Global Trajectory Optimisation Tools for the 21st Centurytrajectory.estec.esa.int/Astro/3rd-astro-workshop... · 2006-09-26 · Global Trajectory Optimisation Tools for the 21st Century

1

Global Global Trajectory Trajectory

Optimisation Optimisation Tools Tools for for

the 21st the 21st CenturyCentury

Dr. Miguel BellDr. Miguel Bellóó MoraMora

OctoberOctober 22ndnd, 2006, 2006

33rdrd International International Workshop on Workshop on Astrodynamics Astrodynamics Tools Tools and and TechniquesTechniques

2

Table of ContentsTable of Contents

• Historical Perspective

• Introduction

• Constrained Parameter Optimisation

Problem Formulation

Solving Methods and Examples

• Constrained Optimal Control

Problem Formulation

Solving Methods and Examples

• Global Trajectory Optimisation Tool:

Requirements

Architecture and Modules

Example

• Conclusions

3

Historical Perspective 1Historical Perspective 1

It uses a generalized Ricattitransformation technique to reduce the TPBVP to an initial value problem

Indirect: Successive Sweep Method (SSM)

Optimum aerospace trajectories

McReynoldsBryson

1965

Lasdon/Waren/Rice (1967): interior penalty functions Ladsdon (1970) : conjugate directions

Second order gradient method

Optimal control of non-lineal systems

Breakwell Speyer Bryson

1963

Bellmann (1957), Bellmann/ Dreyfus (1962), Jacobs (1967)

Dynamic programming


Bellmann1965

Bryson/Denham/Dreyfus (1963), Denham/Bryson (1964), Denham (1966): optimality conditions for constrained case

Steepest ascent –Gradient method

Unconstrained trajectory optimisation with prescribed initial/final state

Bryson, Denham

1962

Kelley (1962): penalty functionsfor path constraints; Kelley/ Kopp/Moyer (1963): second order

Steepest ascent -Gradient method

Unconstrained flight path with prescribed initial and final state

Kelley1960

Continuation / commentsMethodOptimisation Problem

AuthorYear

4


Lasdon/Waren/Rice (1967): interior penalty functions Ladsdon (1970) : conjugate directions

Conjugate gradient method: fast convergence

Optimal control with path constraints

Lasdon, Mitter, Waren

1967

Miele/Levy/Iyer/Well (1970), Miele/Well/Tietze (1973)

Second order methods

Optimal control with equality and inequality path and boundary constraints

Miele Iyer Well

1970

Gradient, separate arcs computation

Optimum flight path with state variable inequality constraint

Speyer, Mehra, Bryson

1967

Additional state and control variables (slack), Jacobson (1968): second order

Gradient, transformation technique

Optimum control with state variable inequality constraint

JacobsonLele

1969

Chernousko/Horwith/Sarachik 1968: Davidon’s method in Hilbert space; Edge/Powers 76

Min-H method (second order to speed convergence)

Constrained flight path with prescribed initial and final state

Gottlieb1967

Järmark (1973), first and second order DDP method for aerospace problems

Differential Dynamics Programming (DDP)

Optimal control with path constraints

JacobsenMayne

1970


AuthorYear

5


Dixon, Hersom, Maany (1984), Zondervan, Bauer, Betts, Huffman (1984) for Space Shuttle re-entry, Dixon, Hersom, Maany, Patel (1985)

Hybrid method with adjoint control transformation

Optimum low thrust interplanetary trajectories

DixonBiggs

1972

The conditions for local control optimality are used to determine the control as an explicit function of the state variables and time

Indirect: Modified Sweep Method (MSM)

Apollo re-entry trajectories

ColungaTapley

1972

Deuflhard (1975), Dickmanns/Well (1975), Maurer (1977), Bock (1978), Oberle (1979): BNDCSO program, Stoer/Bulirsch(1980), Chern (1988), Bulirsch/Stoer/Deuflhard (1990)

Multiple Shooting Algorithm (MSA)


Bulirsch1971

Brusch (1974, 1977): optimisation of Space Shuttle and Atlas/Centaur launchJohnson (1975): solid rocket assisted space shuttle launch

Direct methods, discretisation of control leading to NLP

Optimisation of rocket ascent trajectories

Speyer Kelley Levine Denham

1971


AuthorYear

6


User for the Galileo orbit designMPBVP solverMultiple Swingby Optimisation

D’AmarioByrnesStanford

19801981

Miele (1975), Miele/Damoulakis /Cloutier/Tietze (1974), Miele/ Pritchard/ Damoulakis (1970), Henning/Miele (1973), Heideman/Levy(1975): conjugate SGRA

Sequential Gradient Restoration Algoritm (SGRA)(first order indirect method)


Miele1973

Program TOMPDirect methodOptimum aerospace trajectories

Kraft (DLR)1980

Program POSTDirect methodOptimum aerospace trajectories

Brauer (NASA)

19751977

Friedlander, Feingold, Bender, Hollenbeck: classical Δ-VEGAtechniques

Multiple gravity assist of Venus and Mercury

Optimum trajectories to Mercury

FriedlanderFeingold

1977


AuthorYear

7


First use of Genetic Algorithms in ESA for space problems

Genetic Algorithm


Gómez Tierno(UPM)

1982

Miele, Venkataraman (1984), Miele, Basapur (1985), Miele, Basapur, Lee (1986), M. Belló Mora (1988-1992)

Indirect method: Gradient Restoration

Optimal minimax aeroassisted transfer and re-entry trajectories

MieleMohantyVenkataramanKuo

1982

Horn (1989): STOMP, Hargraves/Paris (1987): direct collocation TROPIC; Jänsch (1990); Jänsch, Kraft, Schnepper, Well (1989): ALTOS;Jänsch, Schnepper (1991): DLR Optimisation Course

Direct multiple shooting method: PROMIS


BockPlitt

1984

Bartholomew-Biggs/Dixon/ Hersom/Maany/Flury/Hechler (1987), Dixon/Bartholomew (1972, 1982): adjoint control transformation

Direct and hybrid methods

Low thrust interplanetary and rocket ascent

Bartholomew – Biggs

1980


AuthorYear

8


V. Companys (1992), M. Belló Mora (1996), M. Hechler (1996-2006)

Parallel shooting, NLP solvers

Optimum Transfer to Libration Point Trajectories

Gomez1985

Multiple gravity assists, including single and double lunar swingby (SLS and DLS)

Optimum ICEE trajectories

FarquharDunhamFolta

19851987


AuthorYear

9


Optimisation of Cluster, Rosetta and Giotto Post-Halley mission

NLP solversMultiple gravity assists optimisation

Belló MoraM. Hechler(ESOC)

1986

Multiple gravity assists with reverse Δ-VEGA techniques and resonant transfers

Optimum trajectories to Mercury

Yen (JPL)19851988


AuthorYear

10


Generic multiple gravity assist solverParallel Shooting formulation and NLP solver

Optimum Multiple Gravity Assists (INTNAV)

Rodriguez-CanabalBelló-Mora(ESOC)

19911994

Weak Stability Boundary Transfers

Optimum lunar transfer trajectories with electric propulsion

Belbruno1987


AuthorYear

11


Enright, Conway (1992): Direct Collocation method (DCNLP), with NPSOL as NLP solver and Hermite cubic interpolation

Optimum trajectory with low thrust

EnrightConway(Illinois)

1991

Direct collocation with equinoctial coordinates (4352 variables, 3484 non-linear constraints)

Optimal Low Thrust Trajectories

Betts1993


AuthorYear

12


Weak Stability Boundary Transfers

Optimal Lunar Transfer

BelbrunoMiller

1993


AuthorYear

13


Direct collocation with modified equinoctial coordinates

Optimal Earth-Venus-Mars Low Thrust Trajectories

Betts1994

Direct collocation with differential inclusion; remove explicit control dependence, reducing the size of the NLP problem

Optimal Interplanetary Low Thrust Trajectories

Coverstone-Carrol (Illinois)Williams (JLP)

1994


AuthorYear

14


Kluever (1995), Kluever(1995), Pierson (1995), Kluever (1996), Kluever, Chang (1996)

Hybrid method, combining Direct (NLP) and indirect methods, using a switching function

Optimum lunar transfer trajectories with chemical and electric propulsion

KlueverPierson

1994

Runge-Kutta parallel shooting to reduce the NLP problem to 1027 variables to optimise

