roberto armellin Œ francesco topputo Œ pierluigi di lizia · x 10 11-1.5-1-0.5 0 0.5 1 x 10 11 x...
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Politecnico di Milano
Dipartimento di Ingegneria Aerospaziale
Via La Masa, 34 20156 Milano, Italy
Roberto Armellin Francesco Topputo Pierluigi Di [email protected] [email protected] [email protected]
Formation Flying: Optimal reconfiguration Maneuver and Optimal Formation Control.
Dynamical model: Linear time-varying equation of motion with respect to eccentric reference orbits.
Propulsion: Continuous and variable electric low-thrust propulsion.
Optimization formulation: State transition matrix and control matrix are used to discretize the differential constraints given by the equation of motion. Optimization variables consist of the control vector at each discretization step. The objective function is the square of the control variables plus a repulsive term to ensure safe maneuvers.
Optimization techniques: SQP algorithm.Perturbative forces: Atmospheric drag, gravitational
up to J4, sensor noise.
Optimization formulation and optimization technique allow the real-time application of the algorithm.
Reference orbit parameters:a=7178 [km] e =0 i=90° Ω=0° ω=0°
461.7509.7535.2∆v [mm/s]
Satellite 3Satellite 3Satellite 1
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radial [m] in-track [m]
cros
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]
Three spacecraft formation reconfiguration
Using eccentric reference orbit increases computational costs but allows:- Smaller modeling errors with respect to
Clohessy-Wiltshire equations.- Formation design on high eccentric orbits like
Molnya.
Three spacecraft formation reconfiguration on high elliptical reference orbit:
a=26000 km e =.74 i=63° Ω=0° ω=0°
824.11293.3 1635.7∆v [mm/s]Satellite 3Satellite 3Satellite 1
After the reconfiguration maneuver has been accomplished the same algorithm can be used for formation station-keeping.
An error-box is defined for each satellite around its nominal position: the station-keeping algorithm forces the satellite to lie inside the specified error-box.
∆v =5.51 [mm/s*orbit]
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-trac
k [m]
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x [m]
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y [m]
Interplanetary transfer to Mars and Mercury using low-thrust propulsion
Dynamical model: Gauss equation written using equinoctial elements.
Ephemeris: Analytical ephemeris model.Propulsion system: Continuous and variable electric
low-thrust propulsion.Optimization formulation: Thrust control vector is
parameterized trough cubic spline. The optimization variables consist of spline coefficients (both for thrust angles and magnitude), escape velocity from Earth sphere of influence, time of departure and time of flight. The optimization function is the propellant mass plus a penalty term to enforce the target planet arrival.
Optimization techniques: Evolutionary algorithm plus SQP.
Combining low thrust arcs and multiple swing-bye maneuvers will be investigated in order to reduce propellant mass
Earth - Mars
Earth - Mercury
.148.168300.17Mars.52.8489.56Mercury
Propellant FractionTMAX [N]TOF [days]
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x (adim., E-M rf)
y (ad
im.,
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rf)
Lambert's 3BP Arc
WsL1p.o.
0.99960.9998
11.0002
1.00041.0006
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y (adim., S-E rf)
x (adim., S-E rf)
z (ad
im.,
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rf)
Transfer Trajectory
Earth
0.99 0.992 0.994 0.996 0.998 1 1.002 1.004
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x 10-3
x (adim., S-E rf)
y (ad
im.,
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rf)
Transfer Trajectory
SE L1 Halo Orbit (Az=100000 km) 0.9885 0.989 0.9895 0.99 0.9905 0.991 0.9915 0.992 0.9925 0.993 -6
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y (adim., S-E rf)x (adim., S-E rf)
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im.,
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rf) L1
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im.,
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rf)
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Earth Moon
Transfers on Halo Orbits in the Sun-Earth System
Transfers on Halo Orbits in the Earth-Moon System
Halos: 3D periodic orbits about L1 or L2 Governed by highly non-linear dynamics Large out-of-plane amplitudes Suitable for a number of new space concepts
In the Earth-Moon system the stable manifolds no longer approach the Earth. A direct injection leaving from low-Earthorbits is forbidden in this case.
A piece of the stable manifold associated to the final halo (blue) has been targeted by using a Lamberts three-body problem arc (green).
The two-impulse maneuver places the spacecraft into a Moon-resonant orbit.
421036L2 (Az=8000 km)
52936L1 (Az=1000 km)
∆t(days)
∆v(m/s)
Leaving from GTO(200 x 35840 km)
In the Sun-Earth system the stable manifolds (blue) associatedto the halos lie in the Earths neighborhood.
The more the out-of-plane amplitudeof the final halo grows, the more thestable manifolds approach the Earth.
A direct injection on these manifolds is possible by leaving from a Keplerian orbit around the Earth.
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2-0.8
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, iner
tial fr
ame)
Departure (Sun-Earth)Conic Link (Sun) Arrival (Sun-Mars)
Sun
Earth
Mars
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ame)
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, iner
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ame)
Sun Jupiter
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(Sun-Jupiter system)
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(Sun-Saturn system)
0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 1.12 1.14-0.1
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im.,
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f)
Forbidden Region
Forbidden Region
Sun
x=1-µ
x=1-µ
Jupiter
Transit Orbit
Asymptotic Orbit
WuL2 p.o.
WsL2 p.o.
Interplanetary Transfers Through Libration Points
Trajectories within the invariant manifold tubes are the only transit orbits through the small allowed region.
If the intersection between two tubes occurs in the configuration space, a small ∆v completes the intersection in the whole phase space and assures the ballistic capture at arrival. Jupiter-Saturn transfer:
intermediate ∆v=931 m/s(but ∆t=3894 days!)
