Progress_on_Understanding_Satellite_Clus

Progress on Understanding Satellite Cluster Flight
About Oblate Planets
Dr. Vladimir Martinusi
Joint Work with Prof. Pini Gurﬁl
Distributed Space Systems Lab
Faculty of Aerospace Engineering
Technion – Israel Institute of Technology
http://dssl.technion.ac.il/
Vladimir Martinusi (DSSL) Cluster Flight About Oblate Planets 02/13/2013 1 / 23

Presentation Outline
1 Introduction
2 Framework
1 Basic Physical Gravitational Models
2 Mathematical Models
3 Spherical Harmonics
4 Classical Approximate Models
5 New Approximation Proposal
3 Analytic solutions
1 Absolute motion
2 Relative motion
4 Applications
1 Cluster initialization
2 Impulsive Maneuvers
5 Additional Results – Open Orbits about Oblate Planets
6 Ongoing & Prospective Research

Introduction: Cluster Flight
Cluster Flight: two or more satellites flying somewhat loose
Min/max range fixed by mission specifications
No strict formation geometry required
Applications:
Geolocation
Space networks
SAR Interferometry
Disaggregated spacecraft
Related technologies:
High-resolution imaging
Data relaying
Interferometry
In-situ gravitometry

Framework
Goal: mitigate the effects of the gravity ﬁeld perturbations
Real problem: the potential of the geoid (see ﬁgure)
V (r) =
Q∈D
dm
r − rQ
=
Q∈D
dm (r)
r2
Q − 2r · rQ + r2
; (1)

Basic Physical Gravitational Models
The homogenous spherical model
leads to the Newtonian potential
analytic equations of motion
inaccurate for realistic scenarios
widely used as a nominal orbit
Ellipsoid
the potential has analytic expression
Legendre polynomials expansion
accurate for realistic scenarios
equations of motion are non-integrable
widely used for numerical propagators

Mathematical Models
The Newtonian potential – homogenous spherical model:
V0 (r) = −
µm
r
(2)
Geoid: Spherical harmonics model:
V (r) = V0 (r) −
∞
∑
k=1
rk
AkPk (cos φ) −
∞
∑
k=1
rk
k
∑
m=1
(Bkm sin mθ + Ckm cos mθ) Pkm (sin φ) (3)
φ = colatitude; θ = argument of latitude
Pk, Pkm = Legendre polynomials
Ak, Bkm, Ckm = spherical harmonics, 1 ≤ m ≤ k

Spherical Harmonics – A Visualization

Classical Approximate Models
Brouwer, Garfinkel, Kozai methods (1959)
Fourier series expansion of the geoid potential
Truncates the series ⇒ averaging
Analytic solution for the equations of motion
The Vinti potential (1959)
Integrable approximation (Stäckel-type dynamical system)
Uses oblate spheroidal coordinates
Drawback: stands only for J2
2 ≈ J4 (Earth)
The two fixed centers (Izsák, Aksenov, Grebenikov)(1960s)
First approached by Euler, Lagrange, Jacobi
Applied in the context of celestial motion
The Cid–Lahulla radial intermediary (1960s)
Integrable approximation
Closed-form solution with elliptic integrals

New Approximation Proposal
The main perturbation is due to the J2 zonal harmonic
Approximating the gravitational model by considering only the J2
term is reasonably accurate
The J2 potential has the expression
VJ2 (r) = −
µm
r
1 − J2
req
r
2
3 cos2
φ − 1 = VJ2 (r, φ) (4)
The term in φ causes the problem to be non-integrable
Ignore the term in φ:
The problem becomes integrable
Physical meaning: motion in the equatorial plane
The J2–central potential has the expression:
VJ2 (r) = −
µm
r
1 + J2
req
r
2
= VJ2 (r) (5)

Analytic Solutions
The Absolute Motion
Falls in the general case of motion in a central force field
Potential of type A/r + B/r3 ⇒ solution with elliptic integrals
Closed form expression for (r,v) [Jezewski 1989, Martinusi and
Gurfil 2011]
Has five independent first integrals ⇒ maximal super-integrability

