Formation Flying in Earth, Libration, and Distant Retrograde Orbits David Folta NASA  Goddard Space Flight Center Advanced Topics in Astrodynamics Barcelona, Spain July 510, 2004. Agenda. I. Formation flying – current and future II. LEO Formations
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
in Earth, Libration,
and Distant Retrograde Orbits
David Folta
NASA  Goddard Space Flight Center
Advanced Topics in Astrodynamics
Barcelona, Spain
July 510, 2004
NASA Themes and Libration Orbits
NASA Enterprises of Space Sciences (SSE) and Earth Sciences (ESE) are a combination of several programs and themes
SSE
ESE
SEU
Origins
SEC
Ref #1
Low Earth Orbit Formations
Ref # 1, 2
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 humanassisted servicing
Membrane telescopes
Very Large Space Telescope (UVOIR): 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
Space Science Launches Possible Libration Orbit missionsRef #1
Future Mission Challenges interferometer
Considering science and operations
Two – Body Motion interferometer
Fundamental Equation of Motion
Ref # 37
Equation Of Motion Propagated. interferometer
External Accelerations Caused By Perturbations
FORCES ONPROPAGATED ORBIT+ accelerations
a = anonspherical+adrag+a3body+asrp+atides+aother
Ref #37
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
Gaussian Lagrange Planetary EquationsRef # 37
Ref # 37
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.
Simplified J2 acceleration model for analysis with acceleration in inertial coordinates
3J2mRe2 ri
5rk2
r2
1
ai =
2r5
df + df + df
dx + dy + dz
a = f =
i j k
D
3J2mRe2 rj
5rk2
r2
1
aj =
2r5
3J2mRe2 rj
5rk2
r2
3
ak =
2r5
Potential AccelerationsRef # 37
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:
HarrisPriester
Models direct heating only
Converts flux value to density
JacchiaRoberts or MSIS
Models both effects
Converts to exospheric temp. And then to atmospheric density
Contains lag heating terms
Atmospheric DragRef # 37
Solar Flux Prediction Solar Effects On The Earth’s Atmosphere
Historical Solar Flux, F10.7cm values
Observed and Predicted (+2s) 19452002
Ref # 37
Acceleration defined as Solar Effects On The Earth’s Atmosphere
Cd A r va2 v
m
Drag Acceleration^
1
2
a =
A = Spacecraft cross sectional area, (m2)
Cd = Spacecraft Coefficient of Drag, unitless
m = mass, (kg)
r = atmospheric density, (kg/m3)
va = s/c velocity wrt to atmosphere, (km/s)
v = inertial spacecraft velocity unit vector
Cd A = Spacecraft ballistic property
^
m
m
Ref # 37
Solar Radiation Pressure Acceleration Solar Effects On The Earth’s Atmosphere
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
Ref # 37



..
..
..
Other PerturbationsRef # 37
Area (A) is calculated based on spacecraft model. Solar Effects On The Earth’s Atmosphere
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
Ballistic CoefficientRef # 37
Varying the mass to area yields different decay rates
Sample: 100kg with area of 1, 10, 25, and 50m2, Cd=2.2
10m2
50m2
25m2
Ref # 37
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
RungeKutta
formula for yn+1, namely:
yn+1 = yn + (1/6)(k1 + 2k2 + 2k3 + k4)
is simply the yvalue of the current point plus a weighted average of four different yjump 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 endpoints.
CowellMoulton
MultiConic (patched)
Matlab ODE 4/5 is a variable step RK
Numerical IntegrationRef # 37
Geocentric Inertial (GCI)
Greenwich Rotating Coordinates (GRC)
Most used systems
GCI – Integration of EOM
ECEF – Navigation
Topographic – Ground station
Ref # 37
Describing Motion Near a Known Orbit the equations of motion.
_
_
r*
v*
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)
_
Chief (reference Orbit)
v
_
r
Deputy
Relative Motion
What equations of motion does the
relative motion follow?
Ref # 5,6,7
Describing Motion Near a Known Orbit the equations of motion.