Direct collocation with equinoctial coordinates

Optimum LEO-GEO transfer with 100 revolutions

ScheelConway(Illinois)

1994

Direct collocation; the problem is divided in 3 phases

Optimal Earth –Mars Low Thrust Trajectories

TangConway(Illinois)

1994


AuthorYear

15


Direct Collocation method, with NZQPT as NLP solver and 7th degree Gauss-Lobato collocation

Optimum lunar transfer with low thrust

HermanConway

1995

Multirevolution method approximating low thrust spiral by analytical expressions of Edelbaum

Optimum lunar transfer trajectories with chemical, electric propulsion and multiple (>100) revolutions

Kluever1995


AuthorYear

16


Coverstone-Carrol (1997): micro-genetic algorithm with a population of only 20 individuals; Hartmann, Coverstone-Carrol, Williams (1998): Pareto Genetic Algorithm with multiobjective optimisation

Genetic Algorithm: withtournament selection, single point crossover, jump mutation and elitism

Optimum Interplanetary transfer with low thrust

RauwolfCoverstone-Carrol

1996

Extremal control accelerationIndirect methodOptimum low thrust lunar transfer

Guelman (Israel)

1995

Direct NLP and hybrid methods using Laplace and Momentum vector transformations

Ballistic multiple gravity assists and Low Thrust optimum trajectories to Mercury

F. HechlerM. Hechler(ESOC)

19951996


AuthorYear

17


Various methods

Optimum New Millenium Trajectories: Deep Space 1, Phobos Sample Return, ...

WilliamsCoverstone-Carrol

1997

Thrust limited path constraintsDirect transcription

Optimum low thrust trajectories

WenzelPrussing(Illinois)

1996

Analytical Keplerian class

Optimal thrust limited transfer between coplanar orbits

Makropoulus1996


AuthorYear

18


Direct method, NLP solver is a Sequential Quadratic Programming(SQP)

Combined multiple gravity assist and solar electric propulsion trajectory to the boundary of the heliosphere

Kluever1997

Eccentric longitude and epoch eccentric longitude formulation

Indirect method

Optimum low thrust trajectories

Kechichian19971999


AuthorYear

19


Hybrid method (OPRQP as NLP solver)

Optimum comet and asteroid rendez vous with combination of multiple gravity assist and electric propulsion

CanoHechler(ESOC)

1998

Hybrid method (OPRQP as NLP solver)

Optimum comet and asteroid rendez vous with electric propulsion for SMART 1 mission

HechlerYañezCano(ESOC)

19971998


AuthorYear

20


Indirect method with “smoothing” averaging technique (Geffroy)

Optimum lunar transfer with electric propulsion for SMART 1 mission

JehnCano(ESOC)

1999

Forwards / backwards scheme, simple thrust laws

Optimum lunar transfer with electric propulsion for SMART 1 mission

SchoenmaekersPulidoJehn(ESOC)

1998


AuthorYear

21


Forwards / backwards scheme, use of lunar resonnace orbits with averaging low thrust optimisation

Optimum lunar transfer with Moon gravity assists and electric propulsion for SMART 1 mission

Schoen-maekersPulidoCano (ESOC)

1999


AuthorYear

22


Direct method, singular 180º arcs, resonant transfers

Ballistic and Combined multiple gravity assist and solar electric propulsion trajectory to Mercury

Langevin1999

Delta launcher, Xenon ion thrusters, thrust law discretised at control nodes and linearly interpolated

Direct method, NLP solver is a Sequential Quadratic Programming (SQP)

Combined multiple gravity assist and solar electric propulsion trajectory to Mercury

Kluever Abu-Saymeh

1998


AuthorYear

23


A WSB transfer type is solved by using a gentic algorithm

Genetic Algorithm

Optimum lunar transfer for the LUNARSAT mission

Biesbroek1999

Systematic method to obtain the initial adjoint variable guess to be applied to the two point boundary problem

Indirect method

Low thrust optimum trajectories

YanWu(China)

1999

Indirect method with a shooting procedure

Combined multiple gravity assist and solar electric propulsion trajectory for solar system escape

CasalinoColasurdoPastrone (P Torino)

1999


AuthorYear

24

Introduction (1)Introduction (1)

• The design of space trajectories leads to two type

of mathematical optimisation problems:

Optimisation of a discrete number of mission parameters

to minimise a cost function subject to equality and

inequality constraints: Non-linear Programming (NLP)

problem

Optimisation of the time evolution of a set of control

variables to minimise a cost function subject to initial, final

and path, equality and inequality constraints:

Constrained Optimal Control (OC) problem

25

Introduction (2)Introduction (2)

•Interplanetary Low

Thrust trajectory

•Atmospheric Re-entry

•Aeroassisted transfer

•Aerobraking

•Launcher ascent

•Solar Electric

propulsion

•Ion engine

•Low Atmosphere flight

•Rocket engine

Constrained

Optimal Control

(OC) Problem

•Optimum orbit transfer

•Optimum orbit control

•Interplanetary design

•Weak Stability

Boundary Transfers

•Chemical propulsion

•Impulsive manoeuvre

•Heliocentric arcs

•Powered swingbies

•Unpowered swingbies

Constrained

Parameter

Optimisation or

Non-Linear (NLP)

Programming

Trajectory Design

Problem

Propulsion system

or Trajectory

Mathematical

Problem

26

Constrained Parameter OptimisationConstrained Parameter Optimisation

• The constrained parameter optimisation problem is

formulated as follows:

Find the minimum of a cost function f(x)

Where x is a vector of parameters with dimension n

Subject to

Equality constraints: gi(x) = 0, i=1, ..., meq

Inequality constraints: gi(x) > 0, i=meq+1,...,m

27


• For equality constraints, the mathematical solution is obtained

by introducing the Lagrange multipliers (1755), and the

augmented cost function:

• The necessary optimality conditions are:

i

m

ii xgxfxL

eq

)()(),(1

∑=

+= λλ

0),(

0),(

=∂

∂

=∂

∂

i

i

xLxxL

λλ

λ

28


• For inequality constraints, the mathematical problem is more

complex. The solution is given by introducing the Karush – Kuhn –

Tacker conditions:

• Where W is the Hessian of the Lagrange function and the columns of

the Z matrix are a base of orthogonal vectors to the Jacobian of the

constraints

0)(),()(

,...,,0

,...,,0)(

,...,,0)(

,...,1,0)(

,...,1,0)()(

1

≥

=≤

=≥

=≥

==

==∂

∂+

∂∂ ∑

=

xZxWxZ

mmi

mmixg

mmixg

mixg

nix

xgxxf

T

eqi

eqii

eqi

eqi

i

jm

jj

i

λ

λ

λ

λ

29


Gradient methods present

good convergence but local

minima can be found

Identify the most suitable

NLP solver: deterministic,

probabilistic or hybrid.

3.- NLP

Solver

Method

•Gradient methods are very

sensitive to the initial guess

•To ensure global minimum,

an exhaustive search must

be done

•Find a first trajectory:

–Systematic scan

–Genetic algorithm

–Simplified methods

(patched conics)

2.- Initial

Guess of

Solution

(most

difficult step)

•Scale factors for:

–variables,

–constraints and

–cost function

•Technical constraints

•Identification of the most

suitable set of parameters

•Identification of equality

and inequality constraints

1. Problem

Formulation

CommentsObjectiveStep

30

NLP NLP Solver MethodsSolver Methods

Search Fibonacci, uniform, asymmetric Gradient OPRQP: recursive quadratic

algorithm with penalty function (Biggs, NOC, Hatfield)

Deterministic

Conjugated Directions

Davidon Fletcher Powell, Fletcher & Reeves, Zongwill

Monte Carlo Pure Random, Chichinazde Random Search

Bremerman, adaptavive search from Matya, Beltrami and Indusi Probabilistic

Genetic Algorithm

Selection, isolation, genes crossover, cromosoms mutation

Hybrid Combination Hartman, Torn, Faginoli, Gaviano

31

NLP Example: Interplanetary TrajectoriesNLP Example: Interplanetary Trajectories

• Classical interplanetary trajectory design is done by

combining different type of arcs:

Launch phase with hyperbolic escape departure

Classical heliocentric arcs between planetary flybies

Powered or unpowered planetary swingbies or gravity

assist trajectories

Singular transfer arcs (180º or 360º transfer angle)

Weak Stability Boundary Transfer techniques

Delta-V Leveraging techniques (Delta-VEGA)

32


TIERRASOL

c 0017-00-INTERP

VENUS

MARTE

• The Multiple Point Boundary Value

Problem (MPBVP) is solved in 3

steps:

– “Patched Conics” (trajectory

with discontinuities)

– “Matched Asymptotic

Expansions” (continuous

trajectory)

– Numerical integration

(parallel shooting + Newton

method)

33


c 0018-00-INTERP

t2

t1

r2

r1F

a

• Lambert problem: find the

Keplerian trajectory joining two

points in a given flight time.

