Dynamics and Stability of Tethered Satellites at Lagrangian Points

T

DG Grupo de Dinámica de Tethers (GDT)

Dynamics and Stability

of Tethered Satellites at Lagrangian Points†

by

J. Peláez,Technical University of Madrid (UPM), 28040 Madrid, Spain

Workshop on Stability and Instability in Mechanical Systems: Applications and Numerical Tools

Thursday, 4 December 2008, Barcelona

† Supported by ESTEC Contract 21259

Dynamics and Stability of Tethered Satellites at Lagrangian Points– p. 1/100

T

DG Grupo de Dinámica de Tethers (GDT-UPM) - Cisas (Padova)

Dynamics and Stability of Tethered Satellites at Lagrangian Points†

by

Claudio Bombardelli, Dario Izzo,

ESTEC-ESA, Noordwijk, 2201 AZ, The Netherlands

Enrico C. Lorenzini, Davide Curreli ,

University of Padova, Padova, 35131, Italy

M. Sanjurjo-Rivo, Fernando R. Lucas, J. Peláez,Technical University of Madrid (UPM), 28040 Madrid, Spain

Daniel J. Scheeres,The University of Colorado, Boulder, CO 80309-0429, USA

M. LaraReal Observatorio de la Armada, 11110 San Fernando, Spain

Final Report for the Advanced Concept Team

† Supported by ESTEC Contract 21259


T

DG Tethers fundamentals

In a tethered system, two masses orbiting at different heights share a common orbital

frequencyΩ0 ⇒ The third Kepler law is broken by the tether tension

m1

m2

r1

r2

r0T

T

G

g1

g2

T

DG Tethers fundamentals

Stable equilibrium along the local vertical

θ

Local Vertical

3mΩ20d

d

G

x

z

Any deviationθ from the local vertical gives

place to a torque. In effect, the gravity gradi-

ent force breaks down in

• one component along the tether, which

is basically balanced by the tether

tension

• one component orthogonal to the tether,

which provides the restoring torque.

This torque leads the tether again to the

local vertical

Thus, the local vertical is a stable equilibrium

position for the tethered system


T

DG Electrodynamic tethers fundamentals

~B

~v~E

~E = ~v × ~Bq+

q−

Conductive rod

In a frame attached to the rod, a motional elec-

tric field appears

~E = ~v × ~B

It is induced by the magnetic field~B inside

which the rod is moving. For a conductive rod

moving in the vacuum, a redistribution of sur-

face charge takes place, leading to a vanishing

electric field inside the rod. Thus, a steady

state is reached with no motion of charged

particles in the rod.

If the rod is moving inside a plasma environ-

ment, this picture changes drastically


T

DG 2007 AAS/AIAA Space Flight Mechanics MeetingSedona, Arizona, January 28 - February 1, 2007

Electrodynamic tethers in a plasma environment

e−e−e−

e−

e−

e−

i+

i+

i+

i+

~B

~v

I

q+

q−

The ionospheric plasma makes the electrons

begin to be attracted by the anodic end of the

rod. Similarly, the ions will be attracted by the

cathodic end. Some of these charges will be

trapped by the rod and they produce a current

I which flows inside the conductive mater-

ial. The amount of currentI can be increased

with the help of plasma contactors placed in

the tethers ends.However, the interaction be-

tween the tether currentI and the magnetic

field ~B gives place to forces acting on the

rod. They will break (or accelerate) its mo-

tion (in the figure they are breaking the mo-

tion). Their resultant is: ~F = ~I × ~B L

A permanent tethered observatory at Jupiter. Dynamical analysis – p.5/24

T

DG 2007 AAS/AIAA Space Flight Mechanics MeetingSedona, Arizona, January 28 - February 1, 2007

