NASA / Goddard Space Flight Center
Goddard Space Flight Center
Formation Flying
in Earth, Libration,
and Distant Retrograde Orbits
David Folta
NASA - Goddard Space Flight Center
Advanced Topics in Astrodynamics
Barcelona, Spain
July 5-10, 2004
1
Agenda
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I. Formation flying – current and future
II. 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 Formations
V. References
•
All references are textbooks and published papers
•
Reference(s) used listed on each slide, lower left, as ref#
2
NASA Themes and Libration Orbits
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NASA Enterprises of Space Sciences (SSE) and Earth Sciences
(ESE) are a combination of several programs and themes
SSE
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ESE
SEU
SEC
Origins
• 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.
3
Ref #1
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Earth Science
Low 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)
4
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
5
Ref #1
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Future Mission Challenges
Considering science and operations
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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
Operational Challenges
Science Challenges
 Interferometers
 Environment
 Data Rates
 Limited Maneuvers
 Servicing of resources in libration orbits
 Minimal fuel
 Constrained communications
 Limited DV directions
 Solar sail applications
 Continuous control to reference trajectories
 Tethered missions
 Human exploration
6
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Background on perturbation
theory / accelerations
• Two Body Motion
• Atmospheric Drag
• Potential Models Forces
• Solar Radiation Pressure
7
Two – Body Motion
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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
8
Ref # 3-7
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FORCES ON
PROPAGATED ORBIT
• Equation Of Motion Propagated.
+ accelerations
• External Accelerations Caused By Perturbations
a = anonspherical+adrag+a3body+asrp+atides+aother
9
Ref #3-7
Gaussian Lagrange Planetary Equations
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• 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
10
Ref # 3-7
Geopotential
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• 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
11
Ref # 3-7
Potential Accelerations
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

 r l 1 Pl m sin  C l m cos m  S l m sin m

m 1
The coordinates of P are now expressed in spherical coordinates r, ,  , where  is the geocentric latitude and  is the
longitude. R is the equatorial radius of the primary body and Pl m sin   is the Legendre’s Associated Functions of degree and 


l
l
1
order m.
The coefficients and are referred to as spherical harmonic coefficients. If m = 0 the coefficients are referred to as zonal
harmonics. If l  m  0 they are referred to as tesseral harmonics, and if l  m  0 , they are called sectoral harmonics.
Simplified J2 acceleration model for analysis with
acceleration in inertial coordinates
ai =
a=
D
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R
  1   J l   Pl sin    
r  l  2  r 
l 1
d + d + d
=
i
j
k
dx + dy + dz
aj =
ak =
Ref # 3-7
-3J2Re2 ri
2r5
-3J2Re2 rj
2r5
-3J2Re2 rj
2r5
1-
5rk2
r2
1-
5rk2
r2
3-
5rk2
r2
12
Atmospheric Drag
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• 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
13
Ref # 3-7
Solar Flux Prediction
Historical Solar Flux, F10.7cm values
Observed and Predicted (+2s) 1945-2002
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F10.7 RADIO FLUX OBSERVED AND PREDICTED
F10.7 - observed
F10.7 - Predicted
300
250
F10.7 RADIO FLUX
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350
200
150
100
50
0
1940
1950
1960
1970
1980
YEAR
Ref # 3-7
1990
2000
2010
14
Drag Acceleration
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• Acceleration defined as
NASA / Goddard Space Flight Center
2 v^
C
A
r
v
d
a
a= 1
2
m
A = Spacecraft cross sectional area, (m2)
Cd = Spacecraft Coefficient of Drag, unitless
m = mass, (kg)
r = atmospheric density, (kg/m3)
va = s/c velocity wrt to atmosphere, (km/s)
^
v = inertial spacecraft velocity unit vector
Cd A = Spacecraft ballistic property
m
• Planetary Equation for semi-major axis decay rate of circular orbit
(Wertz/Vallado p629), small effect in e
Da = - 2pCd A r a2
m
15
Ref # 3-7
Solar Radiation Pressure Acceleration
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1
F   GAs / csˆ , but
c
G 1350 watts / m 2

 4.5x106 watt sec/ m3
8
c
3x10 m / s
 4.5x106 N / m 2  PSR
Therefore,
F  PSR As / csˆ
 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
F  Fs  Fn
Where Fs is the force in the solar direction and Fn is the force normal to the surface
ˆ
F  PSP A s / c [sˆ  2 n]
where
From Lagrange 's planetary equations
  absorptivity coefficient, 0    1,   1  
da
2
a(1  e 2 )

{esin  FR 
FI }
dt n 1  e 2
r
Where FR and FI are the radial and in  track solar pressure forces.
  reflectivity coefficient of specular reflection, 0    1
n̂ is a unit vector normal to the surface, A s / c
As / cis the area normal to the sun direction
16
Ref # 3-7
Other Perturbations
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NASA / Goddard Space Flight Center
• Third Body
a3b =
-/r3
-
r=
-
(rj/rj3
-
– 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
17
Ref # 3-7
Ballistic Coefficient
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NASA / Goddard Space Flight Center
• 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
18
Ref # 3-7
Ballistic Effects
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Varying the mass to area yields different decay rates
Sample: 100kg with area of 1, 10, 25, and 50m2, Cd=2.2
FreeFlyer Plot Window
sat100to1.BL_A (Km)
Sat100to10.BL_A (Km)
Sat100to25.BL_A (Km)
Sat100to50.BL_A (Km)
6750
6500
10m2
6250
50m2
Km
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2/11/2003
25m2
6000
5750
5500
5250
0
Ref # 3-7
5
10
15
20
sat100to1.ElapsedDays (Days)
25
30
19
Numerical Integration
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Goddard Space Flight Center
• 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
20
Ref # 3-7
Coordinate Systems
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• Origin of reference frames:
Planet
Barycenter
Topographic
• Reference planes:
Equator – equinox
Ecliptic – equinox
Equator – local meridian
Horizon – local meridian
Geocentric Inertial (GCI)
Greenwich Rotating Coordinates (GRC)
Most used systems
GCI – Integration of EOM
ECEF – Navigation
Topographic – Ground station
21
Ref # 3-7
Describing Motion Near a Known Orbit
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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)
r
*
 r (t )
*
v *  v * (t )
r  r (t )
v  v (t )
Chief (reference Orbit)
Deputy
_
_v*
r*
_
v
_
r
dr (t )  r (t )  r * (t )
Relative Motion
dv (t )  v (t )  v * (t )
What equations of motion does the
relative motion follow?
d2
r (t )  1 / mF (r , v )  f (r , v )
2
dt
d2
d2
d2
d2 *
*
*
*
d
r

(
r
(
t
)

r
(
t
))

r
(
t
)

r
(
t
)

f
(
r
,
v
)

f
(
r
,
v
)
2
2
2
2
dt
dt
dt
dt
d2
*
*
22
d
r
(
t
)

f
(
r
,
v
)

f
(
r
,
v
)
Ref # 5,6,7
dt 2
Describing Motion Near a Known Orbit
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NASA / Goddard Space Flight Center
As the last equation stands it is exact. However if dr is sufficiently
-- …
small, the term f(r) can be expanded via Taylor’s series
f (r )  f (r *  dr )  f (r * ) 
f
|r r * dr  ...
r
Substituting in yields a linear set of coupled ODEs
d2
f
*
*
*
d
r
(
t
)

f
(
r
)

f
(
r
)

f
(
r
)

|
d
r

f
(
r
)
*
2
r r
dt
r
d2
f
d
r

|r r * dr Lets call this (1)
2
dt
r
This is important since it will be our starting point for everything
that follows
23
Ref # 5,6,7
Describing Motion Near a Known Orbit
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• 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
_
v*
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^
T
_
r*
^
R
^
N
*
r
Rˆ  *
r
Nˆ 
r* v*
r v
*
*
Tˆ  Nˆ  Rˆ
The frame described is known as
- Hill’s
- RTN
- Clohessy-Wiltshire
- RAC
- LVLH
- RIC
Any vector relative to the chief is given by: dr  drRˆ Rˆ  drTˆTˆ  drNˆ Nˆ
24
Ref # 5,6,7
Describing Motion Near a Known Orbit
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First we evaluate in
arbitrary coordinates


f  ( 3 r )  f i  ( 3 xi )
r
r
f
f
 
3

 i 
( 3 xi )  5 xi x j  3 d ij
r x j x j r
r
r
f

 5 ( xi x j  r 2d ij )
r r
 2x2  y 2  z 2

 5
3 xy
r 
3xz

3 xy


3 yz

2 z 2  y 2  x 2 
3xz
2 y2  x2  z 2
3 yz
^
Only in RTN , where the chief has a constant position =Rr*, the matrix takes a form
xRTN = r* , yRTN = zRTN = 0
2 0 0 
2 0 0 




f

2
|r  r *  3  0  1 0   n  0  1 0 
r
r* 

 0 0  1
 0 0  1


25
Ref # 5,6,7
Transforming the Left Hand Side of (1)
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Now convert the left hand side of (1) to RTN frame,
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D d
d
 
