relative Motion
• An Along-track separation remain constant• A 1 m radial position difference yields an along-track motion• A 1 m/s along-track velocity yields an along-track motion• A 1 m/s radial velocity yields a shifted circular motion
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Shuttle Vbar / Rbar
Ref # 7,9
• Shuttle approach strategies•Vbar – Velocity vector direction in an LVLH (CW) coordinate system•Rbar – Radial vector direction in an LVLH (CW) coordinate system
• Passively safe trajectories – Planned trajectories that make use of predictable CW motion if a maneuver is not performed.
• Consideration of ballistic differences – Relative CW motion considering the difference in the drag profiles.
Graphics Ref: Collins, Meissinger, and Bell, Small Orbit Transfer Vehicle (OTV) for On-Orbit Satellite Servicing and Resupply, 15th USU Small Satellite Conference, 2001
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What Goes Wrong with an Ellipse
In state – space notation, linearization is written as
Since the equation is linear
which has no closed form solution if
0
0
x
fA
Adt
dSS 0
),())((),( 0*
0
0
tttrAtdttt
t
0)(),( 21 tAtA
SASdt
d
Ref # 5,6,7
v
rS
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Lambert Problem
• Consider two trajectories r(t) and R(t).
• Transfer from r(t) to R(t) is affected by two Vs– First dVi is designed to match the velocity of a transfer trajectory R(t) at
time t2
– Second dVf is designed to match the velocity of R(t) where the transfer intersects at time t3
• Lambert problem:
Determine the two Vs
Ref # 5,6,7
t
V
dV
dV
i
fVf
i
1
v1
r1
R 1
V1
V2
R 2
r 2
v2
t 2 t 3
R 3 V3
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Lambert Problem
• The most general way to solve the problem is to use to numerically integrate r(t), R(t), and R(t) using a shooting method to determine dVi and then simply subtracting to determine dVf
• However this is relatively expensive (prohibitively onboard) and is not necessary when r(t) and R(t) are close
• For the case when r(t) and R(t) are nearby, say in a stationkeeping situation, then linearization can be used.
• Taking r(t), R(t), and (t3,t2) as known, we can determine dVi and dVf using simple matrix methods to compute a ‘single pass’.
Ref # 5,6,7
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• Enhanced Formation Flying (EFF)
The Enhanced Formation Flying (EFF) technology shall provide the autonomous capability of flying over the same ground track of another spacecraft at a fixed separation in time.
• Ground track Control
EO-1 shall fly over the same ground track as Landsat-7. EFF shall predict and plan formation control maneuvers or a maneuvers to maintain the ground track if necessary.
New Millennium Requirements
• Formation Control
Predict and plan formation flying maneuvers to meet a nominal 1 minute spacecraft separation with a +/- 6 seconds tolerance. Plan maneuver in 12 hours with a 2 day notification to ground.
• Autonomy
The onboard flight software, called the EFF, shall provide the interface between the ACS / C&DH and the AutoConTM system for Autonomy for transfer of all data and tables.
EO-1 GSFC Formation Flying
Ref # 10,11
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EO-1 GSFC Formation Flying
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FF Start
Velocity
FF Maneuver
EO -1 Spacecraft
Landsat-7
In-Track Separation (Km)
Observation Overlaps
Ra
dia
l S
ep
ara
tio
n (
m)
Ideal FF Location
NadirDirection
I-minute separationin observations
Formation Flying Maintenance DescriptionLandsat-7 and EO-1
Different Ballistic Coefficients and Relative Motion
Ref # 10,11
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• Determine (r1,v1) at t0 (where you are at time t0).
• Determine (R1,V1) at t1 (where you want to be at time t1).
• Project (R1,V1) through -t to determine (r0,v0) (where you should be at time t0).
• Compute (r0,v0) (difference between where you are and where you want be at t0).
r1,v1
R1,V1
r0,v0
} r0,v0
Keplerian State K(tF)
Keplerian State (ti)
ti tft0
Keplarian State (t1)
Keplarian State (t0)
t1
t
r(t0)
v(t0)
ManeuverWindow
EO-1 Formation Flying Algorithm
Reference S/CInitial State
Reference S/C
Final State
Formation FlyerInitial State
Formation FlyerTarget State
Reference Orbit
TransferOrbit
Ref # 10,11
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A state transition matrix, F(t1,t0), can be constructed that will be a function of both t1 and t0 while satisfying matrix differential equation relationships. The initial conditions of F(t1,t0) are the identity matrix. Having partitioned the state transition matrix, F(t1,t0) for time t0 < t1 :
We find the inverse may be directly obtained by employing symplectic properties:
F(t0,t1) is based on a propagation forward in time from t0 to t1 (the navigation matrix) F(t1,t0) is based on a propagation backward in time from t1 to t0, (the guidance matrix). We can further define the elements of the transition matrices as follows:
014013
01201101 ,,
,,,
tttt
tttttt
104103
1021011001
1
,,
,,,,
tttt
tttttttt
T
011T
013
T012
T014
011
,,
,,,
tttt
tttttt
1040
*
1030*
1020*
1010*
,)(V
,)(V~
,)(R
,)(R~
ttt
ttt
ttt
ttt
0141
0131
0121
0111
,)(V
,)(V~
,)(R
,)(R~
ttt
ttt
ttt
ttt
)(R~
)(V~
)(R)(V
)(V)(V~
)(R)(R~
11T
11T
0*
0*
0*
0*
tt
tt
tt
tt
State Transition Matrix
Ref # 10,11
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• Compute the matrices [R(t1)], [R(t1)]
according to the following: Given
Compute
• Compute the ‘velocity-to-be-gained’ (v0) for the current cycle.
where F and G are found from Gauss problem and the f & g series and C found through universal variable formulation
v r v0 0 0 R R T T ~t t1 1
~R V v V v R r V r I1 0 1 0 1 0 1 0t C F1 3
11 =
R
rr F + 1 T
00
T T
R R r v V v r V v I1 0 0 1 0 0 1 0tC
G1 1 = r
F 0 T T T
r r r0 0 1 v v v0 0 1
Enhanced Formation Flying Algorithm
~
The Algorithm is found from the STM and is based on the simplectic nature ( navigation and guidance matrices) of the STM)
Ref # 10,11
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Where do I want to be? Where am I ?
AutoConT
M
YesNo
Relative Navigation
• EO-1/LS-7
Decision Rules
• Performance Objectives• Flight Constraints
ThrustersGPS ReceiverGround Uplink
• Landsat-7 Data
C&DH / ACS
• Generate Commands • Execute Burn
Implementation• Compute Burn Attitude• Validate Com. Sequence• Perform CalibrationsHow do I get there?
