Dynamics and Control of Formation Flying Satellites

Texas A&M University - Dept. of Aerospace EngineeringTexas A&M University - Dept. of Aerospace EngineeringTexas A&M University - Dept. of Aerospace EngineeringTexas A&M University - Dept. of Aerospace EngineeringAugust 19, 2003

Dynamics and Control of Dynamics and Control of Formation Flying SatellitesFormation Flying Satellites

byby

S. R. VadaliS. R. Vadali

NASA Lunch & Learn TalkNASA Lunch & Learn Talk

August 19, 2003August 19, 2003


OutlineOutline Formation vs. ConstellationFormation vs. Constellation Introduction to Orbital MechanicsIntroduction to Orbital Mechanics Perturbations and Mean Orbital ElementsPerturbations and Mean Orbital Elements Hill’s EquationsHill’s Equations Initial ConditionsInitial Conditions A Fuel Balancing Control ConceptA Fuel Balancing Control Concept Formation Establishment and MaintenanceFormation Establishment and Maintenance High-Eccentricity OrbitsHigh-Eccentricity Orbits Work in ProgressWork in Progress Concluding RemarksConcluding Remarks


Global Positioning System (GPS) Global Positioning System (GPS) ConstellationConstellation

Difference


Formation Flying: Relative OrbitsFormation Flying: Relative Orbits


Distributed Space Systems- Enabling Distributed Space Systems- Enabling New Earth & Space Science (NASA)New Earth & Space Science (NASA)

Co-observationCo-observation

Multi-point observationMulti-point observation

InterferometryInterferometry

Tethered InterferometryTethered Interferometry

Large Interferometric Space Large Interferometric Space AntennasAntennas


32 optics (300 10 cm) held in phase with 600 m

baseline to give 0.3 micro arc-sec

34 Formation Flying Spacecraft

1 km

Optics

10 km

Combiner Spacecraft

500 km

DetectorSpacecraft

Black hole image!

System is adjustable on

orbit to achieve larger

baselines

The Black Hole Imager: The Black Hole Imager: Micro Arcsecond X-ray Imaging Micro Arcsecond X-ray Imaging

Mission Mission (MAXIM) Observatory Concept(MAXIM) Observatory Concept


Optics

DetectorSpacecraft

Landsat7/EO-1 Formation FlyingLandsat7/EO-1 Formation Flying

450 km in-track and 50m Radial Separation.

Differential Drag and Thrust Used for Formation Maintenance


Motivation for ResearchMotivation for Research

Air Force: Sparse Aperture Radar.Air Force: Sparse Aperture Radar. NASA and ESA: NASA and ESA:

Terrestrial Planet Finder (TPF)Terrestrial Planet Finder (TPF)Stellar Imager (SI)Stellar Imager (SI) LISA, MMS, MaximLISA, MMS, Maxim

Swarms of small satellites flying in precise formations will Swarms of small satellites flying in precise formations will cooperate to form distributed aperture systemscooperate to form distributed aperture systems..

Determine Fuel efficient relative orbits. Do not fight Determine Fuel efficient relative orbits. Do not fight Kepler!!!Kepler!!!

Effect of JEffect of J22? ? How to establish and reconfigure a formation?How to establish and reconfigure a formation? Balance the fuel consumption for each satellite and Balance the fuel consumption for each satellite and

minimize the total fuel. minimize the total fuel.


Introduction to Orbital Introduction to Orbital Mechanics-1Mechanics-1

Formation Flying: Satellites close to each Formation Flying: Satellites close to each other but not necessarily in the same plane.other but not necessarily in the same plane.

Radial (up), x

Along-Track, (y)

Out-of-plane (z)

Deputy

Chief Deputy Dynamics

Communications Navigation

Control


Introduction to Orbital Mechanics-2Introduction to Orbital Mechanics-2

Orbital Elements: Five of the six elements remain constant for the 2-Body Problem.Orbital Elements: Five of the six elements remain constant for the 2-Body Problem. Variations exist in the definition of the elements.Variations exist in the definition of the elements.