As the last equation stands it is exact. However if dr is sufficiently small, the term f(r) can be expanded via Taylor’s series …



Substituting in yields a linear set of coupled ODEs
This is important since it will be our starting point for everything that follows
Lets call this (1)
Ref # 5,6,7
Describing Motion Near a Known Orbit the equations of motion.
_
_
r*
v*
^
T
^
R
^
N
The frame described is known as
 Hill’s  RTN
 ClohessyWiltshire  RAC
 LVLH  RIC
Any vector relative to the chief is given by:
Ref # 5,6,7
Describing Motion Near a Known Orbit the equations of motion.
First we evaluate in
arbitrary coordinates
^
Only in RTN , where the chief has a constant position =Rr*, the matrix takes a form
xRTN = r* , yRTN = zRTN = 0
Ref # 5,6,7
Transforming the Left Hand Side of (1) the equations of motion.
Now convert the left hand side of (1) to RTN frame,
Newton’s law involves 2nd derivatives:
In the RTN frame
Ref # 5,6,7
Transforming the EOM yields the equations of motion.
ClohessyWiltshire Equations
A “balance” form will have no secular growth, k1=0
Note that the ymotion (associated with Tangential) has twice the amplitude of the x motion (Radial)
Ref # 5,6,7
Relative Motion the equations of motion.
A numerical simulation using RK8/9 and point mass
Effect of Velocity (1 m/s) or Position(1 m) Difference
Initial Location 0,0,0
Initial Location 0,0,0
Initial Location 0,0,1
Shuttle Vbar / Rbar the equations of motion.
Graphics Ref: Collins, Meissinger, and Bell, Small Orbit Transfer Vehicle (OTV) for OnOrbit Satellite Servicing and Resupply, 15th USU Small Satellite Conference, 2001
Ref # 7,9
What Goes Wrong with an Ellipse the equations of motion.
In state – space notation, linearization is written as
Since the equation is linear
which has no closed form solution if
Ref # 5,6,7
Consider two trajectories the equations of motion.r(t) and R(t).
Transfer from r(t) to R(t) is affected by two DVs
First dVi is designed to match the velocity of a transfer trajectory R(t) at time t2
Second dVf is designed to match the velocity of R(t) where the transfer intersects at time t3
Lambert problem:
Determine the two DVs
Lambert ProblemRef # 5,6,7
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 F(t3,t2) as known, we can determine dVi and dVf using simple matrix methods to compute a ‘single pass’.
Lambert ProblemRef # 5,6,7
EO1 GSFC Formation Flying numerically integrate
New Millennium Requirements
Ref # 10,11
EO1 GSFC Formation Flying numerically integrate
Formation Flying Maintenance Description numerically integrate
Landsat7 and EO1
Different Ballistic Coefficients and Relative Motion
FF Start
EO 1 Spacecraft
Landsat7
InTrack Separation (Km)
Radial Separation (m)
Velocity
Ideal FF Location
Nadir
Direction
FF Maneuver
Iminute separation
in observations
Observation Overlaps
Ref # 10,11
EO1 Formation Flying Algorithm numerically integrate
Formation Flyer
Initial State
Reference S/C
Initial State
Reference Orbit
r0,v0
}
dr0,dv0
r1,v1
Keplarian State (t1)
Keplerian State K(tF)
dr(t0)
dv(t0)
Keplarian State (t0)
Keplerian State (ti)
Dt
R1,V1
t1
Reference S/C
Final State
ti
tf
Transfer
Orbit
t0
Maneuver
Window
Formation Flyer
Target State
Ref # 10,11
State Transition Matrix numerically integrate
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:
Ref # 10,11
Enhanced Formation Flying Algorithm numerically integrate
The Algorithm is found from the STM and is based on the simplectic nature ( navigation and guidance matrices) of the STM)
~
Ref # 10,11
EO1 AutoCon numerically integrate TM Functional Description
GPS Receiver
Thrusters
EFF
Maneuver
Decision
Inputs
Models
Constraints
AutoConTM
Where do I want to be?
Where am I ?
No
Yes
How do I get there?
