dynamics and control of formation flying satellites in ... · •build on our extensive knowledge...

Alfriend/Barcelona 1

Dynamics and Control ofFormation Flying Satellites

In Earth OrbitTerry Alfriend

Texas A&M University


OutlineLecture 1- Dynamics of relative motion• What is formation flying?• Clohessy-Wiltshire (CW) or Hill’s equations

u Historyu Periodic Solutionsu Assumptions/limitations

• Curvilinear coordinates• Design with differential orbital elements• Elliptic reference orbit• J2 effects• Classification of relative motion orbits• Comparison of theories• Utilizing natural motion


Outline

• Example of the Error Analysis of a LEO satellite• Relative Navigation Error Effect• Constellation Control

Lecture 2- Control concepts and navigation error effect


Texas A&M Team• Faculty

u Alfriendu Vadaliu Junkins

• Postdocsu Schaub (former)

• Graduate Studentsu Gimu Vaddiu Yanu Wilkinsu Naiku Subbarao (former)

• Contractsu AFOSRu NASA Goddard (3)u AFRL (TechSat 21)u AFRL (Power Sail)


What is Formation Flying?

Formation flying is multiple satellites orbiting in closeproximity in a cooperative manner.

Autonomous operation will be necessary.


Example


Background• Interest in the US in the use of swarms of small satellites

flying in precise formationu Mission enabling and mission enhancing technologyu Long baseline interferometers, Virtual antennas

• AR!FL TechSat 21 program - cancelled• NASA/GSFC

u Landsat 7 - EO1u ORIONu MMS

Formation flying satellites is not a difficult concept. Flyingautonomously in precise formation is a new challenge.


Distributed Spacecraft MissionsRepresentative List of Satellite Missions Utilizing Distributed Spacecraft Techniques

ProjectedLaunch Year

Mission Name Mission Type

00 New Millennium Program (NMP) Earth Observing-1 Earth Science01 Gravity Recovery and Climate Recovery (GRACE) Earth Science03 University Nanosats/Air Force Research Laboratory Nanosat 1 Technology Demonstrator03 University Nanosats/Air Force Research Laboratory Nanosat 2 Technology Demonstrator03 NMP ST-5 Nanosat Constellation Trailblazer Space Science04 Techsat-21/AFRL Technology Demo04 Auroral Multiscale Mission (AMM)/APL Space Science/SEC04 ESSP-3-Cena (w/ Aqua) Earth Science05 Starlight (ST-3) Space Science/ASO05 Magnetospheric Multiscale (MMS) Space Science/SEC05 Space Interferometry Mission (SIM) Space Science/ASO06 MAGnetic Imaging Constellation (MAGIC) Space Science06 COACH Earth Science07 Global Precipitation Mission (EOS-9) Earth Science07 Geospace Electrodynamic Connections (GEC) Space Science/SEC08 Constellation-X Space Science/SEU08 Magnetospheric Constellation (DRACO) Space Science/SEC08 Laser Interferometer Space Antenna (LISA) Space Science/SEU09 DARWIN Space Infrared Interferometer/European Space Agency Space Science10 Leonardo (GSFC) Earth Science15 Stellar Imager (SI) Space Science/ASO11 Terrestrial Planet Finder (TPF) Space Science/ASO

Astronomical Low Frequency Array (ALFA)/Explorers Space Science05+ MAXIM Pathfinder Space Science/SEU05+ Living with a Star (LWS) Space Science05+ Soil Moisture and Ocean Salinity Observing Mission (EX-4) Earth Science05+ Time-Dependent Gravity Field Mapping Mission (EX-5) Earth Science05+ Vegetation Recovery Mission (EX-6) Earth Science05+ Cold Land Processes Research Mission (EX-7) Earth Science05+ Hercules Space Science/SEC05+ Orion Constellation Mission Space Science/SEC15 Submillimeter Probe of the Evolution of Cosmic Structure (SPECS) Space Science/SEU15+ Planet Imager (PI) Space Science/ASO15+ MAXIM X-ray Interferometry Mission Space Science/SEU15+ Solar Flotilla, IHC, OHRM, OHRI, ITM, IMC, DSB Con Space Science/SEC15+ NASA Goddard Space Flight Center Earth Sciences Vision Earth Science

15+ NASA Institute of Advanced Concepts/Very Large Optics for the Studyof Extrasolar Terrestrial Planets

