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geoexpress Very Low Thrust Mission Design

3rd ESA Workshop on Astrodynamics Tools and Techniques

Jesús Gil-Fernández

Carlos Corral van Damme

09/23/06 Page 2geoexpress Very Low Thrust Mission Design © GMV, 2006

1. Introduction

2. Optimization Algorithm

3. Trajectory Design

4. Guidance

5. Conclusions

geoexpress Very Low Thrust Mission Design

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Introductiongeoexpress

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Interest of industry/institutions

! Rendezvous with incapacitated GEO

– Refueling, reconfiguration, …

! Cheap mission

– Cheap launch

– Very low-thrust (small system)

! ConeXpress, Orbital Express, SUMO

Extension to other type of missions

! Mission design: comparison vs. chemical propulsion

! Transfers from LEO to high energy orbits! Use of Electric Propulsion in launcher upper

stages

Motivation

SUMO ( 2006 AIAA)

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! Minimum time transfer! Consider rendezvous! Robustness

– Feasible trajectory always provided

! Numerical stability

– Sensitivity of solution

! Flyable trajectory! Flexibility

– Fast computation of new missions (launch date, client)

– Integration with guidance and control

Requirements

SMART-1 (credits ESA)

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Launcher defines initial conditions

! Mass! Orbital elements

‘Client’ SC defines final conditions

! Final orbit ! Longitude or equivalent angle

Path constraint: SC below GEO altitude

! Not applied: super-synchronous transfer! Applied: sub-synchronous transfer

Dynamics main assumptions

! Tmax & ISP constant (No degradation of engines)

! Eclipse: no switch-off (limited restart capacity)– Nevertheless, algorithm copes with eclipses

Constraints

ConeXpress (courtesy of ORL)

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Optimization Algorithm

geoexpress

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! Direct methods

– Possibly best fitted: Betts, Scheel & Conway• Huge number of variables few (expensive) capable NLP solvers

• Requires initial guess of trajectory and control

• Wider convergence radius

Algorithms Review

! Indirect methods

– Geffroy & Epenoy• Averaged optimization reduces sensitivity

of adjoints equation

• Convergence not asured robustness failure

• Rendez-vous problem not considered

– Other interesting algorithms• Suboptimal solutions

– Lyapunov control (Ilgen, Petropoulos)– Blending of decoupled optimal laws

(Kluever)

• Artemis recovery (Circi) Artemis recovery (credits ESA)

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GMV solution: hybrid direct & indirect method

Get the advantages of each method

! Fast-evolution problem solved with optimal control theory (Pontryagin principle)– Low sensitivity to initial values of

adjoints (numerical stability)

– Optimal control analytical representation (uplink & on-board storage)

! Secular trajectory solved via direct method– SQP with line search (standard well-

known algorithm)

– Provides a feasible transfer at each step (robustness)

Hybrid direct/indirect method (1)

) cond itionsality (transvers )(

ec) raj secular t of t (constrain 0)()(

condi tion) (initial

control) (opt imal 0

dyna mics)(adjoint

equa tions ) al (va riation ,,

func tion) (obj ective )(

00

fxx

xKx

x)x(!

u

f

x

f

d

d

uxfd

dx

xJ

f

T

f

T

f

T

T

f

09/23/06 Page 10geoexpress Very Low Thrust Mission Design © GMV, 2006

Fulfill rest of requirements

! Speed-up computation of new missions

! RV problem

Decoupling of second order effects

! Inclination change! Phasing with client! Perturbations

Incremental approach

! Solve more complex problems step-by-step

Verification of algorithm and implementation

! Check against simulator results! Check against known case

Hybrid direct/indirect method (2)

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Trajectory Designgeoexpress

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Comparison with Geffroy & Epenoy case

! Unconstrained GTO to GEO (137.5 days)! Same result within available precision (10-5)! Confirm optimality of solution

Nominal GTO-GEO transfer

! Phasing effect on flight time: few hours

Summary GTO-GEO Transfers

195.62

Super-synchronous

(90ºE)

195.73195.56210.97

Flight Time (days)

Super-synchronous

(90ºW)

Super-synchronous

(no RV)

Sub-synchronous

(no RV)

Trajectory Type

09/23/06 Page 13geoexpress Very Low Thrust Mission Design © GMV, 2006

! Frozen apogee

– No perturbations

! Same control pattern during entire transfer

Sub-synchronous Transfer (1/2)

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! Mean apogee radius constant to leading order

– Osculating apogee radius not fixed

– No GEO belt crossing

Sub-synchronous Transfer (2/2)

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! Small trajectory difference due to RV with different ‘clients’

! Perturbations have significant effect on phasing

Super-synchronous Transfers (1/2)

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! Optimal control

– Initially stress energy increase

– After apogee height maximum, focused on eccentricity control

Super-synchronous Transfers (2/2)

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! Final orbit : 23º-inclination, 11 23 RE

! VEGA launch into 23º-inclination, 3000-km circular orbit

– No inclination change

! Comparison with compatible chemical propulsion:

From LEO to Very High Energy Orbit

Final mass of chemical case includes additional service module or kick-stage

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Guidancegeoexpress

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Feasible schemes

! Full re-optimization

– Re-optimization at predefined epochs

! Optimal-control guidance

– Maintain secular trajectory nodes

– Compute optimal control for next revolution from present state

– Phasing: change last node at the final segments of transfer

! Mixed scheme

– Optimal control guidance for each revolution

– Re-optimize secular trajectory when reaching the nodes

Qualitative trade-off

! Solution: optimal-control guidance

Guidance Schemes Analysis

09/23/06 Page 20geoexpress Very Low Thrust Mission Design © GMV, 2006

! Only deterministic perturbations– 3rd body (Sun, Moon)

– Solar Radiation Pressure

– 4x4 Earth gravity field

! Flight-time increase ~1 day

– Main effect is phasing

Optimal-Control Guidance

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Conclusionsgeoexpress

09/23/06 Page 22geoexpress Very Low Thrust Mission Design © GMV, 2006

New hybrid direct/indirect method

! Numerically stable! Robust! Flyable transfer! Solves rendezvous problem! Fast computation of ‘new’ missions

– Small inclination changes (~10º) and final circular orbits

Validation of algorithm and implementation

! Literature (Geffroy&Epenoy)! Competition: same results than gesop (TTI GmbH)

Guidance scheme and simulator incorporated to geoexpress

! Flight time increase ~1 day– Only deterministic perturbations

– Stochastic perturbations need to be assessed

Conclusions