• Lambert Theorem (1761): the

transfer time depend on the sum of

radii, c and the semimajor axis a.

• Numerical Formulations (iterative):

– Lambert-Euler method

– Gauss method (1801)

– Escobal (1965), C. Simó

(1973), Battin (1987)

34


• Gravity Assists trajectories are extensively used in

interplanetary trajectory design:

35


36

Powered SwingPowered Swing--by by

• If classical powered swing-by is considered:

– Manoeuvres at incoming or outgoing

infinity plus

– Manoeuvre at a finite distance (Type

F)

• Three options are available:

– Full optimisation for each case (CPU

intensive)

– 3D interpolation in tabulated

solutions

– Simplified solution types F, ∞-F or F- ∞

with pericenter kick restriction (cover

> 95% of the optimum solutions and

it is very fast)

R0

μ ΔVm

ΔV1 V1

Rm θm

ΔV2

V2 δ

37

180º Singular Transfer180º Singular Transfer

Vd1 Ve

Vd2

Vd V∞ Ve α γ UV Ur Ve

α V∞ V∞ α = 2 arcsin Ve 2*Ve*cosγ

Vd = Ve± V∞ cos α/2 un - V∞ sin α/2 uv

38

360º Singular Transfer360º Singular Transfer

u2

Vd

Ve

u1V∞

θ

cosβ = Ve

2+V∞2-Vd

2

2 Ve V∞

u2

Vd

u1

V∞

β

Ve

α

Vd = -V∞ cosβ uve + V∞ sinβ (cosθ u1 + sinθ u2)

39


40


41


1 MOONSWINGBY

TARGETARRIVAL

ΔV1

st

2 LUNARSWINGBYnd

ΔV3

PARKINGORBITt , Ω , ω o o o

tA

ΔX = 03ΔV = 03

t , V (3), P (2)p a bs

t , V (3), P (2)2 a2 2

ΔV2

t , V (3), P (2)3 a3 3

ΔX = 0

ΔX = 0ΔV = 0

ΔX = 0

3 LUNARSWINGBYrd

LUNARORBIT

X -X = 0f t

24 optimisation variables: ♦ initial: tp, Ω0, ω0 y ΔV1 ♦ 1er swingby: tps, Va(3), pb(2) ♦ 2º swingby: tp2, Va2(3), pb2(2) ♦ Date of 3ª man. tm ♦ 3er swingby: tp3, Va3(3), pb3(2) ♦ Arrival date TA 15 equality constraints:

♦ X1f - X1b = 0, ♦ V1f - V1b = 0, ♦ X2f - X2b = 0, ♦ X3f - X3b = 0, ♦ Xf - XT = 0, 4 inequality constraints:

♦ Min. Alt. (200 km) 1er swingby ♦ Min. Alt. (200 km) 2º swingby ♦ Min. Alt. (200 km) maniobra ♦ Min. Alt. (200 km) 3er swingby

The cost function is ΔV

42

OPTIMAL CONTROL PROBLEMOPTIMAL CONTROL PROBLEM

The optim al control problem is form ulated as finding the optim um value of the control vector tim e function u and the param eters P in order to m inim ise the function:

∫ +=ft

tf PtxtxgdxPuxLJ

0

)),(),((),,( 0

subject to the boundary and path, equality and inequality constraints:

qkkiPtutxCkkePtutxC

rkiPttxttxD

kePttxttxDPtutxfx

i

e

ffi

ffe

,...,1,0)),(),((,...,1,0)),(),((

,...,1,0),),(,),((

,...,1,0),),(,),(()),(),((

00

00

+=≥==

+=≥

===&

43


Variational Calculus or Pontryagin Principle is applied. The Hamiltonian and the adjoint vector are introduced such that:

0)(

,...,1,0

,...,1,0

,...,1,

)(0

=⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂

==∂∂

+∂∂

==∂∂

=∂∂

−=

+++=

−++= ∫

fT

j

T

j

j

jj

Tt

t

t

TT

tjD

tg

ljPD

Pg

mjuH

njxHp

CLCfpH

dtxpHDjJ f

υ

υ

μλ

υ

&

&

44


The transversatity conditions are:

For the inequality constraints:

There are discontinuities in the adjoints at the “switching points” between constrained and unconstrained regions.

)()(

)()( 00

fT

f

T

txD

xgtp

txD

xgtp

⎥⎦⎤

⎢⎣⎡

∂∂

+∂∂

−=

⎥⎦⎤

⎢⎣⎡

∂∂

+∂∂

−=

υ

υ

qkkituxCqkkituxC

ii

ii

,...,1,0),,(,0,...,1,0),,(,0

+=≤=+==≥

μμ

45


• The resulting general constrained optimal control is

a very complex Multiple Point Boundary Value

problem:

State vector and adjoints differential equations shall be

solved

Some variables are known at the initial, some at the final

and some at the intermediate points of the trajectory

Optimality and transversality conditions provide the

required relations to solve the mahtematical problem

46


Gradient Restauration

Miele (Rice Un.,Houston), re-entry, aeroassisted Indirect: exact

mathematical solution of the MPBVP

“Multiple Shooting”

BNDCSO (Oberle, Grimm), low thrust trajectories

Direct Collocation

TROPIC (Hargraves, Jänsch): Hermite interpolation, ASTOS

“Multiple shooting” direct

PROMIS (Bock, Plitt, Schnepper) direct multiple shooting scheme (ASTOS)

Direct: constrained parameter optimisation NLP (discretisation) Finite

Elements F. Bernelli-Zazzera, M. Vasile Polotecnico Milano

Hybrids Biggs, Dixon (Hatfield)

Ariane 5 ascent , asteroid missions, comets, Mercury

47


Direct methods are simpler

but only approximations of

the optimum

Identify the most suitable

OC method: indirect, direct

or hybrid.

3.- Optimal

Control

Method

•Gradient methods are very

sensitive to the initial guess

•Find a first trajectory:

–Simplified control law

–Use of direct method

to start an indirect

method

2.- Initial

Guess of

Solution

•Scale factors for:

–variables,

–constraints and

–cost function

•Technical constraints

•Identification of state x,

control variables u and

parameters P

•Identification of equality

and inequality constraints

1. Problem

Formulation

CommentsObjectiveStep

48

OCP Method SelectionOCP Method Selection

For the solution of a particular problem (e.g. launcher ascent trajectory orinterplanetary low thrust trajectory) the first step for the selection of the optimisationalgorithm is the trade off between direct and indirect methods:

• Direct methods like the collocation algorithm (program TROPIC) or the direct multiple shooting (program PROMIS) present the followingadvantages: ♦ there is no adjoint differential equations ♦ very easy to implement ♦ short "setup time" for model modifications ♦ very robust with respect to the starting estimates

On the other hand, directs method have the following disadvantages:

♦ the necessary optimality conditions are not satisfied, therefore onlyapproximated solutions are found

♦ local optimality of the solution found cannot be ascertained ♦ very large nonlinear programming problems are generated

49


• Indirect methods like the Gradient Restoration algorithm or the MultipleShooting (program BOUNDSCO) have the following advantages: ♦ they solve the posed optimal control problem exactly, as far as possible

numerically ♦ they reveal the structure of the optimal solution (bang-bang control for

instance) However indirect methods present the following disadvantages: ♦ they involve the solution of the adjoint differential equations ♦ second order methods, such as the Multiple Shooting, requires a good

starting estimation for the algorithm to converge ♦ some algorithms require an analytical pre-analysis to compute the

switching structure and a good initial guess of the adjoint variables

50


In case that the selection problem has been reduced to indirect methods, the secondstep would be a trade off between the most prominent representatives of first(Gradient Restoration) and second (Multiple Shooting) order algorithms:

• The Gradient Restoration algorithm has the following advantages: ♦ it is very robust with respect to the initial estimates ♦ a short "setup time" is required for model modifications ♦ the information on the switching structure may be numerically generated The disadvantage of the Gradient Restoration algorithm is the slow convergence rate close to the final optimum solution, typical of a first ordergradient method.