Space Science

15+ NASA Institute of Advanced Concepts /Ultra-high Throughput X-RayAstronomy Observatory with a New Mission Architecture

Space Science

15+ NASA Institute of Advanced Concepts /Structureless Extremely LargeYet Very Lightweight Swarm Array Space Telescope

Space Science


Distributed Spacecraft MissionsRepresentative List of Satellite Missions Utilizing Distributed Spacecraft Techniques

ProjectedLaunch Year

Mission Name Mission Type

00 New Millennium Program (NMP) Earth Observing-1 Earth Science01 Gravity Recovery and Climate Recovery (GRACE) Earth Science03 University Nanosats/Air Force Research Laboratory Nanosat 1 Technology Demonstrator03 University Nanosats/Air Force Research Laboratory Nanosat 2 Technology Demonstrator03 NMP ST-5 Nanosat Constellation Trailblazer Space Science04 Techsat-21/AFRL Technology Demo04 Auroral Multiscale Mission (AMM)/APL Space Science/SEC04 ESSP-3-Cena (w/ Aqua) Earth Science05 Starlight (ST-3) Space Science/ASO05 Magnetospheric Multiscale (MMS) Space Science/SEC05 Space Interferometry Mission (SIM) Space Science/ASO06 MAGnetic Imaging Constellation (MAGIC) Space Science06 COACH Earth Science07 Global Precipitation Mission (EOS-9) Earth Science07 Geospace Electrodynamic Connections (GEC) Space Science/SEC08 Constellation-X Space Science/SEU08 Magnetospheric Constellation (DRACO) Space Science/SEC08 Laser Interferometer Space Antenna (LISA) Space Science/SEU09 DARWIN Space Infrared Interferometer/European Space Agency Space Science

Notes: ASO-Astronomical Search for Origins, SEC-Sun Earth Connections, SSE-Solar System Exploration, SEU- structure andEvolution of the Universe


TechSat21

• Air Force Technology Mission• 3-170 kg Satellites• Orbit:550 km altitude, 35.4 deg inclination• Relative Navigation: DCGPS

u 1s Rqmt: 10 cm position, 0.5 mm/s velocity

• Formationsu Leader Follower: 30 m-5 kmu In-Plane Elliptic: 60 m-5 km

• Autonomous Orbit Maintenance and Reconfiguration• Challenges/Constraints

u 1 thruster (PPT)-rotate S/C to fire thrusteru Differential dragu Collision avoidance


NMP EO-1 Enhanced Formation Flying (EFF)Experiment

Level - I:Demonstrate the Capability to Fly Over the SameGround Track As LandSat-7 Within 3 Km at a NodalSeparation Interval of Nominally One Minute DuringWhich Time an Image Is Collected.

Level-II:EFF- Shall Provide the Autonomous Capability of FlyingOver the Same Groundtrack of Another S/C at FixedSeparation Times.Autonomy - Shall Provide On-Board AutonomousRelative Navigation and Formation Flying Control forEO-1 and LandSat-7.AutoCon Flight Control System - Shall ProvideAutonomous Formation Flying Control Via AutoCon (toprovide future reusability).Ground Track - EO-1 Shall Fly the Same Ground TrackAs LandSat-7.Separation - EO-1 Shall Remain Within a 1-Minute In-Track Separation from LandSat-7.

FF Start

Velocity

FF Maneuver

Formation Flying Spacecraft

Reference S/C

In-Track Separation (Km)

Observation Overlaps

Rad

ial S

epar

atio

n (

m)

Ideal FF Location

NadirDirection

I-minute separationin observations


Magnetospheric Multi-scale

•5 identical spacecraft in a variably spaced

tetrahedron ( 1 km to several earth radii )•4 orbit phases, orbit adjust•2 year in-orbit (minimum) mission life•Interspacecraft ranging and communication•Advanced instrumentation, integrated payload•Attitude knowledge < 0.1°, spin rate 20 rpm

How do small-scale processes control large-scale phenomenology, such as magnetotaildynamics, plasma entry into themagnetosphere, and substorm initiation?

Phases 1-3, Equatorial - Phase 4, Polar - Determination of Spatial Gradients


The LISA Configuration


LISA Science OverviewLISA Science Overview

• LISA will observe gravitationalwaves from compact galactic binariesand from extragalactic supermassiveblack holes.

