GTOC7�7th Global Trajectory Optimisation Competition�– Multi-Spacecraft Main-Belt Asteroid Tour –�

Organised by Profs. Lorenzo Casalino & Guido Colasurdo

Politecnico di Torino & Università di Roma “Sapienza” 20 May – 17 June, 2014

Results and Methods of the JPL Team (Team 19):

Anastassios Petropoulos, Gregory Lantoine, Damon Landau, Daniel Grebow, Try Lam, Nitin Arora, Drew Jones, Juan Senent, Mark Jesick, Juan Arrieta,

Aline Zimmer, Eric Gustafson, Julie Bellerose, Thomas Pavlak, Mar Vaquero, Jeffrey Stuart, Gregory Whiffen, Paul Finlayson, Jon Sims, Eugene Bonfiglio

Jet Propulsion Laboratory, California Institute of Technology, California, U.S.A.

GTOC7 Workshop

Rome, Italy 06 March 2015

© 2015 California Institute of Technology. Government sponsorship acknowledged

GTOC7 •  Four weeks to solve problem: 20 May – 17 June 2014 •  Rendezvous with as many asteroids as possible, given:

–  16,256 Main-Belt Asteroids –  3 probes, to be dropped from a mothership which they must rejoin –  10-year launch window 2021-2030, 12-year mission duration –  Launch v-infinity <= 6km/s, –  Launch mass = 24,000 kg, composed of

•  6,000 kg Mothership dry •  12,000 kg propellant for Mothership •  3 x 2,000 kg wet-mass probes (daughters) (800 kg dry)

–  Daughter TOF <= 6yrs (from separation to rejoining with Mother)

–  Mother: max 10 impulses total, Isp = 900 s –  Daughters: Low-thrust, Isp = 3000 s, Thrust <= 0.3 N –  Only Daughters’ asteroid rendezvous count, & only if separated

–  Secondary objective: Maximise sum of final Daughter masses

AEP / March 06, 2015 2

Broad Search Methods – Traj Building Blocks •  GTOC5-style optimiser-in-the loop trajectory build up was tried but

was too onerous –  GTOC7 asteroid-to-asteroid legs could be estimated, unlike GTOC5 legs

•  Daughter-Chain Backbone –  Mothership rendezvous with 1st asteroid, drops off 1st daughter

–  Daughter 1 goes to as many asteroids as possible –  Mothership rendezvous with final asteroid of D1 –  D2 and D3 chains then built to go to final asteroid of D1 chain –  … and variations on this theme

•  Mother-Traj Backbone –  Mothership drops off Daughters in asteroid ‘clusters’ (to be defined) –  Daughters pick asteroids that will take them closer to the final ‘cluster’ –  … variations in clusters and drop-off sequence

•  Daughter-Chain Backbone much more successful –  Clusters not populous enough –  Competition for asteroids amongst daughters not severe enough to matter

AEP / March 06, 2015 3

Broad Search Methods – Asteroid Selection

•  Lambert based –  STAR – Lambert arcs between all asteroid pairs at discrete points in time –  Ant Colony Optimisation (ACO) –  GTOC5-style asteroid-chain build-up

•  Phase-free –  Q function (proximity quotient): ‘best-case quadratic minimum time-to-go’, orbit

element coupling –  he metric: ‘distance’ between orbits based on angular momentum and

eccentricity vectors –  Projective cost as well (for AàB, Pick based also on cost BàZ, where Z is a

desired final asteroid)

•  Borg cube –  Asteroid close approaches, plotted as AstNum x AstNum x Date

•  Asteroid Clumping in orbit elements

AEP / March 06, 2015 4

Broad Search Methods – ‘Glue’ and Overview

•  AF (Awesome Filter), Slayer –  Executables to give next asteroids ranked by DV

•  Q & he scripts to give ranking of next asteroids •  BBC (Backbone crusher)

–  Takes a D1 backbone and tries to fit to it precomputed long chains for D2, D3

•  STAR and ACO

AEP / March 06, 2015 5

ACO

STAR Malto ‘Gtool’ Malto

Global Search

Independent optimization

Multiple s/c optimization

Validation & Submission

Malto BBC

Slayer he Q

AF

Malto

AEP / March 06, 2015 6

SQP method SNOPT Analytic derivatives

Ant Colony Optimization (ACO)

•  Classic ACO: –  Biology inspired probabilistic search algorithm –  Finds good paths through graphs

•  Defined by a sequence of discrete variables •  Good for constant edge-cost graphs (e.g. Travelling Salesman

Problem) •  Not so good for: Mixed-Variable Optimisation, path-dependent edge-

costs, NL constraints

AEP / March 06, 2015 7

Modified ACO Highlights

1)  Previous generation information used if and only if the partial sequence exactly matches –  Constraint violating partial sequences added to Tabu list