Electrodynamic tethers in a plasma environment

e−e−e−

e−

e−

e−

i+

i+

i+

i+

~B

~v

I

q+

q−

The ionospheric plasma makes the electrons

begin to be attracted by the anodic end of the

rod. Similarly, the ions will be attracted by the

cathodic end. Some of these charges will be

trapped by the rod and they produce a current

I which flows inside the conductive mater-

ial. The amount of currentI can be increased

with the help of plasma contactors placed in

the tethers ends.However, the interaction be-

tween the tether currentI and the magnetic

field ~B gives place to forces acting on the

rod. They will break (or accelerate) its mo-

tion (in the figure they are breaking the mo-

tion). Their resultant is: ~F = ~I × ~B L

A permanent tethered observatory at Jupiter. Dynamical analysis – p.5/24

T

DG ARIADNA PROGRAM

The two basic tether configurations

Spherical collector

m1

m2

Uni

form

inte

nsity

Insulated cable

Bare tether

m1

m2

Exposed cable

NO

Nun

iform

inte

nsity


T

DG Advanced Topics In Astrodynamics – Tethered Systems

TSS-1R. The Tether is partially deployed

Tether Fundamentals – p.13/48

T

DG Advanced Topics In Astrodynamics – Tethered Systems

STS-46 Tethered Satellite System 1 (TSS-1) satellite deployment from OV-104

The satellite is reeled out via its thin Kevlar tether into the blackness of space during deployment operations from the payload bay of Atlantis.

At the bottom of the frame is the satellite upper boom including (bottom to top) the 12-m deployment boom, tip can, the docking ring, and

concentric ring damper. The Langmuir probe and the dipole-field antenna are stowed at either side of the TSS-1 satellite.

Tether Fundamentals – p.10/48

T

DG Background: Tethered satellites and Lagrangian Points

• The Earth-Moon external point (L2). Excellent scenario for many different missions

• Tethers are useful in some specific missions

* Stabilization at Lagrange points (Colombo, Farquhar, Misra)

• ED tethers: may produce power and propulsion in appropriateenvironments

* Jovian world (Sanmartín et all., Peláez and Scheeres, . . . )

* plasma collection is better with straightened tethers

• Tether’s stabilization needs mechanical tension

* Close to a body: gravity gradient (self balanced EDT)

* Far from body: fast rotating tethers

• Dynamics may be quite different from mass-point satellites

* coupled roto-traslational motion (long tethers)

* varying length


T

DG Outline of Presentation

• Gentil Introduction

• Inert Tethers and Periodic Motions

• Io Exploration with Electrodynamic Tethers

• Stability of tethered satellites at lagrangian collinear points


T

DG Diffusion

• Dynamics of tethered libration-points satellites, by

Manuel Sanjurjo Rivo, Fernando R. Lucas & Jesús Peláez,

XI Jornadas de Trabajo en Mecánica Celeste, 25—27 June 2008,Albergue de la Real Fábrica, Ezcaray, La Rioja, Spain

• Dynamic behavior of a fast-rotating tethered satellite, by

M. Lara & Jesús Peláez,

XI Jornadas de Trabajo en Mecánica Celeste, 25—27 June 2008,Albergue de la Real Fábrica, Ezcaray, La Rioja, Spain

• On the dynamics of a tethered system near the colineal libration points, by

M. Sanjurjo-Rivo, F. R. Lucas, J. Peláez, C. Bombardelli, E.C. Lorenzini, D. Curreli, D. J. Scheeres & M. Lara,

AIAA paper 2008-7380, The2008 AAS/AIAA Astrodynamics Specialist Conference and Exhibit , 18—21 Agosto

2008 Hawaii Convention Center and Hilton Hawaiian Village,Honolulu, Hawaii, USA

• Io exploration with electrodynamic tethers, by

C. Bombardelli, E. C. Lorenzini, D. Curreli, M. Sanjurjo-Ri vo, F. R. Lucas, J. Peláez, D. J. Scheeres & M. Lara,

AIAA paper 2008-7384, The2008 AAS/AIAA Astrodynamics Specialist Conference and Exhibit , 18—21 Agosto