• The Multiple Shooting algorithm presents the following advantages: ♦ it is a second order algorithm with good final convergence rate ♦ control discontinuities are easily treated On the other hand, the Multiple Shooting algorithm has the followingdisadvantages: ♦ a good initial guess of the solution must be provided ♦ the switching structure of the solution must be known in advance

51


The Gradient-Restoration Algorithm has in addition the following advantages:

• This algorithm presents the most general formulation: ♦ The functional to be minimized contains:

An integral part over the path A function of the initial state vector A function of the final state vector

♦ The initial and final state vector may have: A given value A free value Satisfy a set of relations

♦ All kind of constraints may be applied by using suitable transformations: State inequality constraints. State equality constraints. Problems with bounded control. Problems with bounded state.

• The Gradient-Restoration is an algorithm suitable to be implemented in amodular way. The optimal control modules are independent of the functionalmodules which define each particular problem. Many different problems maybe treated with a minimum change in the functional modules.

• An important property of this algorithm is that it produces a sequence offeasible suboptimal solutions; the functions obtained at the end of each cyclesatisfy the constraints to a predetermined accuracy.

52

GRADIENT RESTORATION ALGORITHM (1)GRADIENT RESTORATION ALGORITHM (1)

• The Gradient Restoration algorithm has been developed by A. Miele and the staff of the Aero-Astronautics Group of Rice University, Houston (Texas).

• The technique is a sequence of two-phase cycles, composed of a gradient phase and a restoration phase.

• The gradient phase involves one iteration and is designed to decrease the value of a functional, while the constraints are satisfied to first order.

• The restoration phase involves one or more iterations and is designed to force constraint satisfaction to a predetermined accuracy, while the norm squared of the variations of the control, the parameter, and the missing components of the initial state is minimized.

53


• Let t denote the independent variable, and x(t), u(t) and P

the dependent variables.

• The time t is a scalar, the state x(t) is a vector of dimension

n, the control u(t) is a vector of dimension m and the

parameter P is a vector of dimension p.

• The state x(t) is partitioned into vectors y(t) and z(t), defined

as follows:

– y(t) is a vector of dimension a including those components

of the state that are prescribed at the initial point, and

– z(t) is a vector of dimension b including those components

of the state that are not prescribed at the initial point ( a +

b =N)

54


• the optimal control problem with general boundary conditions

may be stated as follows:

• Minimize the functional:

• with respect to the state x(t), the control u(t), and the

parameter P which satisfy the differential constraints :

• the boundary conditions:

• y(0) = given,

• [w(z,P)]0 = 0,

• [ψ(x,P)]1=0,

( ) ( )[ ] ( )[ ]10

1

0

,,,,, PxgPzhdttPuxfI ++= ∫

( )tPuxtx ,,,)( φ=•

55


• and the nondifferential constraints:

• I, f, g, an h are scalar, the function f is an n-vector, the function w

is a c-vector, the y is a q-vector and the function S is a k vector.

• From calculus of variations, the problem can be recast as that of

minimizing the augmented functional

• J = I + L

• The functional L is defined as:

• l(t) is an n-vector variable Lagrange multiplier r(t) is a k-vector

variable Lagrange multiplier, s is a c-vector constant Lagrange

multiplier and m is a q-vector constant Lagrange multiplier.

( ) 10,0,,, ≤<= ttPuxS

56


=

=

w(z,P)0 = 0,

= 0,

= 0.

57


• As the state vector x(t) is partitioned into an a-vector y(t) and

a b-vector z(t), the multiplier vector λ(t) is partitioned into an

a-vector β(t) and a b-vector γ(t).

• The differential system above defined is in general nonlinear,

approximated methods are employed to find a solution

iteratively; if the norm squared of a vector v is defined as

• N(v) = vT v

• then, the constraints error R can be written as:

58


• the error in the optimality conditions Q is given by

• the exact optimal solution must satisfy

• an approximated solution is obtained if

Q =

59


=

=

=

=

=

A(t) =

B(t) =

C =

D(t) =

E(t) =

• If the gradient or restoration stepsize is defined by the positive number α, the displacement per unit of stepsize is obtained as follows:

60


=

(gxTA + gp

T C)1,

=

=

61


• Gradient Phase:

=

SxTA + Su

TB + SpTC =

D(0) = 0

(wzTE + wp

TC)0 = 0

= 0

B =

C =

E(0) =

62


• In order to solve the boundary value problem presented above, we

employ a forward integration scheme in combination with the

method of particular solutions.

• The technique requires the execution of n + p + 1 independent

sweeps of the differential system, each characterized by a different

value of the (n + p)-vector w, whose components are:

• the n components of the initial multiplier λ(0) and

• the p components of the parameter C.

• The sweep is started by given particular values to w, that is, the

components of the vector λ(0) and C, then the previous equations

constitute a system of b + c linear relations in which the unknowns

are the b + c components of the vectors E(0) σ.

63


=

SxTA + Su

TB + SpTC + S =

D(0) = 0

(wzTE + wp

TC + w)0 = 0

= 0

B =

C =

E(0) =

=

= 0

64


• The present algorithm can be started with nominal functions x(t),

u(t) and P satisfying the given initial value of the state variables and

violating none, some or all of the remaining conditions.

• If the nominal functions are such that the constraint penalization R is

bigger than the preselected limit, the algorithm starts with a

restoration phase, otherwise a gradient phase is started.

• At the end of a gradient phase of any cycle, the constraint error R

must be computed, if it is bigger than the selected tolerance, a

restoration phase is started. Otherwise, the restoration phase is

bypassed, and the next gradient phase of the algorithm is started.

65


• After a restoration phase is completed, the functional I is

computed:

– if it is less than the previous phase, the next cycle of the

sequential gradient-restoration algorithm is started.

– If not, the stepsize of the previous gradient phase is

bisected, and a new restoration phase is started.

• For the restoration phase taken individually, convergence is

achieved whenever the penalization function R is less than

the selected tolerance ε1.

• For the sequential gradient-restoration algorithm taken as a

whole, convergence is achieved whenever restoration and

gradient conditions are satisfied simultaneously.

66


67

GRA EXAMPLE: LISA TRANSFERGRA EXAMPLE: LISA TRANSFER

68

Formation Formation TransferTransfer to HETO: to HETO: Problem DescriptionProblem Description

– Statement of the problem:

• Three satellites to be transferred to a Trailing Orbit 20 degrees behind

the Earth, targeting the LISA operational configuration (a distance

between sats. of 5x106 km, 60° inclined wrt Ecliptic).

– Main assumptions:

• Incremental analysis driving to the final solution;

• No Lunar fly-by assumed;

– Process:

• Impulsive thrusts optimisation (first iteration of the problem solution)

– A 5 manoeuvres transfer strategy designed;

– A Second Order Gradient method selected for optimisation.

• Continuous thrust optimisation (full problem solution)

– Impulsive solution fed as initial guess for the continuous thrust problem;

– A Gradient-Restoration method selected for optimisation.

69

0 50 100 150 200 250 300 350 40059.5

59.6

59.7

59.8

59.9

60

60.1

60.2

60.3

60.4

60.5Evolution of angles in the triangle SC1-SC2-SC3

days

angl

e, d

egre

es

SC1SC2SC3

0 50 100 150 200 250 300 350 4004.975

4.98

4.985

4.99

4.995

5

5.005

5.01

5.015

5.02

5.025Evolution of distances between the satellites

days

dist

ance

, mill

ions

of k

m

SC1-SC2 distanceSC2-SC3 distanceSC1-SC3 distance

0 50 100 150 200 250 300 350 40016

17

18

19

20

21

22

23Evolution of the angular delay wrt the Earth

days

angl

e, d

egre

es

SC1SC2SC3

Formation Formation TransferTransfer to HETO: LISA Orbit Analysisto HETO: LISA Orbit Analysis

70

Formation Formation TransferTransfer to HETO: LISA Launch to HETO: LISA Launch assumptionsassumptions

VEarth V∞ δ

α

z

Sun

Earth

– Launcher data taken for the Impulsive Optimisation and the Full

Optimal Control:

– Upper part mass: 6374 kg;

– Empty mass for upper stage: 1000 kg;

– Specific impulse for impulsive transfer 327 sec;

– Specific impulse for low thrust: 3000 sec;

– Radius of parking orbit: 6578 km;

– Adaptor mass: 10 kg;

– Initial mass after launch equal for the three satellites.

