Application of the Circular Restricted Three-Body problem ...€¦ · OUTLINE Dynamical model CRTBP...
Transcript of Application of the Circular Restricted Three-Body problem ...€¦ · OUTLINE Dynamical model CRTBP...
Application of the Circular Restricted Three-Bodyproblem to mission design.
Roberto Castelli
BCAM - Basque Center for Applied Mathematics
Universita del Salento, Lecce, 9th March 2011
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Introduction
Introduction
In space mission design
I Consider the Force Field acting on the Spacecraft
I Consider Physical and Technical constraints
I Satisfy some mission requirements
I Take care of the fuel consumption and the travelling time
I ....
Genesis Mission
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Introduction
Introduction
N-BODY PROBLEM⇓
First guess trajectoriesdesigned in simplified model
I Two-body model
I Restricted Three-bodyproblem
I Bicircular model
I . . .
Numerical Optimisation in Fullsystem
I Direct/Indirect methods
I Multiple shootingtechnique
I Multiobjectiveoptimisation
Different type of Propulsion (Electric - Chemical)
⇓
Low thrust propulsion – Impulsive manoeuvre
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Introduction
Introduction
N-BODY PROBLEM⇓
First guess trajectoriesdesigned in simplified model
I Two-body model
I Restricted Three-bodyproblem
I Bicircular model
I . . .
Numerical Optimisation in Fullsystem
I Direct/Indirect methods
I Multiple shootingtechnique
I Multiobjectiveoptimisation
Different type of Propulsion (Electric - Chemical)
⇓
Low thrust propulsion – Impulsive manoeuvre
9th March 2011 Dynamical system theory for mission design Roberto Castelli 3 / 48
Introduction
Introduction
N-BODY PROBLEM⇓
First guess trajectoriesdesigned in simplified model
I Two-body model
I Restricted Three-bodyproblem
I Bicircular model
I . . .
Numerical Optimisation in Fullsystem
I Direct/Indirect methods
I Multiple shootingtechnique
I Multiobjectiveoptimisation
Different type of Propulsion (Electric - Chemical)
⇓
Low thrust propulsion – Impulsive manoeuvre
9th March 2011 Dynamical system theory for mission design Roberto Castelli 3 / 48
Introduction
OUTLINE
Dynamical model CRTBPTube DynamicsPatched CRTBP approximation
Examples of mission designEarth to HaloRegions of prevalenceSun-Earth DPO to Earth-Moon DPO
Conclusion
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Dynamical model CRTBP
OUTLINE
Dynamical model CRTBPTube DynamicsPatched CRTBP approximation
Examples of mission designEarth to HaloRegions of prevalenceSun-Earth DPO to Earth-Moon DPO
Conclusion
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Dynamical model CRTBP
Circular Restricted Three-Body problem
I Two Primaries move in circular orbits under the mutual gravitationalattraction
I Massless particle moves under the gravitational influence of twoprimaries
In a rotating, adimensional reference frame, µ = m2/(m1 + m2),
(CRTBP)
x − 2y = Ωx
y + 2x = Ωy
z = Ωz
Ω(x , y , z) =12(x2+y2)+ 1−µ
r1+ µ
r2+ 1
2µ(1−µ)
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Dynamical model CRTBP
Properties of CRTBP
I Non integrable Autonomous Hamiltonian SystemI Symmetry (x , y , z , x , y , z ; t)→ (x ,−y , z ,−x , y ,−z ;−t)I Jacobi Integral: C = 2Ω(x , y , z)− (x2 + y2 + z2) = −2EI Equilibrium points: Lagrangian Points Lj , j = 1, ..., 5.I Hill’s Region: H(C ) = (x , y , z) : 2Ω(x , y , z)− C ≥ 0
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Dynamical model CRTBP Tube Dynamics
OUTLINE
Dynamical model CRTBPTube DynamicsPatched CRTBP approximation
Examples of mission designEarth to HaloRegions of prevalenceSun-Earth DPO to Earth-Moon DPO
Conclusion
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Dynamical model CRTBP Tube Dynamics
Dynamics near periodic orbits
I The periodic orbits separates two necks in the Hill’s region
I Linear Dynamics: saddle × center
I 3 types of orbits: asymptotic, transit, non-transit
[W.S. Koon et al.]
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Dynamical model CRTBP Tube Dynamics
Invariant manifolds
The Stable/Unstable Invariant manifoldsSet of orbits asymptotic to the periodic orbit for t → ±∞
[G. Gomez at al.]