Analytic Solutions (cont’d)
The Absolute Motion (cont’d)
The orbit is contained in a fixed plane, defined by (r0,v0)
The orbit is bounded between two concentric circles
Unique shape defined by: the radial period T; the orbital angle ϕ
There exist several pericenters and apocenters
A main pericenter may be defined based on the ICs [Martinusi and
Gurfil 2012]
A set of 6 orbital elements may be defined (5
are constant)
Regularization of the equations of motion

Analytic Solutions (cont’d)
The Relative Motion
Based on the general solution for the relative motion in a central
force field [Condurache and Martinusi 2007]
Closed form expression for (r, v) [Martinusi and Gurfil 2011]
Explicit periodicity conditions
Particular case – 1:1 commensurability
Theorem
The relative motion in a central force
field is 1:1 periodic if and only if the
radial periods T1,2 and the orbital
angles ϕ1,2 of the absolute motions are
equal.

Application: Cluster Initialization
Algorithm for adjusting the inertial initial conditions [Martinusi and
Gurﬁl 2011]
Equatorial periodic relative motion
Bounded relative motion in inclined orbits
Relative distance remains within
the desired limits for up to one
year
Simulations were performed in
STK® with HPOP propagator

Application: Cluster Initialization (cont’d)

Application: Impulsive Maneuvers
Formation-keeping / Station-keeping impulsive maneuvers
Maneuvers should be made in order to reach the periodicity /
boundedness conditions
Single maneuver correction possible only if the desired orbit is
reachable by only one maneuver
Figure: yellow – actual orbit; orange – desired orbit
the desired orbit must intersect the min/max radii of the actual orbit

Application: Impulsive Maneuvers (cont’d)
Closed form expression for the impulsive maneuver velocity vector
[Martinusi and Gurﬁl 2012]:
∆v =
1
r−
1
2
h+
1 × r−
1 ±
1
r−
1
2E+
1 +
2µ
r−
1
+
µJ2r2
eq
r−
1
3
−
h+
1
r−
1
2
r−
1 − v−
1 .
Algorithm to select the the best maneuver moment
Equatorial orbits: possible anyway within the allowed area
Inclined orbits: at the equatorial plane passage
Maneuver moment depends on: impulse magnitude, drift rate (time
window), other mission speciﬁcations

Additional Results – Open Orbits
Analytic closed-form solution for the open orbits in the equatorial
plane (with J2) [Martinusi and Gurfil 2013]
New orbit revealed: the “fish” orbit
Fly-by’s: explicit computation of:
Deflection angles
Pericenter position
Differences between the J2– central and Keplerian models
Numerical simulations run on actual real missions (Voyager,
Pioneer, Cassini)
A Laplace-Runge-Lenz-like vector revealed

Additional Results – Open Orbits (cont’d)
The “ﬁsh” orbit

Additional Results – Open Orbits (cont’d)
Differences J2–central versus Keplerian fly-by models:
v∞ [km/sec] ∆δ [deg] [deg] ∆rP [km] |∆rP| [km]
Pioneer 10 11.218 0.052 0.026 156.02 181.44
Pioneer 11 14.894 0.164 0.082 275.49 320.32
Voyager 2 5.653 0.003 0.001 39.59 46.05
Ulysses 7.118 0.008 0.004 62.78 73.00
Comparison – Jupiter flybys – Keplerian versus J2 – central
∆δ – difference in deflection angles;
– pericenter angular shift;
∆rP : difference in pericenter altitudes;
|∆rP| : distance between the J2 and the Keplerian pericenters
Could be incorporated into the analytical models of flybys

Ongoing & Prospective Research
1 New set of orbital elements for the J2–central model
Based on the problem super-integrability
Possibility to derive new variational equations
2 Encke-like orbit propagator based on J2 – central nominal orbits
3 Two point boundary value problem in a Cid-Lahulla potential
(closed-form solution for inclined orbits)

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