Advanced Space 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
Advanced Space Propulsion Laboratory
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?
Advanced Space Propulsion Laboratory
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
Advanced Space Propulsion Laboratory
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 #1: Deployable structure
Solution #2: Separated Interferometer
d
d effective Combiner
Collector
Collector
Interferometry Basics
Advanced Space Propulsion Laboratory
• 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 y (x2,y2)
d
x
Single amplitude In Fourier plane v (u1,v1)
u = x2-x1/λ u
(x1,y1)
Inverting Spatial Frequency
Spatial Amplitude
v = y2-y1/λ
Image
To perform the inversion we need to fill the u,v plane
Physical Plane
Fourier Plane
Finite apertures can fill-in Holes in u-v plane
3 Spacecraft Consider single aperture As array of sub-apertures 5 Spacecraft
(xi, yi)
7 Spacecraft
9 Spacecraft
11 Spacecraft
•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.
Advanced Space Propulsion Laboratory
Interferometry for Formations
All separations (u,v points) less Than d are covered by a single aperture d
(xi, yi) (xj, yj)
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
Advanced Space Propulsion Laboratory
Formation Flying Introduction
Optimal imaging configurations yield non-optimal orbital trajectories Rigid Formation
Dynamic Formation Family of inertial orbits
Non-inertial Orbit Inertial Orbit Non-inertial Orbit Time-varying position About center satellite
• Requires constant thrust • Good imaging properties
• Thrust only for error correction • Complicated imaging
Advanced Space Propulsion Laboratory
Propulsion Requirements Hill’s Equations for Formation Fx = &x& − 3Ω 2 x − 2Ωy& m Fy = &y& + 2Ωx& m Fz = &z& + Ω 2 z m
Figure reprinted from Kong, E.M., “Optimal Trajectories and Orbit Design for Separated Spacecraft Interferometry,” Master’s Thesis, MIT Dept. of Aeronautics and Astronautics, November, 1998.
Ω = angular velocity (for GEO Ω =7.3x10-5 rad/sec)
For rigid formation: &x& = x& = &y& = y& = ... = 0 Fx = −3Ω 2 xm Fz = Ω 2 zm
m = 100 kg x, z ~ 10 m
Fx ~ 16 µN Fz ~ 6 µN
Coulomb Control Forces
Advanced Space Propulsion Laboratory
• 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? Maybe! YES!
2 1 Spacecraft 1 at Voltage Vsc1
d
Spacecraft 2 at Voltage Vsc2
F1,2 = 4πε o
10
Coulomb Force (N)
If the plasma Debye length is larger than spacecraft separation, Coulomb forces could be used
−d r1r2Vsc1Vsc 2 exp λ d2 d
-3
10
-4
10
-5
10
-6
Vsc1 = Vsc2 Spacecraft radius = 1 m GEO plasma conditions 6 kV 8 kV 10 kV
10
20
30 40 50 Spacecraft Separation (m)
60
70
Advanced Space Propulsion Laboratory
Spacecraft Charging Isc = Ie + Ii + Iph
Plasma ecurrent
For equilibrium, Mother Nature adjusts spacecraft voltage such that net current is zero.
