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 …---


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


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

^


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

)


-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)


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 ()


7077.60

7077.65

7077.70

7077.75

7077.80

7077.85

0 25 50 75 100 125

Y-A

xis



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


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



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 ()


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

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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


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


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


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


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

Ref # 15

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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

Ref # 15

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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


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

Ref # 16

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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

Ref # 16

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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

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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

Ref # 16

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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


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


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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


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


(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)


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


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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


(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.


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)


(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)


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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

Ref # 1,18

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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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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

Ref # 1,18

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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


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


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


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 [email protected]

[email protected]

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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


1 00 cos i-sin i 0 -sin i -cos i


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 ]

Ref # 3-7

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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

Ref # 19

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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

formation flying in earth, libration, and distant retrograde orbits david folta

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