Strathclyde++

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On the generation and evolution of multipledebris removal missions

Carlos Ortega Absil∗, Lorenzo A. Ricciardi∗, Marilena Di Carlo∗,Cristian Greco†, Romain Serra∗, Mateusz Polnik∗, Aram Vroom†,

Annalisa Riccardi∗, Edmondo Minisci∗, Massimiliano Vasile∗

∗ Department of Mechanical and Aerospace EngineeringUniversity of Strathclyde, Glasgow (United Kingdom)

† Delft Institute of Technology, Delft (The Netherlands)

June 6, 2017

Strathclyde++

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Dynamicalmodels

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The team

Strathclyde++

Team &Resources

Solutionapproach

Dynamicalmodels

Combinatorialsearch

Evolution ofthe solutions

Globaloptimisation

Solutionrefinement

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Tools

• Slack - instant messages, creation of discussion channels(#combinatorial optimisation, #impulsive transfer,#the bakery, ...)

• GitHub - code developing

• Wikispace - sharing of information (problemconsiderations, plots, tutorials on parallel computing)

• GDrive - sharing of data (solution files, databases)

Strathclyde++

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Computing resources

• Archie-West: teaching cluster for the West of Scotland ∼12400 core/hours used

• Torque resource manager: 11 heterogeneous commodityhardware ∼ 60 cores

Strathclyde++

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Final rank

Strathclyde++

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Solutionapproach

Dynamicalmodels

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Solution approach

Three-step process involving:

• Low & high fidelity models

• Global & local optimisers

1 Beam search & sequencepatching method

2 Global evolutionary optimisation

3 Local optimisation

Strathclyde++

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Dynamicalmodels

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Dynamical modelsLow-fidelity cost estimation of a transfer

Assumptions:

• Keplerian motion

• Circular orbits

• No phasing

Combination of two parts:

• Hohmann transfer (2 in-plane maneuvers)

• Change of orbital plane (i and Ωsimultaneously)

The cheapest combination of the two is chosen as the predicted cost.

Strathclyde++

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Solutionapproach

Dynamicalmodels

Combinatorialsearch

Evolution ofthe solutions

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Dynamical modelsTrajectory design for rendezvous

Ideas:

• 5 maneuvers (including 3 ’deep-space’ ones)

• Incremental complexity of the model

1st model: J2 secular perturbations only

• ∆V s applied on velocity obtained from osculating elements

• Propagation performed analytically with mean elements afterconversion (same model as debris except for M)

M = n

[1 +

3

2J2

(R⊕p

)2√1 − e2

(1 − 3

2sin2 i

)]2nd model: J2 complete model

• Numerical integration (predictor-corrector and Runge-Kutta)

Strathclyde++

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Combinatorial searchIncremental construction of launch campaigns

• Itinerary (Dj , tj)• At level k, extend with either:

• transfer to new target in active mission• new launch in available mission time intervals

• Base algorithm: Beam Search• easy to control memory and runtime (Br ,Be)• heuristics to avoid excess of permutations

• J estimated with low-fidelity ∆V model (precomputed).

• Various additional heuristics, corrections to the costfunction, etc. gave solutions with different properties.

Strathclyde++

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Dynamicalmodels

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Future ideas

Combinatorial searchIncremental construction of launch campaigns

• New launch heuristics:• Cyclic exploration of mission timeline (>95% solutions).• Other used: concurrent search, cheap transfer density, etc.

• Naıve initialisation:• Each search from all 123 debris.• Total ∼ 150 searches in a grid of launch times t0.• Keep a database, give priority to promising t0 values.

• Additional heuristics:• Maximise length of shorter mission (→ 13 launches!).• Per-debris rarity bonus, based on:

• Cluster size first and second statistical moments.• Debris frequency in database of unexpensive missions.

Strathclyde++

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Combinatorial searchSequence patching

• Input Feasible sequences

• Output Launch campaign

• AlgorithmFind a clique in theunidirected graph (N,A)

• N sequences• A pairs of compatible

sequences

• Extracted campaignscovering up to 116 debrisfrom a databaseof 2.2 mln sampleswithout single launches

Figure: Partial graph of1500 compatible sequences

Strathclyde++

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Evolution of the solutionsThe Time-shuffler encoding

Idea 1

• each debris associated to a (free)departure time

• times are then sorted in ascendingorder → ID sequence

• ∆t ≤ 30 between sorted timesdefine missions

• within each mission, 5 ≤ ∆t ≤ 30 isimposed

• between missions ∆t ≥ 43 isimposed (3 days safety margin, 10days for removal of first and last)

4.4

2

41.6

3

17

41

69.2 3.1

5

tdeb

ID

4.4

2

17

4

41.6

3 1

69.23.1

5

tsort

ID

8.1

2

17

4

60

3 1

69.23.1

5

tcorr

ID

M1 M2

<5 >5,<30 >30,<43 >5,<30

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Evolution of the solutionsReformulation of the problem

Idea 2

• The Time-shuffler encoding allows working only on continuousvariables → timings are treated explicitly, combinatorial problemis treated implicitly, gradient based refinement possible

• Debris sequence, visit times and mission lengths are optimisedconcurrently → holistic global optimisation of the wholecampaign (transfers evaluated with low fidelity model)

• To reduce number of missions, promising mass-unfeasiblesolutions should be retained → multi-objective problem, withmass constraint violation as second objective

minLt≤t≤Ut

J∗(t), t = (t1, ..., tj , ..., t123)

s.t.