Ref # 10,11
EO1 Formation Flying Subsystem Interfaces numerically integrate
EFF Subsystem
AutoConF
GSFC
JPL
GPS Data Smoother
Stored Command Processor
Cmd Load
GPS
ACE
Subsystem
Subsystem
C&DH
Subsystem
Comm
ACS
Subsystem
Subsystem
EO1 Subsystem LevelCommand and
Telemetry Interfaces
Propellant Data
GPS State Vectors
Thruster Commands
Timed Command Processing
SCP
Thruster Commanding
Uplink
Inertial State Vectors
Downlink
EFF
Subsystem
AutoConF &
GPS Smoother
Orbit Control
Burn Decision and Planning
Mongoose V
Ref # 10,11
Difference in EO1 Onboard numerically integrate
and Ground Maneuver Quantized DVs
Quantized  EO1 rounded maneuver durations to nearest second
Mode Onboard Onboard Ground DV1 Ground DV2 % Diff DV1 % DiffDV2
DV1 DV2 Difference Difference vs.Ground vs.Ground
cm/s cm/s cm/s cm/s % %
AutoGPS 4.16 1.85 4e6 2e1 .0001 12.50
AutoGPS 5.35 4.33 3e7 2e1 .0005 5.883
Auto 4.98 0.00 1e7 0.0 .0001 0.0
Auto 2.43 3.79 3e7 2e7 .0001 .0005
Semiauto 1.08 1.62 6e6 3e3 .0588 14.23
Semiauto 2.38 0.26 1e7 1e7 .0001 .0007
Semiauto 5.29 1.85 8e4 3e4 1.569 1.572
Manual 2.19 5.20 4e7 3e3 .0016 .0002
Manual 3.55 7.93 3e7 3e3 .0008 3.57
Inclination Maneuver Validation: Computed DV at node crossing, of ~ 24 cm/s (114 sec duration), Ground validation gave same results
Ref # 10,11
Difference in EO1 Onboard numerically integrate
and Ground Maneuver ThreeAxis DVs
EO1 maneuver computations in all three axis
Mode Onboard Ground DV1 3axis Algorithm Algorithm
DV1 Difference DV1 vs. Ground DV1 Diff DV1 Diff
m/s cm/s % cm/s %
AutoGPS 2.83 1.122 3.96 0.508 1.794
AutoGPS 8.45 68.33 8.08 0.462 .0054
Auto 10.85 5e4 0000 .0003 0
Auto 11.86 0.178 .0015 .0102 .0008
Semiauto 12.64 0.312 .0024 .00091 .0002
Semiauto 14.76 0.188 .0013 0.000 .0001
Semiauto 15.38 .256 .0016 .0633 .0045
Manual 15.58 10.41 .6682 .0117 .0007
Manual 15.47 0.002 .0001 .0307 .0021
Ref # 10,11
Formation Data from Definitive Navigation Solutions numerically integrate
Radial vs. alongtrack separation over all formation maneuvers
(range of 425490km)
Radial Separation (m)
Groundtrack separation over all formation maneuvers maintained to 3km
Groundtrack
Ref # 10,11
Formation Data from Definitive Navigation Solutions numerically integrate
Alongtrack separation vs. Time over all formation maneuvers
(range of 425490km)
AlongTrack Separation (m)
Semimajor axis of EO1 and LS7 over all formation maneuvers
Ref # 10,11
Formation Data from Definitive Navigation Solutions numerically integrate
Frozen Orbit eccentricity over all formation maneuvers
(range of .001125  0.001250)
eccentricity
Frozen Orbit w vs. ecc. over all formation maneuvers. w range of 90+/ 5 deg.