• LISA will the address the followingquestions;

– What is the structure of space andtime near massive black holes?

– How did the massive black holes atthe centers of galaxies form?

– What happened in the Universebefore light could propagate?

– What are the precursors to Type Iasupernovae?


Relative Motion Equations

r&&r + 2r

w ¥r&r +

r&w ¥rr +

rw ¥

rw ¥

rr( ) + r&&r

c= -

mr3

rr -

mR3

rr

c

ÊËÁ

ˆ¯

�

r r c

r r

x

z

y

q

&&x - 2 &q &y - &&q y - &q 2x = -m(rc + x)

[(rc + x)2 + y2 + z2]32

+m

rc2 + J

2f

xe,x( )

&&y + 2 &q &x + &&q x - &q 2 y = -m y

[(rc + x)2 + y2 + z2]32

+ J2

fy

e,x( )

&&z = -mz

[(rc + x)2 + y2 + z2]32

+ J2

fz

e,x( )

&&rc = rc&q 2 -

m

rc2 1 + J

2f

re( )ÈÎ ˘

&&q = -2 &rc

&qrc

1 + J2

fq e( )ÈÎ ˘


Clohessy-Wiltshire (Hill’s) EquationsAssumptions - Spherical Earth - Circular reference orbit - Equations can be linearized

&&x - 2n&y - 3n2x = 0

&&y + 2n &x = 0

&&z + n2z = 0

x = 2 2x0

+ &y0

/ n( ) - 3x0

+ 2 &y0

/ n( )cosy + &x0

/ n( )siny

y = y0

- 2 &x0

/ n( ) - 3 2x0

+ &y0

/ n( )y + 2 &x0

/ n( )cosy + 2 3x0

+ 2 &y0

/ n( )siny

z = z0cosy + &z

0/ n( )siny

y = nt

With q = argument of latitude and initial conditions at the equator

x = d a - adecos M

y = a dq0

+ dWcos i + 2desinw( ) - 1.5d aq + 2adesin M

z = a d isinq - dWsin icosq( )


Periodic Solutions

Circular relative orbit

B= 3A,a = bx2 + y2 + z2 = 4A2

x

y

z

Horizontal Plane Circular Orbit or Projected Circular Orbit (PCO)

A=B/2 , a =by2 + z2 = B2

x

y

z

x = Asin y + b( ), y = 2Acos y + b( ), z = Bsin y + a( )x2 + y / 2( )2

= A2 ; 2-1 ellipse

Leader-Follower, x=z=0, y=const


CW Equations (cont)

&&x - 2n&y - 3n2x = ux

+ O r / R( ) + O e( ) + O J2( )

&&y + 2n &x = uy

+ O r / R( ) + O e( ) + O J2( )

&&z + n2z = uz

+ O r / R( ) + O e( ) + O J2( )

ux,u

y,u

z( ) are the control forces

Due to J2

:e ≥ O J2( )

For formations of size 1-30 km the neglected nonlinear terms are of

O 10-3( ). Therefore, all the neglected terms in general are of the same order

of magnitude, so for a theory or model to be valid for all types of relative

motion orbits it must include all these effects. If curvilinear coordinates are used

instead of Cartesian coordinates then the nonlinear terms are generally not

needed for orbits less that 30-40 km. As will be seen later the differential J2 effects are small

for some relative motion orbits and can be neglected.


The Problem• Precise relative position knowledge is important, precision

control often is not important. Position maintenance withinbounds is sufficient.

• Minimum fuel control is essential.• Maintenance of nominal relative motion orbits results in

different fuel requirements for each satellite.• System lifetime is defined by the first satellite becoming

non-operational.• Equal fuel consumption/satellite over a long period of

time.• During thruster firings many satellites are not operational

and the relative navigation could be disturbed. Maximizetime between thruster firings.


The Problem (cont)

• Mathematical model is accurate.• System can usually be described by linear

equations.• Control is easy, minimum fuel control for the real

problem with equal fuel consumption for eachsatellite over a long period of time is not easy

This is not a standard Linear-Quadratic-Gaussiancontrol problem.


Philosophy/Principles/Approach• Minimum fuel orbits will result from finding the initial

conditions that yield the orbits that require minimum fuel tomaintain. ˛ Nonlinear analysis

• Minimum fuel control will result from the most accuratemodel.