Plasma H+ current
1 k T 2 eV I e = Asc ene B e exp sc 2πme k BTe
Spacecraft
Photoelectron current e-
Icontrol
1 k T 2 − eVsc I i = − Asc eni B i exp 2πmi k BTi
− eVsc I ph = − Asc eα w I pe exp k BT pe
We can change the spacecraft voltage by creating a current imbalance dVsc I I e + I i + I ph + Icontrol = = =0 dt C 4πε o r
0
Spacecraft Charge Control
Advanced Space Propulsion Laboratory
• Electron emission drives Vsc positive • Ion emission drives Vsc negative • Spacecraft potential control is naturally stable
+
+
Vcontrol
V Vcontrol Vcontrol
+
+
When Vsc = Vcontrol the emission Current is returned (Icontrol = 0)
+ +
+
+
+ Vspacecraft
Vcontrol
Vspacecraft Vspacecraft
+
x
Vplasma (God’s ground)
1-m-radius spacecraft charging analysis for average GEO plasma environment
P = 0.1 W P=1W P = 10 W 3000
Spacecraft Potential (Volts)
2500 2000 1500 1000 500 0 0.00001
0.0001
0.001
Time (sec)
I control =
I control dV sc I = = dt C
0.01
0.1
P Vcontrol
1 1 2 2 k T eV k T + 4 π r 2 en e B e exp sc − en i B i − e α w I ph 2π m i 2π m e k B Te 4 π ε 0r
Coulomb vs. Electric Propulsion
Advanced Space Propulsion Laboratory
Coulomb Control
Thruster Control
1
1
FThruster
2
FThruster
FCoulomb
Comparison of Coulomb Control for 1-m-radius Spacecraft in average GEO plasma with FEEP Thruster technology (Isp = 10,000 sec, η = 0.65)
FCoulomb
FCoulomb =
2 2 πε o r 2 PTotal 2 2 control
d I
FThruster =
ηPTotal 2 gI sp
FCoulomb/FThruster at equivalent power
d
1 mWatt 10 mWatts 100 mWatts 1000 mWatts
7
10
6
10
5
10
4
10
3
10
10
20
30
40
50
Spacecraft Separation (m)
60
70
Advanced Space Propulsion Laboratory
Mission design parameters for two-spacecraft flying in 20-m formation (located on Hill’s z-axis) MEANS OF CONTROL
COULOMB CONTROL
MICROPPT
COLLOID THRUSTER
FEEP
1 x 107
500
1000
10000
0.65
0.026
0.65
0.65
MASS OF FUEL FOR 10 YEARS (kg)
0.00003 (using H2 for Ion source)
0.089
0.045
0.004
INPUT POWER (W)
0.031
0.261
0.021
0.209
MASS/POWER RATIO (kg/W)
0.22
0.37
0.216
0.1125
INERT MASS (kg)
0.0068
0.097
0.005
0.024
TOTAL PROPULSION SYSTEM MASS (kg)
0.00683
0.186
0.050
0.028
SPECIFIC IMPULSE (sec) EFFICIENCY
I sp =
F m& g
Coulomb Orbit Dynamics
Advanced Space Propulsion Laboratory
Do formations exist for forces acting only along position vectors? • • • •
3-spacecraft formations considered 3 canonical orientations Hill’s equations for relative motion GEO orbit with 10-m separation
Figure reprinted from Kong, E.M., “Optimal Trajectories and Orbit Design for Separated Spacecraft Interferometry,” Master’s Thesis, MIT Dept. of Aeronautics and Astronautics, November, 1998.
z
z x
y
y Along-track “leader-follower”
z
x
x
y Zenith-nadir “Coulomb tether”
Z-axis stack
3-Spacecraft Orbital Analysis Advanced Space Propulsion Laboratory
Equilibrium solutions to Hill’s equations Vsc = Parameter Vscr is like equivalent charge:
1 qsc 4πε o r
Vsc r =
z
z
1
z
x
x 0
qsc 4πε o
2
2
y Along-track “leader-follower”
y
1
x 0 1
Zenith-nadir “Coulomb tether”
0 y
2 Z-axis stack
4 collector + 1 combiner Imaging configuration
Spacecraft 1 10 m Spacecraft 4
z y Spacecraft 0 x
10 m
Spacecraft 2
Spacecraft 3
5-Spacecraft Formation Advanced Space Propulsion Laboratory
Solution Family 1
1 2
• Special case of 3-spacecraft z stack • Vehicle 2 and 4 remain neutral
0 4 3
5-Spacecraft Formation Advanced Space Propulsion Laboratory
Solution Family 2 • All 5 Spacecraft charged • Minimum Vscr identified 1 2 0 4 3
Spacecraft 1 & 3 Spacecraft 0
Advanced Space Propulsion Laboratory
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