Maneuver Design• Propagate S/C States• Design Maneuver• Compute V
InputsModels
Constraints
Maneuver Decision
On-Board Navigation
• Orbit Determination• Attitude Determination
EO-1 AutoConTM Functional Description
EFF
Ref # 10,11
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EO-1 Subsystem Level
EO-1 Formation Flying Subsystem Interfaces
EFF Subsystem AutoCon-F
• GSFC• JPL• GPS Data Smoother
Stored Command Processor
Cmd Load
GPSSubsystem
C&DHSubsystem
ACSSubsystem
ACESubsystem
CommSubsystem
Orbit ControlBurn Decision and Planning
Mongoose V
EFFSubsystem
AutoCon-F &GPS Smoother
SCPThruster Commanding
Command and Telemetry Interfaces
Timed Command Processing
Propellant Data
Thruster Commands
GPS State Vectors
UplinkDownlinkInertial State Vectors
Ref # 10,11
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Difference in EO-1 Onboard and Ground Maneuver Quantized DVs
Mode Onboard Onboard Ground V1 Ground V2 % Diff V1 % DiffV2 V1 V2 Difference Difference vs.Ground vs.Ground
cm/s cm/s cm/s cm/s % %
Auto-GPS 4.16 1.85 4e-6 2e-1 .0001 12.50Auto-GPS 5.35 4.33 3e-7 2e-1 .0005 5.883Auto 4.98 0.00 1e-7 0.0 .0001 0.0Auto 2.43 3.79 3e-7 2e-7 .0001 .0005Semi-auto 1.08 1.62 6e-6 3e-3 .0588 14.23Semi-auto 2.38 0.26 1e-7 1e-7 .0001 .0007Semi-auto 5.29 1.85 8e-4 3e-4 1.569 1.572Manual 2.19 5.20 4e-7 3e-3 .0016 .0002Manual 3.55 7.93 3e-7 3e-3 .0008 3.57
Inclination Maneuver Validation: Computed V at node crossing, of ~ 24 cm/s (114 sec duration), Ground validation gave same results
Quantized - EO-1 rounded maneuver durations to nearest second
Ref # 10,11
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Difference in EO-1 Onboard and Ground Maneuver Three-Axis Vs
Mode Onboard Ground V1 3-axis Algorithm Algorithm V1 Difference V1 vs. Ground V1 Diff V1 Diff
m/s cm/s % cm/s %
Auto-GPS 2.83 1.122 3.96 -0.508 -1.794Auto-GPS 8.45 68.33 -8.08 0.462 -.0054Auto 10.85 -5e-4 -0000 .0003 0Auto 11.86 0.178 .0015 -.0102 -.0008Semi-auto 12.64 0.312 .0024 .00091 .0002Semi-auto 14.76 0.188 .0013 0.000 .0001Semi-auto 15.38 -.256 -.0016 -.0633 -.0045Manual 15.58 10.41 .6682 -.0117 -.0007Manual 15.47 0.002 .0001 -.0307 -.0021
Ref # 10,11
EO-1 maneuver computations in all three axis
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-0.125
-0.100
-0.075
-0.050
-0.025
0.000
0.025
0.050
0.075
430 440 450 460 470 480 490 500
EO
1.A
vera
ge ()
EO1.AlongTrackSeparation (Km)
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
0 25 50 75 100 125
Y-Ax
is
EO1.ElapsedDays (Days)
EO1.Average () LS7.Average ()
Radial vs. along-track separation over all formation maneuvers(range of 425-490km)
Ground-track separation over all formation maneuvers maintained to 3km
Formation Data from Definitive Navigation Solutions
Rad
ial S
epar
atio
n (m
)
Gro
undt
rack
Ref # 10,11
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430
440
450
460
470
480
490
500
0 25 50 75 100 125
EO
1.A
vera
ge ()
EO1.ElapsedDays (Days)
7077.60
7077.65
7077.70
7077.75
7077.80
7077.85
0 25 50 75 100 125
Y-A
xis
EO1.ElapsedDays (Days)
EO1.Average () LS7.Average ()
Along-track separation vs. Time over all formation maneuvers(range of 425-490km)
Semi-major axis of EO-1 and LS-7 over all formation maneuvers
Formation Data from Definitive Navigation Solutions
Alo
ngT
rack
Sep
arat
ion
(m)
Ref # 10,11
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0.001125
0.001150
0.001175
0.001200
0.001225
0.001250
0 25 50 75 100 125
Y-A
xis
EO1.ElapsedDays (Days)
EO1.Average () LS7.Average ()
88
89
90
91
92
93
94
95
0.00113 0.00114 0.00115 0.00116 0.00117 0.00118 0.00119 0.00120 0.00121 0.00122 0.00123 0.00124 0.00125
EO
1.A
vera
ge ()
EO1.Average ()
Formation Data from Definitive Navigation Solutions
Frozen Orbit eccentricity over all formation maneuvers(range of .001125 - 0.001250)
Frozen Orbit vs. ecc. over all formation maneuvers. range of 90+/- 5 deg.
ecce
ntri
city
eccentricityRef # 10,11
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EO-1 Summary / Conclusions
o A demonstrated, validated fully non-linear autonomous system
o A formation flying algorithm that incorporates
Intrack velocity changes for semi-major axis ground-track control Radial changes for formation maintenance and eccentricity control Crosstrack changes for inclination control or node changes Any combination of the above for maintenance maneuvers
Ref # 10,11
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o Proven executive flight codeo Scripting language alters behavior w/o flight software changeso I/F for Tlm and Cmdso Incorporates fuzzy logic for multiple constraint checking for maneuver planning and controlo Single or multiple maneuver computations.o Multiple or generalized navigation inputs (GPS, Uplinks).o Attitude (quaternion) required of the spacecraft to meet the V components o Maintenance of combinations of Keperlian orbit requirements
sma, inclination, eccentricity,etc.
Summary / Conclusions
Enables Autonomous StationKeeping, Formation Flying and Multiple Spacecraft Missions
Ref # 10,11
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CONTROL STRATEGIES FOR FORMATION FLIGHTIN THE VICINITY OF THE LIBRATION POINTS
Ref # 11, 12, 13, 14, 15
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NASA Libration Missions
• GEOTAIL(1992) L2 Pseudo Orbit, Gravity Assist• MAP(2001-04) Orbit, Lissajous Constrained, Gravity Assist• JWST (~2012) Large Lissajous, Direct Transfer• CONSTELLATION-X Lissajous Constellation, Direct Transfer?, Multiple S/C• SPECS Lissajous, Direct Transfer?, Tethered S/C• MAXIM Lissajous, Formation Flying of Multiple S/C• TPF Lissajous, Formation Flying of Multiple S/C
• ISEE-3/ICE(78-85) L1 Halo Orbit, Direct Transfer, L2 Pseudo Orbit, Comet Mission• WIND (94-04) Multiple Lunar Gravity Assist - Pseudo-L1/2 Orbit• SOHO(95-04) Large Halo, Direct Transfer • ACE (97-04) Small Amplitude Lissajous, Direct Transfer• GENESIS(01-04) Lissajous Orbit, Direct Transfer,Return Via L2 Transfer• TRIANA L1 Lissajous Constrained, Direct Transfer
L1 Missions
L2 Missions
Ref # 1,12
(Previous missions marked in blue)
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ISEE-3 / ICE
Earth
Lunar OrbitHalo Orbit
L1
Mission: Investigate Solar-Terrestrial relationships, Solar Wind, Magnetosphere, and Cosmic Rays
Launch: Sept., 1978, Comet Encounter Sept., 1985Lissajous Orbit: L1 Libration Halo Orbit, Ax=~175,000km, Ay = 660,000km, Az~
120,000km, Class ISpacecraft: Mass=480Kg, Spin stabilized, Notable: First Ever Libration Orbiter, First Ever Comet Encounter
Solar-Rotating CoordinatesEcliptic Plane Projection
Farquhar et al [1985] Trajectories and Orbital Maneuvers for the ISEE-3/ICE, Comet Mission, JAS 33, No. 3Ref # 1,12,13
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Mission: Investigate Solar-Terrestrial Relationships, Solar Wind, MagnetosphereLaunch: Nov., 1994, Multiple Lunar Gravity AssistLissajous Orbit: Originally an L1 Lissajous Constrained Orbit, Ax~10,000km, Ay ~
350,000km, Az~ 250,000km, Class ISpacecraft: Mass=1254kg, Spin Stabilized, Notable: First Ever Multiple Gravity Assist Towards L1
Solar-Rotating CoordinatesEcliptic Plane Projection
WIND
Ref # 1,12
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MAP
Mission: Produce an Accurate Full-sky Map of the Cosmic Microwave Background Temperature Fluctuations (Anisotropy)Launch: Summer 2001, Gravity Assist TransferLissajous Orbit: L2 Lissajous Constrained Orbit Ay~ 264,000km, Ax~tbd, Ay~ 264,000km,
Class IISpacecraft: Mass=818kg, Three Axis Stabilized,Notable: First Gravity Assisted Constrained L2 Lissajous Orbit; Map-earth Vector
Remains Between 0.5 and 10° off the Sun-earth Vector to Satisfy Communications Requirements While Avoiding Eclipses
Solar-Rotating CoordinatesEcliptic Plane Projection
Ref # 1,12
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JWST
Mission: JWST Is Part of Origins Program. Designed to Be the Successor to the Hubble Space Telescope. JWST Observations in the Infrared Part of the Spectrum.