 
 
Dt dt  moving dt  fixed
Newton’s law involves 2nd derivatives:
D2
d2
d 2
d
d
r

d
r


d
r

2


dr    (  dr )
2
2
2
Dt
dt
dt
dt
D2
d2
d 2
D
dr  2 dr  2  dr  2 
dr    (  dr )
2
Dt
dt
dt
Dt
In the RTN frame
 0
 
   0
 n
 
Ref # 5,6,7
  n2 x 
 x
 x 
  ny 
 2 
 
 


D
D
dr   y  d  D  0

dr   y   
dr   nx    (  dr )    n y 
Dt
Dt
 z  dt
Dt
 0 
 0 
 z 
 


 


2
2
2







2
n
x

n
y

n
x
2






D
2
2
2
dr    n y   2  n x     n y 
2
Dt
  n2 z   0   0 

 
 

26
Transforming the EOM yields
Clohessy-Wiltshire Equations
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d2
f
dr  |r  r * dr
2
dt
r
2
2
 x   3n x  2n y 

  
RTN

  y     2nx 

 z  

nz
  

y  2nx  y  2nx  k1
x  n2 x  2nk1
v0
2k1
x  x0 cos(nt )  sin( nt ) 
n
n
y  2 x0 sin( nt ) 
2v0
2k
cos( nt )  1 t  y0
Note that the y-motion
n
n
z  z0 cos( nt ) 
Ref # 5,6,7
A “balance” form will have no
secular growth, k1=0
0
n
sin( nt )
(associated with Tangential) has
twice the amplitude of the x
27
motion (Radial)
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Relative Motion
A numerical simulation using RK8/9 and point mass
Effect of Velocity (1 m/s) or Position(1 m) Difference
3.5
3.0
2.5
2.0
main.RadialSeparation (Km)
• 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
1.5
1.0
Initial Location 0,0,0
0.5
0.0
-10
-1 m in Radial Position
-5
0
5
10
15
20
main.AlongArcTrackSeparation (Km)
25
30
35
6/4 /2004
+1m/s in Radial Velocity
-0.00100
6/4/2004
1.00
-0.00125
Initial Location 0,0,0
0.75
-0.00150
Initial Location 0,0,-1
-0.00175
0.50
-0.00200
0.25
-0.00225
main.RadialSeparation (Km)
main.RadialSeparation (Km)
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+1 m/s in Alongtrack Velocity
-0.00250
-0.00275
-0.00300
0.00
-0.25
-0.50
-0.75
-0.045
-0.040
-0.035
-0.030
-0.025
-0.020
-0.015
main.AlongArcTrackSeparation (Km)
-0.010
-0.005
0.000
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
main.AlongArcTrackSeparation (Km)
-0.5
0.0
0.5
28
Shuttle Vbar / Rbar
• Shuttle approach strategies
•Vbar – Velocity vector direction in an LVLH (CW) coordinate system
•Rbar – Radial vector direction in an LVLH (CW) coordinate system
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Goddard Space Flight Center
Ref # 7,9
• 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
29
What Goes Wrong with an Ellipse
NASA / Goddard Space Flight Center
Goddard Space Flight Center
In state – space notation, linearization is written as
 0 
 dr 

A   f
dS   

0
 dv 
 x

Since the equation is linear
d
dS  dS 0    A
dt
d
dS  AdS
dt
t
(t , t0 )     dt A(r * (t )) (t , t0 )
t0
which has no closed form solution if
A(t1 ), A(t2 )  0
Ref # 5,6,7
30
Lambert Problem
Goddard Space Flight Center
NASA / Goddard Space Flight Center
• Consider two trajectories r(t) and R(t).
• Transfer from r(t) to R(t) is affected by two DVs
– 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
V
intersects at time t3
f
R3
• Lambert problem:
Determine the two DVs
f
V3
V2
R2
V1
R1
v
r
Vi
2
dV i
1
v
r1
t1
Ref # 5,6,7
dV
t2
2
t3
31
Lambert Problem
NASA / Goddard Space Flight Center
Goddard Space Flight Center
• 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’.
32
Ref # 5,6,7
EO-1 GSFC Formation Flying
New Millennium Requirements
Goddard Space Flight Center
NASA / Goddard Space Flight Center
• 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 Landsat7. EFF shall predict and plan formation control
maneuvers or Da maneuvers to maintain the ground
track if necessary.
• 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.
33
Ref # 10,11
EO-1 GSFC Formation Flying
NASA / Goddard Space Flight Center
Goddard Space Flight Center
34
Formation Flying Maintenance Description
Landsat-7 and EO-1
Goddard Space Flight Center
FF Start
Landsat-7
Velocity
Radial Separation (m)
NASA / Goddard Space Flight Center
Different Ballistic Coefficients and Relative Motion
Nadir
Direction
EO -1 Spacecraft
In-Track Separation (Km)
Ideal FF Location
FF Maneuver
I-minute separation
in observations
Observation Overlaps
35
Ref # 10,11
EO-1 Formation Flying Algorithm
NASA / Goddard Space Flight Center
Goddard Space Flight Center
Reference
Orbit
Formation Flyer
Initial State
Reference S/C
Initial State
• Determine (R1,V1) at t1 (where you want to be at
time t1).
r0,v0
r1,v1
• Determine (r1,v1) at t0 (where you are at time
t0).
dr0,dv0
• Project (R1,V1) through -Dt to determine
(r0,v0) (where you should be at time t0).
• Compute (dr0,dv0) (difference between where you
are and where you want be at t0).
Keplarian State (t1)
Keplarian State (t0)
dv(t0)
Dt
Keplerian State (ti)
Transfer
Orbit
R1,V1
Formation Flyer
Target State
Ref # 10,11
Reference
S/C
Final State
Keplerian State K(tF)
dr(t0)
ti
t1t0
tf
Maneuver
Window
36
State Transition Matrix
Goddard Space Flight Center
NASA / Goddard Space Flight Center
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 :
 t , t   2 t1 , t 0 
t1 , t 0    1 1 0

 3 t1 , t0   4 t1 , t 0 
We find the inverse may be directly obtained by employing symplectic properties:
T






t
,
t
 1 t1 , t 0    4 1 0 T
 3 t1 , t 0 
 2 t1 , t0 T 

1 t1 , t0 T 
1 t0 , t1   2 t 0 , t1 
 t1 , t 0   t0 , t1   

 3 t 0 , t1   4 t 0 , t1 
1
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:
~
~
R (t1 )  1 t1 , t0  R * (t0 )  1 t0 , t1 
R (t1 )   2 t1 , t0  R (t0 )   2 t0 , t1 
~
~
V(t1 )   3 t1 , t0  V* (t0 )   3 t0 , t1 
*
~
R * (t0 ) R * (t0 )  V T (t1 )  R (t1 ) 
~*
   ~T
~

*

V
(
t
)
R
(
t
)
V
(
t
)
V
(
t
)
1
1


0
0 

V(t1 )   4 t1 , t0  V* (t0 )   4 t0 , t1 
37
Ref # 10,11
Enhanced Formation Flying Algorithm
Goddard Space Flight Center
The Algorithm is found from the STM and is based on the
simplectic nature ( navigation and guidance matrices) of the STM)
[
NASA / Goddard Space Flight Center
][
]
Compute the matrices R(t1) , ~R(t1) according to
the following:
dv 0   v 1  v 0 
dr0   r1  r0 
Given
•
Compute
 R  t  =
 R~ t  =
1
1


