Control ofspacecraftswarmsoct01

Advanced Space P Propulsion Laboratory
Control of Spacecraft Swarms Using
Coulomb Forces
Lyon B. King
Gordon G. Parker
Jer-Hong Chong
Satwik Deshmukh
Department of Mechanical Engineering
This research made possible through funding from the
NASA Institute for Advanced Concepts

Motivation: Coulomb Clusters
Laser-cooled trapped ion research at NIST
“How to build a tractor beam
Without gravitons”
• Ions in 1/r2 confining potential
form stable crystal formations
• What would charged spacecraft
do in a gravity potential?

Presentation Overview
• Introduction to formation flying
• Space-based imaging and interferometry
• Formation propulsion requirements
• Spacecraft charging as control force
• Coulomb force metrics
• Coulomb formation orbital dynamics

Space-based Imaging Concepts
Space-based imaging problem:
• Image resolution limited by size of aperture: θ=λ/d
…but…
• Spacecraft size limited by launch vehicle fairing (~ 4m)
Solution #2:
Separated Interferometer
d effective
Combiner
Collector Collector
Solution #1:
Deployable structure
d

Interferometry Basics
• Spatial frequencies in the image are given by u,v points
• Each unique physical separation yields amplitude at one u,v point
Two Spacecraft
In physical plane
(x1,y1)
y
(x2,y2) d
Single amplitude
In Fourier plane
x (u1,v1)
u
v
u = x2-x1/λ
v = y2-y1/λ
Inverting Spatial Frequency Spatial Amplitude Image
To perform the inversion we need to fill the u,v plane

3 Spacecraft
5 Spacecraft
7 Spacecraft
9 Spacecraft
11 Spacecraft
Physical
Plane
Fourier
Plane Finite apertures can fill-in
Holes in u-v plane
Consider single aperture
As array of sub-apertures
(xi, yi)
•Kong, E.M., “Optimal Trajectories and Orbit Design for Separated
Spacecraft Interferometry,” Master’s Thesis, MIT Dept. of Aeronautics
and Astronautics, November, 1998.
•Cornwell, T.J., “A Novel Principle for Optimization of the Instantaneous
Fourier Plane Coverage of Correlation Arrays,” IEEE Trans. On Antennas
and Propagation, Vol. 36, No. 8, 1165-1167.

Interferometry for Formations
All separations (u,v points) less
Than d are covered by a single aperture
(xi, yi)
(xj, yj)
d
Separations (u,v points) greater than d
Must come from separated spacecraft
> d
Implication:
• To provide seamless u,v coverage spacecraft must fly
within close proximity (~ d) of each other

Formation Flying Introduction
Optimal imaging configurations yield non-optimal orbital trajectories
Rigid Formation
Non-inertial Orbit
Inertial Orbit
Non-inertial Orbit
• Requires constant thrust
• Good imaging properties
Dynamic Formation
Family of inertial orbits
Time-varying position
About center satellite
• Thrust only for error correction
• Complicated imaging

Propulsion Requirements
Hill’s Equations for Formation
F
x
F
F
y
= − Ω − Ω
3 2
&& &
= + Ω
&& &
For rigid formation: &x& = x& = &y& = y& = ... = 0
3 2
F x
xm
z
= − Ω Fx ~ 16 μN
2
F = Ω
zm
Fz ~ 6 μN
m = 100 kg
x, z ~ 10 m
z z
m
y x
m
x x y
m
z
2
2
2
= &&
+Ω
Figure reprinted from Kong, E.M., “Optimal Trajectories and Orbit Design for Separated Ω = angular velocity (for GEO Ω =7.3x10-5 rad/sec)

Coulomb Control Forces
• Engineering throttleable thrusters for 10 μN is tough
• Current candidates (FEEP, Colloid) exhaust contaminants
• Collisions are of paramount concern
Is there a better way to control the formation? M YaEySb!e!
10-3
10-4
10-5
10-6
Coulomb Force (N)
10 20 30 40 50 60 70
Spacecraft Separation (m)
6 kV
8 kV
10 kV
If the plasma Debye length is larger
than spacecraft separation, Coulomb
forces could be used
1
2
d
Spacecraft 1 at
Voltage Vsc1
Spacecraft 2 at
Voltage Vsc2
Vsc1 = Vsc2
Spacecraft radius = 1 m
GEO plasma conditions

 
 − =
 
d
F r r V V
sc sc
o
d
1 2 1 2
d
λ
4πε exp 2
1,2

Spacecraft Charging
Spacecraft
Plasma e-current
Plasma H+
current
Photoelectron
current e-
Isc = Ie + Ii + Iph





1
I A en k T exp



=
sc
e sc e k T
B e
B e
e
eV
m
2
2
π


−


1
I A en k T exp



= −
sc
i sc i k T
B i
B i
i
eV
m
2
2
π

 

