asteroid selection and preliminary trajectory …asteroid selection and preliminary trajectory...

Asteroid selection and preliminarytrajectory optimisation in multiple asteroid

rendezvous missionsMarco del Rey Zapatero, Dario Izzo, Tamas Vinko and Claudio Bombardelli,

ADVANCED CONCEPTS TEAM

Asteroid selection and preliminary trajectory optimisation in multiple asteroid rendezvous missions, GTOC3 workshop Turin, 2008

Contents

What went terribly wrong?

The multiple asteroid rendezvous problem

Building an upper bound

Objective function evaluationchemical propulsion

low-thrust propulsion

Results on the GTOC3 data set

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First hour of the competition....

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a e i ! !


Asteroid selection and preliminary trajectory optimisation in multiple asteroid rendezvous missions, GTOC3 workshop Turin, 2008 4

The multiple asteroid rendezvous problem



We consider sets each containing planets/asteroids / comets

We introduce the multiple asteroid rendezvous problem as the problem of selecting one asteroid from each group, one permutation of (the visit order), and the fly-by strategies , so that a set of constraints on on and are satisfied and some objective function is maximized

Dawn, Don Quijote, Marco Polo, GTOC2 and GTOC3 can be all considered as particular instances of this problem

Problem definition

5

n Ni

Ai ! Ai

{0, 1..., n! 1}s

sAivj

vj JN (Ai, s,vj)n! 1

Ai



Given the asteroid , the permutation and the fly-by strategy , the objective function is the global optima of a trajectory problem.

As soon as the number of asteroids or groups increase, and the allowed fly-bys are numerous, the problem complexity soon becomes computationally intractable.

The problem, overall, is a mixed integer programming problem, where discrete variables (the asteroid choice, the possible fly-bys) are mixed together with continuous variables (launch date, thrust magnitude and direction, etc.)

The objective function

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Ai svj JN (Ai, s,vj)



In general:


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JN (Ai, s,vj) = max:!n!1

i=0 m̃i(xi) + Jb(x)subject to: g(xi) <= 0, single phase const.

f(x) <= 0,multi-phase const.




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The following holds:

where,

max: m̃i(xi)subject to: g(xi) <= 0, single phase const.

ji(As(i), As(i+1),vi) =

JN (Ai, s,vj) <= JN (Ai,vj) =n!1!

i=0

ji(As(i), As(i+1),vi) + Jb



The Branch and Prune algorithmAs(0)

v0 v1

As(1) As(2) As(n)

vn!1

j0 > B




v0 v1

As(1) As(2) As(n)

vn!1

j0 > B




v0 v1

As(1) As(2) As(n)

vn!1

j0 < B




v0 v1

As(1) As(2) As(n)

vn!1

j0 < B j0j1 > B




v0 v1

As(1) As(2) As(n)

vn!1

j0 < B j0j1 < B




v0 v1

As(1) As(2) As(n)

vn!1

j0 < B j0j1 < B j0j1..jn!1 > B




v0 v1

As(1) As(2) As(n)

vn!1

j0 < B j0j1 < B j0j1..jn!1 < B



Objective function evaluation



Chemical propulsion

The overall problem dimension is given by: if we let maximum one DSM for each leg

find: x = [x1,x2, ...,xn!1] ! IN1N2...Nn!1

to maximize: mn/m0

subject to: rip >= ri

pmin

time const.

n!1!

i=0

[6 + 4fbi]



Low-thrust propulsionIn direct methods the Optimal Control Problem is transcribed into an Non Linear Programming Problem

Whenever numerical derivatives need to be computed numerical precision is lost, convergence is slow and the algorithm convergence radius is small

Automated differentiation (e.g. based on differential algebra techniques) dramatically improves the performance.

Modeling languages such as AMPL offer the possibility of interfacing to most of the state-of the art NLP solvers using automated differentiation

KNITRO, CONOPT, LANCELOT, SNOPT, DONLP2, IPOPT can all be used by just changing one line of code and they all get fed by AMPL with precise derivative information

For single phase transfers and transfer with one fly-by the problem is solved within a few seconds (also when it implies two or three revolutions), a precision of 1 km on the position 0.0001 km/sec on the velocity and 0.002 on the mass fraction, reducing at the same time the sensitivity to the initial guess



Example (no fly-by)

Computational time:

N = 15, 1000 iter4 sec

N = 60, 1000 iter35 sec



Example (one fly-by)



The GTOC3 case

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We consider 5 sets:

The restriction on the permutation can be set as:

We can force:

Restrictions in the asteroid choice can be set as:

A0 = A4 = {Earth}A1 = A2 = A3 = {Ast1, Ast2...Ast140}

s(0) = 1, s(4) = 4

s = {0, 1, 2, 3, 4}

A1 != A2 != A3

Definition of the problem



We still have to define the flyby strategies

For a direct transfer from to we will have

For a transfer involving a flyby to Earth while going from to , we will agree that

Example

vj = 0Aj Aj+1

Aj Aj+1

vj = {0, 1, 1, 0}Ai = {Earth, 34, 75, 68,Earth}

vj = 1



In the case of GTOC3 (Chemical version):


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JN (Ai,vj) =

Jb(x) = 0.2min !i

10 years

max:!3

i=0 m̃i(xi) + Jb(x)

subject to: rendezvous const.!i > 60Timelength < 10 years



We try to get the expression:

How can we find the bound for and the functions ?A mathematical bound for is This bound is too big for our purposes (as we will see later)A tighter value, that applies to trajectories with a moderate propellant consumption (high values for the mass fraction), has been set to be 0.0219 (400 days).