 



r0
C
 1  F R 1  r0 v T0  V1  v 0 r0T 
V1 v T0  G I


R1
1
T
V1  v 0 V1  v 0  3 r0  1  F R 1r0T  CV1r0T + F I

r0






• Compute the ‘velocity-to-be-gained’ (Dv0) for
the current cycle.
Dv 0 
  R~ t   R t  dr
T
T
1
1
0
 dv 0
where F and G are found from Gauss problem and the f & g
series and C found through universal variable formulation
38
Ref # 10,11
EO-1 AutoConTM Functional Description
Goddard Space Flight Center
GPS Receiver
NASA / Goddard Space Flight Center
Ground Uplink
Thrusters
• Landsat-7 Data
C&DH / ACS
AutoConTM
Where do I want to be?
Relative Navigation
Maneuver
Decision
Decision Rules
• Performance Objectives
• Flight Constraints
• Generate Commands
• Execute Burn
On-Board Navigation
• EO-1/LS-7
No
Where am I ?
Yes
• Orbit Determination
• Attitude Determination
EFF
Maneuver Design
Implementation
• Propagate S/C States
• Design Maneuver
• Compute DV
• Compute Burn Attitude
• Validate Com. Sequence
• Perform Calibrations
How do I get there?
Inputs
Models
Constraints
39
Ref # 10,11
EO-1 Subsystem Level
Goddard Space Flight Center
NASA / Goddard Space Flight Center
GPS
Subsystem
Command and
Telemetry Interfaces
ACE
Subsystem
EO-1 Formation Flying
Subsystem Interfaces
Propellant Data
GPS State Vectors
Thruster Commands
 EFF Subsystem
 AutoCon-F
• GSFC
• JPL
• GPS Data Smoother
 Stored Command
Processor
 Cmd Load
C&DH
Subsystem
Timed Command Processing
SCP
Thruster Commanding
Inertial State Vectors
ACS
Subsystem
EFF
Subsystem
AutoCon-F &
GPS Smoother
Uplink
Downlink
Comm
Subsystem
Orbit Control
Burn Decision and Planning
Mongoose V
40
Ref # 10,11
Difference in EO-1 Onboard
and Ground Maneuver Quantized DVs
Quantized - EO-1 rounded maneuver durations to nearest second
Goddard Space Flight Center
NASA / Goddard Space Flight Center
Mode
Auto-GPS
Auto-GPS
Auto
Auto
Semi-auto
Semi-auto
Semi-auto
Manual
Manual
Onboard Onboard Ground DV1 Ground DV2 % Diff DV1 % DiffDV2
DV1
DV2
Difference Difference vs.Ground vs.Ground
cm/s
cm/s
cm/s
cm/s
%
%
4.16
5.35
4.98
2.43
1.08
2.38
5.29
2.19
3.55
1.85
4.33
0.00
3.79
1.62
0.26
1.85
5.20
7.93
4e-6
3e-7
1e-7
3e-7
6e-6
1e-7
8e-4
4e-7
3e-7
2e-1
2e-1
0.0
2e-7
3e-3
1e-7
3e-4
3e-3
3e-3
.0001
.0005
.0001
.0001
.0588
.0001
1.569
.0016
.0008
12.50
5.883
0.0
.0005
14.23
.0007
1.572
.0002
3.57
Inclination Maneuver Validation: Computed DV at node crossing, of
~ 24 cm/s (114 sec duration), Ground validation gave same results
41
Ref # 10,11
Goddard Space Flight Center
Difference in EO-1 Onboard
and Ground Maneuver Three-Axis DVs
EO-1 maneuver computations in all three axis
NASA / Goddard Space Flight Center
Mode
Auto-GPS
Auto-GPS
Auto
Auto
Semi-auto
Semi-auto
Semi-auto
Manual
Manual
Onboard
DV1
m/s
2.83
8.45
10.85
11.86
12.64
14.76
15.38
15.58
15.47
Ground DV1
3-axis
Algorithm
Difference DV1 vs. Ground DV1 Diff
cm/s
%
cm/s
1.122
68.33
-5e-4
0.178
0.312
0.188
-.256
10.41
0.002
3.96
-8.08
-0000
.0015
.0024
.0013
-.0016
.6682
.0001
-0.508
0.462
.0003
-.0102
.00091
0.000
-.0633
-.0117
-.0307
Algorithm
DV1 Diff
%
-1.794
-.0054
0
-.0008
.0002
.0001
-.0045
-.0007
-.0021
42
Ref # 10,11
Formation Data from Definitive Navigation
Solutions
Goddard Space Flight Center
Radial vs. along-track
separation over all
formation maneuvers
(range of 425-490km)
0.075
0.025
0.000
-0 . 025
-0 . 050
-0 . 075
-0 . 100
E O 1. A v erage ()
3
430
440
450
460
470
E O 1 . A lo n g T r a c k S e p a r a t io n ( K m )
480
490
2
500
1
0
Ground-track
separation over all
formation maneuvers
maintained to 3km
Ref # 10,11
LS 7. A v erage ()
4
-0 . 125
Y - A x is
E O 1 .A v e ra g e
NASA / Goddard Space Flight Center
()
0.050
-1
-2
-3
-4
-5
-6
-7
0
25
50
75
E O 1 . E la p s e d D a y s ( D a y s )
100
125
43
Goddard Space Flight Center
Formation Data from Definitive Navigation
Solutions
Along-track separation
vs. Time over all
formation maneuvers
(range of 425-490km)
500
480
470
460
450
440
E O 1 .A ve ra g e ()
0
25
7 0 7 7 .8 0
50
75
E O 1 .E l a p s e d D a ys (D a ys )
Semi-major axis of
EO-1 and LS-7 over
all formation
maneuvers
Ref # 10,11
L S 7 .A ve ra g e ()
7 0 7 7 .8 5
430
100
Y - A x is
E O 1 .A v e r a g e ( )
NASA / Goddard Space Flight Center
490
125
7 0 7 7 .7 5
7 0 7 7 .7 0
7 0 7 7 .6 5
7 0 7 7 .6 0
0
25
50
75
E O 1 .E l a p s e d D a ys (D a ys )
100
44
125
Goddard Space Flight Center
Formation Data from Definitive Navigation
Solutions
E O 1 .A ve ra g e ()
L S 7 .A ve ra g e ()
0 .0 0 1 2 5 0
Frozen Orbit
eccentricity over all
formation maneuvers
(range of .001125 0.001250)
0 .0 0 1 2 2 5
Y - A x is
NASA / Goddard Space Flight Center
0 .0 0 1 2 0 0
0 .0 0 1 1 7 5
0 .0 0 1 1 5 0
0 .0 0 1 1 2 5
95
0
25
50
75
E O 1 .E l a p s e d D a ys (D a ys )
100
125
Frozen Orbit  vs. ecc.
over all formation
maneuvers.  range of
90+/- 5 deg.
E O 1 .A v e r a g e ( )
94
93
92
91
90
89
88
0 .0 0 1 1 3
0 .0 0 1 1 4
0 .0 0 1 1 5
0 .0 0 1 1 6
0 .0 0 1 1 7
0 .0 0 1 1 8
0 .0 0 1 1 9
0 .0 0 1 2 0
E O 1 .A ve ra g e ()
Ref # 10,11
0 .0 0 1 2 1
0 .0 0 1 2 2
0 .0 0 1 2 3
45
0 .0 0 1 2 4
0 .0 0 1 2 5
EO-1 Summary / Conclusions
Goddard Space Flight Center
NASA / Goddard Space Flight Center
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
46
Ref # 10,11
Summary / Conclusions
NASA / Goddard Space Flight Center
Goddard Space Flight Center
o Proven executive flight code
o Scripting language alters behavior w/o flight software changes
o I/F for Tlm and Cmds
o Incorporates fuzzy logic for multiple constraint checking for
maneuver planning and control
o Single or multiple maneuver computations.
o Multiple or generalized navigation inputs (GPS, Uplinks).
o Attitude (quaternion) required of the spacecraft to meet the
DV components
o Maintenance of combinations of Keperlian orbit requirements
sma, inclination, eccentricity,etc.
Enables Autonomous StationKeeping,
Formation Flying and Multiple Spacecraft Missions
47
Ref # 10,11
NASA / Goddard Space Flight Center
Goddard Space Flight Center
CONTROL STRATEGIES FOR FORMATION FLIGHT
IN THE VICINITY OF THE LIBRATION POINTS
48
Ref # 11, 12, 13, 14, 15
NASA Libration Missions
Goddard Space Flight Center
L1 Missions
NASA / Goddard Space Flight Center
• ISEE-3/ICE(78-85)
• WIND (94-04)
• SOHO(95-04)
• ACE (97-04)
• GENESIS(01-04)
• TRIANA
L1 Halo Orbit, Direct Transfer, L2 Pseudo Orbit,
Comet Mission
Multiple Lunar Gravity Assist - Pseudo-L1/2 Orbit
Large Halo, Direct Transfer
Small Amplitude Lissajous, Direct Transfer
Lissajous Orbit, Direct Transfer,Return Via L2 Transfer
L1 Lissajous Constrained, Direct Transfer
L2 Missions
• GEOTAIL(1992)
• MAP(2001-04)
• JWST (~2012)
• CONSTELLATION-X
• SPECS
• MAXIM
• TPF
L2 Pseudo Orbit, Gravity Assist
Orbit, Lissajous Constrained, Gravity Assist
Large Lissajous, Direct Transfer
Lissajous Constellation, Direct Transfer?, Multiple S/C
Lissajous, Direct Transfer?, Tethered S/C
Lissajous, Formation Flying of Multiple S/C
Lissajous, Formation Flying of Multiple S/C
(Previous missions marked in blue)
Ref # 1,12
49
ISEE-3 / ICE
Goddard Space Flight Center
NASA / Goddard Space Flight Center
Halo Orbit
Lunar Orbit
Earth
L1
Solar-Rotating Coordinates
Ecliptic Plane Projection
Mission:
Launch:
Lissajous Orbit:
Spacecraft:
Notable:
Investigate Solar-Terrestrial relationships, Solar Wind, Magnetosphere,
and Cosmic Rays
Sept., 1978, Comet Encounter Sept., 1985
L1 Libration Halo Orbit, Ax=~175,000km, Ay = 660,000km, Az~
120,000km, Class I
Mass=480Kg, Spin stabilized,
First Ever Libration Orbiter, First Ever Comet Encounter
50
Ref # 1,12,13
Farquhar et al [1985] Trajectories and Orbital Maneuvers for the ISEE-3/ICE, Comet Mission, JAS 33, No. 3
WIND
NASA / Goddard Space Flight Center
Goddard Space Flight Center
Solar-Rotating Coordinates
Ecliptic Plane Projection
Mission:
Launch:
Lissajous Orbit:
Spacecraft:
Notable:
Investigate Solar-Terrestrial Relationships, Solar Wind, Magnetosphere
Nov., 1994, Multiple Lunar Gravity Assist
Originally an L1 Lissajous Constrained Orbit, Ax~10,000km, Ay ~
350,000km, Az~ 250,000km, Class I
Mass=1254kg, Spin Stabilized,
First Ever Multiple Gravity Assist Towards L1
51
Ref # 1,12
MAP
NASA / Goddard Space Flight Center
Goddard Space Flight Center
Solar-Rotating Coordinates
Ecliptic Plane Projection
Mission:
Launch:
Lissajous Orbit:
Spacecraft:
Notable:
Ref # 1,12
Produce an Accurate Full-sky Map of the Cosmic Microwave Background
Temperature Fluctuations (Anisotropy)
Summer 2001, Gravity Assist Transfer
L2 Lissajous Constrained Orbit Ay~ 264,000km, Ax~tbd, Ay~ 264,000km,
Class II
Mass=818kg, Three Axis Stabilized,
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
52
JWST
Goddard Space Flight Center
NASA / Goddard Space Flight Center
Lunar Orbit
L2
Sun
Earth
Solar-Rotating Coordinates
Ecliptic Plane Projection
Mission:
Launch:
Lissajous Orbit:
Spacecraft:
Notable:
Ref # 1,12
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.
JWST~2010, Direct Transfer
L2 large lissajous, Ay~ 294,000km, Ax~800,000km, Az~ 131,000km, Class I
or II
Mass~6000kg, Three Axis Stabilized, ‘Star’ Pointing
Observations in the Infrared Part of the Spectrum. Important That the
53 Sail
Telescope Be Kept at Low Temperatures, ~30K. Large Solar Shade/Solar
A State Space Model
Goddard Space Flight Center
NASA / Goddard Space Flight Center
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,
r
j
x  2ny  U XX x
Y=Y0+y,
, y  2nx  U YY y ,
Z= Z0+z
z  U ZZ z
m
• Pseudopotential:
Y
r
r2
r
r1
r
R
Sun
x
M1
y
X
L1
O
D1
U XX
M2
r
i
 2U
 2
X
 2U
 2U
, U YY  2 , U ZZ  Z 2 .
Y
Earth
D2
D
54
Ref # 1,12,13,17,20
A State Space Model
NASA / Goddard Space Flight Center
Goddard Space Flight Center
x j  A j x j
n
where
G(M 1  M 2 )
D
x j 
 0
 j
 0
y
 