Launch: JWST~2010, Direct TransferLissajous Orbit: L2 large lissajous, Ay~ 294,000km, Ax~800,000km, Az~ 131,000km, Class I
or IISpacecraft: Mass~6000kg, Three Axis Stabilized, ‘Star’ PointingNotable: Observations in the Infrared Part of the Spectrum. Important That the
Telescope Be Kept at Low Temperatures, ~30K. Large Solar Shade/Solar Sail
Earth
Lunar Orbit
Sun
L2
Solar-Rotating CoordinatesEcliptic Plane Projection
Ref # 1,12
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The linearized equations of motion for a S/C close to the libration point are calculated at the respective libration point.
• Linearized Eq. Of Motion Based on Inertial X, Y, Z Using
X = X0 + x, Y=Y0+y, Z= Z0+z
• Pseudopotential:
M1 M2L1
X
R
2r
1r
j
i
x
Y
y
D1 D2
D
O
m
Sun
Earth
A State Space ModelA State Space Model
Ref # 1,12,13,17,20
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jjj Axx where
00000
00200
02000
100000
010000
001000
,
ZZ
YY
XX
j
j
j
j
j
j
j
j
U
nU
nUA
z
y
x
z
y
x
x
Sun
Earth-Moon
Barycenter
L1
Lx
Ly
Lz
Ecliptic Plane
Projection ofHalo orbit
A State Space ModelA State Space Model
D
MMGn
)( 21
Ref # 1,17,20
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Reference Motions
• Natural Formations– String of Pearls – Others: Identify via Floquet controller (CR3BP)
• Quasi-Periodic Relative Orbits (2D-Torus)• Nearly Periodic Relative Orbits• Slowly Expanding Nearly Vertical Orbits
• Non-Natural Formations– Fixed Relative Distance and Orientation
– Fixed Relative Distance, Free Orientation – Fixed Relative Distance & Rotation Rate– Aspherical Configurations (Position & Rates)
+ Stable Manifolds
RLP
Inertial
Ref # 15
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Natural Formations:String of Pearls
x y
z z
y
x
Ref # 15
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CR3BP Analysis of Phase Space Eigenstructure Near Halo Orbit
,0 0RR
R
rx t x
r
Reference Halo Orbit
Chief S/C
Deputy S/C
,0 0
Floquet Modal Matrix:JtE t P t S t E e
6 6
1 1
:Solution to Variational Eqn. in terms of Floquet Modes
j j jj j
x t x t c t e t E t c
1
, 0 0 0
Floquet Decomposition of , 0 :
t Jtt P t e P P t S e P S
t
Ref # 15
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Natural Formations:Quasi-Periodic Relative Orbits → 2-D Torus
y
x
z
Chief S/C Centered View(RLP Frame)
y
x
x
z
y
z
z
xy
Ref # 15
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Floquet Controller(Remove Unstable + 2 of the 4 Center Modes)
1
2 5 6 31 3 4
2 5 6 3
Remove Modes 1, 3, and 4:
0r r r
v v v
x x xx x x
x x x Iv
1
2 3 4 31 5 6
2 3 4 3
Remove Modes 1, 5, and 6:
0r r r
v v v
x x xx x x
x x x Iv
3
2,3,43 or
2,5 6
6
1
,
01
:Find that removes undesired response modes
j jj j
j
j
v xI
x
v
Ref # 15
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Deployment into Torus(Remove Modes 1, 5, and 6)
Origin = Chief S/C
Deputy S/C 0 50 0 0 m
0 1 1 1 m/sec
r
r
Ref # 15
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Deployment into Natural Orbits(Remove Modes 1, 3, and 4)
Origin = Chief S/C
3 Deputies
00 0 0 m
0 1 1 1 m/sec
r r
r
Ref # 15
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Natural Formations:Nearly Periodic Relative Motion
Origin = Chief S/C
10 Revolutions = 1,800 days
Ref # 15
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Evolution of Nearly Vertical OrbitsAlong the yz-Plane
Ref # 15
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Natural Formations:Slowly Expanding Vertical Orbits
Origin = Chief S/C
0r fr t
100 Revolutions = 18,000 days
Ref # 15
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Non-Natural Formations
• Fixed Relative Distance and Orientation – Fixed in Inertial Frame– Fixed in Rotating Frame
• Spherical Configurations (Inertial or RLP)– Fixed Relative Distance, Free Orientation – Fixed Relative Distance & Rotation Rate
• Aspherical Configurations (Position & Rates)– Parabolic– Others
Ref # 15
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X [au]
Y [
au]
Formations Fixed in the Inertial Frame
Y Chief S/C
Deputy S/C
Ref # 15
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Formations Fixed in the Rotating Frame
Chief S/C
Deputy S/C
y
Ref # 15
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X
B
y
Deputy S/C
x
y
2-S/C Formation Modelin the Sun-Earth-Moon System
ˆ ˆ,Z z
cr
x
Chief S/C
ˆd cr r r rr r
Relative EOMs:
r t f r t u t
1L 2L
dr
Ephemeris System = Sun+Earth+Moon Ephemeris + SRP
Ref # 15
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Discrete & Continuous Control
Ref # 15
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Linear Targeter
1 1 11
1 1 1
, k kk k kk k
k kk k k k
A Br r rt t
C Dv v v v
11k k k k k kv B r A r v
0v 0v
0v1v
2v
3v Nominal Formation
Path
1 0,t t 2 1,t t
3 2,t t
1 1r r
Segment of Reference Orbit
Ref # 15
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Discrete Control: Linear Targeter
ˆ10 m
0
Y
Dis
tan
ce E
rro
r R
ela
tiv
e to
No
min
al (
cm)
Time (days)
Ref # 15
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Achievable Accuracy via Targeter Scheme
Max
imu
m D
evia
tio
n f
rom
No
min
al (
cm)
Formation Distance (meters)
ˆr rY
Ref # 15
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Continuous Control:LQR vs. Input Feedback Linearization
r t F r t u t Desired Dynamic Response
Anihilate Natural Dynamics
,u t F r t g r t r t
• Input Feedback Linearization (IFL)
• LQR for Time-Varying Nominal Motions
0
1
, , 0
0
T
T Tf
x t r r f t x t u t x x
P A t P t P t A t P t B t R B t P t Q P t
Nominal Control Input
1
Optimal Control, Relative to Nominal, from LQR
Optimal Control Law:
Tu t u t R B P t x t x t
Ref # 15, 20
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LQR Goals
0
12min
ft T T
d dtJ x t Q x t u t R u t dt
d d d
x t x t x t
u t u t u t
12 12 12 5 5 510 ,10 ,10 ,10 ,10 ,10Q diag
1,1,1R diag
Ref # 14,15
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LQR Process
Step 1: Evaluate the Jacobian matrix, at time , associated with nominal path
evaluated on
Step 2: Numerically integrate the differential Riccatti Equation backwards in time
fr
i
i i
t
A t x t
1
1
1
1 0
om , subject to 0.
