rd E S A W o rk sh o p o n A stro d y n a m ics To o ls a n d Te ch n...
Transcript of rd E S A W o rk sh o p o n A stro d y n a m ics To o ls a n d Te ch n...
![Page 1: rd E S A W o rk sh o p o n A stro d y n a m ics To o ls a n d Te ch n …trajectory.estec.esa.int/Astro/3rd-astro-workshop... · 2006. 9. 25. · R e n d e zv o u s w ith in ca p](https://reader036.fdocuments.in/reader036/viewer/2022081620/611a14013e33dc49e12d4a1f/html5/thumbnails/1.jpg)
© GMV, 2006 Property of GMV
All rights reserved
geoexpress Very Low Thrust Mission Design
3rd ESA Workshop on Astrodynamics Tools and Techniques
Jesús Gil-Fernández
Carlos Corral van Damme
![Page 2: rd E S A W o rk sh o p o n A stro d y n a m ics To o ls a n d Te ch n …trajectory.estec.esa.int/Astro/3rd-astro-workshop... · 2006. 9. 25. · R e n d e zv o u s w ith in ca p](https://reader036.fdocuments.in/reader036/viewer/2022081620/611a14013e33dc49e12d4a1f/html5/thumbnails/2.jpg)
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
![Page 3: rd E S A W o rk sh o p o n A stro d y n a m ics To o ls a n d Te ch n …trajectory.estec.esa.int/Astro/3rd-astro-workshop... · 2006. 9. 25. · R e n d e zv o u s w ith in ca p](https://reader036.fdocuments.in/reader036/viewer/2022081620/611a14013e33dc49e12d4a1f/html5/thumbnails/3.jpg)
© GMV, 2006 Property of GMV
All rights reserved
Introductiongeoexpress
![Page 4: rd E S A W o rk sh o p o n A stro d y n a m ics To o ls a n d Te ch n …trajectory.estec.esa.int/Astro/3rd-astro-workshop... · 2006. 9. 25. · R e n d e zv o u s w ith in ca p](https://reader036.fdocuments.in/reader036/viewer/2022081620/611a14013e33dc49e12d4a1f/html5/thumbnails/4.jpg)
09/23/06 Page 4geoexpress Very Low Thrust Mission Design © GMV, 2006
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)
![Page 5: rd E S A W o rk sh o p o n A stro d y n a m ics To o ls a n d Te ch n …trajectory.estec.esa.int/Astro/3rd-astro-workshop... · 2006. 9. 25. · R e n d e zv o u s w ith in ca p](https://reader036.fdocuments.in/reader036/viewer/2022081620/611a14013e33dc49e12d4a1f/html5/thumbnails/5.jpg)
09/23/06 Page 5geoexpress Very Low Thrust Mission Design © GMV, 2006
! 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)
![Page 6: rd E S A W o rk sh o p o n A stro d y n a m ics To o ls a n d Te ch n …trajectory.estec.esa.int/Astro/3rd-astro-workshop... · 2006. 9. 25. · R e n d e zv o u s w ith in ca p](https://reader036.fdocuments.in/reader036/viewer/2022081620/611a14013e33dc49e12d4a1f/html5/thumbnails/6.jpg)
09/23/06 Page 6geoexpress Very Low Thrust Mission Design © GMV, 2006
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)
![Page 7: rd E S A W o rk sh o p o n A stro d y n a m ics To o ls a n d Te ch n …trajectory.estec.esa.int/Astro/3rd-astro-workshop... · 2006. 9. 25. · R e n d e zv o u s w ith in ca p](https://reader036.fdocuments.in/reader036/viewer/2022081620/611a14013e33dc49e12d4a1f/html5/thumbnails/7.jpg)
© GMV, 2006 Property of GMV
All rights reserved
Optimization Algorithm
geoexpress
![Page 8: rd E S A W o rk sh o p o n A stro d y n a m ics To o ls a n d Te ch n …trajectory.estec.esa.int/Astro/3rd-astro-workshop... · 2006. 9. 25. · R e n d e zv o u s w ith in ca p](https://reader036.fdocuments.in/reader036/viewer/2022081620/611a14013e33dc49e12d4a1f/html5/thumbnails/8.jpg)
09/23/06 Page 8geoexpress Very Low Thrust Mission Design © GMV, 2006
! 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)
![Page 9: rd E S A W o rk sh o p o n A stro d y n a m ics To o ls a n d Te ch n …trajectory.estec.esa.int/Astro/3rd-astro-workshop... · 2006. 9. 25. · R e n d e zv o u s w ith in ca p](https://reader036.fdocuments.in/reader036/viewer/2022081620/611a14013e33dc49e12d4a1f/html5/thumbnails/9.jpg)
09/23/06 Page 9geoexpress Very Low Thrust Mission Design © GMV, 2006
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
![Page 10: rd E S A W o rk sh o p o n A stro d y n a m ics To o ls a n d Te ch n …trajectory.estec.esa.int/Astro/3rd-astro-workshop... · 2006. 9. 25. · R e n d e zv o u s w ith in ca p](https://reader036.fdocuments.in/reader036/viewer/2022081620/611a14013e33dc49e12d4a1f/html5/thumbnails/10.jpg)
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)
![Page 11: rd E S A W o rk sh o p o n A stro d y n a m ics To o ls a n d Te ch n …trajectory.estec.esa.int/Astro/3rd-astro-workshop... · 2006. 9. 25. · R e n d e zv o u s w ith in ca p](https://reader036.fdocuments.in/reader036/viewer/2022081620/611a14013e33dc49e12d4a1f/html5/thumbnails/11.jpg)
© GMV, 2006 Property of GMV
All rights reserved
Trajectory Designgeoexpress