j
z 
 0
j
j
x  j , A 
 x 
U XX
 y j 
 0
 j


z
 0
 
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
2n
U YY
0
0
U ZZ
 2n
0
0
0
0 
0 
1 

0 
0 

0 
ẑL
Sun
ŷL
Projection of
Halo orbit
L1
Ecliptic Plane
x̂L
Earth-Moon
Barycenter
55
Ref # 1,17,20
Reference Motions
Goddard Space Flight Center
NASA / Goddard Space Flight Center
• 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
+ Stable Manifolds
 RLP

 Inertial
– Fixed Relative Distance, Free Orientation
– Fixed Relative Distance & Rotation Rate
– Aspherical Configurations (Position & Rates)
56
Ref # 15
NASA / Goddard Space Flight Center
Goddard Space Flight Center
Natural Formations
57
Natural Formations:
String of Pearls
NASA / Goddard Space Flight Center
Goddard Space Flight Center
ŷ
x̂
ẑ
ẑ
x̂
Ref # 15
ŷ
58
Goddard Space Flight Center
CR3BP Analysis of Phase Space Eigenstructure
Near Halo Orbit
NASA / Goddard Space Flight Center
Reference Halo Orbit
Deputy S/C
 rR 
d x   R    t, 0d x  0
 rR 
Chief S/C
Floquet Decomposition of   t , 0  :
  t , 0   P  t  et P  0   P  t  S  e Jt P  0  S 
1
Floquet Modal Matrix:
E  t   P  t  S    t , 0  E  0  e  Jt
Solution to Variational Eqn. in terms of Floquet Modes:
6
6
j 1
j 1
d x  t    d x j t    c j t  e j t   E t  c
Ref # 15
59
NASA / Goddard Space Flight Center
Goddard Space Flight Center
Natural Formations:
Quasi-Periodic Relative Orbits → 2-D Torus
Chief S/C Centered View
(RLP Frame)
ẑ
ŷ
x̂
ŷ
x̂
ẑ
x̂
ẑ
ẑ
ŷ
x̂
ŷ
60
Ref # 15
Goddard Space Flight Center
Floquet Controller
(Remove Unstable + 2 of the 4 Center Modes)
Find Dv that removes undesired response modes:
03 
d x j    Dv 

j 1
 I3 
NASA / Goddard Space Flight Center
6
 1    d x
j  2,3,4
or
j  2,5, 6
j
j
Remove Modes 1, 3, and 4:
   d x2 r
 Dv   d x
   2v
d x5 r
d x5v
d x6 r
d x6 v
03 
 I 3 
1
d x1  d x3  d x4 
Remove Modes 1, 5, and 6:
   d x2 r
 Dv   d x
   2v
d x3r
d x3v
d x4 r
d x4v
03 
 I 3 
1
d x1  d x5  d x6 
61
Ref # 15
NASA / Goddard Space Flight Center
Goddard Space Flight Center
Deployment into Torus
(Remove Modes 1, 5, and 6)
Deputy S/C
r  0   50 0 0 m
r  0   1 1 1 m/sec
Origin = Chief S/C
62
Ref # 15
NASA / Goddard Space Flight Center
Goddard Space Flight Center
Deployment into Natural Orbits
(Remove Modes 1, 3, and 4)
r  0    r0
3 Deputies
0 0 m
r  0   1 1 1 m/sec
Origin = Chief S/C
63
Ref # 15
NASA / Goddard Space Flight Center
Goddard Space Flight Center
Natural Formations:
Nearly Periodic Relative Motion
10 Revolutions = 1,800 days
Origin = Chief S/C
64
Ref # 15
NASA / Goddard Space Flight Center
Goddard Space Flight Center
Evolution of Nearly Vertical Orbits
Along the yz-Plane
65
Ref # 15
Goddard Space Flight Center
Natural Formations:
Slowly Expanding Vertical Orbits
NASA / Goddard Space Flight Center
100 Revolutions = 18,000 days
r  0
r t f

Origin = Chief S/C
66
Ref # 15
Non-Natural Formations
Goddard Space Flight Center
NASA / Goddard Space Flight Center
• 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
67
Ref # 15
Formations Fixed in the Inertial Frame
Y [au]
NASA / Goddard Space Flight Center
Goddard Space Flight Center
r  rYˆ
Deputy S/C
Chief S/C
X [au]
68
Ref # 15
NASA / Goddard Space Flight Center
Goddard Space Flight Center
Formations Fixed in the
Rotating Frame
Deputy S/C
Chief S/C
r  r ŷ
69
Ref # 15
2-S/C Formation Model
in the Sun-Earth-Moon System
Goddard Space Flight Center
r̂
NASA / Goddard Space Flight Center
Relative EOMs:
r  t   Df  r  t    u  t 
r  rd  rc  rrˆ
Deputy S/C
ŷ

Zˆ , zˆ
rd

x̂
Chief S/C
rc
L1
B
ŷ

x̂
X̂
Ephemeris System = Sun+Earth+Moon Ephemeris + SRP
Ref # 15
L2
70
NASA / Goddard Space Flight Center
Goddard Space Flight Center
Discrete & Continuous Control
71
Ref # 15
Linear Targeter
Goddard Space Flight Center
NASA / Goddard Space Flight Center
Dv3
  t3 , t2 
Dv0
d v0

d v0

Nominal
Formation
Path
Segment of Reference Orbit
Dv2
Dv1
  t2 , t1 
  t1 , t0 
d r1  d r1
 d rk 1 
d rk 1   Ak
      tk 1 , tk      
d vk 1 
d vk 1  Ck
Bk   d rk 1 