w
eccentricity
Ref # 10,11
EO1 Summary / Conclusions numerically integrate
Ref # 10,11
Summary / Conclusions numerically integrate
Enables Autonomous StationKeeping,
Formation Flying and Multiple Spacecraft Missions
Ref # 10,11
Ref # 11, 12, 13, 14, 15
NASA Libration Missions numerically integrate
L1 Missions
L2 Missions
(Previous missions marked in blue)
Ref # 1,12
ISEE3 / ICE numerically integrate
Halo Orbit
Lunar Orbit
Earth
L1
SolarRotating Coordinates
Ecliptic Plane Projection
Mission: Investigate SolarTerrestrial relationships, Solar Wind, Magnetosphere, and Cosmic Rays
Launch: Sept., 1978, Comet Encounter Sept., 1985
Lissajous Orbit: L1 Libration Halo Orbit, Ax=~175,000km, Ay = 660,000km, Az~ 120,000km, Class I
Spacecraft: Mass=480Kg, Spin stabilized,
Notable: First Ever Libration Orbiter, First Ever Comet Encounter
Ref # 1,12,13
Farquhar et al [1985] Trajectories and Orbital Maneuvers for the ISEE3/ICE, Comet Mission, JAS 33, No. 3
WIND numerically integrate
SolarRotating Coordinates
Ecliptic Plane Projection
Mission: Investigate SolarTerrestrial Relationships, Solar Wind, Magnetosphere
Launch: Nov., 1994, Multiple Lunar Gravity Assist
Lissajous Orbit: Originally an L1 Lissajous Constrained Orbit, Ax~10,000km, Ay ~ 350,000km, Az~ 250,000km, Class I
Spacecraft: Mass=1254kg, Spin Stabilized,
Notable: First Ever Multiple Gravity Assist Towards L1
Ref # 1,12
MAP numerically integrate
SolarRotating Coordinates
Ecliptic Plane Projection
Mission: Produce an Accurate Fullsky Map of the Cosmic Microwave Background
Temperature Fluctuations (Anisotropy)
Launch: Summer 2001, Gravity Assist Transfer
Lissajous Orbit: L2 Lissajous Constrained Orbit Ay~ 264,000km, Ax~tbd, Ay~ 264,000km,
Class II
Spacecraft: Mass=818kg, Three Axis Stabilized,
Notable: First Gravity Assisted Constrained L2 Lissajous Orbit; Mapearth Vector Remains Between 0.5 and 10° off the Sunearth Vector to Satisfy Communications Requirements While Avoiding Eclipses
Ref # 1,12
JWST numerically integrate
Lunar Orbit
L2
Earth
Sun
SolarRotating Coordinates
Ecliptic Plane Projection
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 Transfer
Lissajous Orbit: L2 large lissajous, Ay~ 294,000km, Ax~800,000km, Az~ 131,000km, Class I or II
Spacecraft: Mass~6000kg, Three Axis Stabilized, ‘Star’ Pointing
Notable: Observations in the Infrared Part of the Spectrum. Important That the Telescope Be Kept at Low Temperatures, ~30K. Large Solar Shade/Solar Sail
Ref # 1,12
A State Space Model numerically integrate
r
j
m
Y
r
r
r
2
r
1
r
Sun
R
y
r
x
i
X
L
O
M
1
Earth
M
1
2
D
D
1
2
D
The linearized equations of motion for a S/C close to the libration point are calculated at the respective libration point.
Ref # 1,12,13,17,20
A State Space Model numerically integrate
ˆ
z
L
ˆ
y
L
ˆ
x
L
Sun
Projection of
Halo orbit
L
1
EarthMoon
Ecliptic Plane
Barycenter
Ref # 1,17,20
Natural Formations numerically integrate
String of Pearls
Others: Identify via Floquet controller (CR3BP)
QuasiPeriodic Relative Orbits (2DTorus)
Nearly Periodic Relative Orbits
Slowly Expanding Nearly Vertical Orbits
NonNatural Formations
Fixed Relative Distance and Orientation
Fixed Relative Distance, Free Orientation
Fixed Relative Distance & Rotation Rate
Aspherical Configurations (Position & Rates)
Reference Motions+ Stable Manifolds
Ref # 15
Ref # 15
Reference Halo Orbit
Deputy S/C
Chief S/C
Ref # 15
Chief S/C Centered View numerically integrate
(RLP Frame)
Natural Formations:QuasiPeriodic Relative Orbits → 2D TorusRef # 15
Ref # 15
Deputy S/C
Origin = Chief S/C
Ref # 15
3 Deputies
Origin = Chief S/C
Ref # 15
10 Revolutions = 1,800 days
Origin = Chief S/C
Ref # 15
Ref # 15
100 Revolutions = 18,000 days
Origin = Chief S/C
Ref # 15
Fixed Relative Distance and Orientation numerically integrate
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
NonNatural FormationsRef # 15
Y [au] numerically integrate
X [au]
Formations Fixed in the Inertial FrameDeputy S/C
Chief S/C
Ref # 15
Deputy S/C
Chief S/C
Ephemeris System = Sun+Earth+Moon Ephemeris + SRP
Ref # 15
Ref # 15
Distance Error Relative to Nominal (cm)
Time (days)
Ref # 15
Maximum Deviation from Nominal (cm)