2008 Hawaii Convention Center and Hilton Hawaiian Village,Honolulu, Hawaii, USA

Abstracts sent:

• Plasma torus exploration with electrodynamic tethers, by

E. C. Lorenzini, D. Curreli, C. Bombardelli, M. Sanjurjo-Ri vo, F. R. Lucas, J. Peláez, D. J. Scheeres & M. Lara,

19th AAS/AIAA Space Flight Mechanics Meeting, February 8—12, 2009, Savannah, Georgia, USA


T

DG Conclusions

• Gentil Introduction:We have developed the theory that permit the analysis theoretical and

the simulations of tethers, of constant or varying length, inert or electrodynamic, rotating or

non-rotating in the environment provided for a binary system. Some results are completely

new and there is nothing similar in the literature.

• Inert Tethers and Periodic Motions:The influence of the tether length in the existence and

stability of periodic motions associated to Halo orbits hasbeen studied in detail for inert

tethers. Some results are completely new and open interesting research lines and possibilities.

• Io Exploration with Electrodynamic Tethers:All the analysis is an important novelty.

At least two different, and important, research lines have been set up.

• Stability of tethered satellites at lagrangian collinear points: We show that the Hill

approach permits to clarify some obscure details underneath the previous analysis. An

interesting application for the Mars satellites —Deimos orPhobos— can be deduced from our

analysis.


T

DG Deimos and Phobos

Mars - Deimos

Magnitude Value Unit

LT 10 km

xE 22.34 km

∆x 200 m

kξ 4

T 72 hours

m 1000 kg

φ 45 deg

Λ 0.1

Mars - Phobos

Magnitude Value Unit

LT 5 km

xE 16.96 km

∆x 125 m

kξ 3

T 72 hours

m 1000 kg

φ 45 deg

Λ 0.1


T

DG ARIADNA PROGRAM

Gentil Introduction


T

DG Gentil Introduction

GOAL : TO GAIN AN INSIGHT INTO THE DYNAMICS OF FAST ROTATING TS

1. Derive the equations of motion of tethered satellites• CRTBP approximation• inert tethers, constant length• coupled (5-DOF) rotational-traslational motion

2. Fast rotating tethers• Average over the (fast) rotation angle

3. Motion “close” to the smaller primary: Hill’s approach• In some interesting cases, rotational and translational motion decouple!

4. Show relevant characteristics of specific applications


T

DG Inertial motion of an inert tether

• Only gravitational forces

• Dumbbell model ⇒ rigid body dynamics:

T = 12

m (x · x) +1

2Ω · I · Ω, V = −

∫

m

µ

‖y‖dm

• m total mass,I central inertia tensor• x position of c.o.m.,y position of mass elements• Ω tether’s rotation vector• µ gravitational parameter of the attracting body

• Lagrangian approach: L = T − V →d

dt

(

∂L

∂qj

)

−∂L

∂qj= 0,

• qj = dqj/dt, qj : generalized coordinates


T

DG The dumbbell model

• Inertial frameOxyz

• Tether-attached frame: originG, unit vectors(u1, u2, u3),• u1 = u, u2 = k × u/‖k × u‖, u3 = u1 × u2,

• u = Ω × u ⇒ Ω = (Ω · u1,Ω · u2,Ω · u3)T

O

m1

m2

G

x

x

x

yy

z

z

θ

ϕ

u

T

DG Gravitational Potential

V = −

∫

m

µ

‖y‖dm

dm = ρLdsα

G

m2

uG

m1

L1

L2

x yu

r = ‖x‖, cos α = u · x/r

f dm

s

O

• Usual expansion of1

‖y‖in Legendre polynomials

V = −µ

r

∫

m

dm√

1 + 2(s/r) cos α + (s/r)2= −

µm

r

∞∑

n≥0

(

L

r

)n

an Pn(cos α)