•  Probability of re-visiting sequence = 0% –  Increased probability of ‘partially’ repeating Feasible sequences

•  Inversely proportional to the cost of those solutions

2)  ACO divorced from the underlying problem –  Continuous problem evaluated after each generation forms sequences –  ACO ‘knows not’ the meaning of the integer values its selecting

3)  Current ‘Queue’ can depend on partial sequence

AEP / March 06, 2015 8

GTOC7 Sub-Problem

•  Given ‘backbone’ chain (seed), find companion chains –  Having same start & end asteroids –  Completely unique interior asteroids –  Some flexibility on start/end epochs

•  Implementation –  Modified ACO first yields integer sequences –  Each integer = an index to an ordered list of Lambert arcs on given leg

•  Ordered by lowest ΔV –  Lamberts evaluated via Fortran parallel routine “slayer” –  Phase-free metrics (Q-law) used to prune the set of targets

AEP / March 06, 2015 9

N N N

L L L

q

Phase-free-metric Prune

slayer

•  This implementation must be solved in serial… Enumerating the combinations is insane!!

p*L*M Lamberts

p*L*M Lamberts

•  With ACO sequence (e.g. 1-3-2), Lambert evaluations are: Mpq ⋅⋅~

00 t, Af t, fA

p*M Lamberts

p*L Lamberts

MM M

asteroids all ofSet =N

M = Phase-free metric pruned set of asteroids

state starting a from arcsLambert best of #=L

grid TOF of izeSp =

chain ofLength =q

}

} 1

3

2

GTOC7 Sub-Problem

•  Optimisation –  Minimize Total ΔV –  Constrain: total TOF , single leg ΔV, total ΔV –  Adjusted constraints such that ‘feasible’ solutions mostly converge in Malto

•  6.8 years, 3.0 km/sec, 21.0 km/sec

•  Sample result:

–  Seed (14 asteroid-chain):

–  Find D2 chains, excluding interior seed asteroids: (12-ast chains)

–  Repeat for D3, excluding interior asteroids from D2 … –  This seed with D2 and D3 companions, led to 35 point solution

AEP / March 06, 2015 11

STAR Method

•  Compute ‘all’ Lambert-arc transfers between all asteroid pairs and also Earth-asteroid on a time grid: –  Matlab code, vectorised and highly tuned –  Man-in-the-loop still needed (next slide) –  One Linux Workstation, 8 cores (only sometimes fully used)

Comp Time (given previous chains), hrs

Asteroid Chain Length

12* 2 6 7 4 8 8 9 12 10 12 11 8 12 4 13 <1 14

Total Time: <~3 days

* A GPU implementation took about 4-5 hrs

AEP / March 06, 2015 12

How to make half a billion 13-asteroid chains�(STAR)

1)  Don’t worry if it’s possible, “creativity is embarrassing” 2)  Break down the problem

–  All ast-ast transfers are rendezvous, so each leg can be computed independently

–  Grid up encounter times in 10-day increments – only 42 trillion combinations!

–  …but the 13 million time/asteroid nodes can be stored in RAM 3)  Filter, filter, filter

–  d < fudge*0.5*a*t^2 (d is distance between asteroids at arrival) –  If it’s infeasible for 200 days, 190 days won’t fly either –  Derive an analytic approximation for min feasible acceleration

4)  Blow up your hard drive

Flight  (me  on  single  leg,  days  

No.  of  Sol’ns  

AEP / March 06, 2015 13

Constant acceleration approximation

•  Assume gravity is same on Lambert and low-thrust arcs, i.e. solar tide is negligible

•  Solve for accelerations and switch time that nullify V∞ with |a1|=|a2|

•  Iterate to get ΔV (& acceleration) as function of coast time

Lambert  sol’n  

V∞1    

V∞2  

Asteroid  1  Velocity  

Asteroid  2  Velocity  

Constant    accelera(on  with  single  direc(on  switch  

a2  

a1  

•  Derived acceleration and given thrust mag provide initial mass

•  Combine with DV to get appx of final mass as function of initial mass

•  Much better predictor of feasibility and performance than Lambert alone for little additional computation

Lambert ΔV, km/s

Max

Arr

ival

Mas

s on

leg,

kg

AEP / March 06, 2015 14

…13+ links

5)  Now have final mass as function of initial mass for all possible asteroid transfers

6)  Match end asteroid of transfer(s) 1 to beginning asteroid of transfer(s) 2

–  only consider transfers that are feasible for the current masses –  filter out chains with duplicates or low end mass –  Wash, Rinse, Repeat

7)  Problem gets big (TB+) approaching 10-asteroids –  need to use hard drive efficiently –  sort files by asteroid #, only read in one asteroid at a time to link –  could be done in parallel