5 ≤ tsj+1 − tsj ≤ 30 ∀sj ∈ Mi ,∀Mi

ts1,Mi+1 − tsend,Mi ≥ 43 ∀Mi

• solved with MACS, initialised with solutions from Beam Search

Strathclyde++

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Global optimisationMinimisation of the cost of the debris-debris transfer

• Objective: min∑5

i=1 ∆Vi

• Optimisation variables: times of application and magnitude anddirection of impulsive ∆V s

• Population-based algorithm MP-AIDEA (Multi-Population AdaptiveInflationary Differential Evolution Algorithm):https://github.com/strath-ace/smart-o2c

1st GLOBAL OPTIMISATION

Model: J2 secular perturbation

Solver: MP-AIDEA

2nd GLOBAL OPTIMISATION

Model: J2 complete model

Solver: MP-AIDEA

LOCAL OPTIMISATION

Model: J2 complete model

Solver: Matlab fmincon active-set

Strathclyde++

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Solution refinementLocal optimization and tolerance matching

Direct multiple-shooting transcription scheme

• Initial guess: solution from local single-shooting

• J2 complete dynamical model

• Runge-Kutta 4 integration scheme

• WORHP as NLP solver

Accuracy and computational efficiency enhancement

• Augmented variational dynamics to compute first and second-orderderivative information

∂Gx(t,x)∂t

= ∂f(x)∂x· Gx, Gx(t0, x0) = I

where Gx = ∂x(t)∂x0

is the first-order sensitivity matrix.

• Sparsity patterns exploited

• Constraints’ Jacobian andHessian non-zeroelements: < 0.1%

Strathclyde++

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ResultsSubmitted solutions

Solution ID N. Launches J Improvements in solution process7 (The Mistery) 14 918.98 Larger population including diverse features.6 (The One) 14 945.15 Multi-objective formulation of evolution5 (Ghostbuster) 14 967.49 Added evolution algorithm to solution process,

Small population of best submitted solutions.4 (Anibal) 16 1028.72 Further relaxation of search overconstraints.3 (Donald) 16 1059.54 Improved Cyclic Beam Search heuristics.“Make Strathclyde great again” Improved low-fidelity model.

Improved global optimisation on high-fidelity model.2 (Wrappy) 18 1133.94 Cyclic Beam Search.

Improved high-fidelity model.Added single and multiple-shooting refinement.

1 (Wary) 26 1713.07 Beam Search in the first 100 mission days.No thorough trajectory refinement.

Time and mass distribution of the missions of the submitted campaigns:

Strathclyde++

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ResultsFinal solution

Difference in departure time from each debris between final solution and theinitial 14 individuals used by MACS:

RAAN evolution of the spacecraft:

2.35 2.4 2.45 2.5 2.55 2.6

time (days) 10 4

0

50

100

150

200

250

300

350

raan

(de

g)

Strathclyde++

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ResultsLow-fidelity estimation of the ∆V for debris-to-debris transfer

Cost of the transfer from debris 1 to other debris in the database:

23500 24000 24500 25000 25500 26000 26500Time

0

2000

4000

6000

8000

10000

12000

14000

16000

Delta

V

Transfers From Debris 1Destination

2122232425262728292102112

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ResultsExample of transfer between debris

• Black: departure orbital element

• Red: arrival orbital element

• Blue: High-fidelity orbital elements variation during transfer between debris

The optimiser reduces the semi-major axis to exploit the natural J2 drift

Time [hours]

0 10 20 30 40

a [km

]

7000

7050

7100

7150

Time [hours]

0 10 20 30 40

e

0

0.005

0.01

0.015

Time [hours]

0 10 20 30 40

i [d

eg]

98.5

99

99.5

Time [hours]

0 10 20 30 40

Ω [deg]

231.5

232

232.5

233

233.5

Strathclyde++

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Future ideas

• Complete the construction of the surrogate model for the low fidelitytransfer estimation → Multi-layer Perceptron Neural Network with 12inputs (6 departure orbital elements, 6 arrival orbital elements), 2 outputs:time of transfer and ∆V

• Solve the problem as an integer/mixed integer programming problem:example bin-packing problem formulation

• Study the best generation of sequences for sequence patching: allow thebest variety of sequences

• Study the evolutionary (non-) combinatorial approach and its possibleapplications

• Investigate further the best number of impulses for the problem

• Win GTOC next year :)

Strathclyde++

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GTOC last day

while continuing pickling solutions...

Strathclyde++

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GTOC art

Part of the GTOC team enjoying some space debris art... and wine, the day after