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JbJb 0.2/3 = 0.067

ji

JN (Ai,vj) <= JN (Ai,vj) =3!

i=0

ji(Ai, Ai+1,vi) + Jb



The solution to this problem is an upper bound and actually we can drop the i, and let it just be

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No fly-by strategy

We may calculate an upper-bound by finding the global optima of a deep space maneuver transfer between the asteroids:

! mfinal

minitial

"

i

find: xi ! I6

to maximize:

xi = [t, V!, u, v, T, !]

ji

j



Earth fly-by strategyIn the case an Earth fly-by is considered, we may find the bound by optimising a deep space maneuver transfer between the asteroid with the fly-by included

! mfinal

minitial

"

i

xi = [t, V!, u, v, T1, !1, bincl, rp, T2, !2]



Solving global optimisation problemsA large number of optimisation problems need thus to be solved to find the bounds and thus run the branch and prune

ACT DIstributed Global Multi-objective Optimiser (DiGMO) is able to solve efficiently global optimisation problems with a high reliability when a small number of legs is considered

DiGMO uses different population based heuristics (DE, PSO, SA, MC, GA, ACO) cooperatively to solve the same global optimisation problem in a distributed computing environment. The algorithm scales linearly with the number of computer used

The algorithms cooperation create a global optimiser with increased performances and reliability



Particle Swarm Optimisation

Evolutionary Algorithms

Simulated Annealing

Ant Colony Optimisation

Monte Carlo Search

DiGMO



20 40 60 80 100 120 140

20

40

60

80

100

120

140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9



20 40 60 80 100 120 140

20

40

60

80

100

120

140

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

j(Ai, Ai+1, 1)



JN (Ai,vj) =3!

i=0

j(Ai, Ai+1,vi) + Jb

So we execute the branch and prune algorithm. And we calculate the expression:

Breaking out if the j product overpasses a certain value B

This way with a value of B=0.8221 we remain with 2000 possible sequences.

We can guarantee (in the chemical version of the problem) that all sequences with a performance index above 0.8221+0.0219 = 0.8440 are included within those 2000.



200 400 600 800 1000 1200 1400 1600 1800 20000.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

Branch and Bound Rank

J

Without bonus With bonus Upper bound



The low-thrust (real) GTOC3

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We have solved a chemical version of the problem.To solve the low thrust problem we take a big leap by assuming that the best sequences for it, will be the best ones already found to be the best in the chemical model. Now what remains is to obtain an “low thrust” version of those trajectories. To do so, we could think of feeding the chemical solution as an initial guess for a local optimizer.Too complex: it does not usually lead to a good solution, and many times the procedure leads to non convergence.

The low thrust problem

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The low thrust problem - Simplification

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Given a sequence we treat each leg independentlyWe cannot do so rigorously: mass connects different legs.For good trajectories final mass is just 20 % less.But... We cannot trust that a low thrust solution coming from just optimizing locally the legs, is going to be optimal. Actually it may even be unfeasible (due to violation of multiphase constraints)

So we decided to produce lots of solutions for each particular leg, and then try to patch them together.To produce those solutions we took from the chemical model guesses for the departure, arrival and flyby times.



Example

Tdeparture

Leg with flyby

TFlight1

TFlight2



Leg with flyby



The red dots correspond to low thrust optimal solutions for one leg which will be stored.



The Branch and Prune algorithm1st leg 2nd leg 3rd leg 4th leg

Verifies the constraints?mf/mi > B1





YES!





NO!





YES!





YES! Evaluate and store valueof final objective function J!



The results

We tried this idea with the first 20 peaks. For each time value produced by the chemical model T, we generated a grid t={ T +50k, k=-9 , -8, ... 9 }We softened, in a first analysis, the time constraints to avoid problems coming small patching WT = 40 days, Total Timelength = 10 years + 40 days



Peaks



The results

For the best two peaks, we performed several more iterations of the procedure by refining the grid and setting the real time constraints. When the size of each cell was less than 5 days we stopped.



The results: E-E-49-E-37-85-E-E



First leg E-E-49-E-37-85-E-E



Second leg E-E-49-E-37-85-E-E



Third leg E-E-49-E-37-85-E-E



Fourth leg E-E-49-E-37-85-E-E



The results: E-49-E-37-85-E-E

asteroid selection and preliminary trajectory …asteroid selection and preliminary trajectory...

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