71

TransferTransfer to HETO: GRA to HETO: GRA solutionsolution

• Several optimal solutions have been computed, iteratively, in

a sequential process augmenting each one the complexity of

the Full Optimal Control problem solved:

– FOCP1: Optimisation of one thrust arc (no coast arc) at 18 mN;

– FOCP2: Optimisation with variable thrust level (Tmax = 18mN);

– FOCP3: Optimisation with variable thrust level (Tmax = 18mN) and a

geometrical path constraint imposed on the Solar Aspect Angle.

Several cost functions tested and their corresponding solutions

compared.

72

TransferTransfer to HETO: GRA to HETO: GRA solutionsolution, FOCP1 (1), FOCP1 (1)

• State vector (18): satellites positions and velocities

• Control vector (6): orientation angles of the three thrust vectors

(thrust level constant and equal to 18 mN for each SC)

• Parameter vector (7):

– Duration of the transfer for the three satellites;

– Magnitude and orientation of the Earth departure velocity, v∞;

– Phase angle of insertion in the LISA operational circle;

• Cost function composed of the final masses:

• No path constraints.

( )333222111333222111 ,,,,,,,,,,,,,,,,, zyxzyxzyxzyxzyxzyxx &&&&&&&&&r

=

( )332211 ,,,,, δαδαδα=ur

0.100/)( 321 fff MMMJ ++−=

73


0 50 100 150 200 250 300 350 400 4500

5

10

15

Control vector evolution for SC1

Thru

st le

vel,

mN

0 50 100 150 200 250 300 350 400 4500

100

200

300

400

azim

uth,

deg

0 50 100 150 200 250 300 350 400 450-100

-50

0

50

time, days

elev

atio

n, d

eg

0 50 100 150 200 250 300 350 400 4500

5

10

15


Thru

st le

vel,

mN

0 50 100 150 200 250 300 350 400 4500

100

200

300

400

azim

uth,

deg

0 50 100 150 200 250 300 350 400 450-100

-50

0

50

time, days

elev

atio

n, d

eg

0 50 100 150 200 250 300 350 400 4500

5

10

15


Thru

st le

vel,

mN

0 50 100 150 200 250 300 350 400 4500

100

200

300

400

azim

uth,

deg

0 50 100 150 200 250 300 350 400 450-50

0

50

100

time, days

elev

atio

n, d

eg0 50 100 150 200 250 300 350 400 450

415

420

425

430

435

440Evolution of the satellites masses

time, days

sate

llite

s m

asse

s, k

g

SC1SC2SC3 • Mass evolving linearly (one impulsive arc) to

the final values (between 416 and 417 kg);

• Transfer durations ranging from 417 to 433

days.

74


0 50 100 150 200 250 300 350 400 4501.49

1.5

1.51

1.52

1.53

1.54

1.55

1.56x 10

8 Semi major axes evolution

days

sem

i maj

or a

xis,

km

SC1SC2SC3

-0.01 -0.005 0 0.005 0.01 0.015 0.02-0.04

-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01Eccentricity vector evolution

e*cos(raan+pericenter)

e*si

n(ra

an+p

eric

ente

r)

SC1SC2SC3

-0.01 -0.005 0 0.005 0.01 0.015 0.02-0.015

-0.01

-0.005

0

0.005

0.01

0.015Inclination vector evolution

i*sin(raan)

-i*co

s(ra

an)

SC1SC2SC3

50 100 150 200 250 300 350 400 4500

20

40

60

80

100

120

140

160

180Evolution of the angles between the LISA satellites

time, days

angl

es, d

egre

es

SC1SC2SC3

0 50 100 150 200 250 300 350 400 4500

1

2

3

4

5

6x 10

6 Evolution of the distances between the LISA SCs

time, days

dist

ance

s, k

m

SC1-SC2SC2-SC3SC3-SC1

0 50 100 150 200 250 300 350 400 45010

20

30

40

50

60

70

80

90

100Solar Aspect Angle evolution

time, days

SAA

, deg

rees

SC1SC2SC3

75


• State vector (21): satellites positions, masses and velocities

• Control vector (9): thrust levels and the orientation angles of the

three thrust vectors (maximum thrust level equal to 18 mN for each

SC)





• Cost function composed of the final masses and transfer durations:

• No path constraints.

( )333222111321333222111 ,,,,,,,,,,,,,,,,,,,, zyxzyxzyxmmmzyxzyxzyxx &&&&&&&&&r

=

( )3332

max2222

max1112

max ,,·sin,,,·sin,,,·sin δαθδαθδαθ TTTu =r

0.4*)(0.100/)( 321321 ferYearsTransferYearsTransferYearsTransMMMJ fff +++++−=

76


• Two long thrusting arcs for SC1 and SC2;

• Important decrease of final mass, with

respect to the precedent case: values

ranging between 400 and 402 kg;

• Important improvement in the transfer

durations ranging from 355 and 371 days.

0 50 100 150 200 250 300 350 4000

5

10

15

20Control vector evolution for SC1

Thru

st le

vel,

mN

0 50 100 150 200 250 300 350 4000

100

200

300

400

azim

uth,

deg

0 50 100 150 200 250 300 350 400-50

0

50

100

time, days

elev

atio

n, d

eg

0 50 100 150 200 250 300 350 4000

5

10

15


Thru

st le

vel,

mN

0 50 100 150 200 250 300 350 4000

100

200

300

400

azim

uth,

deg

0 50 100 150 200 250 300 350 400-100

-50

0

50

time, days

elev

atio

n, d

eg

0 50 100 150 200 250 300 350 4000

5

10

15


Thru

st le

vel,

mN

0 50 100 150 200 250 300 350 40050

100

150

200

azim

uth,

deg

0 50 100 150 200 250 300 350 400-100

-50

0

50

100

time, days

elev

atio

n, d

eg0 50 100 150 200 250 300 350 400

400

402

404

406

408

410

412

414

416

418


time, days

sate

llite

s m

asse

s, k

g

SC1SC2SC3

77


0 50 100 150 200 250 300 350 4001.49

1.5

1.51

1.52

1.53

1.54

1.55

1.56x 10


days

sem

i maj

or a

xis,

km

SC1SC2SC3

-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005



e*si

n(ra

an+p

eric

ente

r)

SC1SC2SC3

-0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02-0.02

-0.015

-0.01

-0.005

0

0.005

0.01


i*sin(raan)

-i*co

s(ra

an)

SC1SC2SC3

50 100 150 200 250 300 350 4000

30

60

90

120


time, days

angl

es, d

egre

es

SC1SC2SC3

0 50 100 150 200 250 300 350 4000

1

2

3

4

5

6x 10


time, days

dist

ance

s, k

m


0 50 100 150 200 250 300 350 4000

30

60

90


time, daysSA

A, d

egre

es

SC1SC2SC3

78


• State vector (21): satellites positions, masses and velocities

• Control vector (12): thrust levels and orientation angles of the three

thrust vectors, and inequality path constraints related control variables





• Cost function composed of the final masses and transfer durations,

weighted through a parameter K:

• Path constraints on the SAA for the three satellites

( )333222111321333222111 ,,,,,,,,,,,,,,,,,,,, zyxzyxzyxmmmzyxzyxzyxx &&&&&&&&&r

=

( )33332

max22222

max11112

max ,,,·sin,,,,·sin,,,,·sin ωδαθωδαθωδαθ TTTu =r

KferYearsTransferYearsTransferYearsTransMMMJ fff *)(0.100/)( 321321 +++++−=

°≤°− 2590SAA

79


• Several test cases have been run for different values of the K factor in

the cost function

– K = 4: the same cost function as in the FOCP2, for comparison

purposes (equal to FOCP2 with path constraints on the Solar Aspect

Angle);