Step 3: Compute and store the controller gain matrix
Step 4: Repeat steps 1-3 until
Step 5: Numeric
i i N
T Ti i i i i i i
Ti i
i
t t P t
P t A t P t P t A t P t BR B P t Q
K t R B P t
t t
0 0 0
ally integrate the perturbed trajectory forward in time
from , subject to .
Recall from stored data
The new integration step size is defined by the sampled gai
Nt t x t x
K t
n data
Substitute into EOMs and solve for
Apply the computed control input to the perturbed EOMs
d
d d
x t u t
u t u t K t x t x t
Ref # 15
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IFL Process
2
Step 1: Define, analytically, the desired response characteristics
2 ,
Step 2: Begin numerical integration of perturbed path
Step 3: At each point in time, compute
n nr r r r r r g r r
2 2
and apply the control input necessary
to achieve the desired response characteristics:
, ,P D P Ddu t f r r g r r
Annihilate Natural Dynamics
Reflects desired response
Ref # 15
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LQR vs. IFL Comparison
IFL
LQR LQR
IFL
Dynamic Response
Control Acceleration History
5000 km, 90 , 0
0 7 km 5 km 3.5 km 1 mps 1 mps 1 mpsT
x
Dynamic Response Modeled in the CR3BPNominal State Fixed in the Rotating Frame
Ref # 15
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Output Feedback Linearization(Radial Distance Control)
Generalized Relative EOMs
Measured Output
r f r u t
y l r
Formation Dynamics
Measured Output Response (Radial Distance)Actual Response Desired Response
2
2, , ,Tp r r q r
d ly u r g r r
dtr
Scalar Nonlinear Functions of and r r
, 0T
h r t r t u t r t
Scalar Nonlinear Constraint on Control Inputs
Ref # 15
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Output Feedback Linearization (OFL)(Radial Distance Control in the Ephemeris Model)
• Critically damped output response achieved in all cases• Total V can vary significantly for these four controllers
r 2
, Tg r r r r ru t r r f r
r rr
2r
1r
2 2
,1
2
Tg r r r ru t r f r
r r
2, 3
Tr r ru t rg r r r r f r
rr
Control Law ,y l r r
,ˆ
h r ru t r
r
Geometric Approach:Radial inputs only
r
Ref # 15
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OFL Control of Spherical Formationsin the Ephemeris Model
Relative Dynamics as Observed in the Inertial Frame
Nominal Sphere
0 12 5 3 km 0 1 1 1 m/secr r
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OFL Controlled Response of Deputy S/CRadial Distance + Rotation Rate Tracking
2
2
/ /0 0
Radial Error Response (Critically Damped):
2
Rotation Rate Error Response (Exponential Decay):
=
2r
n n
n n
t T t T
n n n n ng
r r r
r
e T
t r r r r r
T
g
e
t
n n nT k
Ref # 15
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OFL Controlled Response of Deputy S/C
2
2
r rr r f u
r r f u
2
2
r rr g tr f u r
r f tu r gr
2
2
constraint
rr r
h h
u t f r
u t r f rg
f
t
t
t
u
g
Equations of Motion in the Relative Rotating Frame
Rearrange to isolate the radial and rotational accelerations:
Solve for the Control Inputs:
ˆˆˆ, ,
ˆˆˆ, ,
r h
r h
f f r f f f f h
u u r u u u u h
Ref # 15
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OFL Control of Spherical FormationsRadial Dist. + Rotation Rate
Quadratic Growth in Cost w/ Rotation Rate
Linear Growth in Cost w/ Radial Distance
Ref # 15
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Inertially Fixed Formationsin the Ephemeris Model
e
100 km
Chief S/C
500 m
ˆ inertially fixed formation pointing vector (focal line)e
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Nominal Formation Keeping Cost(Configurations Fixed in the RLP Frame)
Az = 0.2×106 km
Az = 1.2×106 km
Az = 0.7×106 km
180 days
0
V u t u t dt
5000 kmr
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Max./Min. Cost Formations(Configurations Fixed in the RLP Frame)
x
Deputy S/C
Deputy S/C
Deputy S/C
Deputy S/C
Chief S/C
y
z
Minimum Cost Formations
z
yx
Deputy S/C
Deputy S/CChief S/C
Maximum Cost Formation
Nominal Relative Dynamics in the Synodic Rotating Frame
Ref # 15
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Formation Keeping Cost Variation Along the SEM L1 and L2 Halo Families
(Configurations Fixed in the RLP Frame)
Ref # 15
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Conclusions
• Continuous Control in the Ephemeris Model: – Non-Natural Formations
• LQR/IFL → essentially identical responses & control inputs
• IFL appears to have some advantages over LQR in this case
• OFL → spherical configurations + unnatural rates
• Low acceleration levels → Implementation Issues
• Discrete Control of Non-Natural Formations– Targeter Approach
• Small relative separations → Good accuracy
• Large relative separations → Require nearly continuous control
• Extremely Small V’s (10-5 m/sec)
• Natural Formations– Nearly periodic & quasi-periodic formations in the RLP frame
– Floquet controller: numerically ID solutions + stable manifolds
Ref # 15
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Some Examples from Simulations
• A simple formation about the Sun-Earth L1– Using CRTB based on L1 dynamics– Errors associated with perturbations
• A more complex Fizeau-type interferometer fizeau interferometer.– Composed of 30 small spacecraft at L2– Formation maintenance, rotation, and slewing
Ref # 16, 17, 20
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• Linearized Eq. Of Motion Based on Inertial X, Y, Z Using
X = X0 + x, Y=Y0+y, Z= Z0+z
• Pseudopotential:
•A common approximation in research of this type of orbit models the dynamics using CRTB approximations
•The Linearized Equations of Motion for a S/C Close to the Libration Point Are Calculated at the Respective Libration Point.
