astos for low thrust mission analysis -...
TRANSCRIPT
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AstosSolutions
ASTOS for Low Thrust Mission Analysis
3rd Astrodynamics Workshop, Oct. 2006, ESTEC
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AstosSolutionsOverview
• Low Thrust Trajectory Computation
• Description of the Optimal Control Problem
• Trajectory Optimization and Mission Analysis
– Minimum Time Transfer
– Subsynchronous Transfer
– Restricted Minimum Fuel Transfer
• Summary & Conclusion
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AstosSolutionsElectric Propulsion
• Thrust levels: F < 0.5 N• Exhaust velocity: 3 - 40 km/s
• Can thrust continuously• Thrust direction steerable• First used as AOCS• Then interplanetary travel (deep
space 1, Smart 1, etc.)• Considered for orbit raising to GEO
ùTransfer of system & fuel mass to payload mass
úIncreased transfer duration through Van Allen radiation belt
úIncreased overall complexity of trajectory geometry
ATOS hydrazine arcjet (up) and one of the IMPD thrusters(bottom)
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AstosSolutions
Solution Methods
• Elaborate Control Laws– Koppel, Pollard:
• 1st step: semimajor é• 2nd step: eccentricity ê, inclination ê , a = const.
• Indirect Optimization Methods– Identify relations for adjoint variables, if possible
• Averaging Techniques– Reduce size of optimal control problem
(parameterisation losses)• Direct Optimization Methods
– Requires many parameters (10-100k) => SOCS
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AstosSolutions
Core Objectives for Industrial Mission Analysis
Determination of initial mission specifications is governed by• Varying levels of sophistication
– Earth oblateness effects– Perturbational bodies– Radiation belt modelling, power degradation modelling
• Varying propulsion and system configurations– Propulsion components, thruster characteristics– System driven restrictions, e.g. Solar cells orientation, recharge
cycle• Trade-Off aspects
– Restrictions on orbit geometry, e.g. subsynchronous transfers– Changing objectives, e.g. power output, payload, fuel, trip time
• Mission constraints– Power management– Geometrical path constraints, e.g. subsyncronous transfer– Target orbit definition
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AstosSolutions
Core Objectives for Industrial Mission Analysis
Requirements for mission analysis softwareøAllow quick modification/in-/exclusion of
boundary and path constraints and cost components
øTime economic and reliable computation of transfer trajectories
øRobust with respect to changing dynamicsøProvide optimal results that can easily be
comparedøRelieve user from tuning of optimiser setting
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AstosSolutions
ASTOS© – Work Flow
1. Describe OCP
2. Initialize/Discretize
3. Transcription into NLP is done automatically
4. Optimize & Monitor
5. Simulate
6. Handle Data
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AstosSolutionsControl Law vs Optimization
apo peri
1. with the Control Laws suggested by Pollard with coasts:
0 20 40 60 80 1002.4
2.6
2.8
3
3.2
3.4
3.6
3.8
4
4.2
4.4x 10
4
days
Semimajor Axis [km]
0 20 40 60 80 1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
days
Eccentricity [-]
0 20 40 60 80 1000
5
10
15
20
25
30
days
Inclination [deg]
Cont.
111.9 d191.5 kg
coast
ASTOSPollardf
f
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AstosSolutions
• Low-thrust transfer from a GTO with high inclination (see J.E. Pollard)
• GTO: hapo= 35,786 km; hperi= 185 km; i = 28.5°; mass =1400kg
• 2. with the Control Laws suggested by Pollard without coasts:
0 20 40 60 80 1002.4
2.6
2.8
3
3.2
3.4
3.6
3.8
4
4.2
4.4x 10
4
days
Semimajor Axis [km]
0 20 40 60 80 1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
days
Eccentricity [-]
0 20 40 60 80 1000
5
10
15
20
25
30
days
Inclination [deg]
Control Law vs Optimization
102.3 d218.9 kg
Cont.
