Politecnico di Milano

Dipartimento di Ingegneria Aerospaziale

Via La Masa, 34 20156 Milano, Italy

Roberto Armellin Francesco Topputo Pierluigi Di Liziaarmellin@aero.polimi.it topputo@aero.polimi.it pier_dilizia@hotmail.com

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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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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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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Lambert's 3BP Arc

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Transfer Trajectory

Earth

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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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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.

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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

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∆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.

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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.

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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.

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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

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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

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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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_____

_____ 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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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