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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
Optimum aerospace trajectories
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
Historical Perspective 2Historical Perspective 2
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
Continuation / commentsMethodOptimisation Problem
AuthorYear
5
Historical Perspective 3Historical Perspective 3
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)
Optimal control with equality and inequality path and boundary constraints
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
Continuation / commentsMethodOptimisation Problem
AuthorYear
6
Historical Perspective 4Historical Perspective 4
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)
Optimal control with equality and inequality path and boundary constraints
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
Continuation / commentsMethodOptimisation Problem
AuthorYear
7
Historical Perspective 5Historical Perspective 5
First use of Genetic Algorithms in ESA for space problems
Genetic Algorithm
Optimum aerospace trajectories
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
Optimal control with equality and inequality path and boundary constraints
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
Continuation / commentsMethodOptimisation Problem
AuthorYear
8
Historical Perspective 6Historical Perspective 6
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
Continuation / commentsMethodOptimisation Problem
AuthorYear
9
Historical Perspective 7Historical Perspective 7
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
Continuation / commentsMethodOptimisation Problem
AuthorYear
10
Historical Perspective 8Historical Perspective 8
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
Continuation / commentsMethodOptimisation Problem
AuthorYear
11
Historical Perspective 9Historical Perspective 9
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
Continuation / commentsMethodOptimisation Problem
AuthorYear
12
Historical Perspective 10Historical Perspective 10
Weak Stability Boundary Transfers
Optimal Lunar Transfer
BelbrunoMiller
1993
Continuation / commentsMethodOptimisation Problem
AuthorYear
13
Historical Perspective 11Historical Perspective 11
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
Continuation / commentsMethodOptimisation Problem
AuthorYear
14
Historical Perspective 12Historical Perspective 12
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
Continuation / commentsMethodOptimisation Problem
AuthorYear
15
Historical Perspective 13Historical Perspective 13
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
Continuation / commentsMethodOptimisation Problem
AuthorYear
16
Historical Perspective 14Historical Perspective 14
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
Continuation / commentsMethodOptimisation Problem
AuthorYear
17
Historical Perspective 15Historical Perspective 15
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
Continuation / commentsMethodOptimisation Problem
AuthorYear
18
Historical Perspective 16Historical Perspective 16
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
Continuation / commentsMethodOptimisation Problem
AuthorYear
19
Historical Perspective 17Historical Perspective 17
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
Continuation / commentsMethodOptimisation Problem
AuthorYear
20
Historical Perspective 18Historical Perspective 18
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
Continuation / commentsMethodOptimisation Problem
AuthorYear
21
Historical Perspective 19Historical Perspective 19
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
Continuation / commentsMethodOptimisation Problem
AuthorYear
22
Historical Perspective 20Historical Perspective 20
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
Continuation / commentsMethodOptimisation Problem
AuthorYear
23
Historical Perspective 21Historical Perspective 21
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
Continuation / commentsMethodOptimisation Problem
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
Constrained Parameter OptimisationConstrained Parameter Optimisation
• 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
Constrained Parameter OptimisationConstrained Parameter Optimisation
• 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
Constrained Parameter OptimisationConstrained Parameter Optimisation
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
NLP Example: Interplanetary TrajectoriesNLP Example: Interplanetary Trajectories
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
NLP Example: Interplanetary TrajectoriesNLP Example: Interplanetary Trajectories
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
NLP Example: Interplanetary TrajectoriesNLP Example: Interplanetary Trajectories
• Gravity Assists trajectories are extensively used in
interplanetary trajectory design:
35
NLP Example: Interplanetary TrajectoriesNLP Example: Interplanetary Trajectories
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
NLP Example: Interplanetary TrajectoriesNLP Example: Interplanetary Trajectories
40
NLP Example: Interplanetary TrajectoriesNLP Example: Interplanetary Trajectories
41
NLP Example: Interplanetary TrajectoriesNLP Example: Interplanetary Trajectories
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
OPTIMAL CONTROL PROBLEMOPTIMAL CONTROL PROBLEM
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
OPTIMAL CONTROL PROBLEMOPTIMAL CONTROL PROBLEM
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
OPTIMAL CONTROL PROBLEMOPTIMAL CONTROL PROBLEM
• 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
OPTIMAL CONTROL PROBLEMOPTIMAL CONTROL PROBLEM
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
OPTIMAL CONTROL PROBLEMOPTIMAL CONTROL PROBLEM
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
OCP Method SelectionOCP Method Selection
• 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
OCP Method SelectionOCP Method Selection
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
OCP Method SelectionOCP Method Selection
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
GRADIENT RESTORATION ALGORITHM (2)GRADIENT RESTORATION ALGORITHM (2)
• 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
GRADIENT RESTORATION ALGORITHM (3)GRADIENT RESTORATION ALGORITHM (3)
• 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
GRADIENT RESTORATION ALGORITHM (4)GRADIENT RESTORATION ALGORITHM (4)
• 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
GRADIENT RESTORATION ALGORITHM (5)GRADIENT RESTORATION ALGORITHM (5)
=
=
w(z,P)0 = 0,
= 0,
= 0.
