Optimization of Separated Spacecraft
Interferometer Trajectories in the
Absence of A Gravity
-Well
Gravity-Well
Edmund M. Kong
Prof David W. Miller
MIT Space Systems Laboratory
20th March 1998
Space Systems Laboratory
Massachusetts Institute of Technology
Objective & Approach
Objective : Determine the optimal synthetic imaging trajectory for a
Separated Spacecraft Interferometer
Image Quality Optimization
Trajectory Optimization
Mass Metric
Time Metric
Comparison with Other Alternatives
DS3
Uniformly Spaced
U-V Points
Other Considerations
Performance Metric
Space Systems Laboratory
Trade-offs
Optimal System
Mass
Massachusetts Institute of Technology
Image Quality
Model : 2 Collector and 1 Combiner Interferometer (DS 3)
Physics : Average Image Intensity
⎛ π (1 + cos θ ) D ⎞
⎟
⎜
∑
λ
⎠
k =1 ⎝
N
I (ϕ i , ϕ j ) = 1
N
2
2
⎛ J 1 ( πD λsin θ ) ⎞
⎜ πD sin θ ⎟
⎟
⎜
λ
⎠
⎝
2
π
⎞
⎛
⎜ 2 cos (ϕ i x k + ϕ j y k ) ⎟
λ
⎠
⎝
π
⎞
⎛ 2πD ⎞ ⎛
2
cos
(ϕ i x k + ϕ j y k ) ⎟
⎜
⎟
⎜
∑
λ
⎠
k =1 ⎝ λ ⎠ ⎝
N
1
≈
N
2
2
Combiner
Collector
Space Systems Laboratory
Collector
Massachusetts Institute of Technology
Image Quality - Mean Square Error
MSE =
m
m
∑∑ ( I (ϕ i , ϕ j ) − I o (ϕ i , ϕ j ) )
2
i =1 j =1
1.5
m×m
Approach : Choose N subset of all points
: Compare with Nominal PSF
: Simulated Annealing
Optimization Technique
MSE, (W m-2)2 (x10-3)
Minimize :
Best MSE Performance
1
0.5
0
0
100
200
300
No. of Imaging Points
400
Results : Image quality increases with
no. of imaging points (N)
: Diminishing rate of return
Space Systems Laboratory
Massachusetts Institute of Technology
Point Spread Function Images
41 Imaging Points
121 Imaging Points
201 Imaging Points
281 Imaging Points
Space Systems Laboratory
Massachusetts Institute of Technology
Trajectory Optimization - Mass Metric
:
m& fuel ⎛
N N ⎞
2
⎜ T image ± T image − 4 a ∑ si ⎟
m fuel =
i =1
2 ⎝
⎠
Assumptions : “Stop and Stare”
imaging mode
: Trapezoidal velocity
profile
: Constant acceleration
Parameters : Spacecraft masses
Collector = 150 kg
Combiner = 250 kg
: Cold Gas Propulsion
Isp = 62.5 s, F = 9 mN
Constraint
: Timage = 264 Hours
Approach
: Traveling Salesman
Algorithm
Space Systems Laboratory
1
Fuel Expended Per Image
0.8
Fuel (kg)
Minimize
0.6
0.4
0.2
0
0
100
200
300
No. of Imaging Points
Result : Fuel mass increases with
no. of imaging points (N)
Massachusetts Institute of Technology
400
Trajectory Optimization - Time Metric
Minimize
:
2 N
T =
∑ si
n
a =1
300
Time (Hrs)
Assumptions : “Stop and Stare”
imaging mode
: Triangular velocity
profile
: Small Integration Time
Parameters : Spacecraft masses
Collector = 150 kg
Combiner = 250 kg
: Pulse Plasma Thrusters
Isp = 1000 s, F = 1.4 mN
Constraint : S/C Power 80 W
Approach
: Traveling Salesman
Algorithm
Space Systems Laboratory
Least Maneuvering Time Per Image
250
200
150
100
50
0
0
100
200
300
No. of Imaging Points
Result : Imaging time increases
with no. of imaging points
Massachusetts Institute of Technology
400
Other Alternatives
500
Y (m)
Y (m)