111.9 d191.5 kg
coast
ASTOSPollardf
f
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AstosSolutions
apo peri
3. Optimized with ASTOS/SOCS:
0 20 40 60 80 1002.4
2.6
2.8
3
3.2
3.4
3.6
3.8
4
4.2
4.4x 10
4
days
Semimajor Axis [km]
0 20 40 60 80 1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
days
Eccentricity [-]
0 20 40 60 80 1000
5
10
15
20
25
30
days
Inclination [deg]
Control Law vs Optimization
96.7 d207.0 kg
102.3 d218.9 kg
Cont.
111.9 d191.5 kg
coast
ASTOSPollardf
f
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AstosSolutions
apo peri
4. Optimized with ASTOS/SOCS allowing optimizable thrust:
Control Law vs Optimization
0 20 40 60 80 1002.4
2.6
2.8
3
3.2
3.4
3.6
3.8
4
4.2
4.4
4.6x 10
4
days
Semimajor Axis [km]
0 20 40 60 80 1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
days
Eccentricity [-]
0 20 40 60 80 1000
5
10
15
20
25
30
days
Inclination [deg]
96.7 d207.0 kg
102.3 d218.9 kg
Cont.
111.9 d167.3 kg
111.9 d191.5 kg
coast
ASTOSPollardf
f
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AstosSolutionsCentral Body and Perturbation
Equations of Motion comprehend:– A central gravitational body (Earth)– Perturbation vector in the radial frame – EoM: Equinoctial Elements
The Perturbation Vector can comprise:– Thrust– Oblateness models– Third bodies (Sun, Moon, Mars, Jupiter, etc.)– Solar wind pressure
...+∆+∆+∆=∆ qgT
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AstosSolutionsPower Management
• Thrust and massflow are functions of the provided power P, per thruster:– Thrust : F = f1(P)– Massflow: mdot = f2(P)
• Available power depends on:– Solar array size and characteristics– Power supply for secondary systems– Power management during solar eclipses– Reduction due to radiation damage– Battery capacity and recharge cycles under
consideration of eclipses
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AstosSolutionsBoundary Conditions
• Initial Conditions– Ariane 5 GTO orbit
• Final Conditions– GEO Interbox Gap:
• Semimajor axis: 42,164.169777 km – 500.0 km• Eccentricity: 0.0 °• Inclination: 0.0 °• V-bar excitation• Longitude of GEO box
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AstosSolutions
Radial Frame of the Control Variables
=∆
h
th
r
T
uuu
mT
222)(
),,()(
hthr
Ththr
uuutu
uuutu
++=
=vernal equinox
r
x
y
zih irith
ab
ith
ith
ith
ir
ir
ir
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AstosSolutions
Initial Guess Generation
• Initial guess generation is based on a straight forward simulation• Construction using standard control laws• Generic control history is sufficient to allow steady optimization• Enhanced performance with more sophisticated initial control histories• Use of earlier trajectories of lower-level computations is possible
Benefits:• Non need to compute abstract adjoint variables• Non need to newly generate model equations (see
indirect/hybrid methods)• Pure utilization of physical relations• Preparation of optimization algorithm is not required (0
minutes)è Don‘t waste time on the initial guess
(< 5 minutes)
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AstosSolutions
Optimal overshooting
0 100 2005
10
15
20
25
30
35
40
45
50
55
Time [days]
Radius [10
3 km ]
Minimum Time Transfer
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AstosSolutionsMinimum Time Transfer
0 100-1
-0.5
0
0.5
1
Time [days]
Radial Component
0 100-1
-0.5
0
0.5
1
Time [days]
Tangential Component
0 100-1
-0.5
0
0.5
1
Time [days]
Normal Component
Min tf transfer:
No eclipsesNo phasing
0 100
25
30
35
40
Time [days]
Semimajor Axis [103 km]
0 1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Time [days]
Eccentricity [-]
0 1000
1
2
3
4
5
6
7
8
Time [days]
Inclination [deg]
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AstosSolutionsOptimal Phasing
0 100-1
-0.5
0
0.5
1
Time [days]
Radial Component
0 100-1
-0.5
0
0.5
1
Time [days]
Tangential Component
0 100-1
-0.5
0
0.5
1
Time [days]
Normal Component
0 100-1
-0.5
0
0.5
1
Time [days]
Radial Component
0 100-1
-0.5
0
0.5
1
Time [days]
Tangential Component
0 100-1
-0.5
0
0.5
1
Time [days]
Normal Component
λ0
λ0 + 180°
Time optimal, but not fuel optimal!