• Use, do not fight Kepler.• Must take into account gravitational perturbations.• Build on our extensive knowledge of orbital mechanics and

optimal orbit maneuvers.• To maximize system lifetime all satellites should have near

equal fuel consumption.• The mathematical model should be consistent with the needs

of the mission and the accuracy of the relative navigationsystem.

Know and use the physics


Elliptic Reference Orbit

u = x / rc.v = y / r

c,w = z / r

c

change independent variable; use true anomaly f

d

dt=

df

dt

d

df=

h

rc2

d

df

d 2

dt2=

h

rc2

Ê

ËÁ

ˆ

¯˜

2

d 2

df 2+

2h3

rc5

esin f

p

d

df

¢¢u - 2 ¢v - 3u / 1 + ecos f( ) = 0

¢¢v + 2 ¢u = 0

¢¢z + z = 0

1 . Lawden, D. F., Optimal Trajectories forSpace Navigation, Butterworths, London,1963.

2 . Tschauner, J. and Hempel, P., “Optimale Beschleunigeungsprogramme fur dasRendezvous-Manouever,” Astronautica Acta, Vol. 10, 1964, pp. 296-307.

3 . Carter, T. E., “New Form for the Optimal Rendezvous Equations near a KeplerianOrbit,” AIAA J. of Guidance, Control and Dynamics, Vo l. 13, No. 1, Jan.-Feb.1990.


Nonlinear Effects

�

r r c

r r

x

z

y

qf

dr

l=out-of-plane angle

tan l =z

rThe linearized equations of motion are

d ¢¢r - 2rc ¢f - 3n2dr = 0

rc ¢¢f + 2d ¢r = 0

¢¢l + n2l = 0

Identical to CW equations with x Æ dr,y Æ rcf,z Æ r

cl

Use of curvilinear coordinates reduces the nonlinearity effects.

r = rc

+ dr

rather than

r = rc

+ x( )2+ y2 + z2È

ÎÍ˘˚

1/ 2


Nonlinear Effect References1. Karlgard, C. D. and Lutze, F. H., “Second Order Relative Motion Equations,”

AAS 01-464, AAS/AIAA Astrodynamics Specialist Conference, Quebec City,Quebec, July 2001.

2 . Mitchell, J. W., and Richardson, D. L., “A Th ird Order Analytical Solution forRelative Motion with a Circular Reference Orbit,” Paper AAS 02-147, AAS/AIAA2002 Space Flight Mechanics Conference, San Antonio, TX, January 2002.

3. Vaddi, S. S., Vadali, S. R . and Alfriend, K. T., “Formation Flying:Accommodating Nonlinearity and Eccentricity Perturbations,” AIAA J. ofGuidance, Control and Dynamics, Vol. 26, No. 2, Mar.-Apr. 2002, pp. 214-223.

4 . Alfriend, K. T. and Yan, H., “An Orbital Elements Approach to the NonlinearFormation Flying Problem,” International Formation Flying Conference: Missionsand Technologies, Toulouse, France, October 2002.

Refs. 1-3 treat the adjustment of the initial conditions for theCW equations in Cartesian coordinates to obtain periodic orbits.Ref. 4 is a nonlinear theory using differential orbital elementsthat includes J2 effects.


Design With Differential Orbital Elements


Why?For a circular reference orbit1. The semi-major axis of the in-plane 2x1 ellipse is 0.5ade.2. The maximum out-of-plane distance is a[(di)2+(dWsin)2]1/2.

3. The argument of perigee and initial mean anomaly determine the phasing.These basic concepts continue to hold with non-circular reference orbits. Thus, it is easier to design the relativemotion orbit with differential orbital elements ratherthan trying to find the initial conditions in the relativeCartesian space.

x = d a - adecos M

y = a dq0

+ dWcos i + 2desinw( ) - 1.5d aq + 2adesin M

z = a d isinq - dWsin icosq( )


