Transcript Feasibility of Demonstrating Pulsed Plasma Thrusters on

Feasibility of Demonstrating
PPT’s on FalconSAT-3
C1C Andrea Johnson
United States
Air Force Academy
Outline


Problems encountered with PPT’s
Methods of demonstrating use





Spiral Transfer
Attitude
Model
Experimental Results
Recommendations
Problems Encountered

Low Thrust





160e-6 N maximum thrust
15e-6 second pulse, 2 Hz => 4.8e-9 N average
thrust
Updated data indicates possibly higher average
thrust (50 μN-s)
Power requirements
Inaccuracy of original model


Uncoupled equations of motion
Inaccurate disturbance torque models
Methods of Demonstrating

Spiral Transfer


One PPT yields 1.6 cm change in
semimajor axis with no disturbance torques
No GPS receiver
Methods of Demonstrating Cont.

Attitude Control

Z-axis only possibility for control because
of small moment of inertia (1.31 versus
67.4 kg-m2)
Model





Assumptions
Equations of motion
PPT modeling
Disturbance torques
Validation
Assumptions

Simplified satellite model




Small center of pressure - center of mass
offset
No products of inertia
Constant, known PPT decay rate
Negligible orbital perturbations
Assumptions Cont.
Body Mass:
35.5 kg
Boom Mass (without tip mass):
3.15 kg
Tip Mass:
7.45 kg
Total Mass:
46.1 kg
Inertia Tensor (Stowed Boom):
Inertia Tensor (Deployed Boom):
0 
3.64 0
 0
3.69 0 

 0
0 1.31
kg-m2
0
0 
67.40
 0
67.45 0 

 0
0
1.31
kg-m2
Coefficient of Drag (Cd):
2.6
Spacecraft Dipole:
0.05 A-m2
Orbit:
Altitude = 560 km
Semimajor axis = 6938.137 km
Inclination = 35.4o
Eccentricity = 0
Right Ascension = 0o
Equations of Motion
I11  ( I 3  I 2 )23  T1
I 2 2  ( I1  I 3 )13  T2
I 3 3  ( I 2  I1 )12  T3
PPT Modeling
Actual
160
μN
t
15 usec
4.8 nNs
15 usec
t
4.8
nN
1 sec
Simulation
Disturbance Torques




Gravity Gradient
Magnetic
Drag
Solar Pressure
Gravity Gradient
N
GG
x
 3o ( I z  I y )Tbylo23Tbylo33
GG
y
 3 o ( I x  I z )Tbylo 33Tbylo13
N
N
GG
z
2
2
 3 o ( I y  I x )Tbylo13Tbylo 23
2
Magnetic


13th degree, 13th order
IGRF 10th generation model with secular
terms up to 8th degree and 8th order
Magnetic Cont.
V
a
Br  
  
r n1  r 
k
n 2
m 0
k
1 V
a
B  
   
r 
n 1  r 
1 V
1
B  

r sin  
sin 
n  1 g n,m cosm  h n,m sin m P n,m  
n
n 2 n
 g
cosm  h
m 0
a
 

n 1  r 
k
n,m
n2 n
 m g
m 0
n,m
n,m
P n,m  
sin m



sin m  h n,m cosm P n,m  
Magnetic cont.
z
r
θ
φ
y
x
Bx  Br sin   B cos cos  B sin 
By  Br sin   B cos sin   B cos
Bz  Br cos  B sin 
Magnetic Cont.

ECF to ECI coordinate frame conversion




Precession
Nutation
Sidereal time
Polar motion
Drag
Fdrag , plane
Fdrag ,cylinder


1
  C d V 2 A Nˆ  Vˆ Vˆ
2


2
1
2
ˆ
ˆ
  C d V DL 1  Z  V Vˆ
2
n
N drag   Ri  Fi
i 1
Solar Pressure

1  

Fsolar , plane   Psun A cos sun 1  C s Sˆ  2 C s cos sun  Cd  Nˆ  
3  



1  

Pearth A cos earth 1  C s Eˆ  2 C s cos earth  Cd  Nˆ 
3  





 
 1  
 4
Fsolar ,cylinder   Psun  sin sun 1  C s   C d  ASˆ    C s sin sun  C d  cos sun A1 Zˆ  
6
 3  6 
 3