Mean anomaly:Mean anomaly: 3

:M M na


Orbital Mechanics-3Orbital Mechanics-3 Ways to setup a formation:Ways to setup a formation:

• Inclination difference.• Node difference.• Combination of the two.

Inclination Difference Node Difference


Orbital Mechanics-4Orbital Mechanics-4 JJ22 Perturbation: Perturbation:

• Gravitational Potential:

• J2 is a source of a major perturbation on Low-Earth satellites .

•

Equatorial Bulge Potential of an Aspherical body

2

( , ) 1 (sin )k

ek k

k

Rr J P

r r

2

222

3( , ) (3sin 1)

2e

J

J Rr

r

3

2 1.082629 10J


Orbital Mechanics-5Orbital Mechanics-5 JJ2 2 induces short and long periodic induces short and long periodic

oscillations and secular Drifts in some of oscillations and secular Drifts in some of the orbital elementsthe orbital elements

Secular Drift RatesSecular Drift Rates

• Node:

• Perigee:

• Mean anomaly:

1cos575.0 22

2

in

p

RJ e

1cos3175.0 22

22

in

p

ReJnM e

Drift rates depend on mean a, e, and i

inp

RJ e cos5.1

2

2


Orbital Mechanics-6Orbital Mechanics-6 Analytical theories exist for obtaining Analytical theories exist for obtaining

Osculating elements from the Mean Osculating elements from the Mean elements. elements.

Brouwer (1959)Brouwer (1959) If two satellites are to stay close, their If two satellites are to stay close, their

periods must be the same (2-Body).periods must be the same (2-Body). Under JUnder J2 2 the drift rates must match. the drift rates must match.

Requirements:Requirements:1 2

1 2

1 2M M


Orbital Mechanics-7Orbital Mechanics-7 For small differences in a, e, and iFor small differences in a, e, and i

Except for trivial cases, all the three Except for trivial cases, all the three equations above cannot be satisfied with equations above cannot be satisfied with non-zero a, e, and i elemental differences. non-zero a, e, and i elemental differences.

Need to relax one or more of the Need to relax one or more of the requirements.requirements.

0

0

0

a e i a

ea e i

iM M M

a e i


Orbital Mechanics-8Orbital Mechanics-8 JJ22-invariant Relative Orbits (Schaub and -invariant Relative Orbits (Schaub and

Alfriend, 2001).Alfriend, 2001).

This condition can sometimes lead to large This condition can sometimes lead to large relative orbits (For Polar Reference Orbits) relative orbits (For Polar Reference Orbits) or orbits that may not be desirable.or orbits that may not be desirable.

0( ) ( ) ( )

0

a e i aM M M

ea e i

i


Orbital Mechanics-9Orbital Mechanics-9 JJ22-invariant Relative Orbits (No Thrust Required)-invariant Relative Orbits (No Thrust Required)


Orbital Mechanics-10Orbital Mechanics-10 Geometric Solution in terms of small orbital Geometric Solution in terms of small orbital

element differenceselement differences

For small eccentricityFor small eccentricity

A condition for No Along-track Drift (Rate-A condition for No Along-track Drift (Rate-Matching) is:Matching) is:

( cos ); orbit latitude angle=c cy a i f ( sin sin cos )c c c cz a i i

cos 0cM i

= 2 sin and

2 sin 2 cosc c c

M e M

M e M e M M


Remarks: Our ApproachRemarks: Our Approach

The and constraints result The and constraints result in a large relative orbit for small eccentricity and high in a large relative orbit for small eccentricity and high inclination of the Chief’s orbit. (Jinclination of the Chief’s orbit. (J22-Invariant Orbits)-Invariant Orbits)

Even if the inclination is small, the shape of the relative Even if the inclination is small, the shape of the relative orbit may not be desirable.orbit may not be desirable.

Use the no along-track drift condition (Rate-Matching) only.Use the no along-track drift condition (Rate-Matching) only.

Setup the desired initial conditions and use as little fuel as Setup the desired initial conditions and use as little fuel as possible to fight the perturbations.possible to fight the perturbations.