![Page 12: rd E S A W o rk sh o p o n A stro d y n a m ics To o ls a n d Te ch n …trajectory.estec.esa.int/Astro/3rd-astro-workshop... · 2006. 9. 25. · R e n d e zv o u s w ith in ca p](https://reader036.fdocuments.in/reader036/viewer/2022081620/611a14013e33dc49e12d4a1f/html5/thumbnails/12.jpg)
09/23/06 Page 12geoexpress Very Low Thrust Mission Design © GMV, 2006
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
![Page 13: rd E S A W o rk sh o p o n A stro d y n a m ics To o ls a n d Te ch n …trajectory.estec.esa.int/Astro/3rd-astro-workshop... · 2006. 9. 25. · R e n d e zv o u s w ith in ca p](https://reader036.fdocuments.in/reader036/viewer/2022081620/611a14013e33dc49e12d4a1f/html5/thumbnails/13.jpg)
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)
![Page 14: rd E S A W o rk sh o p o n A stro d y n a m ics To o ls a n d Te ch n …trajectory.estec.esa.int/Astro/3rd-astro-workshop... · 2006. 9. 25. · R e n d e zv o u s w ith in ca p](https://reader036.fdocuments.in/reader036/viewer/2022081620/611a14013e33dc49e12d4a1f/html5/thumbnails/14.jpg)
09/23/06 Page 14geoexpress Very Low Thrust Mission Design © GMV, 2006
! Mean apogee radius constant to leading order
– Osculating apogee radius not fixed
– No GEO belt crossing
Sub-synchronous Transfer (2/2)
![Page 15: rd E S A W o rk sh o p o n A stro d y n a m ics To o ls a n d Te ch n …trajectory.estec.esa.int/Astro/3rd-astro-workshop... · 2006. 9. 25. · R e n d e zv o u s w ith in ca p](https://reader036.fdocuments.in/reader036/viewer/2022081620/611a14013e33dc49e12d4a1f/html5/thumbnails/15.jpg)
09/23/06 Page 15geoexpress Very Low Thrust Mission Design © GMV, 2006
! Small trajectory difference due to RV with different ‘clients’
! Perturbations have significant effect on phasing
Super-synchronous Transfers (1/2)
![Page 16: rd E S A W o rk sh o p o n A stro d y n a m ics To o ls a n d Te ch n …trajectory.estec.esa.int/Astro/3rd-astro-workshop... · 2006. 9. 25. · R e n d e zv o u s w ith in ca p](https://reader036.fdocuments.in/reader036/viewer/2022081620/611a14013e33dc49e12d4a1f/html5/thumbnails/16.jpg)
09/23/06 Page 16geoexpress Very Low Thrust Mission Design © GMV, 2006
! Optimal control
– Initially stress energy increase
– After apogee height maximum, focused on eccentricity control
Super-synchronous Transfers (2/2)
![Page 17: rd E S A W o rk sh o p o n A stro d y n a m ics To o ls a n d Te ch n …trajectory.estec.esa.int/Astro/3rd-astro-workshop... · 2006. 9. 25. · R e n d e zv o u s w ith in ca p](https://reader036.fdocuments.in/reader036/viewer/2022081620/611a14013e33dc49e12d4a1f/html5/thumbnails/17.jpg)
09/23/06 Page 17geoexpress Very Low Thrust Mission Design © GMV, 2006
! 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
![Page 18: rd E S A W o rk sh o p o n A stro d y n a m ics To o ls a n d Te ch n …trajectory.estec.esa.int/Astro/3rd-astro-workshop... · 2006. 9. 25. · R e n d e zv o u s w ith in ca p](https://reader036.fdocuments.in/reader036/viewer/2022081620/611a14013e33dc49e12d4a1f/html5/thumbnails/18.jpg)
© GMV, 2006 Property of GMV
All rights reserved
Guidancegeoexpress
![Page 19: rd E S A W o rk sh o p o n A stro d y n a m ics To o ls a n d Te ch n …trajectory.estec.esa.int/Astro/3rd-astro-workshop... · 2006. 9. 25. · R e n d e zv o u s w ith in ca p](https://reader036.fdocuments.in/reader036/viewer/2022081620/611a14013e33dc49e12d4a1f/html5/thumbnails/19.jpg)
09/23/06 Page 19geoexpress Very Low Thrust Mission Design © GMV, 2006
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
![Page 20: rd E S A W o rk sh o p o n A stro d y n a m ics To o ls a n d Te ch n …trajectory.estec.esa.int/Astro/3rd-astro-workshop... · 2006. 9. 25. · R e n d e zv o u s w ith in ca p](https://reader036.fdocuments.in/reader036/viewer/2022081620/611a14013e33dc49e12d4a1f/html5/thumbnails/20.jpg)
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
![Page 21: rd E S A W o rk sh o p o n A stro d y n a m ics To o ls a n d Te ch n …trajectory.estec.esa.int/Astro/3rd-astro-workshop... · 2006. 9. 25. · R e n d e zv o u s w ith in ca p](https://reader036.fdocuments.in/reader036/viewer/2022081620/611a14013e33dc49e12d4a1f/html5/thumbnails/21.jpg)
© GMV, 2006 Property of GMV
All rights reserved
Conclusionsgeoexpress
![Page 22: rd E S A W o rk sh o p o n A stro d y n a m ics To o ls a n d Te ch n …trajectory.estec.esa.int/Astro/3rd-astro-workshop... · 2006. 9. 25. · R e n d e zv o u s w ith in ca p](https://reader036.fdocuments.in/reader036/viewer/2022081620/611a14013e33dc49e12d4a1f/html5/thumbnails/22.jpg)
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