I are topologically equivalent toN − 2 dimensional cylinders inthe N − 1 dim. energy manifold
I act as separatrices in the phasespace between transit andnon-transit orbit
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Dynamical model CRTBP Tube Dynamics
Invariant manifolds
The Stable/Unstable Invariant manifoldsSet of orbits asymptotic to the periodic orbit for t → ±∞
I are topologically equivalent toN − 2 dimensional cylinders inthe N − 1 dim. energy manifold
I act as separatrices in the phasespace between transit andnon-transit orbit
I approach the smaller primary
I tangent to the eigenspace of thelinearized system (monodromymatrix)
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Dynamical model CRTBP Tube Dynamics
Box covering of the part of unstable manifold of an Halo orbit in theSun-Earth CRTBP.
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Dynamical model CRTBP Patched CRTBP approximation
OUTLINE
Dynamical model CRTBPTube DynamicsPatched CRTBP approximation
Examples of mission designEarth to HaloRegions of prevalenceSun-Earth DPO to Earth-Moon DPO
Conclusion
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Dynamical model CRTBP Patched CRTBP approximation
Mission Design: dynamical system theory
Dynamical system theory in low energy trajectory designPatched 3-body problem
• The 4-Body system isapproximated with thesuperpositions of two RestrictedThree-Body problems
• The invariant manifold structuresare exploited to design legs oftrajectory
• The design restricts to theselection of a connection point ona suitable Poincare section
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Dynamical model CRTBP Patched CRTBP approximation
Some examples
Low energy transfer to the Moon (Fig. from [W.S. Koon et al.])
Petit Grand Tour of the moons of Jupiter, (Fig. from [G. Gomez at al.])
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Examples of mission design
OUTLINE
Dynamical model CRTBPTube DynamicsPatched CRTBP approximation
Examples of mission designEarth to HaloRegions of prevalenceSun-Earth DPO to Earth-Moon DPO
Conclusion
9th March 2011 Dynamical system theory for mission design Roberto Castelli 16 / 48
Examples of mission design Earth to Halo
OUTLINE
Dynamical model CRTBPTube DynamicsPatched CRTBP approximation
Examples of mission designEarth to HaloRegions of prevalenceSun-Earth DPO to Earth-Moon DPO
Conclusion
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Examples of mission design Earth to Halo
Leo to Halo mission design
I Scientific purposes: Solar observer [ISEE, SOHO, Genesis], Lunarfar-side data relay
I Low energy ballistic transfers made up of impulsive manoeuvres.
I two coupled Restricted Three-Body Problem Planar + Spatial
I Statement of the problem: Optimisation theory, with dynamics
described by the Restricted Four-Body model - bicircular, spatial -
with the Sun gravitational influence (Sun perturbed CRTBP).
[R. Castelli et al.]
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Examples of mission design Earth to Halo
Mission Design
Earth escape stage:
Planar Sun-Earth modelLaunch point on LEO (167 km)Tangential manoeuvre (∆V )
Halo orbit arrival
Spatial Earth-Moon modelStable manifoldBallistic capture to the Halo
Poincare section along a line in configuration space
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Examples of mission design Earth to Halo
Transfer Points
I Properties of transfer points:
I Necessary condition for a feasible transfer:
the pair of points on the section must have
the same location in configuration space.
I The discontinuity in terms of ∆v has to be small.
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Examples of mission design Earth to Halo
Poincare maps →Transfer Points
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Examples of mission design Earth to Halo
Sample first guess trajectory
I First guess trajectories with JEM = 3.159738 (Az = 8000 km)
and JEM = 3.161327 (Az = 10000 km) are later optimized
in the bicircular Sun-perturbed EM model.
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Examples of mission design Earth to Halo
Designed trajectories
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Examples of mission design Earth to Halo
SOLUTION PERFORMANCES
Name Type ∆vi [m/s] ∆vf [m/s] ∆vt [m/s] ∆t [days]
sol.1.1 Two-Imp. 3110 214 3324 106
sol.1.2 Sing-Imp. 3161 0 3161 105
sol.2.1 Two-Imp. 3150 228 3378 128
sol.2.2 Sing-Imp. 3201 0 3201 134
Mingotti Two-Imp. – – 3676 65
Parker Two-Imp. 3132 618 3750 –
Parker Sing-Imp. 3235 – 3235 –
Mingtao Three-Imp. 3120 360 3480 17
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Examples of mission design Regions of prevalence
OUTLINE
Dynamical model CRTBPTube DynamicsPatched CRTBP approximation
Examples of mission designEarth to HaloRegions of prevalenceSun-Earth DPO to Earth-Moon DPO
Conclusion
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Examples of mission design Regions of prevalence
Choice of the Poincare section
How to chose the Poincare section in the Patched CRTBP approximation?