8)  34 million unique 13-asteroid sequences –  filtered from 480M with unique times

9)  175 thousand unique 14-asteroid sequences 10) No feasible 15-asteroid sequences found

AEP / March 06, 2015 15

‘Single-Spacecraft’ formulation

•  Mother rendezvous with asteroids A0 and Af •  Daughters start at A0 and end at Af •  Mother and daughter trajectories are optimized independently

M  

Di  

E  

Ai1  

A0   Af  

A0   Af  

i=1,2,3  

AEP / March 06, 2015 16

Multiple s/c formulation

•  Mother no longer needs to rendezvous with A0 and Af •  All daughter sequences start and end with their mother •  Mother is coasting when daughters are released and picked up •  Problem can be solved using ‘Gtool / OptiFor’ (with SNOPT solver)

–  Different discretization and control scaling between mother & daughters –  Objective function is the sum of the daughter final masses

•  Initial guess from standard formulation (Malto or ConstantAcc/Lambert)

M  

Di  

E   A0   Af  

Af  

Xm20  =  Coast(Xm1f,  tm20-­‐tm1f)  tm1f  <  tm20      

Xd10  =  Coast(Xm1f,  td10-­‐tm1f)  tm1f  <  td10  <  tm20  

Xdff  =  Coast(Xm2f,  tdff-­‐tm2f)  tdff  >  tm2f  

Ai1   Ai2  AEP / March 06, 2015 17

Daughter asteroid sequences

•  All daughter sequences start and end with their mother (-1) •  The daughters should not visit the same first and last asteroids

–  But that is the case with the standard formulation •  Several combinations must be run to decide the best way to chop off

the unnecessary repeats from the daughter sequences –  Mini-combinatorial problem !

•  Example: our submitted 36-pt solution –  Initial guess from STAR & Malto

800  kg  834  kg  777  kg  

810  kg  827  kg  804  kg  

800  kg  868  kg  800  kg  

14-13-13-length ast chains

13-12-11

12-13-11 è J=36

Gtool masses

AEP / March 06, 2015 18

35-pt solution bumpy road

•  34-pt –  14-ast backbone D1 from STAR & BBC, 12-ast D2 and D3 from ACO + Malto

•  35-pt

–  One asteroid (12538) added to D2 with close approach tool –  Bug: one repeated asteroid !

•  True 35-pt, J’~2555kg –  New D2 sequence from ACO using 12538 and without repeated asteroid

808  kg  792  kg  785  kg  

804  kg  869  kg  903  kg  

795  kg  818  kg  928  kg  

801  kg  831  kg  923  kg  

14-12-12

14-11-11

14-11-10

14-11-10

AEP / March 06, 2015 19

JPL’s Submitted Solution

J = 36, J’ = 2,450.257 kg

AEP / March 06, 2015 20

−4 −3 −2 −1 0 1 2 3 4−3

−2

−1

0

1

2

3

NBCP: Return of probe 3 (5/30/2034) tof: 3704.9 days

X (AU)

mother

Depart:Earth (4/12/2024) tof: 0.0 days v'

: 6.00 km/s Y (A

U)

Mother Mf = 8027.16 kg 8 impulses

1st release (D2) Mf = 826.62 kg

JPL’s Submitted Solution

J = 36, J’ = 2,450.257 kg

AEP / March 06, 2015 21

2nd release (D1) Mf = 816.93 kg

3rd release (D3) Mf = 806.70 kg

−4 −3 −2 −1 0 1 2 3 4

−3

−2

−1

0

1

2

3

X (AU)

daughter3_new

Y (A

U)

Asteroid or MotherShipmatchpoint

A Brief Timeline of J At JPL

AEP / March 06, 2015 22

Day 0 Day 28

EAAA STAR & Malto

5

EA7

GTOC5-style Malto

7 9

EA

12

11

EA

13

EA13 ConstAcc EA14 Prelim. STAR

14

J=28 BBC

18

J=32 ACO

20

STAR +

J = ’35’ (34, bug!)

ACO, Gtool, Malto

24

EA14 Full STAR

25 26

J =

35

Day 27.5 J = 36 Gtool

Day 28 J = 36

Gtool, Malto

Gtool hiatus STAR hiatus

Trajectory Animation���

Legend��

Yellow-Red Central Body – Sun�Blue Line – Earth’s orbit�

Big Blue Dot – Earth�Big Magenta Dot – Mothership�

�Small Red Dots (fading) – Unvisited Asteroids�

Green/Red/Blue squares – Asteroids Visited by 1st/2nd/3rd probe releases�Green/Red/Blue ‘X’ – 1st/2nd/3rd Probe (release order)�

AEP / March 06, 2015 23

Thank You to�Profs. Casalino and Colasurdo !

AEP / March 06, 2015 24