– K = {0.1, 0.5, 1, 2} to test the effect of a weighted cost function

containing the final mass and the transfer durations on the final

masses, the Earth departure velocity and the transfer durations

– K = 0: only the final mass is subject of optimisation;

80

TransferTransfer to HETO: GRA to HETO: GRA solutionsolution, FOCP3, K=4, FOCP3, K=4

0 50 100 150 200 250 300 350 4000

5

10

15


Thru

st le

vel,

mN

0 50 100 150 200 250 300 350 4000

100

200

300

400

azim

uth,

deg

0 50 100 150 200 250 300 350 400-100

-50

0

50

100

time, days

elev

atio

n, d

eg

0 50 100 150 200 250 300 350 4000

5

10

15


time, days

Thru

st le

vel,

mN

0 50 100 150 200 250 300 350 4000

100

200

300

400

azim

uth,

deg

0 50 100 150 200 250 300 350 400-100

-50

0

50

100

time, days

delta

, deg

0 50 100 150 200 250 300 350 4000

5

10

15


Thru

st le

vel,

mN

0 50 100 150 200 250 300 350 4000

50

100

150

200

azim

uth,

deg

0 50 100 150 200 250 300 350 400-100

-50

0

50

100

time, days

delta

, deg

0 50 100 150 200 250 300 350 400400

405

410

415

420


time, days

sate

llite

s m

asse

s, k

g

SC1SC2SC3 • Two long thrusting arcs for each satellite;

• SAA Constraint improves final mass with

respect to the precedent FOCP2 (values

from 404 to 406.5) but penalises transfer

duration (between 367 and 382).

81


0 50 100 150 200 250 300 350 4001.49

1.5

1.51

1.52

1.53

1.54

1.55

1.56

1.57x 10


days

sem

i maj

or a

xis,

km

SC1SC2SC3

-0.04 -0.035 -0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005



e*si

n(ra

an+p

eric

ente

r)

SC1SC2SC3

-0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02-0.02

-0.015

-0.01

-0.005

0

0.005

0.01


i*sin(raan)

-i*co

s(ra

an)

SC1SC2SC3

50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

160


time, days

angl

es, d

egre

es

SC1SC2SC3

0 50 100 150 200 250 300 350 4000

1

2

3

4

5

6x 10


time, days

dist

ance

s, k

m


0 50 100 150 200 250 300 350 40060

70

80

90

100

110


time, days

SAA

, deg

rees

SC1SC2SC3

82


0 50 100 150 200 250 300 350 4000

5

10

15


Thru

st le

vel,

mN

0 50 100 150 200 250 300 350 4000

100

200

300

400

azim

uth,

deg

0 50 100 150 200 250 300 350 400-100

-50

0

50

100

time, days

elev

atio

n, d

eg

0 50 100 150 200 250 300 350 4000

5

10

15


Thru

st le

vel,

mN

0 50 100 150 200 250 300 350 4000

100

200

300

400

azim

uth,

deg

0 50 100 150 200 250 300 350 400-100

-50

0

50

100

time, days

elev

atio

n, d

eg

0 50 100 150 200 250 300 350 4000

5

10

15


Thru

st le

vel,

mN

0 50 100 150 200 250 300 350 4000

100

200

300

azim

uth,

deg

0 50 100 150 200 250 300 350 400-100

-50

0

50

100

time, days

elev

atio

n, d

eg0 50 100 150 200 250 300 350 400

405

410

415

420

425


time, days

sate

llite

s m

asse

s, k

g

SC1SC2SC3

• Three thrusting arcs for SC1 and SC2; SC3

remains with only one coast arc;

• Transfer durations slightly worsened wrt K=4

case, values ranging from 376 to 392 days;

• Final mass improved around 6 kg per satellite wrt

K=4 case (values between 409 and 412 kg)

83


0 50 100 150 200 250 300 350 4001.49

1.5

1.51

1.52

1.53

1.54

1.55

1.56

1.57x 10


days

sem

i maj

or a

xis,

km

SC1SC2SC3

-0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005



e*si

n(ra

an+p

eric

ente

r)

SC1SC2SC3

-0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02-0.02

-0.015

-0.01

-0.005

0

0.005

0.01


i*sin(raan)

-i*co

s(ra

an)

SC1SC2SC3

50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

160


time, days

angl

es, d

egre

es

SC1SC2SC3

0 50 100 150 200 250 300 350 4000

1

2

3

4

5

6x 10


time, days

dist

ance

s, k

m


0 50 100 150 200 250 300 350 40060

70

80

90

100

110


time, days

SAA

, deg

rees

SC1SC2SC3

84


0 50 100 150 200 250 300 350 4000

5

10

15


Thru

st le

vel,

mN

0 50 100 150 200 250 300 350 4000

100

200

300

400

azim

uth,

deg

0 50 100 150 200 250 300 350 400-100

-50

0

50

100

time, days

elev

atio

n, d

eg

0 50 100 150 200 250 300 350 400 4500

5

10

15


Thru

st le

vel,

mN

0 50 100 150 200 250 300 350 400 4500

100

200

300

azim

uth,

deg

0 50 100 150 200 250 300 350 400 450-100

-50

0

50

time, days

elev

atio

n, d

eg

0 50 100 150 200 250 300 350 4000

5

10

15


Thru

st le

vel,

mN

0 50 100 150 200 250 300 350 4000

100

200

300

400

azim

uth,

deg

0 50 100 150 200 250 300 350 400-100

-50

0

50

100

time, days

elev

atio

n, d

eg0 50 100 150 200 250 300 350 400 450

416

418

420

422

424

426

428

430

432


time, days

sate

llite

s m

asse

s, k

g

SC1SC2SC3

• Coast arcs widened wrt precedent K=2 case;

• Transfer durations slightly longer (around 15

days) than precedent K=2 case (values between

390 and 410 days);



85


0 50 100 150 200 250 300 350 400 4501.49

1.5

1.51

1.52

1.53

1.54

1.55

1.56x 10


days

sem

i maj

or a

xis,

km

SC1SC2SC3

-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005



e*si

n(ra

an+p

eric

ente

r)

SC1SC2SC3

-0.01 -0.005 0 0.005 0.01 0.015 0.02-0.015

-0.01

-0.005

0

0.005

0.01


i*sin(raan)

-i*co

s(ra

an)

SC1SC2SC3

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

160


time, days

angl

es, d

egre

es

SC1SC2SC3

0 50 100 150 200 250 300 350 4000

1

2

3

4

5

6x 10


time, days

dist

ance

s, k

m


0 50 100 150 200 250 300 350 400 45060

70

80

90

100

110


time, daysSA

A, d

egre

es

SC1SC2SC3

86

TransferTransfer to HETO: GRA to HETO: GRA solutionsolution, FOCP3, K=0.5, FOCP3, K=0.5

0 50 100 150 200 250 300 350 400 4500

5

10

15


Thru

st le

vel,

mN

0 50 100 150 200 250 300 350 400 4500

100

200

300

400

azim

uth,

deg

0 50 100 150 200 250 300 350 400 450-100

-50

0

50

100

time, days

elev

atio

n, d

eg

0 50 100 150 200 250 300 350 400 4500

5

10

15


Thru

st le

vel,

mN

0 50 100 150 200 250 300 350 400 4500

100

200

300

azim

uth,

deg

0 50 100 150 200 250 300 350 400 450-100

-50

0

50

time, days

elev

atio

n, d

eg

0 50 100 150 200 250 300 350 400 4500

5

10

15


Thru

st le

vel,

mN

0 50 100 150 200 250 300 350 400 4500

100

200

300

400

azim

uth,

deg

0 50 100 150 200 250 300 350 400 450-100

-50

0

50

100

time, days

elev

atio

n, d

eg0 50 100 150 200 250 300 350 400 450

420

422

424

426

428

430

432

434


time, days

sate

llite

s m

asse

s, k

g

SC1SC2SC3

• Thrusting arcs tend to decrease as K decreases;

• Transfer durations slightly longer (around 20

days) than precedent K=1 case (values between

405 and 426 days);



87


0 50 100 150 200 250 300 350 400 4501.49

1.5

1.51

1.52

1.53

1.54

1.55

1.56x 10


days

sem

i maj

or a

xis,

km

SC1SC2SC3

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005



e*si

n(ra

an+p

eric

ente

r)