Dk  d vk1  Dvk 
Dvk  Bk 1 d rk 1  Akd rk   d vk
72
Ref # 15
Discrete Control: Linear Targeter
Distance Error Relative to Nominal (cm)
NASA / Goddard Space Flight Center
Goddard Space Flight Center
r  10 m  Yˆ
r0
Time (days)
73
Ref # 15
Achievable Accuracy
via Targeter Scheme
Maximum Deviation from Nominal (cm)
NASA / Goddard Space Flight Center
Goddard Space Flight Center
r  rYˆ
Formation Distance (meters)
74
Ref # 15
NASA / Goddard Space Flight Center
Goddard Space Flight Center
Continuous Control:
LQR vs. Input Feedback Linearization
• LQR for Time-Varying Nominal Motions
x  t    r
r   f  t , x  t  , u  t  
 x  0   x0
T
P   AT  t  P  t   P  t  A  t   P  t  B  t  R 1B T  t  P  t   Q  P t f   0
Optimal Control Law:
Nominal Control Input
u t  
u t 




   R 1 BT P  t   x  t   x  t   
Optimal Control, Relative to Nominal, from LQR 
• Input Feedback Linearization (IFL)
Desired Dynamic Response
r t   F  r t    u t 
u t  
F  r t 
 g  r t  , r t  
Anihilate Natural Dynamics
75
Ref # 15, 20
LQR Goals
Goddard Space Flight Center
NASA / Goddard Space Flight Center
min J 
1
2

tf
t0
d x  t T Qd x  t   d ud  t T Rd ud  t   dt


d x t   x t   x t 
d ud  t   ud  t   ud  t 
Q  diag 1012 ,1012 ,1012 ,105 ,105 ,105 
R  diag 1,1,1
76
Ref # 14,15
LQR Process
Goddard Space Flight Center
Step 1: Evaluate the Jacobian matrix, at time ti , associated with nominal path
 A  ti  evaluated on x  ti 
NASA / Goddard Space Flight Center
Step 2: Numerically integrate the differential Riccatti Equation backwards in time
from ti  ti 1 , subject to P  t N   0.
P  ti    AT  ti  P  ti   P  ti  A  ti   P  ti  BR 1B T P  ti   Q
Step 3: Compute and store the controller gain matrix
K  ti   R 1 BT P  ti 
Step 4: Repeat steps 1-3 until ti 1  t0
Step 5: Numerically integrate the perturbed trajectory forward in time
from t0  t N , subject to x  t0   x0 .
 Recall K  t  from stored data
 The new integration step size is defined by the sampled gain data
 Substitute x  t  into EOMs and solve for ud  t 
 ud  t   ud  t   K  t   x  t   x  t  
 Apply the computed control input to the perturbed EOMs
Ref # 15
77
IFL Process
Goddard Space Flight Center
NASA / Goddard Space Flight Center
Step 1: Define, analytically, the desired response characteristics
 r  r  2n  r  r
   r  r   g r , r 
2
n
Step 2: Begin numerical integration of perturbed path
Step 3: At each point in time, compute and apply the control input necessary
to achieve the desired response characteristics:
ud  t    f  r P2 D , r P2 D   g  r , r 
Annihilate Natural Dynamics
Reflects desired response
78
Ref # 15
LQR vs. IFL Comparison
Goddard Space Flight Center
r  5000 km,   90 ,   0
d x  0   7 km 5 km 3.5 km 1 mps 1 mps 1 mps 
T
Dynamic Response
NASA / Goddard Space Flight Center
Control Acceleration History
LQR
LQR
IFL
IFL
Dynamic Response Modeled in the CR3BP
Nominal State Fixed in the Rotating Frame
Ref # 15
79
Output Feedback Linearization
(Radial Distance Control)
NASA / Goddard Space Flight Center
Goddard Space Flight Center
Formation Dynamics
r  Df  r   u  t 
 Generalized Relative EOMs
y  l r 
 Measured Output
Measured Output Response (Radial Distance)
Actual Response
Desired Response
d 2l
y  2  p r , r   q r , r u T r  g r , r 
dt
Scalar Nonlinear Functions of r and r
Scalar Nonlinear Constraint on Control Inputs
h  r t  , r t   u t  r t   0
T
80
Ref # 15
Output Feedback Linearization (OFL)
(Radial Distance Control in the Ephemeris Model)
NASA / Goddard Space Flight Center
Goddard Space Flight Center
y  l r , r 
Control Law
hr , r 
Geometric Approach:
Radial inputs only
r
u t  
r
 g  r , r  r T r 
r
u t   
 2  r    r  Df  r 
r 
r
 r
r
1
2
r
r
rˆ
 1 g  r , r  r T r 
u t   
 2  r  Df  r 
2
2
r
r 


rTr 
r
u  t   rg  r , r   2  r  3   r  Df  r 
r 
r

• Critically damped output response achieved in all cases
• Total DV can vary significantly for these four controllers
81
Ref # 15
Goddard Space Flight Center
OFL Control of Spherical Formations
in the Ephemeris Model
NASA / Goddard Space Flight Center
r  0  12 5 3 km
r  0  1 1 1 m/sec
Nominal Sphere
Relative Dynamics as Observed in the Inertial Frame
Ref # 15
82
NASA / Goddard Space Flight Center
Goddard Space Flight Center
OFL Controlled Response of Deputy S/C
Radial Distance + Rotation Rate Tracking
Radial Error Response (Critically Damped):
d r  2nd r  n2d r
r  g r  t   rn  2n  r  rn   n2  r  rn 
Rotation Rate Error Response (Exponential Decay):
d =d 0e t / T 
d   d 0 T  et / T   d T
  g  t    n     n  T   n  kn    n 
83
Ref # 15
OFL Controlled Response of Deputy S/C
Goddard Space Flight Center
Equations of Motion in the Relative Rotating Frame
NASA / Goddard Space Flight Center
r  r 2  f r  ur
r  2r  f  u
f r  Df  rˆ,
f  Df  ˆ,
u  u  ˆ,
f h  Df  hˆ
u  u  hˆ
ur  u  rˆ,

h
Rearrange to isolate the radial and rotational accelerations:
r  f r  ur  r 2  g r  t 
r  f  u  2r  rg  t 
Solve for the Control Inputs:
ur  t   g r  t   f r  r 2
u  t   rg  t   f  2r
uh  t    f h  constraint 
84
Ref # 15
Goddard Space Flight Center
OFL Control of Spherical Formations
Radial Dist. + Rotation Rate
NASA / Goddard Space Flight Center
Quadratic Growth in Cost w/ Rotation Rate
Linear Growth in Cost w/ Radial Distance
85
Ref # 15
Goddard Space Flight Center
Inertially Fixed Formations
in the Ephemeris Model
NASA / Goddard Space Flight Center
ê
500 m
100 km
eˆ  inertially fixed formation pointing vector (focal line)
Chief S/C
Ref # 15
86
Goddard Space Flight Center
Nominal Formation Keeping Cost
(Configurations Fixed in the RLP Frame)
NASA / Goddard Space Flight Center
180 days
DV 
r  5000 km

u  t   u  t dt
0
Az = 0.2×106 km
Az = 0.7×106 km
Az = 1.2×106 km
87
Ref # 15
Max./Min. Cost Formations
(Configurations Fixed in the RLP Frame)
NASA / Goddard Space Flight Center
Goddard Space Flight Center
Maximum Cost Formation
Minimum Cost Formations
ẑ
ẑ
Deputy S/C
ŷ
ŷ
x̂
x̂
Deputy S/C
Chief S/C
Deputy S/C
Chief S/C
Deputy S/C
Deputy S/C
Deputy S/C
Nominal Relative Dynamics in the Synodic Rotating Frame
88
Ref # 15
NASA / Goddard Space Flight Center
Goddard Space Flight Center
Formation Keeping Cost Variation
Along the SEM L1 and L2 Halo Families
(Configurations Fixed in the RLP Frame)
89
Ref # 15
Conclusions
Goddard Space Flight Center
• Continuous Control in the Ephemeris Model:
NASA / Goddard Space Flight Center
– 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 DV’s (10-5 m/sec)
• Natural Formations
– Nearly periodic & quasi-periodic formations in the RLP frame
– Floquet controller: numerically ID solutions + stable manifolds
90
Ref # 15
Some Examples from Simulations
NASA / Goddard Space Flight Center
Goddard Space Flight Center
• 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
91
Ref # 16, 17, 20
A State Space Model
NASA / Goddard Space Flight Center
Goddard Space Flight Center
•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.
• Linearized Eq. Of Motion Based on Inertial X, Y, Z Using
CRTB problem rotating frame
X = X0 + x,
r
j
x  2ny  U XX x
m
Y
r
r2
r
r1
r
R
y
x
Sun
M1
X
L1
O
D1
r
i
M2
D2
Earth
D
x j  A j x j
n
Ref # 16, 17, 20
Y=Y0+y,
G(M 1  M 2 )
D
where
• Pseudopotential:
 2U
U XX 
X 2
x j 
 0
 j
 0
y 

j
z 
 0
x j  j , Aj 
 x 
U XX
 y j 
 0
 j

 0
 z 
Z= Z0+z
, y  2nx  U YY y ,
, U YY
z  U ZZ z
 2U
 2U
 2 , U ZZ  2 .
Z
Y
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
2n
U YY
0
0
U ZZ
 2n
0
0
0
0 
0 
1 