Orbital Element Design -In-Plane

-3000

-2000

-1000

0

1000

2000

3000

-6000 -4000 -2000 0 2000 4000 6000

In-Track Displacement - m

Rad

ial D

ispla

cem

ent -

m

e=0e=.1e=0.2e=0.4e=0.6

a=26,608 km, de=0.0001, vary e


Orbital Element Design-Out-of-Plane

a=26,608 km, de=0.0001, di=0.0002, vary e

-10000

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

-8000 -6000 -4000 -2000 0 2000 4000 6000 8000

In-Track Displacement - m

Out

of P

lane

Disp

lace

men

t - m

e=0e=.1e=0.2e=0.4e=0.6


Gravitational Perturbation Effects


Out of Plane Motion


Gravitational Perturbation Effects• J2 is the dominant effect

u Nodal precession

u Perigee drift

u Mean motion

˙ W = -1.5J 2 Re / p( ) 2 ncosi

˙ w = 0.75J2 Re / p( )2 n 5cos2 i - 1( )dn = -0.75J2 Re / p( )2

h 1- 3cos2 i( )n

d &W = 1.5J2

Re

p

Ê

ËÁˆ

¯

2

nsin id i - 3J2

Re

p

Ê

ËÁˆ

¯

2ncos i

1 - e2( ) d e2( ) + 3J2

Re

p

Ê

ËÁˆ

¯

2

ncos id a

a

Similar effects for perigee rotation and change of period


Gravitational Perturbation Effects

• Both absolute and differential gravitational effectsmust be considered.

• For two satellites to remain close (quasi-periodicrelative motion orbits) the differential mean anglerates must be zero.

• Differential mean rates are created by changes in(a,e,i).

• Design orbits in mean element space, notosculating and transform to osculating for the ICs.


Mean Elements

Angular Rates˙ l = 1

L3 + e 34L7

LG

Ê Ë Á ˆ

¯ ˜

31 - 3 H2

G2Ê

Ë Á

ˆ

¯ ˜ = 1

L3 + e 34L7h3 1- 3cos2 i( )

˙ g = e3

4L7LG

Ê Ë Á ˆ

¯ ˜

41 - 5 H2

G2Ê

Ë Á

ˆ

¯ ˜ = e

34L7h4 1 - 5cos2 i( )

˙ h = e3

2L7LG

Ê Ë Á ˆ

¯ ˜

4 HG

Ê Ë Á ˆ

¯ ˜ = e

32L7h4 cosi

l = mean anomaly, g = arg.of perigee ,h = rt.ascension

L = ua ,G = Lh, H = Gcosi, h = 1 - e2( )1/ 2

�

M = - 1

2L2+ J2

m2r

Rer

Ê Ë Á

ˆ ¯ ˜

2

1- 3sin 2 i sin2 q( )M = -

1

2L2 - J21

4L6

LG

Ê Ë Á

ˆ ¯ ˜

3

1- 3H 2

G2

Ê

Ë Á

ˆ

¯ ˜

Delaunay variables

Hamiltonian (normalized variables)

Averaged

Non-Averaged


Bounded Relative Motion Orbits

For two satellites to not drift apart

d &l = d &g = d &h = 0 or d &M = d &w = d &W = 0

d L = dG = d H = 0,d a = de = d i = 0

In the circular orbit, spherical Earth problem this means

d a = 0fiequal energy


Possible ConstraintsPeriod Matching Constraint (In-track drift)

l = l + g = M + w

d &l + d &Wcos i = -3

L4d L + e 3

4L7h53h + 4( ) h sin2id i + 3cos2 i - 1( )dhÈ

Î˘˚ = 0

d a = -J

2

2h5

Re

a

Ê

ËÁˆ

¯

2

3h + 4( ) h sin2id i + 3cos2 i - 1( )dhÈÎ

˘˚ a

d &W = -e 3

2L7h54cos idh + h sin id i( ) = 0 Out-of Plane Drift

d &w = e 3

4L7h55h sin2id i + 4 5cos2 i - 1( )dhÈ

Î˘˚ = 0 Radial Drift


Relative Motion Orbit Classification


Type 1 - Three Constraints d a = de = d i = 0

Equal energy, angular momentum and polarcomponent of angular momentum

Type 1A: Leader Follower dW=0,dl,dw constant

Type 1B Linear Out of Plane, dWπ0

No fuel for maintenance.