10 1100 MM

End of Phase-1


Hill-Clohessey-Wiltshire Hill-Clohessey-Wiltshire Equations-1Equations-1

( )223 2

2 2 2 2[( ) ]

223

2 2 2 2[( ) ]

32 2 2 2[( ) ]

r xcx y y xrcr x y zc

yy x x y

r x y zcz

z

r x y zc

Eccentric reference orbit relative motion dynamics (2-Body) :

Assume zero-eccentricity and linearize the equations:


Hill-Clohessey-Wiltshire Hill-Clohessey-Wiltshire Equations-2Equations-2

HCW Equations:

Bounded Along-Track Motion Condition

2

2

2 3 0

2 0

0

c c

c

c

x n y n x

y n x

z n z

where .3cnac

Velocity vector

Along orbit normal

zChief

y

Deputy

2 0cy n x


Bounded HCW SolutionsBounded HCW Solutions

Projected Circular Re. Projected Circular Re. Orbit.Orbit.

General Circular Re. General Circular Re. Orbit.Orbit.

10

1 0 3

2 0

sin( )2

cos( )

sin( )

cx n t

y c n t c

z c n t

1 2

3

2 2 2

0

c c

c

y z

1 2

3

2 2 2 2

3

20

c c

c

x y z


PCO and GCO Relative OrbitsPCO and GCO Relative Orbits

Projected Circular Orbit (PCO) General Circular Orbit (GCO)


Initial Conditions in terms of Mean Initial Conditions in terms of Mean Element Differences :Element Differences :

General Circular Relative Orbit. General Circular Relative Orbit.2

224 2

2

1

2

2

12 3

1

3 4(1 3cos ) sin 2

2

1

0.5 sin( ) /

cos( ) /

sin( )/

sin

([ cot sin( ) cos( )] ) /2

cos( ) /2

e c cc c

c c c c

c c

c c

c

cc

c c cc

c cc

R e eJai i i

a a

e

e c a

i c a

c ai

cc i c a

e

cM a

e

Eccentricity Difference

Inclination Difference

Node Difference

Mean Anomaly Difference

Perigee Difference

Semi-major axis Difference


Simulation ModelSimulation ModelEquations of motion for one satellite

ur r

rnr

7

2

55

22

3153

ˆ6

2 r

z

rr

zRJ

re

r

xr

T

r r0

0

yT

r H r

H r

( )0 0

0 0

zH

T

r H0

0

r r r1 0 v v v1 0 Rotating frame coordinates

Inertial Relative Displacement Inertial Relative Velocity

Initial conditions: Convert Mean elements to Osculating elements and then find position and velocity.


Hill’s Initial Conditions with Rate-Hill’s Initial Conditions with Rate-MatchingMatching

Chief’s orbit is eccentric: e=0.005Chief’s orbit is eccentric: e=0.005 Formation established using inclination difference only.Formation established using inclination difference only.

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Along-Track (km)

Out

-of-P

lane

(km

)

Relative Orbits in the y-z plane, (2 orbits shown)

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Along-Track (km)

Out

-of-P

lane

(km

)

150 Relative Orbits

0 0


Drift Patterns for Various Initial Drift Patterns for Various Initial ConditionsConditions

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5150 orbits, Sat-2, phase=90

Out

-of-p

lane

In track-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Out

-of-p

lane

In track

150 orbits, Sat-3, phase=120

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Out

-of-p

lane

In track

150 orbits, Sat-4, phase=-120

The above pattern is for a deputy with no inclination difference, only node difference.

End of Phase-2


Fuel Requirements for a Circular Projection Fuel Requirements for a Circular Projection Relative Orbit FormationRelative Orbit Formation

Sat #1and 4 have max and zero

Sat #3 and 6 have max but zero

1 and 4 will spend max fuel; 3 and 6 will spend min fuel to fight J2.

y

z

i

i

Snapshot when the chief is at the equator. Pattern repeats every orbit of the Chief

1

3

4

2

5

6


Fuel Balancing Control ConceptFuel Balancing Control Concept

Balance the fuel consumption over a certain period by rotating all the deputies by an additional rate

y

z

Snapshot when the chief is at the equator.