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AstosSolutionsYaw and Pitch over Anomaly
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AstosSolutionsThrust Direction in each Revolution
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AstosSolutionsInfluence of Perturbations
0 100-1
-0.5
0
0.5
1
Time [days]
Radial Component
0 100-1
-0.5
0
0.5
1
Time [days]
Tangential Component
0 100-1
-0.5
0
0.5
1
Time [days]
Normal Component
0 100
25
30
35
40
Time [days]
Semimajor Axis [103 km]
0 1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Time [days]
Eccentricity [-]
0 1000
1
2
3
4
5
6
7
8
Time [days]
Inclination [deg]
0 50 100 150 2005
10
15
20
25
30
35
40
45
50
55
Time [days]
A po gee/Peri gee [10
3 km ]
Opt control unperturbed caseOpt control perturbed case
Unperturbed dynamicsPerturbed dynamics
Is this influence much higher than that of thrust vector error?
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AstosSolutions
0 50 100 150 200 2505
10
15
20
25
30
35
40
45
Time [days]
Radius [10
3 km ]
Time optimal subsynchronous transfer
Time optimal but not fueloptimal because of permanent constant thrust!
Subsynchronous Transfer
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AstosSolutions
Subsynchronous Transfer Optus B3, December Launch
Subsync. transfer:
With eclipsesWith phasing
0 100 200-1
-0.5
0
0.5
1
Time [days]
Radial Component
0 100 200-1
-0.5
0
0.5
1
Time [days]
Tangential Component
0 100 200-1
-0.5
0
0.5
1
Time [days]
Normal Component
0 100 200
25
30
35
40
Time [days]
Semimajor Axis [103 km]
0 100 2000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Time [days]
Eccentricity [-]
0 100 2000
1
2
3
4
5
6
7
8
Time [days]
Inclination [deg]
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AstosSolutionsThrust Direction in each Revolution
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AstosSolutionsRestricted Minimum Fuel Transfer I
0 50 100 150 200 2500
20
40
60
80
100
120
140
160
180
200
Time [days]
Pro pellant Mass [k g]
0 50 100 150 200 2505
10
15
20
25
30
35
40
45
Time [days]
Peri gee/A po gee [10
3 km ]
Upper bound: Max availablepower
Lower bound: Min continuous thrust
Fueloptimal
Time optimal
Cost: 17.6 daysBenefit:
24.6 kg
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AstosSolutionsRestricted Minimum Fuel Transfer II
Transfer Costs & Benefits for Arrival on 01-Mar-2009
140
150
160
170
180
190
200
12-M
ay-20
08
22-M
ay-20
08
1-Jun
-2008
11-Ju
n-200
8
21-Ju
n-200
8
1-Jul-
2008
11-Ju
l-200
8
21-Ju
l-200
8
31-Ju
l-200
8
10-A
ug-20
08
20-A
ug-20
08
30-A
ug-20
08
Launch Date
Fu
el C
on
sum
pti
on
[kg
]
Subsynchronous
Overshooting
210.5 days
283.2 days
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AstosSolutionsTime Consumption
Generation of an initial guess– New missions with similar configurations: 1min– New missions: 5-15 min
Optimization of the trajectory– Reliable solution for a standard mission: 10 min– Converged solution for standard mission: 1-2 hours– Complex cases: Up to a few hours CPU time– Research on new scenarios: takes days
èIt all depends on the level of sophistication and the tolerances
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AstosSolutionsSummary & Conclusion
The NLP setup for GTO-GEO transfers in ASTOS/SOCS has proven to- be easily extendable/adaptable to new mission requirements- be reliable/stable with respect to the sophistication of the dynamics- produce solutions that are superior to control law applications
NLP optimization is a well appropriate alternative for low-thrust GTO-GEO mission analysis
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AstosSolutions
Thank you!