0 
0 

0 
92
Periodic Reference Orbit
NASA / Goddard Space Flight Center
Goddard Space Flight Center
x   Ax sin(  xy t   xy )
x   Ax xy cos( xy t   xy )
y   Ay cos( xy t   xy )
y  Ay xy sin(  xy t   xy )
z  Az sin(  z t   z )
z  Az z cos( z t   z )
A = Amplitude
 = frequency
 = Phase angle
93
Ref # 16, 17, 20
Centralized LQR Design
NASA / Goddard Space Flight Center
Goddard Space Flight Center
State Dynamics
x j  A j x j  B j u j  a solpres  a FB
Performance Index
to Minimize
1
J   {( x  x R )T Q (x  x R )  uT R u}dt
20

j
  R  B  S x
j 1
j T
Control
u
Algebraic Riccatic Eq.
time invariant system
S ( B R 1 B T )S  SA  AT S  Q  0
B Maps Control Input From
Control Space to State Space
O33 
B 

I
 3 
j
Q Is Weight of State Error
1
I3

p
Q j 
 O33


O33 

1 
I3
q 
R Is Weight of Control
1
R  I3
r
j
94
Ref # 16, 17, 20
Centralized LQR Design
NASA / Goddard Space Flight Center
Goddard Space Flight Center
Sample LQR Controlled Orbit
DV budget with
Different Rj Matrices
95
Ref # 16, 17, 20
Disturbance Accommodation Model
Goddard Space Flight Center
NASA / Goddard Space Flight Center
•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.
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
96
Ref # 16, 17, 20
Disturbance Accommodation Model
Goddard Space Flight Center
State Error
With Disturbance Accommodation
Disturbance
accommodation
state is out of
phase with state
error, absorbing
unnecessary
control effort
Control Effort
NASA / Goddard Space Flight Center
Without Disturbance Accommodation
97
Ref # 16, 17, 20
Motion of Formation Flyer With Respect to
Reference Spacecraft, in Local (S/C-1) Coordinates
Goddard Space Flight Center
NASA / Goddard Space Flight Center
Reference Spacecraft Location
Reference Spacecraft Location
Formation Flyer Spacecraft Motion
Reference Spacecraft Location
98
Ref # 16, 17, 20
DV Maintenance in Libration Orbit Formation
NASA / Goddard Space Flight Center
Goddard Space Flight Center
99
Ref # 16, 17, 20
Stellar Imager Concept
NASA / Goddard Space Flight Center
Goddard Space Flight Center
(Using conceptual distances and control requirements
to analyze formation possibilities)
 Stellar Imager (SI) concept for a spacebased, 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
Ref # 16, 17, 20
Drones
500 m
Hub
Sphere Origin
500 m
500 m
100
Stellar Imager
z (km)
• 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.
SI Slewing Geometry
Formation DV Cost per
Formation
0.5
0.5
slewing maneuver
y (km)
NASA / Goddard Space Flight Center
Goddard Space Flight Center
0
-0.5
-0.5
0.5
0
x (km)
0
-0.5
-0.5
0.5
Drones at beginning
0
x (km)
0.5
Focal
Length
(km)
0.5
0.5
4
4
Slew
Angle
(deg)
30
90
30
90
Hub
DDVV
(m/s)
1.0705
1.1355
1.2688
1.8570
Drone 2
Drone 31
DVDV (m/s) DD
VV
0.8271
0.9395
1.1189
2.1907
(m/s)
0.8307
0.9587
1.1315
2.1932
z (km)
z (km)
0.5
0
0
-0.5
0.5
-0.5
-0.5
Ref # 16
0
y (km)
0.5
0
y (km) -0.5 -0.5
0.5
0
x (km)
101
Stellar Imager Mission Study Example Requirements
Goddard Space Flight Center
NASA / Goddard Space Flight Center
 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 Intermediate Precision -
RF ranging, star trackers, and thrusters
~ centimeters
Laser ranging and micro-N thrusters control ~ 50 microns
Optics adjusted, phase diversity, wave front. ~ nanometers
102
Ref # 16
State Space Controller Development
Goddard Space Flight Center
NASA / Goddard Space Flight Center
 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 and
the state vector.
•Origin in figure is Earth
•Solar rotating coordinates
103
Ref # 16
State Space Controller Development, LQR Design
Goddard Space Flight Center
NASA / Goddard Space Flight Center
• 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
x  x y z
A ( t t 0 )
x  Ax
x
y
T
z 
A k (t  t 0 ) k

k!
k 0

• The STM is created from the
(t  t 0 )  e
dynamics partials output from
Generator and assumes to be constant over an analysis time period
State Dynamics and Error
Performance Index
to Minimize
x  Ax  Bu
tf
J
~

x
T
Ref # 16

( )W~
x ( )  u T ( )Vu( ) d
t0
Control and Closed Loop Dynamics
Algebraic Riccati Eq.
time invariant system
~
x (t )  x(t )  x ref (t )
u   K (t )~
x ~x  ( A  BK (t )) ~x
S ( B R 1 B T ) S  SA  AT S  Q  0
104
State Space Controller Development, LQR Design
Goddard Space Flight Center
NASA / Goddard Space Flight Center
• Expanding for the SI collector (hub) and mirrors (drones) yields
a controller
~
~
x 2  Ax 2  Bu 2  Bu1
• Redefine A and B such that
 A1

A



B Maps Control Input From
Control Space to State Space
O33 
B 

I
 3 
j
Ref # 16
A2





A j 
 B1
 B
 1
B   B1


 B1

Bi
W Is Weight of State Error
1
I3

p
Q j 
W
 O33


O33 

1 
I3
q 
Bi






B j 
V Is Weight of Control
1
j
V  I3
R
r
105
Simplified extended Kalman Filter
Goddard Space Flight Center
NASA / Goddard Space Flight Center
 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
~
x k 1  Ad ~
x k  Bd u k  w
~
 The non-linear measurement model is y k  m(x k )  v
and the covariances of process and measurement noise are
 
Evv   R
E wwT  Q
T
 Hub measurements are range(r) and azimuth(az) / elevation(el) angles from Earth
Drone measurements are r, az, el from drone to hub
z
Hub
y
b3
r
x
el
Drone
r  x2  y2  z2
az
el  sin 1 (
z
x
), and az  sin 1 (
)
r
r cos(el )
b2
b1
Ref # 16
106
Simulation Matrix Initial Values
Goddard Space Flight Center
NASA / Goddard Space Flight Center
• Continuous state weighting and
control chosen as
1



V  1


1 
•The process and measurement noise
Covariance (hub and drone) are
0

 0





0
Qc  

1e  6




1e  6


1e  6

• Initial covariances
Ref # 16

0.12


R




 0.3 


 1500000 
1
 1


1
P1 ()  
86.4 2




86.4 2
2





2
 0.3  


 1500000  








86.4 2 
1e6



1e6




1e6
W 

1e3




1e3


1e3


0.00012


R




.001 2



P1 ()  




.001
 0.0003


 0.5 
2





2
 0.0003 


 0.5  
2
.001 2
.0864 2
.0864 2
107








.0864 2 
Results – Libration Orbit Maintenance
• 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
NASA / Goddard Space Flight Center
Goddard Space Flight Center
108
Ref # 16
Results – Libration Orbit Maintenance
Goddard Space Flight Center
NASA / Goddard Space Flight Center
• Estimation errors for 12 simulations for 1 year: Position and Velocity
• Estimation errors of 250 meters and 2e-4 m/s in each component
109
Ref # 16
Results – Formation Slewing
NASA / Goddard Space Flight Center
Goddard Space Flight Center