Type 2 - Two Constraints

Period Matching d a = -

J2

2h5

Re

a

Ê

ËÁˆ

¯

2

3h + 4( ) h sin2id i + 3cos2 i - 1( )dhÈÎ

˘˚ a

Type 2A: J2 Invariant with Perigee Drift

h sin id i = -4cos idh,de = -

he

dhProblems with a prescribed di and near circular or near polar

Dv / orbit = 6ep

Re

a

Ê

ËÁˆ

¯

2nae

h61 + 5cos2 i( )dh

Circular orbit or circular projection orbit withprescribed dW, de=r/2a


Class 2A Required deFormation Flying

Equal Nodal Precession Rates

0

0.007

0.014

0.021

0.028

0 0.02 0.04 0.06 0.08 0.1

Chief Eccentricity

Del

ta e

I=50I=60I=70I=80I=85

a = 7500 km

1 km out-of-plane by inclination difference


Class 2 - Two ConstraintsType 2B: J2 Invariant with Right Ascension Drift

5h sin2id i = -4 5cos2 i - 1( )dh

Dv / orbit = 1.2ep na

Re

a

Ê

ËÁˆ

¯

2

1 + 5cos2 i( ) tan idh

Same problems as 2A with prescribed di


Class 3 One ConstraintUse Period Matching constraint.

Right Ascension Drift

Dv / orbit = 3p J

2nasin i

Re

a

Ê

ËÁˆ

¯

2

4cos idh + h sin id i( )Perigee Drift

Dv / orbit = 1.5p J

2nae

Re

a

Ê

ËÁˆ

¯

2

5h sin2id i + 4 5cos2 i - 1( )dhÈÎ

˘˚

It may also be necessary to control the absolute perigee drift. To maintainthe PCO and GCO this is required because of the different in-plane andout-of-plane frequencies


Example 1a = 7153, e = 0.05, i = 48 deg, l = h = 0, g = 30 degdh = 0.005, de = 0.0001

Osculating Elements Mean Elementsda = 0.351 m, di = 0.001 deg


Example 1 - (cont)Osculating

Mean

dW dq


J2 Changes the Physics• With no gravitational perturbations the initial conditions that yield

bounded relative motion are constrained to a 5-dimensional manifoldor in momenta space a 2-dimensional manifold, the sphere of radius a0,da=0.

• The addition of J2 changes the dimension of the constraint manifoldaccording to the number of constraints

J2=0 constraint surfaceClass 1 orbits

Class 3 orbits are a distortedsphere in the neighborhood ofthe point

Class 2 orbits J2Invariant constraint surfaceao

2i0

y0=cos-1e


Generalized Relative Motion Solution

Need:

A solution to the relative motion that is validfor all eccentricities and includes at theminimum 1st order absolute and differentialgravitation perturbation effects.


Two Possible Approaches

• Relative Orbit Theory - Geometric Methodu Use existing analytic orbit theories and expand about the Chief

orbit to obtain relative orbital elements.u Transform from relative orbital element space to relative motion

space (x,y,z).u Does not consider nonlinear terms

• Generalized Hill’s equationsu Using relative motion variables obtain Hamiltonian including

gravitational perturbations, chief orbit ellipticity and nonlinearterms.

u Transform to action-angle variable space.u Use Lie transforms to obtain perturbation solution


Geometric Method

xT = x, &x, y, &y, z, &z( ),eT = a,q ,i,q1,q

2,W( ),q

1= ecosw ,q

2= esinw

x t( ) = A ec( )de

osc= A e

c( ) Ddem

= A ec( ) DF

emt( )de

mt0( )

x t( ) = A ec( ) DF

emt( ) D-1 t

0( ) A-1 t0( )x t

0( )x t( ) = F

xt( )x

0,F

xt( ) = A e

c( ) DFem

t( ) D-1 t0( ) A-1 t

0( )D =

∂eosc

∂em

Objective: Obtain a state transition matrix that isvalid for all eccentricities and includes the 1st orderabsolute and differential J2 effects.