57
GRADIENT RESTORATION ALGORITHM (6)GRADIENT RESTORATION ALGORITHM (6)
• 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
GRADIENT RESTORATION ALGORITHM (7)GRADIENT RESTORATION ALGORITHM (7)
• the error in the optimality conditions Q is given by
• the exact optimal solution must satisfy
• an approximated solution is obtained if
Q =
59
GRADIENT RESTORATION ALGORITHM (8)GRADIENT RESTORATION ALGORITHM (8)
=
=
=
=
=
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
GRADIENT RESTORATION ALGORITHM (9)GRADIENT RESTORATION ALGORITHM (9)
=
(gxTA + gp
T C)1,
=
=
61
GRADIENT RESTORATION ALGORITHM (10)GRADIENT RESTORATION ALGORITHM (10)
• Gradient Phase:
=
SxTA + Su
TB + SpTC =
D(0) = 0
(wzTE + wp
TC)0 = 0
= 0
B =
C =
E(0) =
62
GRADIENT RESTORATION ALGORITHM (11)GRADIENT RESTORATION ALGORITHM (11)
• 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
GRADIENT RESTORATION ALGORITHM (12)GRADIENT RESTORATION ALGORITHM (12)
=
SxTA + Su
TB + SpTC + S =
D(0) = 0
(wzTE + wp
TC + w)0 = 0
= 0
B =
C =
E(0) =
=
= 0
64
GRADIENT RESTORATION ALGORITHM (13)GRADIENT RESTORATION ALGORITHM (13)
• 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
GRADIENT RESTORATION ALGORITHM (14)GRADIENT RESTORATION ALGORITHM (14)
• 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
GRADIENT RESTORATION ALGORITHM (15)GRADIENT RESTORATION ALGORITHM (15)
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
TransferTransfer to HETO: GRA to HETO: GRA solutionsolution, FOCP1 (2), FOCP1 (2)
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
Control vector evolution for SC2
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
Control vector evolution for SC3
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
TransferTransfer to HETO: GRA to HETO: GRA solutionsolution, FOCP1 (3), FOCP1 (3)
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
TransferTransfer to HETO: GRA to HETO: GRA solutionsolution, FOCP2 (1), FOCP2 (1)
• 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)
• 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 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
TransferTransfer to HETO: GRA to HETO: GRA solutionsolution, FOCP2 (2), FOCP2 (2)
• 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
20Control vector evolution for SC2
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
20Control vector evolution for SC3
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
420Evolution of the satellites masses
time, days
sate
llite
s m
asse
s, k
g
SC1SC2SC3
77
TransferTransfer to HETO: GRA to HETO: GRA solutionsolution, FOCP2 (3), FOCP2 (3)
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
8 Semi major axes evolution
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
0.01Eccentricity vector evolution
e*cos(raan+pericenter)
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
0.015Inclination vector evolution
i*sin(raan)
-i*co
s(ra
an)
SC1SC2SC3
50 100 150 200 250 300 350 4000
30
60
90
120
150Evolution of the angles between the LISA satellites
time, days
angl
es, d
egre
es
SC1SC2SC3
0 50 100 150 200 250 300 350 4000
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 4000
30
60
90
120Solar Aspect Angle evolution
time, daysSA
A, d
egre
es
SC1SC2SC3
78
TransferTransfer to HETO: GRA to HETO: GRA solutionsolution, FOCP3 (1), FOCP3 (1)
• 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
• 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 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
TransferTransfer to HETO: GRA to HETO: GRA solutionsolution, FOCP3 (2), FOCP3 (2)
• 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
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 50 100 150 200 250 300 350 400-100
-50
0
50
100
time, days
elev
atio
n, d
eg
0 50 100 150 200 250 300 350 4000
5
10
15
20Control vector evolution for SC2
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
20Control vector evolution for SC3
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
425Evolution of the satellites masses
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
TransferTransfer to HETO: GRA to HETO: GRA solutionsolution, FOCP3, K=4, FOCP3, K=4
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
8 Semi major axes evolution
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
0.01Eccentricity vector evolution
e*cos(raan+pericenter)
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
0.015Inclination vector evolution
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
180Evolution of the angles between the LISA satellites
time, days
angl
es, d
egre
es
SC1SC2SC3
0 50 100 150 200 250 300 350 4000
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 40060
70
80
90
100
110
120Solar Aspect Angle evolution
time, days
SAA
, deg
rees
SC1SC2SC3
82
TransferTransfer to HETO: GRA to HETO: GRA solutionsolution, FOCP3, K=2, FOCP3, K=2
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 50 100 150 200 250 300 350 400-100
-50
0
50
100
time, days
elev
atio
n, d
eg
0 50 100 150 200 250 300 350 4000
5
10
15
20Control vector evolution for SC2
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
20Control vector evolution for SC3
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
430Evolution of the satellites masses
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
TransferTransfer to HETO: GRA to HETO: GRA solutionsolution, FOCP3, K=2, FOCP3, K=2
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
8 Semi major axes evolution
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
0.01Eccentricity vector evolution
e*cos(raan+pericenter)
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
