- 1 - 3rd Global Trajectory Optimisation Competition Workshop Aula Magna del Lingotto, Turin...

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3rd Global Trajectory Optimisation Competition Workshop

Aula Magna del Lingotto, Turin (Italy), June 27, 2008© 2008 DEIMOS Space, S.L. – www.deimos-space.com

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TO THE 3rd GLOBAL

TRAJECTORY OPTIMISATION

COMPETITION (GTOC3)

Miguel Belló, Juan L. CanoMariano Sánchez, Francesco Cacciatore

DEIMOS Space S.L., Spain

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ContentsContents

• Problem statement

• DEIMOS Space team

• Asteroid family analysis

• Solution steps:– Step 0: Asteroid Database Pruning

– Step 1: Ballistic Global Search

– Step 2a: Gradient Restoration Optimisation

– Step 2b: Local Direct Optimisation

• DEIMOS solution presentation

• Conclusions

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Problem StatementProblem Statement

• Escape from Earth, rendezvous with 3 asteroids and

rendezvous with Earth

• Depature velocity below 0.5 km/s

• Launch between 2016 and 2025

• Total trip time less than 10 years

• Minimum stay time of 60 days at each asteroid

• Initial spacecraft mass of 2,000 kg

• Thrust of 0.15 N and Isp of 3,000 s

• Only Earth GAMs allowed (Rmin = 6,871 km)

• Minimise following cost function:

max

321

i

f ),,min(K

mm

J

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DEIMOS Space TeamDEIMOS Space Team

• Miguel Belló Mora, Managing Director of DEIMOS Space,

in charge of the systematic analysis of ballistic solutions

and the reduction to low-thrust solutions by means of the

gradient-restoration algorithm

• Juan L. Cano, Senior Engineer, has been in charge of the

low-thrust analysis of solution trajectories making use of a

local optimiser (direct method implementation)

• Francesco Cacciatore, Junior Engineer, has been in

charge of the analysis of preliminary low-thrust solutions

by means of a shape function optimiser

• Mariano Sánchez, Head of Mission Analysis Section, has

provided support in a number of issues

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• Semi-major axis range: [0.9 AU-1.1 AU]• Eccentricity range: [0.0-0.9]• Inclination range: [0º-10º]

• Solution makes use of low eccentricity, low inclination asteroids

Asteroid Family AnalysisAsteroid Family Analysis

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• To reduce the size of the problem, a preliminary analysis

of earth-asteroid transfer propellant need is done by

defining a “distance” between two orbits

• This distance is defined as the minimum Delta-V to

transfer between Earth and the asteroid orbits

• By selecting all asteroids with “distance” to the Earth

bellow 2.5 km/s, we get the following list of candidates:

– 5, 11, 16, 19, 27, 30, 37, 49, 61, 64, 66, 76, 85, 88, 96, 111,

114, 122 & 129

• In this way, the initial list of 140 asteroids is reduced

down to 19

• Among them numbers 37, 49, 76, 85, 88 and 96 shall be

the most promising candidates

Step Step 00: : Asteroid Database PruningAsteroid Database Pruning

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• The first step was based on a Ballistic Scanning Process

between two bodies (including Earth swingbys) and saving

them into databases of solutions

• Assumptions:

– Ballistic transfers

– Use of powered swingbys

– Compliance with the problem constrains

• This process was repeated for all the possible phases

• As solution space quickly grew to immense numbers, some

filtering techniques were used to reduce the space

• The scanning procedure used the following search values:

– Sequence of asteroids to visit

– Event dates for the visits

• An effective Lambert solver was used to provide the

ballistic solutions between two bodies

Step 1: Ballistic Global SearchStep 1: Ballistic Global Search

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• Due to the limited time to solve the problem, only

transfer options with the scheme were tested:

E-E–A1–E–E–A2–E–E–A3–E–E

• All possible options with that profile were

investigated, including Earth singular transfers of

180º and 360º

• The optimum sequence found is:

E–49–E–E–37–85–E–E

• Cost function in this case is: J = 0.8708

• This step provided the clues to the best families of

solutions

Step 1: Ballistic Global SearchStep 1: Ballistic Global Search

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• A tool to translate the best ballistic solutions into

low-thrust solutions was used

• A further assumption was to use prescribed thrust-

coast sequences and fixed event times

• The solutions were transcribed to this formulation

and solved for a number of promising cases

• Optimum thrust directions and event times were

obtained in this step

• A Local Direct Optimisation Tool was used to

validate the solution obtained

Step 2Step 2aa: Gradient Restoration Optimisation: Gradient Restoration Optimisation

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• Final spacecraft mass: 1716.739 kg

• Stay time at asteroids: 135.2 / 60.0 / 300.3 days

• Minimum stay time at asteroid: 60 days

• Cost function

• Solution structure:

• Mission covers the 10 years of allowed duration

• Losses from ballistic case account to a 0.05%

Best Solution FoundBest Solution Found

0.861655365.25*10

60.00*0.2

2000.0001716.739

J

E – TCT – 49 – TC – E – C – E – TCT – 37 – TCT – 85 – TC – E – CTCT – E

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Best Solution FoundBest Solution Found

Celestial Body Arrival /

Start (MJD) Departure / Stop (MJD)

Duration / Stay (days)

Mass (Kg) Excess velocity (km/s)

Perigee radius (km)

Earth - 60963.53 - 2000.00 0.500 -

Thrust 60963.53 60987.23 23.70

Coast 60987.23 61933.52 946.29

Thrust 61933.52 62000.31 66.79

49 (2000 SG344) 62000.31 62135.48 135.17 1960.14 0.000 -

Thrust 62135.48 62162.49 27.01

Coast 62162.49 62407.63 245.14

Earth 62407.63 - - 1948.24 1.818 64967.0

Coast 62407.63 62772.88 365.25

Earth 62772.88 - - 1948.24 1.818 62484.0

Thrust 62772.88 62796.30 23.42

Coast 62796.30 62916.02 119.72

Thrust 62916.02 62987.56 71.54

37 (2004 QA22) 62987.56 63047.56 60.00 1906.41 0.000 -

Thrust 63047.56 63096.34 48.78

Coast 63096.34 63328.57 232.23

Thrust 63328.57 63479.62 151.05

85 (2006 BZ147) 63479.62 63779.95 300.33 1818.38 0.000 -

Thrust 63779.95 63916.63 136.68

Coast 63916.63 64144.40 227.77

Earth 64144.40 - - 1758.17 1.349 160054

Coast 64144.40 64402.20 257.80

Thrust 64402.20 64449.44 47.24

Coast 64449.44 64569.22 119.78

Thrust 64569.22 64616.03 46.81

Earth 64616.03 - - 1716.74 - -

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Best solution: Full trajectoryBest solution: Full trajectory

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Best solution: DistancesBest solution: Distances

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Best solution: MassBest solution: Mass

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Best solution: Thrust componentsBest solution: Thrust components

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Best solution: From Earth to asteroid 37Best solution: From Earth to asteroid 37

Segment Earth to asteroid 49:– E–TCT–49– 2½ revolutions about Sun– Duration of 1,047 days

Segment asteroid 49 to 37:– 49-TC-E-C-E-TCT-37– 2½ revolutions about Sun– Duration of 852 days

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Best solution: From asteroid 37 to EarthBest solution: From asteroid 37 to Earth

Segment asteroid 37 to 85:– 37–TCT–85 – 1¼ revolutions about Sun– Duration of 450 days

Segment asteroid 85 to Earth:– 85–TC–E–CTCT–E– 2½ revolutions about Sun– Duration of 836 days

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• Use of ballistic search algorithms seem to be still applicable to provide good initial guesses to low-thrust trajectories even in these type of problems

• Such approach saves a lot of computational time by avoiding the use of other implementations with larger complexity (e.g. shape-based functions)

• Transcription of ballistic into low-thrust trajectories by using a GR algorithm has shown to be very efficient

• Failure to find a better solution is due to:– The a priori imposed limit in the number of Earth

swingbys (best solution shows up to 3 Earth-GAMs)– Non-optimality of the assumed thrust-coast structures

between phases

ConclusionsConclusions

- 1 - 3rd Global Trajectory Optimisation Competition Workshop Aula Magna del Lingotto, Turin...

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