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
0

 0





0
Qc  

1e  24




1e  24


1e  24

Formation Slewing:
900 simulation shown
Purple - Begin
Red - End
Ref # 16
110
Results – Formation Slewing
• 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
DRONE
HUB
drone 2 tracking error
xdot error (m/s)
x error (km)
0.02
0
-0.02
0
500
1000
0
-0.01
ydot error (m/s)
y error (km)
0.01
1500
0.2
0
-0.2
0
500
1000
zdot error (m/s)
0
-1
0
500
1000
1500
0
500
1000
1500
0
500
1000
1500
0
500
1000
1500
0.1
0
-0.1
1500
1
z error (km)
NASA / Goddard Space Flight Center
Goddard Space Flight Center
0.2
0
-0.2
time
time
Represents both 0.5 and 4 km focal lengths
111
Ref # 16
Results – Formation Slewing
Goddard Space Flight Center
NASA / Goddard Space Flight Center
• Estimation errors;12 simulations for 24 hours: position and velocity
• Estimation 3s 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
112
Ref # 16
Results – Formation Slewing
NASA / Goddard Space Flight Center
Goddard Space Flight Center
Formation Slewing Average DVs (12 simulations)
Focal
Length (km)
Slew
Angle (deg)
Hub
(m/s)
Drone 2
(m/s)
0.5
0.5
4
4
30
90
30
90
1.0705
1.1355
1.2688
1.8570
0.8271
0.9395
1.1189
2.1907
Drone 31
(m/s)
0.8307
0.9587
1.1315
2.1932
Formation Slewing Average Propellant Mass
Focal
Length (km)
Slew
Angle (deg)
0.5
0.5
4
4
30
90
30
90
Hub
mass-prop
(g)
6.0018
6.3662
7.1135
10.4112
Drone 2
mass-prop
(g)
0.8431
0.9577
1.1406
2.2331
Drone 31
mass-prop
(g)
0.8468
0.9773
1.1534
2.2357
113
Ref # 16
Results – Formation Slewing
NASA / Goddard Space Flight Center
Goddard Space Flight Center
Formation Slewing Average DVs (without noise)
Focal
Length (km)
Slew
Angle (deg)
Hub
(m/s)
Drone 2
(m/s)
Drone 31
(m/s)
0.5
0.5
4
4
30
90
30
90
0.0504
0.1581
0.4420
1.3945
0.0853
0.2150
0.5896
1.9446
0.0998
0.2315
0.6441
1.9469
Formation Slewing Average Propellant Mass (without noise)
Focal
Length (km)
Slew
Angle (deg)
0.5
0.5
4
4
30
90
30
90
Hub
mass-prop
(g)
0.2826
0.8864
2.4781
7.8182
Drone 2
mass-prop
(g)
0.0870
0.2192
0.6010
1.9822
Drone 31
mass-prop
(g)
0.1017
0.2360
0.6566
1.9846
114
Ref # 16
Results – Formation Reorientation
NASA / Goddard Space Flight Center
Goddard Space Flight Center
• 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
115
Ref # 16
Results – Formation Reorientation
NASA / Goddard Space Flight Center
Goddard Space Flight Center
• 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
116
Ref # 16
Results – Formation Reorientation
NASA / Goddard Space Flight Center
Goddard Space Flight Center
• The steady-state estimation 3s values are: x ~30 meters, y and z ~ 50 meters
• The steady-state estimation 3s velocity values are about 1 millimeter per second.
• For any drone and either focal length, the steady-state 3s position values
are less than 0.1 meters, and the steady-state velocity 3s values are
less than 1e-6 meters per second.
Ref # 16
Formation Reorientation Average DVs
Focal
Length (km)
0.5
4
Slew
Angle (deg)(m/s)
90
90
Hub
(m/s)
1.0126
1.0133
Drone 2
Drone 31
(m/s)
0.8421
0.8095
0.8496
0.8190
Formation Reorientation Average Propellant Mass
Focal
Length (km)
Slew
Angle (deg)
0.5
4
90
90
Hub
mass-prop
(g)
5.6771
5.6811
Drone 2
mass-prop
(g)
0.8584
0.8661
Drone 31
mass-prop
(g)
0.8252
0.8349
117
Results – Formation Reorientation
Goddard Space Flight Center
NASA / Goddard Space Flight Center
Formation Reorientation Average DVs (without noise)
Focal
Length (km)
0.5
4
Slew
Angle (deg)(m/s)
90
90
Hub
(m/s)
0.0408
0.0408
Drone 2
Drone 31
(m/s)
0.1529
0.1496
0.1529
0.1495
Formation Reorientation Average Propellant Mass ( without noise)
Focal
Length (km)
Slew
Angle (deg)
0.5
4
90
90
Hub
mass-prop
(g)
0.2287
0.2287
Drone 2
mass-prop
(g)
0.1623
0.1623
Drone 31
mass-prop
(g)
0.1525
0.1524
118
Ref # 16
Summary
Goddard Space Flight Center
(using example requirements and constraints)
NASA / Goddard Space Flight Center
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.
119
Ref # 16
Summary
Goddard Space Flight Center
(using example requirements and constraints)
NASA / Goddard Space Flight Center
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
Ref # 16
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
120
NASA / Goddard Space Flight Center
Goddard Space Flight Center
- A Distant Retrograde Formation
- Decentralized control
121
DRO Mission Metrics
Goddard Space Flight Center

Earth-constellation distance: > 50 Re (less interference) and< 100 Re (link margin).
NASA / Goddard Space Flight Center


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.
122
Ref # 1,18
Distant Retrograde Orbit (DRO)
Goddard Space Flight Center
NASA / Goddard Space Flight Center
Why DRO?
Direction of Orbit is Retrograde
L1
L2
Sun-Earth Line
Earth
Lunar Orbit
Solar Rotating Coordinate System ( Earth-Sun line is fixed)
• Stable Orbit
• No Skp DV
• Not as distant
as L1
• Mult. Transfers
• No Shadows?
• Good
Environment
Really a Lunar
Periodic Orbit
Classified as a
Symmetric Doubly
Asymptotic Orbit in
the Restricted ThreeBody Problem
123
Ref # 1,18
Earth Distant Retrograde Orbit (DRO) Orbit
NASA / Goddard Space Flight Center
Goddard Space Flight Center
124
Ref # 1,18
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
NASA / Goddard Space Flight Center
Goddard Space Flight Center
X-Y view
3-D view
Y-Z view
SIRA spacecraft
125
Ref # 1,18
DRO Formation Control Analysis
NASA / Goddard Space Flight Center
Goddard Space Flight Center
126
Ref # 1,18
DRO Formation Control Analysis
NASA / Goddard Space Flight Center
Goddard Space Flight Center
127
Ref # 1,18
DRO Formation Control Analysis
NASA / Goddard Space Flight Center
Goddard Space Flight Center
128
Ref # 1,18
Formation Control Analysis
Goddard Space Flight Center
NASA / Goddard Space Flight Center
How much DV to initialize, maintain, and resize?
Phase
a
b
c
d
e
f
Max
[m/s]
0.635
1.323
0.757
0.679
1.201
0.679
Mean
[m/s]
0.607
0.792
0.674
0.616
0.721
0.608
Min
[m/s]
0.592
0.392
0.541
0.582
0.367
0.503
Std
[m/s]
0.014
0.293
0.069
0.031
0.263
0.056
Phase Description
a) Init 25km sphere
b) Maintain 25km sphere (strict PD control) one month
c) Maintain 25km sphere (loose control) one month
d) Resize from 25km to 50km
e) Maintain 50km sphere (strict PD control) one month
f) Maintain 50km sphere (loose control) one month
Examples:
Initialize & maintain 2 yr:
= 33 m/s
Initialize, Maintain 2yr,
& four resizes:
= 36 m/s
129
Ref # 1,18
NASA / Goddard Space Flight Center
Goddard Space Flight Center
Earth - Moon L4 Libration Orbit
an alternate orbit location
130
Ref # 1,18
DRO Formation Control Analysis
Goddard Space Flight Center
 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
ImpulsiveBurn.X-Component (Km/s)
ImpulsiveBurn.Y-Component (Km/s)
ImpulsiveBurn.Z-Component (Km/s)
0.015
0.010
0.005
0.000
FreeFlyer Plot Window
Km/s
NASA / Goddard Space Flight Center
FreeFlyer Plot Window
4/30/2003
Impulsive Maneuver
of 16th s/c
Radial Distance from Center
4/30/2003
-0.005
-0.010
-0.015
Formation.Range (Km)
Formation.Range (Km)
Formation.Range (Km)
Formation.Range (Km)
Formation.Range (Km)
Formation.Range (Km)
Formation.Range (Km)
25
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
20
0.045
0.050
0.055
A1.ElapsedDays (Days)
Km
15
10
5
0
131
0.000
Ref # 1,18
0.025
0.050
0.075
0.100
A1.ElapsedDays (Days)
0.125
0.150
0.175
General Theory of Decentralized Control
Goddard Space Flight Center
MANY NODES IN A NETWORK CAN COOPERATE
NASA / Goddard Space Flight Center
TO BEHAVE AS SINGLE VIRTUAL PLATFORM:
•REQUIRES A FULLY CONNECTED NETWORK
OF NODES.
NODE 2
NODE 1
•EACH NODE PROCESSES ONLY ITS OWN
MEASUREMENTS.
•NON-HIERARCHICAL MEANS NO LEADS OR
NODE 3
MASTERS.
•NO SINGLE POINTS OF FAILURE MEANS
DETECTED FAILURES CAUSE SYSTEM TO
DEGRADE GRACEFULLY.
•BASIC PROBLEM PREVIOUSLY INVESTIGATED
BY SPEYER.
•BASED ON LQG PARADIGM.
NODE 5
•DATA TRANSMISSION REQUIREMENTS ARE
MINIMIZED.
NODE 4
132
Ref # 19
References, etc.
NASA / Goddard Space Flight Center
Goddard Space Flight Center
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
NASA Web Sites, www.nasa.gov, Use the find it @ nasa search input for SEC, Origin, ESE, etc.
Earth Science Mission Operations Project, Afternoon Constellation Operations Coordination Plan, GSFC, A. Kelly May 2004
Fundamental of Astrodynamics, Bate, Muller, and White, Dover, Publications, 1971
Mission Geometry; Orbit and Constellation design and Management, Wertz, Microcosm Press, Kluwer Academic Publishers, 2001
An Introduction to the Mathematics and Methods of Astrodynamics, Battin
Fundamentals of Astrodynamics and Applications, Vallado, Kluwer Academic Publishers, 2001
Orbital Mechanics, Chapter 8, Prussing and Conway
Theory of Orbits: The Restricted Problem of Three Bodies V. Szebehely.. Academic Press, New York, 1967
Automated Rendezvous and Docking of Spacecraft, Wigbert Fehse, Cambridge Aerospace Series, Cambridge University Press, 2003
“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.
“Results of NASA’s First Autonomous Formation Flying Experiment: Earth Observing-1 (EO-1)”, Folta, AIAA/AAS Astrodynamics
Specialist Conference, Monterey, CA, 2002
“Libration Orbit Mission Design: Applications Of Numerical And Dynamical Methods”, Folta, Libration Point Orbits and
Applications, June 10-14, 2002, Girona, Spain
“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, 1970
“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.
“Formation Flight near L1 and L2 in the Sun-Earth/Moon Ephemeris System including solar radiation pressure”, Marchand and
Howell, paper AAS 03-596
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.
“Formation Flying Satellite Control Around the L2 Sun-Earth Libration Point”, Hamilton, Folta and Carpenter, Monterey, CA,
AIAA/AAS, 2002
“Formation Flying with Decentralized Control in Libration Point Orbits”, Folta and Carpenter, International Space Symposium
Biarritz, France, 2000
SIRA Workshop, http://lep694.gsfc.nasa.gov/sira_workshop
“Computation and Transmission Requirements for a Decentralized Linear-Quadratic-Gaussian Control Problem,” J. L. Speyer, IEEE
133
Transactions on Automatic Control, Vol. AC-24, No. 2, April 1979, pp. 266-269.
References, etc.
Goddard Space Flight Center
NASA / Goddard Space Flight Center
Questions on formation flying? Feel free to contact:
kathleen howell howell@ecn.purdue.edu
david.c.folta@nasa.gov
134
NASA / Goddard Space Flight Center
Goddard Space Flight Center
Backup and other slides
135
DST/Numerical Comparisons
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Numerical Systems
• Limited Set of Initial
Conditions
• Perturbation Theory
• Single Trajectory
• Intuitive DC Process
• Operational
Dynamical Systems
• Qualitative Assessments
• Global Solutions
• Time Saver / Trust Results
• Robust
• Helps in choosing numerical
methods
(e.g., Hamiltonian =>
Symplectic Integration Schemes?)
136
Libration Point Trajectory Generation Process
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NASA / Goddard Space Flight Center
 Phase and Lissajous Utilities Universe/User Control and
Generate Lissajous of Interest
Patch Pts
Output / Intermediate
Data
Lissajous
Patch Points
and Lissajous
Monodromy
. Stable
Fixed Points and
and Unstable Manifold
.
Approximations
 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.
Manifold
1-D Manifold
Transfer
Transfer Trajectory to
Earth Access Region
137
MAP Mission Design: DST Perspective
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Goddard Space Flight Center
• MAP Manifold and Earth Access
• Manifold Generated Starting
with Lissajous Orbit
• Swingby Numerical Propagation
• Trajectory Generated Starting
with Manifold States
Starting Point
138
JWST DST Perspective
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Goddard Space Flight Center
• JWST Manifold and Earth Access
• Manifold Generated Starting
with Halo Orbit
• Swingby Numerical Propagation
• Trajectory Generated Starting
with Manifold States
Starting Point
139
Two – Body Motion
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Q
y
NASA / Goddard Space Flight Center
•Motion of spacecraft in elliptical orbit
•Counter-clockwise
•x and y correspond to P and Q-axes in PQW
frame
Spacecraft
Two angles are defined
E is Eccentric Anomaly
is True Anomaly
x and y coordinates are
x = r cos 
y = r sin 
a
Center of Orbit
E
r