1

3

4

2

5

6


Modified Hill’s Equations to Account Modified Hill’s Equations to Account for Jfor J22

2

20

2

22

2 3

2

2 cos sin

3 sin

2

c c x

c y

c z c c

ec

c

x n y n x u

y n y u

z n z u An

RA J n i

a

Analytical solution

The near-resonance in the z-axis is detuned by

Assume no in-track drift condition satisfied.

nn

c c cn M

( sin sin cos )c c c cz a i i


Balanced Formation Control Saves Balanced Formation Control Saves FuelFuel

0.5 sin( )

cos( )

sin( )

R

R

R

x

y

z

Ideal Trajectory Ideal Control for perfect cancellation of the disturbance and for

2

2 2 cos sin

x R

y R

z R

u n x

u n y

u n z An

Optimize over time and an infinite number of satellites

nAJopt2

3

2

3

Aopt nAJ 2

0

0

2

0

2

0

2222

4

1

dduuu

nJ zyx


Analytical ResultsAnalytical ResultsBenefits of Rotation (Circular Projection Orbit)Benefits of Rotation (Circular Projection Orbit)

0 20 40 60 80 100 120 140 160 180 2000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10

-5

Time (days)

cost

(m

2 /sec

3 )

(0)= 0(0)= /4(0)= /2(0)= 3/4

Fuel Balanced in 90 days


Nonlinear Simulation ResultsNonlinear Simulation Results

= -2.47 / day

0 20 40 60 80 100 120 140 160 1800

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6x 10

-5

Time (days)

Cost

(m2 /s

ec3 )

0 20 40 60 80 100 120 140 160 1800

2

4

6x 10

-5

Time (days)

Cost

(m2 /s

ec3 )

0 = 0,

0 = /2,3/2

Co s

t (m

2 /se

c3 )

2.5 / day = 0

0

0 0

=1 km 7,100

0.005, 70

a km

e i

Equivalent to 28 m/sec/yr/satEquivalent to 52 m/sec/yr

Formation cost equivalent to 32 m/sec/yr/sat

Co s

t (m

2 /se

c3 )

Benefits of Rotation (Circular Projection Orbit)Benefits of Rotation (Circular Projection Orbit)

= 0,

= /2, 3 /2


Nonlinear Simulation ResultsNonlinear Simulation Results

Orbit Radii over one year(8 Satellites)Orbit Radii over one year(8 Satellites)


Disturbance AccommodationDisturbance Accommodation

Do not cancel JDo not cancel J22 and and

Eccentricity induced periodic Eccentricity induced periodic disturbances above the orbit disturbances above the orbit rate.rate.

Utilize Filter StatesUtilize Filter States No y-bias filterNo y-bias filter LQR DesignLQR Design Transform control to ECI and Transform control to ECI and

propagate orbits in ECI propagate orbits in ECI frame.frame.

The Chief is not controlled.The Chief is not controlled.

0

22 0 2

23 0 2

22 0 2

23 0 3

0

22 0 2

23 0 3

(2 )

(3 )

(2 )

(3 )

(2 )

(3 )

x x

x x x

x x x

y y y

y y y

z z

z z z

z z z

z u

z n z u

z n z u

z n z u

z n z u

z u

z n z u

z n z u

End of Phase-3


Formation Establishment and Formation Establishment and ReconfigurationReconfiguration

Changing the Size and Shape of the Changing the Size and Shape of the Relative Orbit.Relative Orbit.

Can be Achieved by a 2-Impulse Can be Achieved by a 2-Impulse Transfer.Transfer.

Analytical solutions match Analytical solutions match numerically optimized Results.numerically optimized Results.

Gauss’ Equations Utilized for Gauss’ Equations Utilized for Determining Impulse magnitudes, Determining Impulse magnitudes, directions, and application times. directions, and application times.

Assumption: The out-of-plane cost Assumption: The out-of-plane cost dominates the in-plane cost. Node dominates the in-plane cost. Node change best done at the poles and change best done at the poles and inclination at the equator crossings.inclination at the equator crossings.