 − = −
I A eα I exp eV
 
ph sc w pe k T

sc
B pe
For equilibrium, Mother
Nature adjusts spacecraft
voltage such that net
current is zero.
Icontrol
We can change the spacecraft voltage by creating a current imbalance
0
I I I
sc e i ph
πε
4
=
+ +
I
= =
r
C
dV
dt
o
+ Icontrol 0

Spacecraft Charge Control
• Electron emission drives Vsc positive
• Ion emission drives Vsc negative
• Spacecraft potential control is naturally stable
+ +
+
+
Vcontrol
+ +
Vspacecraft
V
x
Vplasma (God’s ground)
Vspacecraft
Vcontrol
+
+
+
Vcontrol
Vspacecraft
When Vsc = Vcontrol the emission
Current is returned (Icontrol = 0)
Vcontrol
+


1-m-radius spacecraft charging analysis
for average GEO plasma environment
3000
2500
2000
1500
1000
500
I = P
control V
en k T eV
en k T
α
e e I
4 π ε r
2
exp
2
I 4 π r
I
C
dV
dt
0
1
2
1
2
2
control
sc







− 



− 





+
= =
w ph
i
B i
i
sc
B e
B e
e
m
k T
m
π π
0
0.00001 0.0001 0.001 0.01 0.1
Time (sec)
Spacecraft Potential
(Volts)
P = 0.1 W
P = 1 W
P = 10 W
control

1
2
2 2
F = πε r P
o Total
2 2
control
Coulomb d I
F P
Total
sp
=η
Thruster 2
gI
d
FCoulomb
FCoulomb
1
2
FThruster
FThruster
Coulomb
Control
Thruster
Control
Coulomb vs. Electric Propulsion
107
106
105
104
103
FCoulomb/FThruster at equivalent power
Comparison of Coulomb Control for 1-m-radius
Spacecraft in average GEO plasma with FEEP
Thruster technology (Isp = 10,000 sec, η = 0.65)
1 mWatt
10 mWatts
100 mWatts
1000 mWatts
10 20 30 40 50 60 70
Spacecraft Separation (m)

Mission design parameters for two-spacecraft flying in
20-m formation (located on Hill’s z-axis)
COLLOID FEEP
THRUSTER
COULOMB MICROPPT
CONTROL
MEANS OF
CONTROL
SPECIFIC 1 x 107 500 1000 10000
IMPULSE (sec)
I F sp &
mg
=
EFFICIENCY 0.65 0.026 0.65 0.65
0.00003 0.089 0.045 0.004
(using H2 for
Ion source)
MASS OF FUEL FOR
10 YEARS (kg)
INPUT POWER (W) 0.031 0.261 0.021 0.209
MASS/POWER 0.22 0.37 0.216 0.1125
RATIO (kg/W)
INERT MASS (kg) 0.0068 0.097 0.005 0.024
TOTAL PROPULSION 0.00683 0.186 0.050 0.028
SYSTEM MASS (kg)

Coulomb Orbit Dynamics
Do formations exist for forces acting only along position vectors?
Figure reprinted from Kong, E.M., “Optimal Trajectories and Orbit Design for Separated
• 3-spacecraft formations considered
• 3 canonical orientations
• Hill’s equations for relative motion
• GEO orbit with 10-m separation
x
z
y
x
z
y
x
z
y
Along-track
“leader-follower”
Zenith-nadir
“Coulomb tether”
Z-axis stack

3-Spacecraft Orbital Analysis
Equilibrium solutions to Hill’s equations
V 1
q
πε
sc
o
sc
sc
o
sc
4
=
V r q
r
πε
4
=
x
z
y
x
z
y
x
z
y
Along-track
“leader-follower”
Zenith-nadir
“Coulomb tether”
Z-axis stack
0
2
1
0
1
2
0
1
2
Parameter Vscr is like equivalent charge:

x
y
z
10 m
10 m
Spacecraft 1
Spacecraft 3
Spacecraft 2
Spacecraft 4
Spacecraft 0
4 collector + 1 combiner
Imaging configuration

5-Spacecraft Formation
1
2
3
4
Solution Family 1
0 • Special case of 3-spacecraft z stack
• Vehicle 2 and 4 remain neutral

1
2
3
4
0
5-Spacecraft Formation
Solution Family 2
Spacecraft 1 & 3
Spacecraft 0 • All 5 Spacecraft charged
• Minimum Vscr identified

Phase I Summary
Conclusions
• Coulomb forces comparable with best thrusters
• Continuous force dither/variation is possible
• Required charge control demonstrated as early as 1979 (SCATHA)
• Rich family of orbital solutions possible
• Particularly suited to Fizeau interferometry (visible GEO imager?)
• Coulomb control works best where thrusters work worst => synergistic control
• Coulomb control can help with collision avoidance
• Even if Coulomb is not used for control……
natural charging will be significant perturbation that must be addressed!
On-going tasks
• Examine formations for stability
• Develop dynamic simulation
• Formulate control laws
• Search for more complicated formation solutions
• Perform vehicle sizing analysis for canonical mission

Control ofspacecraftswarmsoct01

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