x
P
ae
Focus
In terms of Eccentric anomaly, E
a cosE = ae + x
x = 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
a
Coordinates of spacecraft in plane of motion are then
r = a( 1 - ecosE)
x = a( cosE - e)
y = a( 1 - e2)1/2 sinE
Ref # 3-7
= a(1- e2) / (1 + e cos
= r cos 
= r sin 
vx = (/p)1/2 [-sin ]
vy = (/p)1/2 [e + cos ] where p = a (1 - e2) 140
Two – Body Motion
Goddard Space Flight Center
NASA / Goddard Space Flight Center
Differentiate
- x, y, and r wrt time in terms of E to get
dx/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
141
Ref # 3-7
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Coordinate system transformation
Euler Angle Rotations
NASA / Goddard Space Flight Center
Y’
Y
Suppose we rotate the x-y plane about x’ = x cos  + y sin 
the z-axis by an angle  and call the
y’ = -x sin  + y cos 
z’ = z
new coordinates x’,y’,z’
X’

X
Z
Rotate about Z
x’
y’
z’
Rotate about Y
x’
y’
z’
Rotate about X
x’
y’
z’
=
cos 
-sin 
0
sin 
cos 
0
0
0
1
x
y
z
=
cos 
0
-sin
0
1
0
sin
0
cos
x
y
z
=
1
0
0
0
-sin 
-cos 
x
y
z
0
cos 
-sin 
= Az’()
x
y
z
142
Ref # 3-7
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Coordinate system transformation
Orbital to inertial coordinates
NASA / Goddard Space Flight Center
Inertial to/from Orbit plane
r- = 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
x
y
z
x
y
z
=
Px
Py
Pz
cos  sin 0
-sin  cos 0
0
0
1
x
y
z
=
=
cos W cos  - sin Wsin cos i
sin W cos  - cos Wsin cos i
sin Wsin i
Qx
Qy
Qz
Wx
Wy
Wz
sin W0
cos W0
0
1
rcosf
rsinf
0
-cos W sin  - sin Wcos cos i sin Wsin i
-sin W sin  - cos Wcos cos i -cos Wsin i
cos W sin i
cos i
rcosf
rsinf
0
1
0
0 cos i
0 -sin i
0
-sin i
-cos i
rcosf
rsinf
0
cos W
-sin W
0
For Velocity transformation, vx = (/p)1/2 [-sin ] and vy = (/p)1/2 [e + cos ]
143
Ref # 3-7
Principles Behind Decentralized Control
NASA / Goddard Space Flight Center
Goddard Space Flight Center
• THE STATE VECTOR IS DECOMPOSED INTO TWO PARTITIONS:
•
•
x C DEPENDS ONLY ON THE CONTROL.
x Dj DEPENDS ONLY ON THE LOCAL MEASUREMENT DATA NODE-j.
A LOCALLY OPTIMAL KALMAN FILTER OPERATES ON x D. j
C
A GLOBALLY OPTIMAL CONTROL IS COMPUTED, USING x AND GLOBALLY OPTIMAL
DATA THAT IS RECONSTRUCTED LOCALLY USING TWO ADDITIONAL VECTORS:
A “DATA VECTOR”
h j, IS MAINTAINED LOCALLY IN ADDITION TO THE STATE.
A TRANSMISSION VECTOR
 lj
THAT MINIMIZES THE DIMENSIONS OF THE
DATA WHICH MUST BE EXCHANGED BETWEEN NODES.
• EACH NODE-j
COMPUTES AND TRANSMITS
OTHER NODES IN THE NETWORK.
 lj TO AND RECEIVES  jl FROM ALLTHE
144
Ref # 19
The Hohmann Transfer
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NASA / Goddard Space Flight Center
2 1
V 2    
 r a
where:
Vp 
a

rp
ra  rp
2
2ra
ra  rp
Vc 
DV1 
DV2 

r1

r2
2r2

r1  r2

r1

2r1



r1  r2
r2

2r2
 1

r1  r1  r2

Va 

2rp
ra
ra  rp

r


2r1 
1



r2 
r1  r2 
DVT = DV1 + DV2
145
Ref # 3,7
The LQG Decentralized Controller Overview
Goddard Space Flight Center
NASA / Goddard Space Flight Center
FROM NETWORK
CONTROL VECTORS
TRANSMISSION VECTORS
z -1
GLOBALLY
OPTIMAL CONTROL
z -1
LOCALLY
OPTIMAL FILTER
CONTROL VECTOR
TO NETWORK
& LOCAL PLANT
LOCAL SENSOR
z -1
GLOBAL DATA
RECONSTRUCTION
TRANSMISSION VECTOR
TO NETWORK
z -1
146
Ref # 19