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Concurrent Trajectory and Vehicle Optimization for an
Orbit Transfer
Christine Taylor May 5, 2004
Presentation Overview
z Motivation z Single Objective Optimization z Problem Description z Mathematical Formulation z Design Variable Bounds z Verification
z Multi-Objective Optimization z Convergence z Sensitivity Analysis
z Vehicle Selection z Problem Formulation z Optimization z Sensitivity Analysis
z Future Work
Motivation
z Traditionally, orbit transfers are optimized with respect to a selected vehicle
z Competing objectives of initial mass and time of flight are weighted by the preference of the customer z High priority Æ minimize time of flight z Low priority Æ minimize initial mass
z Vehicle selection is as important as the trajectory z The choice of vehicle drastically impacts both performance and cost z Differing priorities would impact vehicle choice
z Evaluate both the trajectory and vehicle selection for different preferences to show ‘optimal’ orbit transfer configurations
Single Objective Optimization Problem Formulation
z Consider a co-planar orbit transfer from low Earth orbit (LEO) to geosynchronous Earth orbit (GEO) using a two-stage chemical rocket
z Minimize the initial mass of the system for a given payload mass z Define three design variables
z Transfer angle (ν): Angle between first and second burn z Specific impulse of first engine (Isp1) z Specific impulse of second engine (Isp2)
z Parameters z Payload mass (mp) = 1000 kg
z Structural factor (α) = 0.1 for both engines
z Initial radius (r0) = 6628 km (250 km altitude)
z Final radius (rf) = 42378 km (36000 km altitude) z Define two disciplinary models
z Orbit transfer calculation: Assumes first burn is tangent to initial orbit z Input: the transfer angle, and the initial and final radii z Output: ∆V of each burn and time of flight
z Rocket equation: Assumes each burn is impulsive z Input: the ∆V of each burn, the specific impulse of each engine, the structural factor, and the payload mass z Output: the initial mass
Mathematical Problem
Orbit Equations for One Tangent Burn
Formulation
z Minimize J(x) = mi
z Subject to the disciplinary modelequations z Orbit transfer Equations z Rocket equation
z And subject to the variable bounds z 135 deg <=ν<=180 deg z 300 sec<=Isp1,Isp2 <=450 sec
Rocket Equation for Two stages
5
Transfer Angle Bounds
Initial Mass vs. Delta V for an Isp of 450 sec Time of flight vs. transfer angle5 x 10
0
0.5
1
1.5
2
2.5
3
3.5
time
(sec
)
−8
−6
−4
−2
0
2
4
6 x 10
Initi
al M
ass
(kg)
Time of flight goes to infinity as ν approaches 133 degrees
Initial mass goes negative as ∆V becomes large
0 5 10 15 110 120 130 140 150 160 170 180
DV (km/s) nu (deg)
Model Verificationz Minimum initial mass solution
is a Hohmann transferz Transfer angle = 180 degz Specific Impulse = 450 secz Time of flight is half the
transfer orbital period
z Using a SQP method, from any initial guess, model is verified
z Initial mass = 2721.2 kg
z Time of flight = 19086 sec (5.3 hours)
−5
0
5
x 104
−5
0
5
x 104
−5
−4
−3
−2
−1
0
1
2
3
4
5
x 104
Y
Z
X
Z E
CI
Multi-Objective Optimization
0.8 1 1.2 1.4 1.6 1.8 2
x 104
2500
3000
3500
4000
4500
5000
5500
6000
6500
7000
time(sec)
mi(k
g)
Initial mass vs. Time of flight
Equal
initial mass
Minimum time solution*
weights Minimum
solution Utopia point
* Indicates that the solution is independent of engine selection
Minimum time of flight solution: ν = 135 deg
Time of flight = 8078.5 sec (2.2 hrs) Initial mass = 6598 kg
z The two objective to beminimized are initial mass and time of flight
z Scale each objective to benon-dimensional and approximately the same order of magnitude z Scale factor for mass is the
payload mass (1000 kg) z Scale factor for time of flight is
period of initial orbit (5370sec) J1 = m /mpi
J2 = time/P0
z Use weighted sum approach J = λJ1 + (1-λ)J2
Comparison of SQP and SA Convergence
0 500 1000 1500 2000 2500 3000 3500 4000 4500 50002
4
6
8
10
12
14
Iteration Number
Sys
tem
Ene
rgy
SA convergence historycurrent configurationnew best configuration
SA parameters: exponential cooling schedule,
dT = 0.75, neq = 50, nfrozen = 40
ν = 3.14 deg
Isp1 = 449.1 sec
Isp2 = 449.7 sec
mi = 2725 kg
Convergence time for SA = 33.48 sec
0 5 10 15 20 25 302
3
4
5
6
7
8
9
10
Iteration Number
Obj
ectiv
e F
unct
ion
SQP Convergence History
ν = 3.14 deg
Isp1 = 450 sec
Isp2 = 450 sec
mi = 2721 kg
Convergence time for SQP = 0.32 secSQP parameters: objective function tolerance = 10-7
Sensitivity Analysisz Sensitivity to design variables
z Minimum initial mass is most sensitive to specific impulse of first engine:
z Minimum time of flight is only sensitive to transfer angle
mi(k
g)
• Sensitivity to Scale Factors As the scale factor for initial mass increases, the magnitude of J1 decreases
Initial mass vs. Time of flight 7000
6500 Minimum time of flight*
6000
5500
Equal weights
Minimum initial mass solution Utopia
point
5000
4500
4000
3500
3000
25000.8 1 1.2 1.4 1.6 1.8 2
4time(sec) x 10
z Define three design variables z Transfer angle (ν) z Engine for first and second
burns z Select one of three different
engines for each burn z Engine A: α = 0.08, Isp = 300
sec z Engine B: α = 0.10, Isp = 400
sec z Engine C: α = 0.12, Isp = 450
sec z Use SA, determine Pareto
front
Vehicle Selection
0.8 1 1.2 1.4 1.6 1.8 2
x 104
2000
3000
4000
5000
6000
7000
8000
9000
10000
11000
12000 Pareto front
Time of flight (sec)
Initi
al m
ass
(kg)
initial mass
Minimum time of flight*
Minimum
z Engine C is chosen for both * Indicates that the solution is independent of
engines and for each set of engine selection
weights, except λ = 0
Sensitivity to Engine Change
Pareto front Choice of Engines vs. Varying Structure Ratio for Minimum Initial Mass
3000
4000
5000
6000
7000
8000
9000
10000
11000
12000
Initi
al m
ass
(kg)
Second stage uses Engine C
Second stage uses Engine B
Second stage uses Engine B
2950
3000
3050
3100
3150
3200
Initi
al m
ass
(kg
)
Switch point from Engine C
First stage uses Engine C
Minimum time solution First stage uses Engine A
First stage uses Engine B
to Engine B
0.8 1 1.2 1.4 1.6 1.8 2 0.12 0.13 0.14 0.15 0.16 0.17 0.18 4 Varying Structure Ratio for Engine CTime of flight (sec) x 10
Engine C: α = 0.148, Isp = 425 sec
Conclusions and Future Work
z Global Optimality? z Single objective optimization z Multi objective optimization z Discrete engine selection
z Expand the model to include different types of engines z Allow for low thrust, high impulse engines z Modify model to include constant thrust trajectories
z Analyze more complicated problems z Allow for out of plane maneuvers z Examine different origin/destination pairs
z Examine the relationship between scaling factors and the Pareto front
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