For a mean element solution D=I


Geometric Method - A Matrix

rR

d=

rR

c+

rr = R + x( ) r

exc

+ yre

yc+ z

re

zc

rR

dC = T CET ED

rR

dD

rR

dC = T CE T EC + dT EC( )

Rc

+ d Rc

0

0

Ê

Ë

ÁÁÁ

ˆ

¯

˜˜˜

= I + T CEdT EC( )R

c+ d R

c

0

0

Ê

Ë

ÁÁÁ

ˆ

¯

˜˜˜

rR

dC =

Rc

+ d Rc

0

0

Ê

Ë

ÁÁÁ

ˆ

¯

˜˜˜

+ RcT CE

dT11

dT12

dT13

Ê

Ë

ÁÁÁ

ˆ

¯

˜˜˜

=R

c+ d R

c

0

0

Ê

Ë

ÁÁÁ

ˆ

¯

˜˜˜

+ Rc

0

dq + dWcos ic

- cosqcsin i

cdW + sinq

cd i

Ê

Ë

ÁÁÁ

ˆ

¯

˜˜˜


A Matrix

rV = ( &R) e

x+ (Rv

n) e

y+ (- Rv

t) e

z∫ (V

rJ2) e

x+ (V

tJ2) e

y+ (V

nJ2) e

z

rV

d= (V

rJ2+ &x - yv

n+ zv

t) e

x+ V

tJ2+ &y + xv

n- zv

r( ) ey

+ VnJ2

+ &z - xvt

+ yvr( ) e

z

rV

d=

VrJ2

+ dVrJ2

VtJ2

+ dVtJ2

VnJ2

+ dVnJ2

Ê

Ë

ÁÁÁÁ

ˆ

¯

˜˜˜˜

+ VrJ2

0

dq + ci

dWsq d i - cq s

idW

Ê

Ë

ÁÁÁ

ˆ

¯

˜˜˜

+ VtJ2

-dq - ci

dW0

cq d i + sq si

dW

Ê

Ë

ÁÁÁ

ˆ

¯

˜˜˜

+ VnJ2

-sq d i + cq si

dW

-cq d i - sq si

dW

0

Ê

Ë

ÁÁÁ

ˆ

¯

˜˜˜

The angular velocity is different formean and osculating elements.

r

v = vr,v

t,v

n( )


Example

Circular horizontal plane orbit with 500 m radius

ac = 7100kmec = 0.005ic = 70degWc = w = f = 0q c = 0q1c = 0.005 q2c = 0

n = ma3 = 1.05531¥10-3 r / s

x = 0˙ x = nc / 4y = 0.5km˙ y = 0z = 0˙ z = n / 2

d a = 0

dq = 0.004055deg

d i = 0.0040148 deg

dq1

= 0

dq2

= -3.556 ¥ 10-5

dW = 0

Chief Elements Deputy Elements


Resultsa=7100 km, e=0.005, i=70 deg

Circular projected orbit, r=500 m

CW equations Geometric Method


References

1. Gim, D-W and Alfriend, K.T., “The State Transition Matrix of Relative Motion for thePerturbed Non-Circular Reference Orbit,” AIAA J. of Guidance, Control and Dynamics, Vol.26, No. 6 Nov-Dec 2003, pp 956-971.

2. Gim, D-W and Alfriend, K.T., “Satellite Relative Motion Using Differential EquinoctialElements,” Paper No. AAS 02-186,2002 AAS/AIAA Space Flight Mechanics Conference,San Antonio, January 26-29, 2002, to appear in Int. J. of Celestial Mechanics and DynamicalAstronomy. Same as 1 but with differential equinoctial elements to avoid all elementsingularities.


Theory Comparison

or

Which relative motion theory do you use?


Problem

• For a given problem how do you selectwhich theory or model to use?

• How do you compare accuracy?• What is acceptable accuracy?


Modeling Error Index

Given the system &x = f x,t( ), x 0( ) = x0

The initial conditions xio,i = 1, N are initial conditions on the desired relative motion orbit, e.g., projected circular orbit.Let xi t( ) be the exact solution for the initial condition xi 0( ) and let xi t( ) be the solution for the model. Non-dimensionalize the variables byy = Wx.

The modeling error index is defined to be

n t( ) = supi =1,N

yiT t( ) yi t( )

yiT t( ) yi t( ) - 1

For a one dimensional system y = y 1 + e( )n = 2e


Theories Compared

• Hill, C-W• Linear, spherical Earth (non-J2)

u Lawden, Tschauner-Hempel, Carter, Garrisonu Obtained by setting J2=0 in Gim-Alfriend

• Small eccentricity.u Contains O(e) for non-J2 terms and e=0 J2 terms

• Gim-Alfriend• Vadali Unit Sphere• Yan-Alfriend nonlinear theory


Comparison

• a=8000 km, i=50 deg• Projected circular orbit (PCO)• Vary e and r (size of PCO)• 100 equally spaced points on the PCO, calculate

error index after 1 day.


