Global Optimization of Impulsive Interplanetary Transfers R. Armellin, Dipartimento di Ingegneria Aerospaziale,Politecnico di Milano

Taylor Methods and Computer Assisted Proofs Barcelona, June, 3 – 7, 2008

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Motivation

‣ Space activities are expensive:

‣ The goal of the trajectory design is to find the best solution in terms of propellant consumption while still achieving the mission goals

we want to reduce the required propellant

Ariane 5 launch cost: 200 M$ ÷Allowed Spacecraft Mass: 10000 kg =Cost per kilogram: 20000 $/kg

‣ Propellant represents the main contribution to s/c mass:• Propellant is on average 40% of spacecraft mass

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Outline

‣ Dynamical Model ‣ Patched-Conics Approximation ‣ Two-Impulse Transfers

• Ephemerides Evaluation• Lambert´s Problem Solution‣ Differential Algebra Based Global Optimization ‣ Rigorous Global Optimization with COSY-GO‣ Multiple Gravity Assist (MGA) space pruning

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‣ The 2-Body Problem considers two point masses in mutual orbit about each other

E.g.m1 Sunm2 Spacecraft

Dynamical Model: 2-Body Problem

m1

m2

The relative motion of the two masses is governed by:

!̈r = − kr3

!r

Analytical solutions exist for the 2-Body Problem: Conic Arcs • explicit• implicit (Kepler´s equation)

!r = !r(θ)t = t(θ)

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Patched-Conics Approximation

‣ The whole interplanetary transfer is divided in several arcs

‣ Each arc is the solution of a 2-Body Problem considering the spacecraft and only one other planet at a time

E.g.: 2-impulse Earth-Mars transfer 3 conic arcs

Earth escape Heliocentric phase Mars capture

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2-Impulse Planet-to-Planet Transfer

‣ 2-impulse Earth-Mars transfer has been selected as first benchmark problem• Applied for preliminary

design of Earth-Mars (any planet-to-planet transfer) interplanetary transfers

• Objective function characterized by several comparable local minima

‣ Future benchmark problems• Multiple Gravity Assist

interplanetary transfers E.g.: Cassini-Huygens

-8 -6 -4 -2 0 2 4-2

0

2

4

6

8

10

X [AU]

Cassini-Huygens trajectory (Alternative solution)

Y [AU]

Earth 25/10/1997

Venus (GA1)19/05/1998

Venus (GA2)24/06/1999

Earth (GA) 18/08/1999

Jupiter 17/02/2001

Saturn 01/12/2005

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Optimization Problem

‣ The optimization variables are the time of departure t0 and the time of flight ttof

‣ The positions of the starting and arrival planets are computed through the ephemerides evaluation:

(rE, vE) = eph(t0, Earth) and (rM, vM) = eph(t0 +ttof, Mars)‣ The starting velocity v1 and the final one v2 are computed by

solving the Lambert´s problem

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Optimization Problem

‣ The parking velocity and desired final velocities are

‣ The pericenter velocities of the escape and arrival hyperbolavcE =

õE/rcE v

cM =

õM/rcM

vp2 =√

2µM/rcM + v∞22vp1 =

√2µE/rcE + v∞1

2

‣ Objective function

‣ Constraint: ∆V1 < ∆V1,max∆V = ∆V1 + ∆V2

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Ephemerides Evaluation

‣ Polynomial interpolations of accurate planetary ephemerides (JPL-Horizon) are used for the preliminary phase of the space trajectory design

‣ Given an epoch and a celestial body, its orbital parameters can be analytically evaluated

‣ The nonlinear equation (Kepler’s Eq) is solved for the eccentric anomaly E

‣ The relation delivers θ‣ The position and the velocity (r, v) of the celestial body in

inertial frame reference frame are computed

(a, e, i,Ω,ω,M)

M = E − e sinE

tanθ

2=

√1 + e1− e tan

E

2

We have to solve an implicit equation: Kepler´s equation

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Lambert´s Problem (1/2)