M 1 M 2L1
X
R
2r
1r
j
i
x
Y
y
D 1 D 2
D
O
m
Sun
Earth
A State Space ModelA State Space Model
jjj Axx where
00000
00200
02000
100000
010000
001000
,
ZZ
YY
XX
j
j
j
j
j
j
j
j
U
nU
nUA
z
y
x
z
y
x
x
D
MMGn
)( 21
CRTB problem rotating frame
Ref # 16, 17, 20
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)cos(
)sin(
)cos(
)sin(
)cos(
)sin(
zzzz
xyxyxyy
xyxyxyx
zzz
xyxyy
xyxyx
tAz
tAy
tAx
tAz
tAy
tAx
Periodic Reference Orbit
A = Amplitude = frequency = Phase angle
Ref # 16, 17, 20
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FBsolpresjjjjj BA aauxx
dtRQJ TRTR })(){(2
1
0
uuxxxx
xu SBRTj1jj
0)( QSASASBRBS TT1
Centralized LQR Design
3
33
I
OB j
333
333
1
1
Iq
O
OIpQ j 3
1I
rR j
State Dynamics
Performance Index to Minimize
Control
Algebraic Riccatic Eq.time invariant system
B Maps Control Input From Control Space to State Space
Q Is Weight of State Error R Is Weight of Control
Ref # 16, 17, 20
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Centralized LQR Design
Sample LQR Controlled Orbit
V budget with Different Rj Matrices
Ref # 16, 17, 20
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•The A Matrix Does Not Include the Perturbation Disturbances nor Exactly Equal the Reference
• Disturbance Accommodation Model Allows the States to Have Non-zero Variations From the Reference in Response to the Perturbations Without Inducing Additional Control Effort
•The Periodic Disturbances Are Determined by Calculating the Power Spectral Density of the Optimal Control [Hoff93] To Find a Suitable Set of Frequencies.
Disturbance Accommodation Model
Unperturbed x,y,z = 4.26106e-7 rad/s
Perturbed x = 1.5979e-7 rad/s 2.6632e-6 rad/s y = 2.6632e-6 rad/s z = 2.6632e-6 rad/s
Ref # 16, 17, 20
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Without Disturbance Accommodation With Disturbance Accommodation
Stat
e E
rror
Con
trol
Eff
ort
Disturbance accommodation state is out of phase with state error, absorbing unnecessary control effort
Disturbance Accommodation Model
Ref # 16, 17, 20
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Motion of Formation Flyer With Respect to Reference Spacecraft, in Local (S/C-1) Coordinates
Reference Spacecraft Location Reference Spacecraft Location
Reference Spacecraft Location
Formation Flyer Spacecraft Motion
Ref # 16, 17, 20
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V Maintenance in Libration Orbit Formation
Ref # 16, 17, 20
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Stellar Imager (SI) concept for a space-based, UV-optical interferometer, proposed by Carpenter and Schrijver at NASA / GSFC (Magnetic fields, Stellar structures and dynamics)
500-meter diameter Fizeau-type interferometer composed of 30 small drone satellites
Hub satellite lies halfway between the surface of a sphere containing the drones and the sphere origin.
Focal lengths of both 0.5 km and 4 km, with radius of the sphere either 1 km or 8 km.
L2 Libration orbit to meet science, spacecraft, and environmental conditions
Drones
Hub
Sphere Origin500 m
500 m
500 m
Stellar Imager Concept(Using conceptual distances and control requirements
to analyze formation possibilities)
Ref # 16, 17, 20
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-0.50
0.5
-0.50
0.5-0.5
0
0.5
x (km)y (km)
z (k
m)
-0.5 0 0.5-0.5
0
0.5Formation
x (km)
y (k
m)
-0.5 0 0.5-0.5
0
0.5
x (km)
z (k
m)
-0.5 0 0.5-0.5
0
0.5
y (km)
z (k
m)
Focal Length (km)
Slew Angle (deg)
Hub
V(m/s)
Drone 2
V (m/s)
Drone 31
V (m/s)
0.5 30 1.0705 0.8271 0.8307 0.5 90 1.1355 0.9395 0.9587 4 30 1.2688 1.1189 1.1315 4 90 1.8570 2.1907 2.1932
Stellar Imager
• Three different scenarios make up the SI formation control problem; maintaining the Lissajous orbit, slewing the formation, and reconfiguring • Using a LQR with position updates, the hub maintains an orbit while drones maintain a geometric formation
The magenta circles represent drones at the beginning of the simulation, and the red circles represent drones at the end of the simulation. The hub is the black asterisk at the origin.
Drones at beginning
Formation V Cost per slewing maneuver
SI Slewing Geometry
Ref # 16
V V V
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Stellar Imager Mission Study Example Requirements
Maintain an orbit about the Sun-Earth L2 co-linear point
Slew and rotate the Fizeau system about the sky, movement of few km and attitude adjustments of up to 180deg
While imaging ‘drones’ must maintain position ~ 3 nanometers radially from Hub ~ 0.5 millimeters along the spherical surface
Accuracy of pointing is 5 milli-arcseconds, rotation about axis < 10 deg
3-Tiered Formation Control Effort:
Coarse - RF ranging, star trackers, and thrusters ~ centimetersIntermediate - Laser ranging and micro-N thrusters control ~ 50 micronsPrecision - Optics adjusted, phase diversity, wave front. ~ nanometers
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This analysis uses high fidelity dynamics based on a software named Generator that Purdue University has developed along with GSFC
Creates realistic lissajous orbits as compared to CRTB motion. Uses sun, Earth, lunar, planetary ephemeris data Generator accounts for eccentricity and solar radiation pressure. Lissajous orbit is more an accurate reference orbit. Numerically computes and outputs the linearized dynamics matrix, A, for a single satellite at each epoch. Data used onboard for autonomous
computation by simple uploads or onboard computation as a background task of the 36 matrix elements andthe state vector.
•Origin in figure is Earth•Solar rotating coordinates
State Space Controller Development
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• Rotating Coordinates of SI: X = X0 + x, Y=Y0+y, Z= Z0+z
where the open-loop linearized EOM about L2 can be expressed as and the A matrix is the Generator Output
• The STM is created from the dynamics partials output from Generator and assumes to be constant over an analysis time period
xx A Tzyxzyx x
)()()(~ ttt refxxx
xu ~)(tK
State Space Controller Development, LQR Design
0
0)(0 !
)()( 0
k
kkttA
k
ttAett
State Dynamics and Error
Performance Index to Minimize
Control and Closed Loop Dynamics
Algebraic Riccati Eq.time invariant system
uxx BA
ft
t
TT dVWJ0
)()()(~)(~ uuxx
0)( QSASASBRBS TT1
xx ~))((~ tBKA
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3
33
I
OB j
333
333
1
1
Iq
O
OIpQ j
3
1I
rR j
B Maps Control Input From Control Space to State Space
W Is Weight of State ErrorV Is Weight of Control
State Space Controller Development, LQR Design
VW
• Expanding for the SI collector (hub) and mirrors (drones) yieldsa controller
• Redefine A and B such that
1222~~ uuxx BBA
jA
A
A
A
2
1
j
i
i
BB
BB
BB
B
B
1
1
1
1
Ref # 16
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Simplified extended Kalman Filter
Using dynamics described and linear measurements augmented by zero-mean white Gaussian process and measurement noise
Discretized state dynamics for the filter are: where w is the random process noise
The non-linear measurement model is and the covariances of process and measurement noise are
Hub measurements are range(r) and azimuth(az) / elevation(el) angles from Earth Drone measurements are r, az, el from drone to hub
wBA kdkdk uxx ~~1
vm kk )~(xy QwwE T
RvvE T
y
x
z
Hub
b2
r
az
el
b1
b3
Drone
r x y z 2 2 2
elz
rand az
x
r el
sin ( ), sin (
cos( ))1 1
Ref # 16
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• Continuous state weighting and control chosen as
•The process and measurement noiseCovariance (hub and drone) are
• Initial covariances
Simulation Matrix Initial Values
31
31
31
61
61
61
e
e
e
e
e
e
W
1
1
1
V
61
61
61
0
0
0
e
e
eQc
2
2
2
1500000
3.0
1500000
3.0
1.0
R
2
2
2
5.0
0003.0
5.0
0003.0
0001.0
R
2
2
21
4.86
4.86
4.86
1
1
1
)(P
2
2
2
2
2
2
1
0864.
