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

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3.5

time

(sec

)

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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

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x 104

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CI

Multi-Objective Optimization

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x 104

2500

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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

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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

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7

8

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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

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3500

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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

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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

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Initi

al m

ass

(kg)

Second stage uses Engine C

Second stage uses Engine B

Second stage uses Engine B

2950

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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