Given:‣ initial position r1‣ final position r2‣ time of flight ttof

Find the initial velocity, v1, the spacecraft must have to reach r2 in ttof

• The solution of the BVP exploits the analytical solution of the 2-body problem

• Given r1, r2 and ttof there exists only one conic arc connecting the two points in the given time

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Lambert´s Problem (1/2)

‣ Several algorithms have been developed for the identification and characterization of the resulting conic arc

‣ A nonlinear equation must be solved (Lagrange´s equation for the time of flight): in which , , and

‣ The value of s and c depend on r1 and r2, so the nonlinear equation depends both on t0 and ttof

a(x) =s

2(1− x2)

f(x) = log(A(x))− log(ttof ) = 0

A(x) = a(x)3/2((α(x)− sin(α(x)))− (β(x)))

β(x) = 2 arcsin(

s− c2a(x)

)

‣ We used an algorithm developed by Battin (1960)

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DA Solution of Parametric Implicit eqs

‣ Search the solution of for p belonging to

‣ Use classical methods (e.g., Newton) to compute x0 solution of

‣ Initialize and as DA variables and expand

‣ Build the following map and invert it:

‣ Force ∆f = 0 so obtaining the Taylor expansion of of the solution w.r.t. the parameter:

f(x, p) = 0p ∈ [pl, pu]

∆x = ∆x(∆p)

f(x, p0) = 0[x] = x0 + ∆x [p] = p0 + ∆p

∆f =M(∆x,∆p)

(∆f∆p

)=

([M][Ip]

) (∆x∆p

) (∆x∆p

)=

([M][Ip]

)−1 (∆f∆p

)

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Example: Mars Ephemerides

Errors on position, (a), and velocity, (b), between the DA and the point-wise evaluation of Mars ephemerides

(a) (b)

Epoch interval: 40 days

Errors drastically decrease when the order of the Taylor series increases

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Example: Objective Function

‣ The DA evaluation of the planetary ephemerides and the Lambert´s problem solution enables the Taylor expansion of the objective function Taylor representation

of the objective functionTaylor representation error w.r.t. point-wise evaluation

Box width: 40 days

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Earth-Mars Direct Transfer

16

[1000, 6000]× [100, 600]Search space: Maximum departure impulse: ∆V1 < 5 km/sPlatform: Pentium IV 3.06 GHz laptop

Objective function overview

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‣ Bound the value of on IF discard

For each subinterval :

‣ Initialize t0 and ttof as DA variables and compute a Taylor expansion of the objective function ∆V and the constraint ∆V1 on

DA Based Global Optimizer (1/2)

DA based global optimization algorithm:

‣ Subdivide the search space in subintervals‣ Suitably initialize the value of

t0

ttof

‣ Bound the value of on IF discard

∆V

∆Vopt

min∆V > ∆Vopt

!X

!X

!X

!X!X

!X

!X

∆V1min∆V1 > ∆V1,max

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‣ EvaluateIF update , and store and

DA Based Global Optimizer (2/2)

‣ Build and invert the map of the objective function gradient:

‣ Localize the zero-gradient point IF discard

!x∗ = (t∗0, t∗tof )

!x∗ /∈ !X !X

∆V ∗ = ∆V (!x∗)

∆V ∗ < ∆Vopt ∆Vopt !x∗ !X

‣ If necessary, a more accurate identification of the actual optimum can be finally achieved using a higher order DA computation on the last stored subinterval

!x∗

!X

(∇t0∆V∇ttof ∆V

)=M

(t0

ttof

) (t0

ttof

)=M−1

(∇t0∆V∇ttof ∆V

)

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Earth-Mars Direct Transfer

[1000, 6000]× [100, 600]Search space: Maximum departure impulse: ∆V1 < 5 km/sPlatform: Pentium IV 3.06 GHz laptop

Solution 1:• 10-day boxes + 5th order• Pruning + Global Opt: 59.98 s• = 5.6973 km/s• x* = [3573.188, 324.047]∆Vopt