Results - e=0.001


Results r=160 m


Comparison Summary

• Need to include chief eccentricity in the model.• Need to include J2 in the model if the relative motion orbit

involves a differential inclination.• Nonlinear effects are small for relative motion orbits up to

40 km in size if curvilinear coordinates are used.• Small e theory with J2 is good for e<0.005


Natural DynamicsPurpose: Investigate utilizing the natural dynamics of the relativemotion as best as possible.

Example: For leader-follower in an elliptic orbit one can maintain aconstant angular separation with a differential argument of perigeebut not a constant distance.


Natural Dynamics

Constant Dimensionless Radius for PCO

uy

= uz

= 0

ux

= 1.5R &q 2 ecos f

1 + ecos f

ÊËÁ

ˆ¯

ra

ÊËÁ

ˆ¯

sin q + a( ) = 1.5ra

ÊËÁ

ˆ¯

R

p

ÊËÁ

ˆ¯

R &q 2ecos f sin q + a( )Constant Radius

ux

=re

p

ÊËÁ

ˆ¯

R &q 2 sin 2 f + b( )

uy

=re

2 p

ÊËÁ

ˆ¯

R &q 2 cos b + cos 2 f + b( )ÈÎ ˘

uz

= -re

2 p

ÊËÁ

ˆ¯

R &q 2 3sin 2 f + b( ) - sin bÈÎ ˘


Natural Dynamics

J =1

Tu

x2 + u

y2 + u

z2( )

0

T

Ú dt

JND

=9e2

64h5r2n4 4 + 3e2( ) - 2 1 + e2( )cos2bÈ

Î˘˚

JD

=n4r2e2

4h98 + 21e2 -

1

8e4Ê

ËÁˆ¯

+ e2 6 +5

8e2Ê

ËÁˆ¯

cos2bÈ

ÎÍ

˘

˚˙

JND

JD

=9

16

4 + 3e2( ) - 2 1 + e2( )cos2bÈÎ

˘˚

8 + 21e2 - 18

e4ÊËÁ

ˆ¯

+ e2 6 + 58

e2ÊËÁ

ˆ¯

cos2bÈ

ÎÍ

˘

˚˙

JND

JD

ª9

642 - cos2b( )


Natural Dynamics

JND

= 0

uy

= - R &q 2( ) rp

ecos f

ux

= 2 R &q 2( ) rp

esin f

JD

=r2e2n4

3240 + 42e2 + 9e4( )

Leader-Follower Circular Relative Orbit

uy

= uz

= 0

ux

= 1.5ra

ÊËÁ

ˆ¯

R

p

ÊËÁ

ˆ¯

R &q 2ecos f sin q + a( )Constant Radius

ux

=re

p

ÊËÁ

ˆ¯

R &q 2 sin 2 f + b( )

uy

=re

2 p

ÊËÁ

ˆ¯

R &q 2 cos b + cos 2 f + b( )ÈÎ ˘

uz

= -3re

4 p

Ê

ËÁ

ˆ

¯˜ R &q 2 3sin 2 f + b( ) - sin bÈÎ ˘

JND

JD

=9h4

4

4 + 3e2( ) - 2 1 + e2( )cos2bÈÎ

˘˚

51 + 147e2 + 714

e4ÊËÁ

ˆ¯

+ 1 + 42e2 + 17516

e4ÊËÁ

ˆ¯

cos2bÈ

ÎÍ

˘

˚˙

\

JND

JD

=9

2

2 - cos2b( )51 + cos2b( )


Dynamics Summary• Include perturbation effects in defining initial conditions for relative

motion• Define relative motions in mean elements, transform to osculating.

u (a,e,i) or (L,G,H) define relative drift rates• Conditions derived for (da, de, di) for J2 invariant relative orbits

u Can only constrain perigee drift or differential nodal precessionu Near circular Master orbit and near polar Master orbit require large,

possibly undesirable d e for J2 relative motion invariant orbits• Look for periodic solutions and determine their stability

u Establish attractive/stable reference motionsu Establish/validate optimal navigation and control concepts to maintain

these relative motions• For more precision higher order gravity terms and transformation

terms can be included.• Utilize the natural dynamics to the maximum extent possible

dynamics and control of formation flying satellites in ... · •build on our extensive knowledge...

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