Formation Distance (meters)
Ref # 15
Ref # 15, 20
Ref # 14,15
Ref # 15
Dynamic Response
Control Acceleration History
LQR
LQR
IFL
IFL
Dynamic Response Modeled in the CR3BP
Nominal State Fixed in the Rotating Frame
Ref # 15
Formation Dynamics
Measured Output Response (Radial Distance)
Desired Response
Actual Response
Scalar Nonlinear Constraint on Control Inputs
Ref # 15
Critically damped output response achieved in all cases numerically integrate
Total DV can vary significantly for these four controllers
Geometric Approach:
Radial inputs only
Output Feedback Linearization (OFL)(Radial Distance Control in the Ephemeris Model)Control Law
Ref # 15
Nominal Sphere
Relative Dynamics as Observed in the Inertial Frame
Ref # 15
Ref # 15
Equations of Motion in the Relative Rotating Frame
Rearrange to isolate the radial and rotational accelerations:
Solve for the Control Inputs:
Ref # 15
Quadratic Growth in Cost w/ Rotation Rate
Linear Growth in Cost w/ Radial Distance
Ref # 15
Az = 0.2×106 km
Az = 0.7×106 km
Az = 1.2×106 km
Ref # 15
Maximum Cost Formation
Minimum Cost Formations
Deputy S/C
Deputy S/C
Deputy S/C
Chief S/C
Chief S/C
Deputy S/C
Deputy S/C
Deputy S/C
Nominal Relative Dynamics in the Synodic Rotating Frame
Ref # 15
Ref # 15
Continuous Control in the Ephemeris Model: numerically integrate
NonNatural 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 NonNatural Formations
Targeter Approach
Small relative separations → Good accuracy
Large relative separations → Require nearly continuous control
Extremely Small DV’s (105 m/sec)
Natural Formations
Nearly periodic & quasiperiodic formations in the RLP frame
Floquet controller: numerically ID solutions + stable manifolds
ConclusionsRef # 15
A simple formation about the SunEarth L1 numerically integrate
Using CRTB based on L1 dynamics
Errors associated with perturbations
A more complex Fizeautype interferometer fizeau interferometer.
Composed of 30 small spacecraft at L2
Formation maintenance, rotation, and slewing
Some Examples from SimulationsRef # 16, 17, 20
A State Space Model numerically integrate
r
j
m
Y
r
r
r
2
r
1
r
R
y
r
Sun
x
i
X
L
O
M
1
M
1
2
Earth
D
D
1
2
D
CRTB problem rotating frame
Ref # 16, 17, 20
Periodic Reference Orbit numerically integrate
A = Amplitude
w = frequency
f = Phase angle
Ref # 16, 17, 20
Centralized LQR Design numerically integrate
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
Centralized LQR Design numerically integrate
Sample LQR Controlled Orbit
DV budget with
Different Rj Matrices
Ref # 16, 17, 20
Disturbance Accommodation Model numerically integrate
Unperturbed
wx,y,z = 4.26106e7 rad/s
Perturbed
wx = 1.5979e7 rad/s
2.6632e6 rad/s
wy = 2.6632e6 rad/s
wz = 2.6632e6 rad/s
Ref # 16, 17, 20
Disturbance Accommodation Model numerically integrate
With Disturbance Accommodation
Without Disturbance Accommodation
Disturbance accommodation state is out of phase with state error, absorbing unnecessary control effort
State Error
Control Effort
Ref # 16, 17, 20
Motion of Formation Flyer With Respect to numerically integrate
Reference Spacecraft, in Local (S/C1) Coordinates
Reference Spacecraft Location
Reference Spacecraft Location
Formation Flyer Spacecraft Motion
Reference Spacecraft Location
Ref # 16, 17, 20
D numerically integrate V Maintenance in Libration Orbit Formation
Ref # 16, 17, 20
Stellar Imager Concept numerically integrate
(Using conceptual distances and control requirements
to analyze formation possibilities)
Ref # 16, 17, 20
Stellar Imager numerically integrate
SI Slewing Geometry
Formation DV Cost per
slewing maneuver
DV
DV
DV
Drones at beginning
Ref # 16
Stellar Imager Mission Study Example Requirements numerically integrate
attitude adjustments of up to 180deg
~ 3 nanometers radially from Hub
~ 0.5 millimeters along the spherical surface
Coarse  RF ranging, star trackers, and thrusters ~ centimeters
Ref # 16
State Space Controller Development numerically integrate
Ref # 16
State Space Controller Development, LQR Design numerically integrate
State Dynamics and Error
Performance Index
to Minimize
Control and Closed Loop Dynamics
Algebraic Riccati Eq.
