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Aerospace Dynamics and Optimal Control workshopParis, 23 May 2008 - 1
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Application of Optimal Control to Space Transportation System Design
Aerospace Dynamics and Optimal Control WorkshopParis, May 23rd 2008
Nicolas BérendLong-Term Design and System Integration Department
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Aerospace Dynamics and Optimal Control workshopParis, 23 May 2008 - 3
Summary
• Introduction about space transportations applications and optimal control.
• Presentation of Onera’s FLOP/OLGA solver and applications.
• Interior point method.
• Other methods and applications.
• Needs in optimal control.
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Aerospace Dynamics and Optimal Control workshopParis, 23 May 2008 - 4
Applications
Space Transportation Systems
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Aerospace Dynamics and Optimal Control workshopParis, 23 May 2008 - 5
Main types of trajectory optimization problems
• Ascent.• Reentry.• Orbital Transfer.
Load factor
Equilibrium glide boundary
Heat Flux
Entry point
Drag deceleration
Velocity
Dynamic pressure
[Drag, Velocity] profile for a reentry phase
Dynamic pressure : 1/2.ρ.Vr2
Fairing Heat Flux : Cq.ρ1/2.Vr3.15
Heat flux peak : 1/2.ρ.Vr3
Verticality barrier
Re-entry
Visibility
Fairing Jettisoning
AOA ~ 0°
Altitude
Velocity
[Altitude, Velocity] profile for an ascent phase
Low-thrust orbital transfer
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Aerospace Dynamics and Optimal Control workshopParis, 23 May 2008 - 6
Special needs for RLV studies (1/2):Branching trajectories
• In branching cases:• 2 simultaneous controlled phases
follows a single one, starting from the same position (e.g. staging).
• the simultaneous phases have their own constraints and/or performance index.
Branching cases are complex to study.
May lead to multi-objective optimization problems.
The problem of initialization is more
acute.
Using existing solvers may not be the best solution.
Range
Altitude Orbiter trajectory
Booster return trajectory
Joint flight
Staging point
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Aerospace Dynamics and Optimal Control workshopParis, 23 May 2008 - 7
Special needs for RLV studies (2/2):abort trajectories
• RLV: specifications of high probability for vehicle recovery (10-4).
study of trajectories in case of a non-catastrophic failure (e.g. 1 engine out of n).
trajectories may be different from one case to another (e.g. RTLS, TAL, ATO).
complex problem: choice of the abort strategy as a function of the failure time.
many cases to study.
R a n g e
A l t i t u d e N o m i n a l t r a j .
R T L S
T A L
A T O
E n g i n e f a i l u r e
Need for a very robust tool, fast convergence.
Example of abort strategies for differentfailure times
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Aerospace Dynamics and Optimal Control workshopParis, 23 May 2008 - 8
General statement of a trajectory optimization problem
• Trajectory dynamics:
Different state vector possible (orbital parameters, position+velocity vectors, etc.)
• Performance index. Examples: max(payload), max(range), ...
• Local constraints: Examples: altitude, velocity, thermal flux (fairing separation).
• Path constraints: Examples: mechanical and thermal loads.
x¿
t = f x t , u t , a
J = φ x t f , a ,t f ∫0
tfL x t , u t , a ⋅dt
ψ i x t i , a , t i = 0 or ≤ 0
γ i x t , u t , a = 0 or ≤ 0
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Aerospace Dynamics and Optimal Control workshopParis, 23 May 2008 - 9
Generalized Projected GradientFLOP/OLGA: Future Launcher Optimization Program
(C. Aumasson, Onera)• Main features:
• Adapted to both expendable and reusable LV.• Multi-phases trajectories: ascent and/or reentry.• 3 d.o.f. dynamics (controls= attitude angles).• Solves for both controls (u(t)) and design parameters (a).• Library of:
• Vehicles characteristics (aerodynamics, propulsion, etc.)• Performance indexes (payload mass, thermal flux, etc.).• Constraints (thermal flux, transverse load factor, etc).
• Optimization method• OLGA optimization core: generalized projected gradient (direct method).• Main features:
• Management of multi-phases dynamics.• two-level optimization problem using a fixed “global step”.• Explicit expression of “current” Lagrange multipliers instead of the solving NLP
problem.
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Aerospace Dynamics and Optimal Control workshopParis, 23 May 2008 - 10
OLGA optimization coreFormulation of the global optimization problem
• Find the optimal controls u*(t) and parameters a*(t) such that:
• The following performance index is minimized:
• The following constraints are satisfied:
• for i = 1 to q with:
• Possible constraints:• Final, intermittent constraints (ϕi)• Final/integral constraints (Li)• Path constraints, formulated as integral constraints (Li)
J = φ0 x t f , a , t f ∫0
tfL0 x t , u t , a ⋅dt
ψ i ≤ 0
ψ i = φi x t f , a , t f ∫0
tfL i x t , u t , a ⋅dt
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Aerospace Dynamics and Optimal Control workshopParis, 23 May 2008 - 11
OLGA optimization corePrinciple of the method
• At each iteration: calculation of controls variations δu(t) and parameters variations δa for given objectives of performance index variations δJ and constraints variations δΨ
• First order sensibility equations are used in the process:
calculation of the Hamiltonians and adjoint vectors.