• an = an(m1, m2, mT ) are functions of tether & end masses (a0 = 1, a1 = 0)


T

DG Kinetic analysis

• Kinetic energy: T =1

2m (x · x) +

1

2Ω · I · Ω

• Rotational motion (tether attached frame):

• Inertia tensor (null moment of inertia aroundu):

I =

0 0 0

0 I 0

0 0 I

I = mL2a2 =1

12mL2 (3 sin2 2φ − 2Λ

)

• φ = φ(m1, m2, mT ), Λ = mT /m

• Ω · u = 0 ⇒ Ω · I · Ω = I ‖u‖2

• Position of the tether:u = (cos ϕ cos θ, cos ϕ sin θ, sin ϕ)

• Kinetic energy: T =1

2m(

x2 + y2 + z2)

+1

2I(

ϕ2 + θ2 cos2 ϕ)


T

DG Circular Restricted Three Body Problem

m

x1

x2GP

ℓ

mP1

mP2

x

y

zω

~i

~j

~k

O

ℓ2 = (1 − ν)ℓ

νℓ

• + Perturbations acting on the c.o.m (ED forces, size, varyinglength. . .)

• + Attitude dynamics


T

DG Motion relative to the synodic frame

1. Two attractive bodies (primaries). Gravitational forces calculated under two

assumptions: 1) spherical attracting bodies and, 2) small ratios for Lri

• Gravitational potentialVi (i = 1, 2) of each primary:

Vi = −mµi

ri

[

1 + (L

ri)2 a2 P2(cos αi) + O(

L

ri)3]

, cos αi = u·xi/ri

2. Rotating frame ⇒ inertia forces• Constant rotation rateω in thez axis direction• Generalized potential includes

• Coriolis term:m ω (x y − y x)• Rotation term:m (ω2/2) (x2 + y2)


T

DG Lagrangian function and non-dimensional variables

1. Lagrangian function

L

m=

1

2(x2 + y2 + z2) + ω (x y − y x) +

ω2

2(x2 + y2) +

µ1

r1+

µ2

r2

+L2 a2

[

µ1

r31

P2(cos α1) +µ2

r32

P2(cos α2) +1

2[ϕ2 + θ2 cos2 ϕ]

]

+ O(L3/r3)

Here,a2 = I/(m L2), 0 < a2 < 1/4

2. Generalized coordinates:x, y, z, ϕ, θ

3. We use non-dimensional variables with characteristics values: total mass of the system,

distanceℓ between primaries,τ = ω t (primaries’ orbit period= 2 π)


T

DG Governing equations for a constant length inert librating tether

x − 2y − (1 − ν)(1 −1 + x

ρ31

) − x(1 −ν

ρ3) = ǫ

20

(1 − ν)

ρ51

[3(ρ1 · u)(~i · u) − (1 + x)S2(α1)] +ν

ρ5[3(ρ · u)(~i · u) − xS2(α2)]

y + 2x + y(1 − ν)

ρ31

− y(1 −ν

ρ3) = ǫ

20

(1 − ν)

ρ51

[3(ρ1 · u)(~j · u) − yS2(α1)] +ν

ρ5[3(ρ · u)(~j · u) − yS2(α2)]

z + zν

ρ3+ z

(1 − ν)

ρ31

= ǫ20

(1 − ν)

ρ51

[3(ρ1 · u)(~k · u) − zS2(α1)] +ν

ρ5[3(ρ · u)(~k · u) − zS2(α2)]

θ cos ϕ − 2(1 + θ)ϕ sin ϕ = 3(1 − ν)

ρ51

(ρ1 · u)(ρ1 · u2) + 3ν

ρ5(ρ · u)(ρ · u2)

ϕ + (1 + θ)2

sin ϕ cos ϕ = 3(1 − ν)