I Usually: on a straight line
I Sometimes: on the boundary of the sphere of influence
I Here: The PS is set according with the prevalence of each CRTBP[R. Castelli]
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Examples of mission design Regions of prevalence
The regions of prevalence
Comparison: Bicircular model ⇔ two CRTBP
∆SE (z) =‖ BCP − CR3BPSE ‖, ∆EM(z) =‖ BCP − CR3BPEM ‖
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Examples of mission design Regions of prevalence
The regions of prevalence
The Regions of Prevalence of each CR3BP is defined according with bythe sign of
∆E (z) = (∆SE −∆EM)(z)
For a choice of the relative phase of the primaries θ
RPEM(θ) = z ∈ C : ∆E (z) > 0 EM Region of Prevalence
RPSE (θ) = z ∈ C : ∆E (z) < 0 SE Region of Prevalence
The curve Γ(θ) = z ∈ C : ∆E (z) = 0• is a closed, simple curve
• is defined implicitly as a function of (x , y)
• depends on θ → changes in time.
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Examples of mission design Regions of prevalence
Coupled CR3BP Approximation
Choice of the Poincare section as the boundary of the Regions of Prevalence
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Examples of mission design Regions of prevalence
Detection of the connection points
Box Covering Approach, GAIOI Compute the Poincare map W s
EM,2(γ1) ∩ Γ(θ) and cover it with BoxStructures
I Cover the transfer region
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Examples of mission design Regions of prevalence
Detection of the connection points
Box Covering Approach, GAIOI Compute the Poincare map W s
EM,2(γ1) ∩ Γ(θ) and cover it with BoxStructures
I Cover the transfer regionI Intersect the Box Covering with W u
SE ,1,2(γ2)
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Examples of mission design Regions of prevalence
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Examples of mission design Sun-Earth DPO to Earth-Moon DPO
OUTLINE
Dynamical model CRTBPTube DynamicsPatched CRTBP approximation
Examples of mission designEarth to HaloRegions of prevalenceSun-Earth DPO to Earth-Moon DPO
Conclusion
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Examples of mission design Sun-Earth DPO to Earth-Moon DPO
First stage: Leo to SE-DPO
Look for impulsive manoeuvre transfer from Leo to DPO in SE-CRTBP
I Integrate backwards thestable manifold
I Intersect the manifoldwith Leo
I Select those intersectionsthat are tangent to Leow.r.t geocentriccoordinates
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Examples of mission design Sun-Earth DPO to Earth-Moon DPO
∆V at Leo
Remark A manoeuvre |v | appliedin the same direction of the motionproduces the maximal change ofJacobi constant. It holds
∆J = |v |2 + 2|C ||v ||z |
where C = Vt
|z| − 1 depends on the
Leo altitude and |z | is the
geocentric distance, Vt orbital
velocity
The Jac.const on a Leo ( 167 Km alt.) is about 3.070352The Jac.const. of family g is in the range [3.00014;3.00092]
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Examples of mission design Sun-Earth DPO to Earth-Moon DPO
Jacobi- DPO h-LEO (Km) ∆V (m/s)
3.000464798057 220 3190
3.000464798057 160 3212
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Examples of mission design Sun-Earth DPO to Earth-Moon DPO
Second stage: SE-DPO to EM-DPO
Procedure for design the transfer:
1) Select two DPOs
2) Compute the Poincare map of (un)-stable manifold on a section( line through the Earth with slope θSE and θEM).
Left: Stable manifold in the interior region for a DPO in the EM-CRTBP.