• Validity of sensibility equations is ensured by fixing a « global variation step » N:
Γ = global sensibility matrix.
δJ = [∂ K0
∂ a∫0
tf ∂ H 0
∂ a⋅dt ]⋅δa∫0
tf [ ∂H 0
∂ u⋅δu ]⋅dt δΨ i = [ ∂K i∂ a
∫0
tf ∂H i
∂ a⋅dt ]⋅δa∫0
tf [∂ H i
∂ u⋅δu]⋅dt
N 2 = [δJ δψ T ]⋅Γ−1⋅[δJδψ ]
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Aerospace Dynamics and Optimal Control workshopParis, 23 May 2008 - 12
OLGA optimization coreOrganization of the optimization process (1/2)
• At each iteration:
Subsidiary optimization problem
Find the objectives for constraint variations δΨ*i such that:
• Norm of the new constraints values is minimized:
• Variations stay within the « normalized constraint step » NΨ≤N :• (to ensure the validity of the calculus of variations)
Solution:
η = Lagrange multiplierΓΨ= constraints sensibility matrix.
min ∥ψδψ ¿∥S ∥δψ¿∥=N ψ
δψ ¿ =−Γ ψ⋅Γ ψη⋅S−1 −1⋅ψ
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Aerospace Dynamics and Optimal Control workshopParis, 23 May 2008 - 13
OLGA optimization coreOrganization of the optimization process (2/2)
• Local optimization problemFind control variations δu*(t) and parameters variations δa such that:
• Variations on performance index is minimized:(if <0: the performance index is reduced)
• Variations on constraints are equals to objectives:for i =1 to q
• Solution:
min δJ
δu¿ t =W−1⋅[ ∂H 0T
∂ u. ..
∂ HjT
∂ u ]⋅µ δa¿ = B−1⋅[ ∂K 0
∂ a∫0
tf ∂H 0
∂ a⋅dt
T
.. . ∂ Kq∂a∫0
tf ∂ H q
∂a⋅dt
T ]⋅µµ = Γ−1⋅[δJ ¿
δψ¿ ] = [ µ0 µ1 . . . µq ]T ν i=µ iµ0
δψ i = δψi¿
= “current” Lagrange multipliers (i = 1 to q)
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Aerospace Dynamics and Optimal Control workshopParis, 23 May 2008 - 14
« Classical » Projected Gradient Method
Normalized step domain (N)
Domain for zero constraint variations
Projected gradient
Gradient of the performance
index
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Aerospace Dynamics and Optimal Control workshopParis, 23 May 2008 - 15
Generalized Projected Gradient Method
Gradient of the performance
index
Modified projected gradient
Domain for desired constraint variations
Domain for satisfied
constraints
Normalized step domain (N)
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Aerospace Dynamics and Optimal Control workshopParis, 23 May 2008 - 16
FLOP/OLGA – Generalized Projected GradientAdvantages and Drawbacks
• Advantages:• Starts with unsatisfied constraints easier initialization process
(no adjoint/Lagrange multiplier initialization).• Full support of multi-phases problems.• General formulation of performance index and constraint function easier adaptation to new problems.
• Drawbacks:• Slow convergence (1st order method).• Yields local optimal solution (like others gradient-based methods).• Convergence sometimes difficult because of the formulation of
path constraints as final-integral constraints.
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Aerospace Dynamics and Optimal Control workshopParis, 23 May 2008 - 17
Example of trajectory optimization using FLOP/OLGA:Ascent trajectory for a classical (expendable) launcher
Ariane 5 trajectory
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Aerospace Dynamics and Optimal Control workshopParis, 23 May 2008 - 18
Example of trajectory optimization using FLOP/OLGA:SOH RLV ascent and reentry trajectory
• Performance index:• max(payload mass)
• Path constraints: - thermal flux - dynamic pressure - load factor• • Local constraints: - orbital parameters at
injection - final altitude and
velocity
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Aerospace Dynamics and Optimal Control workshopParis, 23 May 2008 - 19
Example of trajectory optimization using FLOP/OLGA:booster return trajectories for a TSTO concept
Booster return trajectories for variants of a TSTO concept with different staging Mach number
• Performance index:min(range to landing site)• Path constraints: - thermal flux - dynamic pressure - load factor• Local constraints: - final altitude and
velocity.
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Aerospace Dynamics and Optimal Control workshopParis, 23 May 2008 - 20
Interior point optimization method
• J. Laurent-Varin PhD Thesis (2005): « RLV Trajectory Optimization using an Interior Point Method » (Cnes/Onera/Inria, PhD supervisor F. Bonnans).
• Interior point formulation of the optimal control problem:• Augmented performance index with log-barrier penalization:
Original problem Unconstrained problem withwith path constraints penalized performance index
• Solving of the unconstrained problem with decreasing penalization factor ε.• Other features of the PhD work:
• Runge-Kutta discretization with analysis of the discretization error and mesh refinement.• Dedicated linear algebra solvers (QR factorization for band matrices).