Invariant manifolds do not intersect when inner planets are considered. In this case the departure (red) and arrival (blue) legs are linked together by means of an intermediate two-body Lamberts arc (green). This technique lead to a multi-burn transfer trajectory.
8063559Earth-Mars
4902964Earth-Venus
∆t(days)
∆v(m/s)
The high transfer times are due to the asymptotic nature of the manifolds.
The solutions do not take into account departing and arrival maneuvers.
-0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08-0.1
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im.,
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rf)
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im.,
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Earth
Forbidden Region
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Forbidden Region
WsL1
Lunar Transfers Through L1
Moon-Assisted Orbital Transfers
~
At L1 a negligible maneuver closes the Hills curves and assures the permanent (theoretical) capture by the Moon.
47924
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∆t (days)∆v (m/s)Departing from a
200 x 35840 km GTO
The classical orbital transfers have been computed from the three-body problem point of view.
Results show that Moon-assisted orbital transfers can be obtained if the right lunar phase is chosen and the Moon gravitational attraction is exploited.
The cost for a transfer betweena LEO (200 km) and the GEO is∆v-∆vH = -117 m/s where ∆vH isthe cost for the Hohmann solution.
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15
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adim
., EM
rf)
Forbidden Region
Moon
WsL1
WuL1
L1
Direct Earth-to-Moon transfers (through L1) have been obtained by targeting the L1 stable manifold (blue) with a Lamberts three-body arc.
The Moon approach occurs at the minimum energy level that allows the transfers (C=C1).
Why Global Optimization in Space Mission Design:
The objective function is typically non-convex: e.g. the ∆V requirements show quasi-periodical features on the date of departure, associable to the synodic periods of the planetary system.
Why Multiobjective Optimization in Space Mission Design:
Mission RequirementsMission Objectives
Technological requirementsCost minimization
Presence of several evaluation criteria
Why Evolutionary Algorithms (EAs) in Global Multiobjective Optimization:They handle a set of solutions simultaneously and can then identify several points approximating the Pareto front in a single runThey need little a priori knowledge about the problem to solveThey are less sensitive to the shape of the Pareto front
-8 -6 -4 -2 0 2 4-2
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X [AU]
Cassini-Huygens trajectory (Alternative solution)
Y [A
U]
Earth 25/10/1997
Venus (GA1)19/05/1998
Venus (GA2)24/06/1999
Earth (GA) 18/08/1999
Jupiter 17/02/2001
Saturn 01/12/2005
-12 -10 -8 -6 -4 -2 0 2 4 6-6
-4
-2
0
2
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X [AU]
Cassini-Huygens trajectory (best solution)
Y [A
U] Earth 20/11/1997
Venus (GA1)18/05/1998
Venus (GA2)29/06/1999
Earth (GA)21/08/1999
Jupiter 17/09/2001
Saturn 16/09/2011
Choice of the Evolutionary Algorithm:Main EAs branches:
Genetic Algorithms
Evolutionary Strategies
Evolutionary Programming
Mutation exaltationAutomatic archive maintenance
Due to:
Fast Evolutionary Programming (FEP) [Yao, Liu, Lin,1999] has been used, together with:
Debs constraint handling method [Deb, 2000]Stopping criterion based on Kernels density estimate
Applications:Single objective optimization (Cassini-Huygens mission)
Using the ∆V requirements as the only evaluation criterion could lead to time consuming transfer trajectory (right hand figure). In order to avoid the loss of good solutions one has to use global multiobjective optimization techniques.
∆V= 6368.2 m/s ∆t=13.82 y ∆V= 7154.6 m/s ∆t=8,1 y
Multi objective optimization (Mars-Express mission)
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ass [
kg]
_____
_____ Space System design optimization
Simultaneous optimization of Space system design and mission analysis
Pareto fronts corresponding to System Design optimization (fixed mission analysis) and to the simultaneous optimization of System Design and Mission Analysis are compared: these two design phases, classically separately optimized, must be part of a unique optimization process, due to their strong interaction.
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Uncertainties in Space System Design Optimization
________ Pareto front with nominal parameters value
________ Interval Pareto front with uncertain parameters value
[5.4,6.6] hDaylight phase[1.35,1.65] hEclipse phase[227.7,232.3] dTransfer time[2970,3030] m/s∆V[108,132] WPayload power[180,220] kgPayload mass
Pareto front approximation in case of uncertain design parameters (reference mission Mars Express):
The use of interval analysis for flexible optimization levels (reference mission Mars Express):
The effects of the uncertainties on both the mission objective and constraints are evaluated and considered during the whole optimization process.
The decision process is supported by sensitivity information on the objective functionsCrisp information on local Pareto fronts can be obtained in a second level optimization process
A better search space covering is gained
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Comparison between Crisp and Interval Pareto front (first optimization level)
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________
Crisp Pareto front Interval Pareto front
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Interval information and local crisp Pareto front (second optimization level)
selected individual
Crisp Pareto front ________
________ Interval Pareto front ________ Crisp local Pareto front
Individual selection
Interval Evolutionary Programming:
is often asked to handle uncertain informationneeds support tools providing intervals on the design variables rather than crisp information
The designer:
MethodMotivation Relevant problemsDependency problem on:
Intervals computation: given X=[a,b], X-X = [a-b,b-a] ≠ [0,0]
Individuals selection: all the objective functions may depends on the same design variables
Real vectors on the search space can be substituted by interval vectorsDesign parameters can be set as intervals in case of uncertainty on their valuesInterval analysis is used for the computation of intervals on the objective function values