0.015Inclination vector evolution
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
180Evolution of the angles between the LISA satellites
time, days
angl
es, d
egre
es
SC1SC2SC3
0 50 100 150 200 250 300 350 4000
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 40060
70
80
90
100
110
120Solar Aspect Angle evolution
time, days
SAA
, deg
rees
SC1SC2SC3
84
TransferTransfer to HETO: GRA to HETO: GRA solutionsolution, FOCP3, K=1, FOCP3, K=1
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 50 100 150 200 250 300 350 400-100
-50
0
50
100
time, days
elev
atio
n, d
eg
0 50 100 150 200 250 300 350 400 4500
5
10
15
20Control vector evolution for SC2
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
20Control vector evolution for SC3
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
434Evolution of the satellites masses
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);
• Final mass improved around 6 kg per satellite wrt
K=2 case (values between 416 and 418 kg)
85
TransferTransfer to HETO: GRA to HETO: GRA solutionsolution, FOCP3, K=1, FOCP3, K=1
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
-0.02 -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
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
0 50 100 150 200 250 300 350 4000
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 4000
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 45060
70
80
90
100
110
120Solar Aspect Angle evolution
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
20Control 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
100
time, days
elev
atio
n, d
eg
0 50 100 150 200 250 300 350 400 4500
5
10
15
20Control vector evolution for SC2
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
20Control vector evolution for SC3
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
436Evolution of the satellites masses
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);
• Final mass improved around 3 kg per satellite wrt
K=1 case (values between 420 and 421 kg)
87
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 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
-0.015 -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
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
0.015
0.02Inclination 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
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 45060
70
80
90
100
110
120Solar Aspect Angle evolution
time, daysSA
A, d
egre
es
SC1SC2SC3
88
TransferTransfer to HETO: GRA to HETO: GRA solutionsolution, FOCP3, K=0.1, FOCP3, K=0.1
0 50 100 150 200 250 300 350 400 450 5000
5
10
15
20Control vector evolution for SC1
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
20Control vector evolution for SC2
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
20Control vector evolution for SC3
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
440Evolution of the satellites masses
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
TransferTransfer to HETO: GRA to HETO: GRA solutionsolution, FOCP3, K=0.1, FOCP3, K=0.1
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
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 0.02-0.025
-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
-0.015 -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
0.015Inclination vector evolution
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
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
6
7x 10
6 Evolution of the distances between the LISA SCs
time, days
dist
ance
s, k
m
SC1-SC2SC2-SC3SC3-SC1
0 50 100 150 200 250 300 350 400 450 50060
70
80
90
100
110
120Solar Aspect Angle evolution
time, daysSA
A, d
egre
es
SC1SC2SC3
90
TransferTransfer to HETO: GRA to HETO: GRA solutionsolution, FOCP3, K=0, FOCP3, K=0
0 100 200 300 400 500 600 700 800 9000
5
10
15
20Control vector evolution for SC1
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
20Control vector evolution for SC2
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
20Control vector evolution for SC3
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
440Evolution of the satellites masses
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);
• Final mass improved around 5 kg per satellite wrt
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
TransferTransfer to HETO: GRA to HETO: GRA solutionsolution, FOCP3, K=0, FOCP3, K=0
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
8 Semi major axes evolution
days
sem
i maj
or a
xis,
km
SC1SC2SC3
-0.01 -0.005 0 0.005 0.01 0.015-14
-12
-10
-8
-6
-4
-2
0
2
4
6x 10
-3 Eccentricity vector evolution
e*cos(raan+pericenter)
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
0.01Inclination vector evolution
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
180Evolution of the angles between the LISA satellites
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
6 Evolution of the distances between the LISA SCs
time, days
dist
ance
s, k
m
SC1-SC2SC2-SC3SC3-SC1
0 100 200 300 400 500 600 700 800 90060
70
80
90
100
110
120Solar Aspect Angle evolution
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
TransferTransfer to HETO: GRA to HETO: GRA solutionsolution, , conclusionsconclusions (2)(2)
• 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
TransferTransfer to HETO: GRA to HETO: GRA solutionsolution, , conclusionsconclusions (3)(3)
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
TransferTransfer to HETO: GRA to HETO: GRA solutionsolution, , conclusionsconclusions (4)(4)
• 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
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