time invariant system
Ref # 16
State Space Controller Development, LQR Design numerically integrate
W Is Weight of State Error
B Maps Control Input From
Control Space to State Space
V Is Weight of Control
W
V
Ref # 16
Simplified extended Kalman Filter numerically integrate
zeromean white Gaussian process and measurement noise
where w is the random process noise
and the covariances of process and measurement noise are
Ref # 16
Simulation Matrix Initial Values numerically integrate
Ref # 16
Results – Libration Orbit Maintenance numerically integrate
Ref # 16
Results – Libration Orbit Maintenance numerically integrate
Ref # 16
Results – Formation Slewing numerically integrate
Formation Slewing:
900 simulation shown
Purple  Begin
Red  End
Ref # 16
Results – Formation Slewing numerically integrate
DRONE
HUB
Represents both 0.5 and 4 km focal lengths
Ref # 16
Results – Formation Slewing numerically integrate
Hub estimation 0.5 km
separation / 90 deg slew
Drone estimation 0.5 km
separation / 90 deg slew
Ref # 16
Results – Formation Slewing numerically integrate
Formation Slewing Average DVs (12 simulations)
Focal Slew Hub Drone 2 Drone 31
Length (km) Angle (deg) (m/s) (m/s) (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
Formation Slewing Average Propellant Mass
Focal Slew Hub Drone 2 Drone 31
Length (km) Angle (deg) massprop massprop massprop
(g) (g) (g)
0.5 30 6.0018 0.8431 0.8468
0.5 90 6.3662 0.9577 0.9773
4 30 7.1135 1.1406 1.1534
4 90 10.4112 2.2331 2.2357
Ref # 16
Results – Formation Slewing numerically integrate
Formation Slewing Average DVs (without noise)
Focal Slew Hub Drone 2 Drone 31
Length (km) Angle (deg) (m/s) (m/s) (m/s)
0.5 30 0.0504 0.0853 0.0998
0.5 90 0.1581 0.2150 0.2315
4 30 0.4420 0.5896 0.6441
4 90 1.3945 1.9446 1.9469
Formation Slewing Average Propellant Mass (without noise)
Focal Slew Hub Drone 2 Drone 31
Length (km) Angle (deg) massprop massprop massprop
(g) (g) (g)
0.5 30 0.2826 0.0870 0.1017
0.5 90 0.8864 0.2192 0.2360
4 30 2.4781 0.6010 0.6566
4 90 7.8182 1.9822 1.9846
Ref # 16
Results – Formation Reorientation numerically integrate
Ref # 16
Results – Formation Reorientation numerically integrate
DRONE
HUB
Ref # 16
Results – Formation Reorientation numerically integrate
Formation Reorientation AverageDVs
Focal Slew Hub Drone 2 Drone 31
Length (km) Angle (deg) (m/s) (m/s) (m/s)
0.5 90 1.0126 0.8421 0.8095
4 90 1.0133 0.8496 0.8190
Formation Reorientation AveragePropellant Mass
Focal Slew Hub Drone 2 Drone 31
Length (km) Angle (deg) massprop massprop massprop
(g) (g) (g)
0.5 90 5.6771 0.8584 0.8252
4 90 5.6811 0.8661 0.8349
Ref # 16
Results – Formation Reorientation numerically integrate
Formation Reorientation Average DVs (without noise)
Focal Slew Hub Drone 2 Drone 31
Length (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 AveragePropellant Mass ( without noise)
Focal Slew Hub Drone 2 Drone 31
Length (km) Angle (deg) massprop massprop massprop
(g) (g) (g)
0.5 90 0.2287 0.1623 0.1525
4 90 0.2287 0.1623 0.1524
Ref # 16
Summary numerically integrate
(using example requirements and constraints)
Ref # 16
Summary numerically integrate