SC1SC2SC3

-0.01 -0.005 0 0.005 0.01 0.015 0.02-0.015

-0.01

-0.005

0

0.005

0.01

0.015


i*sin(raan)

-i*co

s(ra

an)

SC1SC2SC3

50 100 150 200 250 300 350 400 4500

20

40

60

80

100

120

140

160


time, days

angl

es, d

egre

es

0 50 100 150 200 250 300 350 400 4500

1

2

3

4

5

6x 10


time, days

dist

ance

s, k

m


0 50 100 150 200 250 300 350 400 45060

70

80

90

100

110


time, daysSA

A, d

egre

es

SC1SC2SC3

88


0 50 100 150 200 250 300 350 400 450 5000

5

10

15


Thru

st le

vel,

mN

0 50 100 150 200 250 300 350 400 450 5000

100

200

300

400

azim

uth,

deg

0 50 100 150 200 250 300 350 400 450 500-100

-50

0

50

100

time, days

elev

atio

n, d

eg

0 50 100 150 200 250 300 350 400 450 5000

5

10

15


Thru

st le

vel,

mN

0 50 100 150 200 250 300 350 400 450 5000

100

200

300

azim

uth,

deg

0 50 100 150 200 250 300 350 400 450 500-100

-50

0

50

time, days

elev

atio

n, d

eg

0 50 100 150 200 250 300 350 400 4500

5

10

15


Thru

st le

vel,

mN

0 50 100 150 200 250 300 350 400 4500

100

200

300

azim

uth,

deg

0 50 100 150 200 250 300 350 400 450-100

-50

0

50

100

time, days

elev

atio

n, d

eg0 50 100 150 200 250 300 350 400 450 500

422

424

426

428

430

432

434

436

438


time, days

sate

llite

s m

asse

s, k

g

SC1SC2SC3

• Former second thrusting arc of SC3 generates two

new arcs with a coast arc between them;

• Transfer duration diverges greatly wrt precedent

K=0.5 case (values between 445 and 499 days)

• Final mass improved around 3.5 kg per satellite

wrt K=0.5 case (values between 424 and 426 kg)

89


0 50 100 150 200 250 300 350 400 450 5001.49

1.5

1.51

1.52

1.53

1.54

1.55

1.56x 10


days

sem

i maj

or a

xis,

km

SC1SC2SC3

-0.01 -0.005 0 0.005 0.01 0.015 0.02-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005



e*si

n(ra

an+p

eric

ente

r)

SC1SC2SC3

-0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02-0.02

-0.015

-0.01

-0.005

0

0.005

0.01


i*sin(raan)

-i*co

s(ra

an)

SC1SC2SC3

0 50 100 150 200 250 300 350 400 4500

20

40

60

80

100

120

140

160


time, days

angl

es, d

egre

es

SC1SC2SC3

0 50 100 150 200 250 300 350 400 4500

1

2

3

4

5

6

7x 10


time, days

dist

ance

s, k

m


0 50 100 150 200 250 300 350 400 450 50060

70

80

90

100

110


time, daysSA

A, d

egre

es

SC1SC2SC3

90


0 100 200 300 400 500 600 700 800 9000

5

10

15


Thru

st le

vel,

mN

0 100 200 300 400 500 600 700 800 9000

100

200

300

400

azim

uth,

deg

0 100 200 300 400 500 600 700 800 900-100

-50

0

50

100

time, days

elev

atio

n, d

eg

0 100 200 300 400 500 600 700 800 9000

5

10

15


Thru

st le

vel,

mN

0 100 200 300 400 500 600 700 800 9000

100

200

300

azim

uth,

deg

0 100 200 300 400 500 600 700 800 900-100

-50

0

50

100

time, days

elev

atio

n, d

eg

0 100 200 300 400 500 600 7000

5

10

15


Thru

st le

vel,

mN

0 100 200 300 400 500 600 7000

100

200

300

400

azim

uth,

deg

0 100 200 300 400 500 600 700-100

-50

0

50

100

time, days

elev

atio

n, d

eg0 100 200 300 400 500 600 700 800 900

428

430

432

434

436

438


time, days

sate

llite

s m

asse

s, k

g

SC1SC2SC3

• Transfer duration doubles wrt precedent K=0.1

case (values between 700 and 810 days);


K=0.1 case (values between 428 and 430 kg);

• Thrusting arcs of the first half of the transfer for

precedent cases almost disappear for K=0

91


0 100 200 300 400 500 600 700 800 9001.495

1.5

1.505

1.51

1.515

1.52

1.525

1.53

1.535x 10


days

sem

i maj

or a

xis,

km

SC1SC2SC3

-0.01 -0.005 0 0.005 0.01 0.015-14

-12

-10

-8

-6

-4

-2

0

2

4

6x 10

-3 Eccentricity vector evolution


e*si

n(ra

an+p

eric

ente

r)

SC1SC2SC3

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-0.02

-0.015

-0.01

-0.005

0

0.005


i*sin(raan)

-i*co

s(ra

an)

SC1SC2SC3

150 200 250 300 350 400 450 500 550 600 650 7000

20

40

60

80

100

120

140

160


time, days

angl

es, d

egre

es

SC1SC2SC3

0 100 200 300 400 500 600 7000

1

2

3

4

5

6

7

8

9

10x 10


time, days

dist

ance

s, k

m


0 100 200 300 400 500 600 700 800 90060

70

80

90

100

110


time, daysSA

A, d

egre

es

SC1SC2SC3

92

TransferTransfer to HETO: GRA to HETO: GRA solutionsolution, , conclusionsconclusions (1)(1)

• The Gradient Restoration Algorithm has shown to be valid for finding optimal

solutions for any generic cost function; moreover:

– Adaptable for executing cases with fixed and variable thrust levels;

– The switching structure for the low thrust control vector is autonomously

generated by the algorithm;

– It enables the inclusion of equality and inequality path constraints;

– Initial solution (in terms of x, state vector; u, control vector; and P,

parameters vector) can violate none, some or even all of the problem

constraints (path constraints, boundary conditions).

– In the cases here presented, the solution provided by the impulsive

optimisation was used to start the GRA, showing rapid convergence in

the first optimisation stages.

93

0 0.5 1 1.5 2 2.5 3 3.5 4400

405

410

415

420

425

430

Fina

l mas

s, k

g

0 0.5 1 1.5 2 2.5 3 3.5 4300

400

500

600

700

800

900Effect of cost function on the transfer duration and final mass

Relative weight between the duration and the final mass in the cost function

Transfer duration, days

SC1 transfer durationSC2 transfer durationSC3 transfer duration

SC1 final massSC2 final massSC3 final mass


• Method allows to assess the

changing evolution of the

optimal switching structure as

the cost functions changes:

– For high K values, optimal

transfer durations and final

mass decrease;

– For low ratios, the transfer

durations increases

exponentially to around

800 days for K=0, while

mass stabilises at values

around 430 kg per SC.

94


0 0.5 1 1.5 2 2.5 3 3.5 4400

410

420

430

440

Relative weight between the transfer duration and the final mass in the cost function

Fina

l mas

s, k

g

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2Effect of cost function on the final mass and the Earth hyperbolic departure velocity

Earth hyperbolic departure velocity, km/s

Earth hyp. dep. velocity

SC1 final massSC2 final massSC3 final mass

• Effect on Hyperbolic Earth

departure velocity:

– The higher the K

parameter, the higher

the optimal v∞;

• Solution provided by MAS

Working paper 424 in the

range of obtained with the

GRA:

– Transfer duration of

410.6 days for a final

mass of 415 kg

95


• Table of results for the K parametric analysis, compared against the

two FOCP1 and FOCP2 (with no path constraints):Final mass (kg) Transfer duration (days)

SC1 SC2 SC3 Σ(Final mass) SC1 SC2 SC3

V∞ (km/s)

FOCP1 416.387 416.415 417.249 1250.051 433.475 432.944 417.153 0.38335

FOCP2 401.912 401.315 400.183 1203.41 355.1308 371.5140 366.8972 1.40472

K=0 429.621 429.458 428.610 1287.689 810.5846 805.3231 698.1632 0.29532

K=0.1 425.827 424.071 423.736 1273.634 498.5141 468.3126 444.8585 0.50277

K=0.5 421.425 420.915 420.308 1262.648 421.9704 426.3874 405.4204 0.68288

K=1.0 418.312 417.148 416.630 1252.090 398.1342 410.2358 391.8869 0.84099

K=2.0 412.599 410.446 409.654 1232.699 376.5257 392.1169 380.5438 1.07464

FOCP3

K=4.0 406.495 404.791 403.948 1215.234 367.3369 381.9740 373.6440 1.27733

• Addition of constraints, for the same cost function (comparison of

FOCP2 against FOCP3, K=4) penalises the transfer duration in some

10 to 15 days, but improving the final mass around 4 kg.