Formation EstablishmentFormation Establishment

1 km GCO Established with

0 45

0 90

1 km PCO Established with


Formation ReconfigurationFormation Reconfiguration

1 km, PCO to 2 km, PCO

0 0

0 0 1 km, PCO to 2 km, PCO

0 0

0 90

Chief is at the Asc. Node at the Beginning.


Reconfiguration CostReconfiguration Cost

Cost vs. Final Phase Angle

This plot helps in solving the slot assignment problem. The initial and final phase angles should be the same for fuel optimality for any initial phase angle.


Optimal AssignmentOptimal Assignment

Objectives: (i) Minimize Overall Fuel Consumption (ii) Homogenize Individual Fuel Consumption

PCO 1 to 2i f GCO 1 to 2i f

End of Phase-4


Relative Motion on a Unit SphereRelative Motion on a Unit Sphere

1

[ ] 0

0

TC D

x

y C C I

z

Chief

Deputy

Unit Sphere

x

y

ECI

: Attitude Matrix of the Chief

: Attitude Matrix of the DeputyC

D

C

C

Relative Position Vector


Relative Motion Solution on the Relative Motion Solution on the Unit SphereUnit Sphere

2 2 2 20 1 0 1

2 2 2 20 1 0 0 1 0

0 1 0

1 ( / 2) ( / 2) ( ) ( / 2) ( / 2) ( )

( / 2) ( / 2) (2 ) ( / 2) ( / 2) (2 )

1/ 2 ( ) ( )[ ( ) (2 )]

x c i c i c s i s i c

s i c i c c i s i c

s i s i c c

2 2 2 20 1 0 1

2 2 2 20 1 0 0 1 0

0 1 0

( / 2) ( / 2) ( ) ( / 2) ( / 2) ( )

- ( / 2) ( / 2) (2 ) ( / 2) ( / 2) (2 )

1/ 2 ( ) ( )[ ( ) (2 )]

y c i c i s s i s i s

s i c i s c i s i s

s i s i s s

0 1 0 1 0 1 1( ) ( ) ( ) [ ( ) ( ) ( ) ( ) ( )] ( ) z s i s c s i c i c c i s i s

Valid for Large Angles


Analytical Solution using Analytical Solution using Mean Orbital Elements-1Mean Orbital Elements-1

Mean rates are constant.

2 2 221 0.75 1 ( / ) 3cos 1j j j e j jM n J e R p i

221.5 ( / ) cos( )j e j j jJ R p n i

2 220.75 ( / ) 5cos 1j e j j jJ R p n i

2(1 )j j jp a e


Analytical Solution using Analytical Solution using Mean Orbital Elements-2Mean Orbital Elements-2

Actual Relative Motion.

2[1- cos( ) -1/ 2 (cos(2 ) -1) .....], 0,1j j j j j jr a e M e M j


High Eccentricity Reference OrbitsHigh Eccentricity Reference Orbits

Eccentricity expansions do not converge for high e.

Use true anomaly as the independent variable and not time.

Need to solve Kepler’s equation for the Deputy at each data output point.


Formation Reconfiguration for High-Formation Reconfiguration for High-Eccentricity Reference OrbitsEccentricity Reference Orbits


High Eccentricity Reconfiguration High Eccentricity Reconfiguration CostCost

Cost vs. Final Phase Angle

Impulses are applied close to the apogee. No symmetry is observed with respect to phase angle.


Research in ProgressResearch in Progress

Higher order nonlinear theory and period matching conditions for large relative orbits.

Continuous control Reconfiguration (Lyapunov Functions).

Nonsingular Elements (To handle very small eccentricity)

Earth-moon and sun-Earth Libration point Formation Flying.


Concluding RemarksConcluding Remarks

Discussed Issues of Near-Earth Formation Flying and methods for formation design and maintenance.

Spacecraft that have similar Ballistic coefficients will not see differential drag perturbations.

Differential drag is important for dissimilar spacecraft (ISS and Inspection Vehicle).

Design of Large Near-Earth Formations in high-eccentricity orbits pose many analytical challenges.

Thanks for the opportunity and hope you enjoyed your lunch!!

Documents

Dynamics and Control of Formation Flying Satellites