(using example requirements and constraints)
Ref # 16
DRO Mission Metrics numerically integrate
Ref # 1,18
Distant Retrograde Orbit (DRO) numerically integrate
Periodic Orbit
Direction of Orbit is Retrograde
L1
L2
Earth
SunEarth Line
Lunar Orbit
Solar Rotating Coordinate System ( EarthSun line is fixed)
Ref # 1,18
Earth Distant Retrograde Orbit (DRO) Orbit numerically integrate
Ref # 1,18
3D view numerically integrate
SIRA spacecraft
DRO Formation Sphere
XY view
YZ view
Ref # 1,18
DRO Formation Control Analysis numerically integrate
Ref # 1,18
DRO Formation Control Analysis numerically integrate
Ref # 1,18
DRO Formation Control Analysis numerically integrate
Ref # 1,18
Formation Control Analysis numerically integrate
How much DV to initialize, maintain, and resize?
Examples:
Initialize & maintain 2 yr:
= 33 m/s
Initialize, Maintain 2yr,
& four resizes:
= 36 m/s
Ref # 1,18
DRO Formation Control Analysis numerically integrate
Impulsive Maneuver of 16th s/c
Radial Distance from Center
Ref # 1,18
General Theory of Decentralized Control numerically integrate
MANY NODES IN A NETWORK CAN COOPERATE
TO BEHAVE AS SINGLE VIRTUAL PLATFORM:
NODE 2
NODE 1
NODE 3
NODE 5
NODE 4
Ref # 19
NASA Web Sites, numerically integrate www.nasa.gov, Use the find it @ nasa search input for SEC, Origin, ESE, etc.
Earth Science Mission Operations Project, Afternoon Constellation Operations Coordination Plan, GSFC, A. Kelly May 2004
Fundamental of Astrodynamics, Bate, Muller, and White, Dover, Publications, 1971
Mission Geometry; Orbit and Constellation design and Management, Wertz, Microcosm Press, Kluwer Academic Publishers, 2001
An Introduction to the Mathematics and Methods of Astrodynamics, Battin
Fundamentals of Astrodynamics and Applications, Vallado, Kluwer Academic Publishers, 2001
Orbital Mechanics, Chapter 8, Prussing and Conway
Theory of Orbits: The Restricted Problem of Three Bodies V. Szebehely.. Academic Press, New York, 1967
Automated Rendezvous and Docking of Spacecraft, Wigbert Fehse, Cambridge Aerospace Series, Cambridge University Press, 2003
“A Universal 3D 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 1012, Boston, MA.
“Results of NASA’s First Autonomous Formation Flying Experiment: Earth Observing1 (EO1)”, Folta, AIAA/AAS Astrodynamics Specialist Conference, Monterey, CA, 2002
“Libration Orbit Mission Design: Applications Of Numerical And Dynamical Methods”, Folta, Libration Point Orbits and Applications, June 1014, 2002, Girona, Spain
“The Control and Use of LibrationPoint Satellites”. R. F. Farquhar. NASA Technical Report TR R346, National Aeronautics and Space Administration, Washington, DC, September, 1970
“Stationkeeping at the Collinear Equilibrium Points of the EarthMoon System”. D. A. Hoffman. JSC26189, NASA Johnson Space Center, Houston, TX, September 1993.
“Formation Flight near L1 and L2 in the SunEarth/Moon Ephemeris System including solar radiation pressure”, Marchand and Howell, paper AAS 03596
Halo Orbit Determination and Control, B. Wie, Section 4.7 in Space Vehicle Dynamics and Control. AIAA Education Series, American Institute of Aeronautics and Astronautics, Reston, VA, 1998.