J =∫0
tfL x t , u t ⋅dt
a ≤ g x t , u t ≤ b
J ε =∫0
tfLε x t , u t ⋅dt
Lε x , u = L x , u −ε⋅∑i=1
ng
[ log g i x , u −ai log b i−g i x , u ]
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Aerospace Dynamics and Optimal Control workshopParis, 23 May 2008 - 21
Example of RLV trajectory optimization with the interior point algorithm
Joint booster +orbiter ascent
Orbiter ascent
1
4
2 Booster return
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Aerospace Dynamics and Optimal Control workshopParis, 23 May 2008 - 22
Trajectory optimization for an Aeroassisted Orbital Transfer Vehicle (1/2)
• Studies in the frame of Onera’s CENTOR project (2004-2007)
• Aeroassistance = use of atmospheric forces in the upper atmosphere to perform an orbit transfer with no propellant consumption.
• Problem similar to atmospheric reentry but:
• Different objective function.• Different flight domain.
Orbit before aeroassistance
Orbit after aeroassistance
α
Drag
Lift
Velocity
µ
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Aerospace Dynamics and Optimal Control workshopParis, 23 May 2008 - 23
Trajectory optimization for an Aeroassisted Orbital Transfer Vehicle (2/2)
• For precise optimization of a given concept and mission analysis:
« Heavy » optimization tool (FLOP/OLGA).
• For performance calculation in the frame of multidisciplinary design optimization:
Simplified optimization tool OCTAVES:
• Direct approach.• Simplified control law defined by
a few parameters ( sub-optimal method).
• SQP solver.
AOTV trajectory optimization with OCTAVES
AOTV trajectory optimization with FLOP
Concept Cryo - Mission 2 - avec et sans borne sur gîte
1 113
1 465
0
20
40
60
80
100
120
140
160
0 100 200 300 400 500 600 700 800 900
Temps (s)
Alti
tude
(km
)
0
200
400
600
800
1000
1200
1400
1600
Flux
ther
miq
ue (k
W/m
²)
alt itude alt itude 2 flux therm flux therm 2
Rn = 1.622m
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Aerospace Dynamics and Optimal Control workshopParis, 23 May 2008 - 24
Optimization of an airlaunch system(Cnes/Onera DEDALUS UAV concept)
• Need for a rapid and robust trajectory optimization method (context: joint UAV+launcher vehicule design)
Simplified optimization tool OSTRAL:
• Direct approach.• Simplified control law defined by a
few parameters ( sub-optimal method).
• SQP solver.
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Aerospace Dynamics and Optimal Control workshopParis, 23 May 2008 - 25
Needs in optimal control applied to launchers(cf. 2003 Onera preparatory study to the creation of Cnes’
OPALE research group)
• Initialisation methods.• Global ascent+reentry optimization and branching
trajectories ( RLV).• Global optimization method (search of the « best »
optimum).• Stochastic optimization ( safety requirements for manned
or unmanned systems).• OC in the context of Multidisciplinary Design Optimization
(MDO)
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Aerospace Dynamics and Optimal Control workshopParis, 23 May 2008 - 26
2008 update: needs in optimal control applied to space transportation systems
• Currently drop of interest for RLV studies less need for global ascent+reentry optimization and branching trajectories (#2).
• Main need: OC in the context of Multidisciplinary Design Optimization (#5).
• Easy integration to a MDO process.• No (or reduced) user intervention for initialization.• Robustness.
• Other needs still existing but with a lower priority:• Initialisation method (#1, linked to #5).• Initialisation method (#3, linked to #5).• Stochastic optimization (#4)
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Aerospace Dynamics and Optimal Control workshopParis, 23 May 2008 - 27
Example of simplified MDO process for a multi-stage launcher
Staging optimization
M ? / Max(Φ(M)) ΨV(M,α,Vc)=0
Convergence ? |Vf-Vtarget|ε
Trajectory optimisation
A ? / Max(Vf(M,A)) Ψhf(M,A)=0 Ψγf(M,A)=0
Volume & geometry design
- Loss coefficients α - Velocity gap : (Vf-Vtarget)
Masses M
Masses M
Surface Sref
Fin
Yes
No
Update : - Loss coefficients α - Caract velocity :
Vc = f(Vf-Vtarget)
- α - (Vf-Vtarget)
- Loss coefficients α - Caracteristic velocity VC
V0 , h0 , γ0
Mu
geometry
ML
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Aerospace Dynamics and Optimal Control workshopParis, 23 May 2008 - 28
Conclusion
• Their exist today different OC methods and tools, for different contexts and applications (reentry, ascent, ...).
• No OC problem in space transportation applications seems out of reach with existing tools
but ...
• No optimization method is perfect in every aspect:• Easiness of initialization.• Rapidity and radius of convergence.• Quality of the optimum found.
• Today, the main challenge is to find easily and rapidly an optimal trajectory for a new launcher ( initialization and convergence domain issues).