0864.
0864.
001.
001.
001.
)(P
Ref # 16
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• Only Hub spacecraft was simulated for maintenance • Tracking errors for 1 year: Position and Velocity• Steady State errors of 250 meters and .075 cm/s
Results – Libration Orbit Maintenance
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• Estimation errors for 12 simulations for 1 year: Position and Velocity• Estimation errors of 250 meters and 2e-4 m/s in each component
Results – Libration Orbit Maintenance
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Length of simulation is 24 hours Maneuver frequency is 1 per minute Using a constant A from day-2 of the previous simulation Tuning parameters are same but strength of process noise is
241
241
241
0
0
0
e
e
eQc
Formation Slewing: 900 simulation shown
Results – Formation Slewing
Purple - Begin Red - End
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• Tracking errors for 24 hours: Position and Velocity• Steady State errors of 50 meters - hub, 3 meters - drone and 5 millimeters/sec – hub, and 1 millimeter/s - drone
0 500 1000 1500-0.02
0
0.02
x er
ror
(km
)
drone 2 tracking error
0 500 1000 1500-0.2
0
0.2
y er
ror
(km
)
0 500 1000 1500-1
0
1
z er
ror
(km
)
time
0 500 1000 1500-0.01
0
0.01
xdot
err
or (
m/s
)
0 500 1000 1500-0.1
0
0.1
ydot
err
or (
m/s
)
0 500 1000 1500-0.2
0
0.2
zdot
err
or (
m/s
)
time
HUB DRONE
Results – Formation Slewing
Represents both 0.5 and 4 km focal lengths
Ref # 16
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• Estimation errors;12 simulations for 24 hours: position and velocity• Estimation 3 errors of ~50km and ~1 millimeter/s for all scenarios
Hub estimation 0.5 km separation / 90 deg slew
Drone estimation 0.5 km separation / 90 deg slew
Results – Formation Slewing
Ref # 16
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Focal Slew Hub Drone 2 Drone 31Length (km) Angle (deg) (m/s) (m/s) (m/s)
0.5 30 1.0705 0.8271 0.83070.5 90 1.1355 0.9395 0.95874 30 1.2688 1.1189 1.13154 90 1.8570 2.1907 2.1932
Formation Slewing Average Vs (12 simulations)
Focal Slew Hub Drone 2 Drone 31Length (km) Angle (deg) mass-prop mass-prop mass-prop
(g) (g) (g)
0.5 30 6.0018 0.8431 0.84680.5 90 6.3662 0.9577 0.97734 30 7.1135 1.1406 1.15344 90 10.4112 2.2331 2.2357
Formation Slewing Average Propellant Mass
Results – Formation Slewing
Ref # 16
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Focal Slew Hub Drone 2 Drone 31 Length (km) Angle (deg) (m/s) (m/s) (m/s)
0.5 30 0.0504 0.0853 0.09980.5 90 0.1581 0.2150 0.23154 30 0.4420 0.5896 0.64414 90 1.3945 1.9446 1.9469
Focal Slew Hub Drone 2 Drone 31Length (km) Angle (deg) mass-prop mass-prop mass-prop
(g) (g) (g)
0.5 30 0.2826 0.0870 0.10170.5 90 0.8864 0.2192 0.23604 30 2.4781 0.6010 0.65664 90 7.8182 1.9822 1.9846
Formation Slewing Average Vs (without noise)
Results – Formation Slewing
Formation Slewing Average Propellant Mass (without noise)
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• Rotation about the line of sight• Length of simulation is 24 hours • Maneuver frequency is 1 per minute• Using a constant A from day-2 of the previous simulation• Tuning parameters are same as slewing • Reorientation of 4 drones 900
Results – Formation Reorientation
Ref # 16
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• Tracking errors: Position and Velocity• Steady State errors of 40 meters - hub, 4 meters - drone and 8 millimeters/sec – hub, and 1.5 millimeter/s - drone
HUB DRONE
Results – Formation Reorientation
Ref # 16
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Focal Slew Hub Drone 2 Drone 31Length (km) Angle (deg) (m/s) (m/s) (m/s)0.5 90 1.0126 0.8421 0.80954 90 1.0133 0.8496 0.8190
Formation Reorientation Average Vs
Focal Slew Hub Drone 2 Drone 31Length (km) Angle (deg) mass-prop mass-prop mass-prop
(g) (g) (g)0.5 90 5.6771 0.8584 0.82524 90 5.6811 0.8661 0.8349
Formation Reorientation Average Propellant Mass
• The steady-state estimation 3 values are: x ~30 meters, y and z ~ 50 meters• The steady-state estimation 3 velocity values are about 1 millimeter per second. • For any drone and either focal length, the steady-state 3 position values are less than 0.1 meters, and the steady-state velocity 3 values are less than 1e-6 meters per second.
Results – Formation Reorientation
Ref # 16
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Focal Slew Hub Drone 2 Drone 31Length (km) Angle (deg) (m/s) (m/s) (m/s)0.5 90 0.0408 0.1529 0.1496 4 90 0.0408 0.1529 0.1495
Formation Reorientation Average Vs (without noise)
Focal Slew Hub Drone 2 Drone 31Length (km) Angle (deg) mass-prop mass-prop mass-prop
(g) (g) (g)0.5 90 0.2287 0.1623 0.15254 90 0.2287 0.1623 0.1524
Formation Reorientation Average Propellant Mass ( without noise)
Results – Formation Reorientation
Ref # 16
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o The control strategy and Kalman filter using higher fidelity dynamics provides satisfactory results.
o The hub satellite tracks to its reference orbit sufficiently for the SI mission requirements. The drone satellites, on the other hand, track to only within a few meters.
o Without noise, though, the drones track to within several micrometers.
o Improvements for first tier control scheme (centimeter control) for SI could be accomplished with better sensors to lessen the effect of the process and measurement noise.