“Formation Flying Satellite Control Around the L2 SunEarth Libration Point”, Hamilton, Folta and Carpenter, Monterey, CA, AIAA/AAS, 2002
“Formation Flying with Decentralized Control in Libration Point Orbits”, Folta and Carpenter, International Space Symposium Biarritz, France, 2000
SIRA Workshop, http://lep694.gsfc.nasa.gov/sira_workshop
“Computation and Transmission Requirements for a Decentralized LinearQuadraticGaussian Control Problem,” J. L. Speyer, IEEE Transactions on Automatic Control, Vol. AC24, No. 2, April 1979, pp. 266269.
References, etc.References, etc. numerically integrate
Questions on formation flying? Feel free to contact:
kathleen howell [email protected]
DST/Numerical Comparisons numerically integrate
Numerical SystemsDynamical Systems
Libration Point Trajectory Generation Process numerically integrate
Generate Lissajous of Interest
And Eigenvalues/Eigenvectors
For Half Manifold of Interest
a Differential Corrector Step
To Achieve Mission Constraints.
Output / Intermediate
Data
Universe/User Control and
Patch Pts
Patch Points
and Lissajous
Lissajous
.
Fixed Points and Stable
and Unstable Manifold
Approximations
Monodromy
.
Manifold
1D Manifold
Transfer
Transfer Trajectory to
Earth Access Region
MAP Mission Design: DST Perspective numerically integrate
Starting Point
JWST DST Perspective numerically integrate
Starting Point
Q numerically integrate
y
Spacecraft
a
r
E
q
Center of Orbit
x
P
ae
Focus
a
Two – Body Motion


Ref # 37
Two – Body Motion numerically integrate





Ref # 37
Coordinate system transformation numerically integrate
Euler Angle Rotations
Y
Y’
X’
Suppose we rotate the xy plane about the zaxis by an angle a and call the new coordinates x’,y’,z’
x’ = x cos a + y sin a
y’ = x sin a + y cos a
z’ = z
a
X
Z
x
y
z
x’
y’
z’
x
y
z
cos a sin a 0
sin a cos a 0
0 0 1
=
Az’(a)
=
Rotate about Z
Rotate about Y
Rotate about X
x
y
z
x’
y’
z’
cos b 0sinb
0 1 0
sinb 0 cosb
=
x’
y’
z’
x
y
z
1 0 0
0 cos g sin g
0 sin g cos g
=
Ref # 37
Coordinate system transformation numerically integrate
Orbital to inertial coordinates
Inertial to/from Orbit plane
r = r(x,y,z) r = r(P,Q,W)




In orbit plane system, position r= (rcosf)P + (rsinf)Q + (0)W where P,Q,W are unit vectors

Px Qx Wx
Py Qy Wy
Pz Qz Wz
x
y
z
rcosf
rsinf
0
=
x
y
z
cos w sin w 0
sin w cos w 0
0 0 1
1 0 0
0 cos isin i
0 sin i cos i
cos W sin W 0
sin W cos W 0
0 0 1
rcosf
rsinf
0
=
x
y
z
rcosf
rsinf
0
cos W cos w  sin W sin w cos i cos W sin w  sin W cos w cos i sin W sin i
sin W cos w  cos W sin w cos i sin W sin w  cos W cos w cos i cos W sin i
sin W sin i cos W sin i cos i
=
For Velocity transformation, vx = (m/p)1/2 [sin q] and vy = (m/p)1/2 [e + cos q]
Ref # 37
Principles Behind Decentralized Control numerically integrate
Ref # 19
The LQG Decentralized Controller Overview numerically integrate
CONTROL VECTOR
TRANSMISSION VECTORS
LOCAL SENSOR
TRANSMISSION VECTOR
z 1
FROM NETWORK
GLOBALLY
OPTIMAL CONTROL
CONTROL VECTORS
TO NETWORK
& LOCAL PLANT
z 1
LOCALLY
OPTIMAL FILTER
z 1
GLOBAL DATA
RECONSTRUCTION
TO NETWORK
z 1
Ref # 19