 
 1  
 4
Pearth  sin earth 1  C s   C d  AEˆ    C s sin earth  C d  cos earth A1 Zˆ 
6
 3  6 
 3



Validation




Integrator: Attitude and orbital energy
and momentum should be constant
Gravity gradient: Should match C
program data
Magnetic field: Should match C
program data
Drag and solar pressure validated using
hand calculations
Integrator

Energy and momentum constant if no
external torques

Attitude
  I

h
Orbit
V2 


2 R

1 T
 I
2
h  R V
Normalized error
 o   t 
o
ho  ht 
ho
Integrator: Attitude
Energy
Momentum
Maximum error: 3e-14
Maximum error: 1.5e-14
Integrator: Orbit
Energy
Momentum
Maximum error: 2.5e-14
Maximum error: 7.5e-15
Gravity Gradient Validation
U o Sat: A ttitu d e Lo g File
R oll A ngle
1 .5
P i tc h A n g l e
Y aw A ngle
1 .0
0 .5
0 .0
-0 .5
-1 .0
-1 .5
1 J an
J an 70
2 Fri
Ti m e
Gravity Gradient Validation
Cont.
Magnetic Field Validation

Magnetic field in ECF matched C
program numerical output



8th degree, 8th order
With secular terms
ECF to ECI conversion output matched
C program
Estimation Theory

Kalman filter




Truncate results
Statistical mean smoother
Batch estimator
Data used by filters comes from attitude
determination Kalman filter
Estimation Theory Cont.
I z z  3o I y  I x Tbylo1 3Tbylo2 3  M x By  M y Bx 
2
 I y  I x x y  N zdrag  N zsolar  NoPPT
 gk
z

ˆz  I z  z  I y  I x  3o 2Tbylo1 3Tbylo2 3   x y

z  M x By  M y Bx   N zdrag  N zsolar  NoPPT
 gk
z
N zPPT  NoPPT
 gk
z
Estimation Theory Cont.
 N oPPT

z


X   Mx 
 My 


 z
H 
 N

z
M x
H  1 By
z 

M y 
 Bx

Batch Filter Algorithm

H  1 By
 Bx

T
H
T
k Hk
H WH  
R
k
T
ˆk  z k 
H
T
k z
H W O  C   
R
k
Batch Filter Algorithm Cont.

X  H WH
T

1
H W O  C 
T
If X   (user defined), then exit the loop. If not,
X  X  X
Experimental Results
No Noise
Percent Error Kalman
w/o Smoothing
Actual
PPT torque
0.0000
Dipole (x)
0.0000
Dipole (y)
0.0500
Percent Error Kalman w/
Smoothing
Percent Error
Batch
N/A
N/A
N/A
N/A
N/A
N/A
3.4001E-10
8.6001E-10
0.0000E+00
Experimental Results Cont.
0.3E-6 on B field Actual
PPT torque
0.0000
Dipole (x)
0.0000
Dipole (y)
0.0500
Percent Error Kalman
w/o Smoothing
Percent Error Kalman
w/ Smoothing
Percent Error Batch
N/A
N/A
N/A
N/A
N/A
N/A
2.0709
2.2331
0.0318
Experimental Results Cont.
No PPT's
Noise
0.3E-3 on w
1.33E-6 on
wdot
With PPT's
Percent error
Noise
Percent error
0.379488
0.3E-3 on w
10.07
0.379488
1.33E-6 on
wdot
10.07
Experimental Results Cont.


Batch filter is more accurate with and
without noise for longer firing times
Kalman filter converges faster for short firing
times, but has comparatively poor accuracy
Recommendations





24 hour firing
Magnetorquers and non-essential systems off
Magnetometer readings are taken or IGRF
data provided
Attitude data for the entire firing period is
taken
Initialize attitude determination Kalman filter
at the start of firing and provide batch filter
data only after convergence
Questions?