Summary (using example requirements and constraints)
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o Tuning the controller and varying the maneuver intervals should provide additional savings as well. Future studies must integrate the attitude dynamics and control problem
o The propellant mass and results provide a minimum design boundary for the SI mission. Additional propellant will be needed to perform all attitude maneuvers, tighter control requirement adjustments, and other mission functions.
o Other items that should be considered in the future include;• Non-ideal thrusters, • Collision avoidance, • System reliability and fault detection• Nonlinear control and estimation• Second and third control tiers and new control strategies and algorithms
Ref # 16
Summary (using example requirements and constraints)
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- Decentralized control
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DRO Mission Metrics
Earth-constellation distance: > 50 Re (less interference) and< 100 Re (link margin). Closer than 100 Re would be desirable to improve the link margin requirement A retrograde orbit of <160 Re (10^6 km), for a stable orbit would be ok
The density of "baselines" in the u-v plane should be uniformly distributed. Satellites randomly distributed on a sphere will produce this result.
Formation diameter: ~50 km to achieve desired angular resolution
The plan is to have up to 16 microsats, each with it's own "downlink".
Satellites will be "approximately" 3-axis stabilized.
Lower energy orbit insertion requirements are always appreciated.
Eclipses should be avoided if possible.
Defunct satellites should not "interfere" excessively with operational satellites.
Ref # 1,18
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Lunar Orbit
Earth
L1 L2
Sun-Earth Line
Direction of Orbit is Retrograde
Solar Rotating Coordinate System ( Earth-Sun line is fixed)
Distant Retrograde Orbit (DRO)
Why DRO?
• Stable Orbit• No Skp V• Not as distant as L1• Mult. Transfers• No Shadows?• Good Environment
Really a Lunar Periodic OrbitClassified as a Symmetric Doubly Asymptotic Orbit in the Restricted Three-Body Problem
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Earth Distant Retrograde Orbit (DRO) Orbit
Ref # 1,18
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DRO Formation Sphere
•Matlab generated sphere based on S03 algorithm Uniform distribution of points on a unit sphere 16 points at vertices represents spacecraft locations
X-Y view
Y-Z view
3-D view
SIRA spacecraft
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DRO Formation Control Analysis
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DRO Formation Control Analysis
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DRO Formation Control Analysis
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Formation Control Analysis
Phase Max Mean Min Std[m/s] [m/s] [m/s] [m/s]
a 0.635 0.607 0.592 0.014b 1.323 0.792 0.392 0.293c 0.757 0.674 0.541 0.069d 0.679 0.616 0.582 0.031e 1.201 0.721 0.367 0.263f 0.679 0.608 0.503 0.056
Phase Descriptiona) Init 25km sphereb) Maintain 25km sphere (strict PD control) one monthc) Maintain 25km sphere (loose control) one monthd) Resize from 25km to 50kme) Maintain 50km sphere (strict PD control) one monthf) Maintain 50km sphere (loose control) one month
How much V to initialize, maintain, and resize?
Examples:
Initialize & maintain 2 yr:= 33 m/s
Initialize, Maintain 2yr, & four resizes:
= 36 m/s
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Earth - Moon L4 Libration Orbitan alternate orbit location
Ref # 1,18
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-0.015
-0.010
-0.005
0.000
0.005
0.010
0.015
0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.055
FreeFlyer Plot Window4/30/2003
Km
/s
A1.ElapsedDays (Days)
ImpulsiveBurn.X-Component (Km/s) ImpulsiveBurn.Y-Component (Km/s)
ImpulsiveBurn.Z-Component (Km/s)
0
5
10
15
20
25
0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175
FreeFlyer Plot Window4/30/2003
Km
A1.ElapsedDays (Days)
Formation.Range (Km) Formation.Range (Km) Formation.Range (Km) Formation.Range (Km)
Formation.Range (Km) Formation.Range (Km) Formation.Range (Km)
Earth/Moon L4 Libration Orbit Spacecraft controlled to maintain only relative separations Plots show formation position and drift (sphere represent 25km radius) Maneuver performed in most optimum direction based on controller output
Impulsive Maneuver of 16th s/c
Radial Distance from Center
DRO Formation Control Analysis
Ref # 1,18
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NODE 3
NODE 4
NODE 5
NODE 1
NODE 2
MANY NODES IN A NETWORK CAN COOPERATETO BEHAVE AS SINGLE VIRTUAL PLATFORM:
General Theory of Decentralized ControlGeneral Theory of Decentralized Control
•REQUIRES A FULLY CONNECTED NETWORKOF NODES.
•EACH NODE PROCESSES ONLY ITS OWNMEASUREMENTS.
•NON-HIERARCHICAL MEANS NO LEADS ORMASTERS.
•NO SINGLE POINTS OF FAILURE MEANSDETECTED FAILURES CAUSE SYSTEM TODEGRADE GRACEFULLY.
•BASIC PROBLEM PREVIOUSLY INVESTIGATEDBY SPEYER.
•BASED ON LQG PARADIGM.
•DATA TRANSMISSION REQUIREMENTS AREMINIMIZED.
Ref # 19
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References, etc.
1. NASA Web Sites, www.nasa.gov, Use the find it @ nasa search input for SEC, Origin, ESE, etc.2. Earth Science Mission Operations Project, Afternoon Constellation Operations Coordination Plan , GSFC, A. Kelly May 20043. Fundamental of Astrodynamics, Bate, Muller, and White, Dover, Publications, 19714. Mission Geometry; Orbit and Constellation design and Management, Wertz, Microcosm Press, Kluwer Academic Publishers, 20015. An Introduction to the Mathematics and Methods of Astrodynamics, Battin6. Fundamentals of Astrodynamics and Applications, Vallado, Kluwer Academic Publishers, 20017. Orbital Mechanics, Chapter 8, Prussing and Conway8. Theory of Orbits: The Restricted Problem of Three Bodies V. Szebehely.. Academic Press, New York, 19679. Automated Rendezvous and Docking of Spacecraft, Wigbert Fehse, Cambridge Aerospace Series, Cambridge University Press, 200310. “A Universal 3-D Method for Controlling the Relative Motion of Multiple Spacecraft in Any Orbit,” D. C. Folta and D. A. Quinn Proceedings of the AIAA/AAS
Astrodynamics Specialists Conference, August 10-12, Boston, MA.11. “Results of NASA’s First Autonomous Formation Flying Experiment: Earth Observing-1 (EO-1)”, Folta, AIAA/AAS Astrodynamics Specialist Conference,
Monterey, CA, 2002 12. “Libration Orbit Mission Design: Applications Of Numerical And Dynamical Methods”, Folta, Libration Point Orbits and Applications, June 10-14, 2002, Girona,
Spain 13. “The Control and Use of Libration-Point Satellites”. R. F. Farquhar. NASA Technical Report TR R-346, National Aeronautics and Space Administration, Washington,
DC, September, 197014. “Station-keeping at the Collinear Equilibrium Points of the Earth-Moon System”. D. A. Hoffman. JSC-26189, NASA Johnson Space Center, Houston, TX, September
1993.15. “Formation Flight near L1 and L2 in the Sun-Earth/Moon Ephemeris System including solar radiation pressure”, Marchand and Howell, paper AAS 03-59616. Halo Orbit Determination and Control, B. Wie, Section 4.7 in Space Vehicle Dynamics and Control. AIAA Education Series, American Institute of Aeronautics and
Astronautics, Reston, VA, 1998.17. “Formation Flying Satellite Control Around the L2 Sun-Earth Libration Point”, Hamilton, Folta and Carpenter, Monterey, CA, AIAA/AAS, 200218. “Formation Flying with Decentralized Control in Libration Point Orbits”, Folta and Carpenter, International Space Symposium Biarritz, France, 200019. SIRA Workshop, http://lep694.gsfc.nasa.gov/sira_workshop20. “Computation and Transmission Requirements for a Decentralized Linear-Quadratic-Gaussian Control Problem,” J. L. Speyer, IEEE Transactions on Automatic
Control, Vol. AC-24, No. 2, April 1979, pp. 266-269.