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Earth-Mars Direct Transfer

[1000, 6000]× [100, 600]Search space: Maximum departure impulse: ∆V1 < 5 km/sPlatform: Pentium IV 3.06 GHz laptop

Solution 1:• 10-day boxes + 5th order• Pruning + Global Opt: 59.98 s• = 5.6973 km/s• x* = [3573.188, 324.047]

Solution 2:• 100-day boxes + 5th order• Pruning + Global Opt: 0.55 s• = 5.6974 km/s• x* = [3573.530, 323.371]

∆Vopt

∆Vopt

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Verified GO of Earth-Mars Transfer

‣ Number of steps: 216911‣ Computation time: 4954.39 s‣ Enclosure of the minimum:

[5.6974155, 5.6974159] km/s

‣ Enclosure of the solution:t0 ∈ [3573.176, 3573.212]ttof ∈ [324.034, 324.088]

‣ Implicit equations can be solved in a verified way enabling the Taylor Model evaluation of the objective function

‣ COSY-GO is applied for the global optimization of an impulsive Earth-Mars transfer

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Planet-to-Planet Transfer

‣ Number of steps: 91447‣ Computation time: 2393.81 s‣ Enclosure of the minimum:

[7.0827043, 7.0827061] km/s

‣ Enclosure of the solution:

0 1 2 3 4 5 6 7 8 9 10x 104

5

10

15

20

25

30

35

Iteration #

CutO

ff [k

m/s]

t0 ∈ [3262.544, 3262.603]ttof ∈ [163.281, 163.369]

‣ Most of the computational time is spent in splitting box containing discontinuities

‣ Boxes containing discontinuities with size lower than a given threshold are rejected

‣ The solution is mathematically not rigorous

2800 3000 3200 3400 3600 3800 4000 4200150

200

250

300

350

400

450

500

550

Earth-Venus

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MGA Space Pruning

‣ MGA transfers problems are characterized by • High number of variables

(+1 for each GA)

• Constraints ongravity assist radiiand ∆VGA

• Increased multimodalityand more discontinuities

‣ “Validated” solution is almost impractical (we have obtained solution for 1 GA)

‣ The objective function is simply given by ∆V = ∆V1 + ∑∆VGAi + ∆Vn

‣ IDEA: use DA expansion and polynomial bounders just to prune the search space

Earth, T1, ∆V1

Mars, T2, ∆VGA

Jupiter, T3, ∆V3

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‣ For the first arc: • Subdivide and in subintervals• For each subinterval :

- Compute the expansion of on- Bound the polynomial expansion of on- IF discard and all its

subsequent combinations

DA Based Pruning of MGA

!X

!X

!Xmin∆V1 > ∆V1,max

∆V1

!X

!X∆V1

‣ For each gravity assist:• Subdivide in subintervals• For each subinterval remaining from the previous arc and for each

combination with the subinterval on ,

- Compute the expansion of and on- Bound the polynomial expansion of and on

!X

T1 T2

i = 1

i = 2, ..., n− 1

Ti+1

Ti+1 !X

∆VGAi rpi!X∆VGAi rpi

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DA Based Pruning of MGA

- IF or discard and all its subsequent combinations

min∆VGAi > ∆VGAi,max!X

‣ For the last arc: • For each subinterval remaining from the previous arc

- Compute the expansion of on- Bound the polynomial expansion of on- IF discard

∆Vn

i = n

!X

∆Vn !X

min∆Vn > ∆Vn,max !X

max rpi < rpi,min

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Cassini-like transfer

Planet Search Space[day]Int. size

[day]∆V Cut-off

[km/s]

E Dep. Epoch[MJD2000] -1000 - 0 50 4

V tof1 10 - 410 25 2V tof2 100 - 500 25 2E tof3 10 - 410 25 2J tof4 400 - 2000 200 2S tof5 1000 - 6000 200 6

‣ Problem Definition:

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Tot. boxes 32768000

Remaining 1085 (0.003%)