500
Proposed DS3 Trajectory
0
-500
-500
0
X (m)
Proposed DS3
500
67 Uniformly Spaced Imaging Points
0
-500
-500
0
X (m)
500
Uniformly Spaced
Reference : Linfield, JPL
Space Systems Laboratory
Massachusetts Institute of Technology
PSF Comparison
231 Uniformly Spaced
Imaging Points
Proposed DS3
(261 Points)
281 Optimal MSE
Imaging Points
Space Systems Laboratory
Massachusetts Institute of Technology
Fuel and Time Metrics vs MSE
350
1.2
300
1
250
Time (Hours)
Fuel (kg)
Fuel Expended vs Image Quality
Proposed DS3
0.8
0.6
Uniform Spacing
0.4
0.2
0 -5
10
-4
10
Proposed DS3
Uniform Spacing
200
150
100
Optimized MSE
Maneuvering Time vs Image Quality
Optimized MSE
50
-3
10
MSE (W m-2)2
-2
10
0 -5
10
-4
10
-3
10
MSE (W m-2)2
Result : Better MSE with lower fuel consumption or shorter imaging
time
Space Systems Laboratory
Massachusetts Institute of Technology
-2
10
Other Considerations
Image Quality versus Fuel Trade-off
Main Lobe Comparison
2
1
MSE, (W m-2)2 (x10-3)
Psiy
1
0.5
Fuel (kg)
Trade-off
Main Lobe
Comparison
DS3 (261 Points)
Uniformly Spaced (231 Points)
Optimized MSE (281 Points)
0
0
Psix
Imaging Time = 74 Hours
150
200
250
No. of Imaging Points
0
350
300
1500
Mass (kg)
Y (m)
0
100
Total System Mass Required (500 Images)
“Image on
the Fly”
500
50
Extension to
N spacecraft
1000
500
11
9
7
-500
-500
Space Systems Laboratory
0
X (m)
500
No. of Cornwell Points
5
2
3
4
5
6
7
No. of S/C
Reference : Linfield, JPL
Massachusetts Institute of Technology
Conclusion
•
Determined the optimal imaging locations
•
Determined the optimal trajectories
– Mass metric
– Time metric
1
Fuel Expended Per Image
250
200
0.6
Time (Hrs)
Fuel (kg)
0.8
0.4
0.2
0
0
150
100
50
100
200
300
No. of Imaging Points
0
0
400
100
200
300
No. of Imaging Points
400
Compared with other alternatives
350
Maneuvering Time vs Image Quality
•
300
Time (Hours)
250
Proposed DS3
Uniform Spacing
200
150
100
Optimized MSE
50
0 -5
10
-4
10
-3
10
MSE (W m-2)2
-2
10
Future considerations
– MSE versus Mass trade-off
– MSE versus Time trade-off
– Extension to N spacecraft
– “Image on the Fly” mode
– Other Metrics
Total System Mass Required (500 Images)
1500
Mass (kg)
•
Least Maneuvering Time Per Image
300
1000
500
11
9
7
No. of Cornwell Points
Space Systems Laboratory
5
2
3
4
5
6
7
No. of S/C
Massachusetts Institute of Technology
Simulated Annealing
•
•
•
•
•
•
Statistical Approach
Randomly select a configuration and calculate cost, Cr
– If Cr < Cr-1
accept rth configuration
accept r only if exp(-Cr/T) > Random(0,1)
– If Cr > Cr-1
– when Cr is accepted, decrease T (system temperature)
– Continue until system is frozen (no new solution accepted in N trials)
Does not guarantee global minimum
Quick and easy implementation
Reasonable solution achieved in short computation time
Reference:
–
S. Kirkpatrick, C. D., Gelatt, Jr., M. P. Vecchi, “Optimization by Simulated Annealing”,
Science, Volume 220, Number 4598, 13th May 1983.
Space Systems Laboratory
Massachusetts Institute of Technology