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References, etc.
Questions on formation flying? Feel free to contact:
kathleen howell howell@ecn.purdue.edu
david.c.folta@nasa.gov
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Backup and other slides
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Numerical Systems Dynamical Systems
• Limited Set of Initial Conditions• Perturbation Theory• Single Trajectory• Intuitive DC Process• Operational
• Qualitative Assessments• Global Solutions• Time Saver / Trust Results• Robust• Helps in choosing numerical methods (e.g., Hamiltonian => Symplectic Integration Schemes?)
DST/Numerical Comparisons
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Libration Point Trajectory Generation Process
Phase and Lissajous Utilities
Generate Lissajous of Interest
Compute Monodromy Matrix
And Eigenvalues/Eigenvectors
For Half Manifold of Interest
Globalize the Stable Manifold
Use Manifold Information for
a Differential Corrector Step
To Achieve Mission Constraints.Transfer
Manifold
Monodromy
Patch Points and Lissajous
Fixed Points and Stable and Unstable Manifold
Approximations
.
.
1-D Manifold
Transfer Trajectory to Earth Access Region
Universe/User Control and Patch Pts
Output / Intermediate Data
Lissajous
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MAP Mission Design: DST Perspective
• MAP Manifold and Earth Access• Manifold Generated Starting with Lissajous Orbit
• Swingby Numerical Propagation• Trajectory Generated Starting with Manifold States
Starting Point
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JWST DST Perspective
• JWST Manifold and Earth Access• Manifold Generated Starting with Halo Orbit
• Swingby Numerical Propagation• Trajectory Generated Starting with Manifold States Starting Point
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Two – Body Motion
•Motion of spacecraft in elliptical orbit•Counter-clockwise•x and y correspond to P and Q axes in PQW frame
Two angles are defined E is Eccentric Anomalyis True Anomaly
x and y coordinates are x = r cos y = r sin
In terms of Eccentric anomaly, Ea cosE = ae + xx = a ( cosE - e)From eqn. of ellipse: r = a(1 - e2)/(1 + e cos)Leads to: r = a(1 – e cosE)Can solve for y: y =a(1 - e2)1/2 sinE
Coordinates of spacecraft in plane of motion are then
r = a( 1 - ecosE) = a(1- e2) / (1 + e cos x = a( cosE - e) = r cos vx = (/p)1/2 [-sin ] y = a( 1 - e2)1/2 sinE = r sin vy = (/p)1/2 [e + cos ] where p = a (1 - e2)
- -
E
a
ae
ar
Center of Orbit
Focus
Spacecraft
x
y
P
Q
Ref # 3-7
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Differentiate x, y, and r wrt time in terms of E to getdx/dt = -asinE de/dt dy/dt = a(1-e2)1/2 cosE dE/dt dr/dt = ae sinE dE/dt
From the definition of angular momentum h = r cross dr/dt and expand to get h = a2(1-e2)1/2 dE/dt[1-ecosE] in direction perpendicular to orbit plane
Knowing h2 =ma(1-e2), equate the expressions and cancel common factor to yield()1/2/a3/2 = (1-ecosE) dE/dt
Multiple across by dt and integrate from the perigee passage time yields n(t0 - tp) = E – esinE
Where n = ()1/2/a3/2 is the mean motion
We can also compute the period: P=2p(a3/2 / ()1/2 ) which can be associated with Kepler’s 3rd law
---
- -
Two – Body Motion
Ref # 3-7
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Coordinate system transformationEuler Angle Rotations
x’ = x cos + y sin y’ = -x sin + y cos z’ = z
cos sin -sin cos 0 0 1
x’ y’z’
=x yz
Az’()x yz
=
cos 0sin0 1-sin 0 cos
x’ y’z’
=x yz
1 00 cos -sin 0 -sin -cos
x’ y’z’
=x yz
X
Y X’Y’
Suppose we rotate the x-y plane about the z-axis by an angle and call the new coordinates x’,y’,z’
Rotate about Z
Rotate about Y
Rotate about X
Z
Ref # 3-7
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Coordinate system transformationOrbital to inertial coordinates
Inertial to/from Orbit planer = r(x,y,z) r = r(P,Q,W)-- - -
In orbit plane system, position r= (rcosf)P + (rsinf)Q + (0)W where P,Q,W are unit vectors
Px Qx Wx
Py Qy Wy
Pz Qz Wz
x yz
=
rcosf rsinf0
cos sin -sin cos 0 0 1
1 00 cos i-sin i 0 -sin i -cos i
cos sin -sin cos 0 0 1
rcosf rsinf0
=x yz
cos cos - sin sin cos i -cos sin - sin cos cos i sin sin isin cos - cos sin cos i -sin sin - cos cos cos i -cos sin i sin sin i cos sin i cos i
rcosf rsinf0
=x yz
-
For Velocity transformation, vx = (/p)1/2 [-sin ] and vy = (/p)1/2 [e + cos ]
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Principles Behind Decentralized ControlPrinciples Behind Decentralized Control
• THE STATE VECTOR IS DECOMPOSED INTO TWO PARTITIONS:
DEPENDS ONLY ON THE CONTROL.
DEPENDS ONLY ON THE LOCAL MEASUREMENT DATA NODE-j.
• A LOCALLY OPTIMAL KALMAN FILTER OPERATES ON .
• A GLOBALLY OPTIMAL CONTROL IS COMPUTED, USING AND GLOBALLY OPTIMAL DATA THAT IS RECONSTRUCTED LOCALLY USING TWO ADDITIONAL VECTORS:
A “DATA VECTOR” , IS MAINTAINED LOCALLY IN ADDITION TO THE STATE.
A TRANSMISSION VECTOR THAT MINIMIZES THE DIMENSIONS OF THE DATA WHICH MUST BE EXCHANGED BETWEEN NODES.
• EACH NODE-j COMPUTES AND TRANSMITS TO AND RECEIVES FROM ALLTHE OTHER NODES IN THE NETWORK.
xC
x jD
x jD
xC
h j
lj
lj jl
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The Hohmann Transfer
Vr a
2 2 1
ar ra p2
where:
Vr
r
r rpp
a
a p
2V
r
r
r raa
p
a p
2
Vrc
Vr
r
r r r r
r
r r11
2
1 2 1 1
2
1 2
2 21
Vr
r
r r r r
r
r r22
1
1 2 2 2
1
1 2
21
2
VT = V1 + V2
Ref # 3,7
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CONTROL VECTORS
TRANSMISSION VECTORS
z -1
TRANSMISSION VECTOR
z -1
z -1
z -1
GLOBAL DATA
RECONSTRUCTION
CONTROL VECTORGLOBALLY
OPTIMAL CONTROL
FROM NETWORK
TO NETWORK& LOCAL PLANT
TO NETWORK
LOCAL SENSORLOCALLY
OPTIMAL FILTER
The LQG Decentralized Controller OverviewThe LQG Decentralized Controller Overview
Ref # 19
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