96

Global Optimisation Tool: RequirementsGlobal Optimisation Tool: Requirements

• Ease of Use: The tool shall provide user interface and wizards

• Modularity: The tool shall be based on modules that can run independently

• Expandability: The modularity feature of the tool shall allow the easy addition

of new modules

• Connectivity: The tool will allow the user to process data files generated by a

number of external mission analysis tools

• Standarisation of mission data: The tool will allow to:Create a project associated to a mission under study with all the data relevant for the

mission definition

Generate the input data files for the execution of the ‘connectable’ mission analysis

tools

Import the data resulting from the execution of those external tools with the purpose

of storage, visualization or further processing

• Platform Portability: All developments will ensure the portabitlity of the tool

between different platforms

• Open Tool: New modules to deal with new models or optimisation problems

must be easily incorporated by the user

97

Global Optimisation Tool: Architecture Global Optimisation Tool: Architecture

EXTERNAL MISSION ANALYSIS TOOLS

OPTIMISATION TOOL

PROJECTS

INTNAV

LOTNAV

SEPNAV

NAVELIP LODATO

GREST/ GRETCHEN

STREAM

USOC

IMAT

Utilities

98

Global Optimisation Tool: Analyst ModeGlobal Optimisation Tool: Analyst Mode

99

New Scheme

Global Optimisation Tool: Developer ModeGlobal Optimisation Tool: Developer Mode

100

Global Optimisation Tool: ArchitectureGlobal Optimisation Tool: Architecture

Basic Mission Constructor

Initial Condition Generator

Mission FailureSimulation SW

Direct User Inputof the state vector

Initial Conditions:Tini - MJD2000)

Xini(6) - MEE2000

"BASCON1"1st BodyMissions

Output

WSB

Level Line Plots

"ADD"Add Body Fly-by

Mission List

"OPT"Mission Optimiser(Patched Conics)

Mission List

"mission.ini"

Real Case

Test Case

JPLEPH COMETS ASTNUM ASTUNN

EphemeridesDVm

DVc

DVt

Va

DET

bascon1.inp Basic Mission Database"bas.mis"

OutputSHT

Mission List

Additional Body MissionDatabase"add.mis"

add.inp

New Body inMission List

opt.inpOutput

SHT

Optimised Mission List

OPR

Database of FinalOptimised Missions

"opt.mis"

"VERIF"Verification &

Plotting Program

ver.inpOutput

SHT

DET

SMA ECC INC RPER RAPH PER

TrajectoryAngle Plot

Distance PlotB-plane Level Lines

"ADDINT"Add Intermediate

Body Fly-by

addint.inp

Output

SHT

Additional IntermediateBody Mission Database

"addint.mis"

New MissionList

DVd

strandorb.det

SHT

New IntermediateBod y in Miss ion Lis t

101

Trajectory Propagation Utility 1Trajectory Propagation Utility 1

• It shall be based on a Runge-Kutta integrator. State vector propagation will be performed

in Mean Earth Equator 2000 although reference frame transformation will be available for

input /output purposes

• Air drag perturbation. The user shall be able to select among several models:Simple exponential air density model (reference density and scale height as user input)

Complete atmospheric models for Earth, Venus, Mars, Jupiter, Saturn, Uranus, Neptune, and Titan

• Planet’s or Moon’s gravity field perturbation. The gravity field of the central body shall

be defined by means of the well known gravity potential in Spherical Harmonics

• Third-body perturbation. Gravity field associated to third bodies, assumed spherical

• Solar Radiation Pressure Perturbation shall be an option as a function of a user defined

Area to mass ratio. Spherical and cubic spacecraft are proposed for accounting for the

effect of the SRP. Moreover a S/C with solar array shall also be an option.

• Low Thrust accelerations. The user shall be able to define a thrust profile. The input

data for the definition of the thrust law will contain a set of time intervals and coefficients

by which the three components of the thrust direction law are approximated by means of

Chebyshev polynomials of a given degree.

• Impulsive Manoeuvres will also be available, inserting a Delta-V in a known direction in

a celestial reference frame

102

Trajectory Propagation Utility 2Trajectory Propagation Utility 2

• It shall be considered in the scope of the tool the inclusion of other induced

interactions as: Relativistic accelerations

Albedo induced accelerations

Accelerations due to time varying gravity fields

• Regarding the interactions considered, many of them depend on features and

models of the global Solar System constants and dynamics, which need to be

either defined, either modelled. Those include the definition of: Relativistic accelerations

Time and coordinates reference frames

Solar System bodies gravity constants

Solar System bodies geometrical radiuses and sphere of influence radiuses

Solar activity parameters

Solar System bodies atmospheric data

Ephemerides of planets, satellites, comets and asteroids

• The Trajectory propagation Utility shall work in two modes: Initial Value Pronblem (IVP)

Multiple Point Boundary value problem (MPBVP)

103

Optimisation ModuleOptimisation Module

• The optimisation module shall implement the following functionalities:

NLP Constrained Parameter Optimisation Solvers, including the

following options:Deterministic Methods like Gradient first and second order algorithms

(e.g. OPXRQP, NPSOL, NZQPT, …)

Probabilistic methods like Genetic Algorithms, Evolutionary Methods, …

Mixed methods: probabilistic method to initialise and deterministic

method for final convergence

Full Optimal Control problem (OCP) Solvers, including the following

options:Indirect methods:

Gradient Restoration Algorithm

Indirect Multiple Shooting

Direct Methods:Direct Multiple Shooting

Direct Collocation

Hybrid method with adjoint control transformation

104

Basic Astrodynamics CalculatorBasic Astrodynamics Calculator

• Calendar Date Transformation

• Coordinate Frame

Transformations

• State Vector / Orbital

Parameters Transformations

• Graphical Lambert Solver Utility

• Orbital Transfer Utility

• Gravity Loss Utility

• Powered and Unpowered

Swing-By Parameters Utility

• Celestial Body Ephemeris Utility

Departure 1 at time t1

Arrival 1 at time t2

Departure 2 at time t2

12

3

1

2

3 1

23

Arrival 2 at time t3

105

Orbit Profile CalculatorOrbit Profile Calculator

• Eclipse Profiles Utility

• Ground Station Visibility Utility

• Planetary Lander Visibility

Utility

• Occultation of Orbiter

• Minimum Altitude

• Relative Motion between two

satellites:

– Rendez vous and Docking

Utility

– Formation Flying Utility

106

Auxiliary CalculationsAuxiliary Calculations

• Coverage Utility:

– Sky Coverage Utility

– Planetary Surface Coverage

utility

• Environment Monitoring Utility:

– Radiation Analysis Utility

– Meteoroid and Space Debris

Flux Utility

• Launcher Performance Module:

– Soyuz type: Parking Orbit

+ Upper Stage

– Ariane-Vega type

• 3D Interplanetary Visualisation

Tool

107

ConclusionsConclusions

• A historical perspective of space trajectory optimisation during the last

40 years of the XXth century has been presented

• An analysis of the most typical optimisation problems is summarised

• Problem formulation, numerical methods and typical examples are

presented for:Constrained Parameter Optimisation Problem (NLP):

Deterministic methods

Probabilistic methods

Full Optimal Control Problem (OCP):Indirect methods

Direct methods

Hybrid methods

• A Global Trajectory Optimisation Tool is presented:High Level Requirements

Tool Architecture

Tool Main Modules and Utilities

• A powerful 3D Interplanetary Visualisation Tool has been presented, it

is a key element to perform a complete results exploitation

Documents

Global Trajectory Optimisation Tools for the 21st Centurytrajectory.estec.esa.int/Astro/3rd-astro-workshop... · 2006-09-26 · Global Trajectory Optimisation Tools for the 21st Century