Right: Unstable manifold of a DPO in SE-CRTBP
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Examples of mission design Sun-Earth DPO to Earth-Moon DPO
Design
Procedure for design the transfer:
I Select two DPOsI Compute the Poincare map of (un)-stable manifold on a section
3) Write the two maps in the same system of coordinates, beingθ = θSE − θEM the relative phase of the primaries at the transfer time
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Examples of mission design Sun-Earth DPO to Earth-Moon DPO
Design
Procedure for design the transfer:
I Select two DPOsI Compute the Poincare map of (un)-stable manifold on a sectionI Write the two maps in the same system of coordinates, beingθ = θSE − θEM the relative phase of the primaries
4) Look for possible connections on the Poicare section
Projection of the Poincare maps onto the (x , vx) plane and (x , vy ) plane, in EM-rf
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Examples of mission design Sun-Earth DPO to Earth-Moon DPO
Results: Interior connection
9th March 2011 Dynamical system theory for mission design Roberto Castelli 40 / 48
Examples of mission design Sun-Earth DPO to Earth-Moon DPO
Results: Exterior connection
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Examples of mission design Sun-Earth DPO to Earth-Moon DPO
SE-Jacobi EM-Jacobi Time, (days) ∆V (m/s)?
Interior 3.0004647980 3.02599 115 339
Exterior 3.00043012418 3.026764 116 8? Connection between two DPOs
Type Start Target Time ∆V (m/s)
Mingotti Ext- Optim. LEO DPO Jac=? 90 3160
Ming Exterior LEO Retr. DPO 101 3207
Ming Interior LEO Retr. DPO 33 3802
[G. Mingotti et al.] : Earth to EM-DPO with low thrust propulsion[X. Ming at el.]: Earth to retrograde stable orbit around the Moon
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Conclusion
OUTLINE
Dynamical model CRTBPTube DynamicsPatched CRTBP approximation
Examples of mission designEarth to HaloRegions of prevalenceSun-Earth DPO to Earth-Moon DPO
Conclusion
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Conclusion
Conclusion
I Dynamical model
I The CRTBP is introduced to model the dynamics
I The invariant manifolds provide low energy transfers in the phase space
I Design technique:
I The patched CRTBP approximation has been formalized.
I Immediate definition of the transfer points in the phase space throughthe box covering approach.
I A technique to design impulsive transfers has been developed.
I Designed trajectory:
I Efficient trajectories in terms of ∆v have been designed in the planarand spatial case
I A non-classical Poincare section has been presented in terms of Regionsof prevalence.
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Conclusion
REFERENCES
W. S. Koon, M. W. Lo, J. E. Marsden, E. Jerrold, and S. D. Ross. Heteroclinicconnections between periodic orbits and resonance transitions in celestial mechanicsChaos, 10(2):427–469, 2000.
G. Gomez, W.S. Koon, M.W. Lo, J.E. Marsden, J. Masdemont and S.D. Ross –Invariant Manifolds, the Spatial Three-Body Problem and Space Mission Design –Advances in the Astronautical Sciences, 2002
W. S. Koon, M. W. Lo, J. E. Marsden and S. D. Roos, – Low energy transfer tothe Moon – Celestial Mech. Dynam. Astronom, Vol 81, pp 63-73, 2001
R. Castelli, G. Mingotti, A. Zanzottera and M. Dellnitz, Intersecting InvariantManifolds in Spatial Restricted Three-Body Problems: Design and Optimization ofEarth-to-Halo Transfers in the Sun–Earth–Moon Scenario, submitted to Commun.Nonlinear Sci. Numer. Simulat.
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Conclusion
REFERENCES
R. Castelli Regions of Prevalence in the Coupled Restricted Three-Body ProblemsApproximation Submitted to Commun. Nonlinear Sci. Numer. Simulat.
M. Dellnitz; G. Froyland; O. Junge The algorithms behind GAIO – Set orientednumerical methods for dynamical systems B. Fiedler (ed.): Ergodic Theory,Analysis, and Efficient Simulation of Dynamical Systems, pp. 145-174, Springer,2001
J.S. Parker– Families of low-energy lunar halo trasfer – Proceedings of theAAS/AIAA Space Flight Mechanics Meeting, pp 483–502, 2006.
G.Mingotti, F. Topputo, and F. Bernelli-Zazzera, Exploiting Distant Periodic Orbitsand their Invariant Manifolds to Design Novel Space Trajectories to the Moon,Proceedings of the 20th AAS/AIAA Space Flight Mechanics Meeting, San Diego,California, 14-17 February, 2010
Ming X. and Shijie X. Exploration of distant retrograde orbits around Moon, ActaAstronautica, Vol.65, pp. 853–850, 2009
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Conclusion
Thank you
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Conclusion
⇒ Dynamical model⇒ Mission Design
⇒ References
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