Obj Fcn 4.9357 km/s

Cassini-like transfer

‣ This work has been done under ESA Ariadna contract

‣ Final report available athttp://www.esa.int/gsp/ACT/ariadna/completed.htm

http://www.esa.int/gsp/ACT/ariadna/completed.htmhttp://www.esa.int/gsp/ACT/ariadna/completed.htmhttp://www.esa.int/gsp/ACT/ariadna/completed.htmhttp://www.esa.int/gsp/ACT/ariadna/completed.htm

Global Optimization of Impulsive Interplanetary Transfers R. Armellin, Dipartimento di Ingegneria Aerospaziale,Politecnico di Milano

Taylor Methods and Computer Assisted Proofs Barcelona, June, 3 – 7, 2008

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Conclusions and Future Work

‣ DA and TM global optimizers are effective tools for the global optimization of planet-to-planet transfers

‣ Efficient management of regions with singularities is needed for TM global optimization with COSY-GO

‣ Validated management of nonlinear constraints will be required to apply COSY-GO to MGA transfers

‣ DA is a promising technique for search space pruning of high dimensional problems such Multiple Gravity Assist (MGA) interplanetary transfers

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Verified Implicit Eq Solution - 1D

‣ Suppose to have the (n +1) differentiable function f over the domain D = [-1, 1] and its n-th order Taylor model P(x) + I so that

for all

‣ Consider the enclosure R of P(x) + I over D and suppose P´(x) >d >0 on D with P(0) = 0

‣ Find the Taylor Model C(y) + J on R so that any solution of the problem f(x) = y lies in C(y) + J

f(x) ∈ P (x) + I x ∈ D

Algorithm:

‣ First compute C(y), the n-th order polynomial inversion of P(x), so that

‣ Using Taylor model computation, obtain where includes the terms of order exceeding n in P(C(y)), and thus scales with at least order n +1

P (C(y)) =n y

P (C(y)) ∈ y + J̃ J̃

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‣ Use the consequences of small correction ∆x to C(y) to find the rigorous reminder J for C(y) so that all the solutions of f (x) = y lie in C(y) +J. According to the mean value theorem:

for suitable ξ ∈ [C(y), C(y) + ∆x]‣ Since is bounded below by d, the set

will never contain the zero except for the interval

which is the desired interval

Verified Implicit Eq Solution - 1D

f(C(y) + ∆x)− y ∈ P (C(y) + ∆x)− y + I= P (C(y)) + ∆x · P ′(ξ)− y + I⊂ y + J̃ + ∆x · P ′(ξ)− y + I= ∆x · P ′(ξ) + I + J̃

P ′

J = −I + J̃d

∆x · P ′(ξ) + I + J̃

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Verified Implicit Eq Solution - vD

‣ Let P(x) + I be a n-th order Taylor model of the (n +1) times differentiable function f over the domain so that:D = [−1, 1]v

• Indicate with L(x) the linear part of P(x)• Instead of the original problem and in analogy with the 1D

case, consider the problem of finding a verified enclosure of the inverse of where L is analytically inverted

f (x) ∈ P(x) + I for all x ∈ D

L−1◦ f

• The Taylor model enclosure of over D is:P̄ + J = L−1 ◦ (P + I)

P̄(x) + J L−1◦ f

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Verified Implicit Eq Solution - vD

‣ It is worth observing that:P̄1 = x1 + h.o.t. we can bound from below

P̄2 = x2 + h.o.t. we can bound from below

etc.

∂P̄1∂x1∂P̄2∂x2

Consequently we can proceed as in the 1D case on

‣ When the solution has been obtained for right-compose with

L−1◦ f

L−1◦ fL−1

‣ Application to the solution of f (x, p) = 0: y = f (x, p) p = p

Once a validated inversion of the system is achieved, just set y = 0{

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Orbital parameters

• The orbital parameters are: (a, e, i,Ω,ω, θ)

• The position and the velocity (r, v) in cartesian coordinates are obtained from the orbital parameters by simple algebraic relations