ρ51

(ρ1 · u)(ρ1 · u3) + 3ν

ρ5(ρ · u)(ρ · u3)

ν ≡ reduced mass of the small primary, ε20 = (

L

ℓ)2a2, S2(αi) =

3

2(5 cos

2αi − 1)

cos α1 =ρ1

ρ1

· u, cos α2 =ρ

ρ· u


T

DG Rotating Tether Attitude Dynamics

The Attitude Dynamics of the tether can

be analyzed more intuitively using the

Newton-Euler formulation. The core of the

analysis is the angular momentum equa-

tion:dHG

dt= M G

HereMG is the resultant of theexternaltorques applied to the center of massG of

the tethered system andHG is the angu-

lar momentum of the system, atG, in the

motion relative to the center of mass.

u1 ≡ u

u2

u3

G

HG

m1

m2


T


In theDumbbell Model the angular velocity of the tether and its angular momentumHG are:

~Ω = u × u + u (u · Ω ), HG = I Ω = I(u × u)

Attached to the tether we take a reference frameGu1u2u3 where the unit vectors are given by:

u1 = u, u2 =u

‖u‖, u3 = u1 × u2

In this body frame the angular momentum is:

HG = IΩ⊥u3, where Ω⊥ = ‖u × u‖ = ‖u‖

and the angular momentum equation takes the form:

dΩ⊥

dtu3 + Ω⊥

du3

dt=

1

IM G


T


This way we obtain the following equa-

tions:

du1

dt= Ω⊥u2

du3

dt=

M2

Ω⊥Iu2

dΩ⊥

dt=

M3

I

We introduce the Tait-Bryant angles (or

Cardan angles) as generalized coordinates.

Notice that, from a mathematical point

of view, we have a four-orden system of

ODE’S. u1 ≡ u

u2

u3

G

x0

y0

z0

φ3

φ2

φ1

φ1

s


T


In terms of the Bryant angles the unit vectorsu1 andu3 are given by

u1 = (cos φ2 cos φ3, cos φ1 sin φ3 + sin φ1 sin φ2 cos φ3, sin φ1 sin φ3 − cos φ1 sin φ2 cos φ3)

u3 = (sin φ2, − sin φ1 cos φ2, cos φ1 cos φ2)

The equations governing the time evolution of the Bryant angles are:

dφ1

dτ= − M2

ω2I · 1

Ω⊥

· cos φ3

cos φ2

dφ2

dτ= − M2

ω2I · 1

Ω⊥

· sin φ3

dφ3

dτ= Ω⊥ +

M2

ω2I · 1

Ω⊥

· cos φ3 tan φ2

dΩ⊥

dτ=

M3

ω2I

whereΩ⊥ = Ω⊥/ω is the non-dimensional form ofΩ⊥. These equations should be integrated from the

initial conditions:

at τ = 0 : φ1 = φ10, φ2 = φ20, φ3 = φ30, Ω⊥ = Ω⊥ 0


T

DG Fast Rotating Tether

For afast rotating tether the value ofΩ⊥ is large,Ω⊥ ≫ 1. There are two characteristic times:

1) the period of the orbital dynamics of both primaries,τ = O(1) and

2) the period of the intrinsic rotation of the tetherτ1 = Ω⊥τ = O(1)

For example, consider the governing equation

dφ

dτ= f(φ1, φ2, φ3, Ω⊥)

Its averaged equations form is:

<dφ

dτ>=

1

2π

∫ 2π

0f(φ1, φ2, φ3, Ω⊥)dτ1

To integrate the functionf(φ1, φ2, φ3, Ω⊥) theslow

variables(φ1, φ2, Ω⊥) take constant values and the

fast variableφ3 is approximated byφ3 ≈ τ1 + φ30.

u1 ≡ u

u2

v1v2

u3 ≡ v3

Gφ3

φ3

Stroboscopic frame


T

DG Averaged equations for an inert fast rotating tether

x − 2y − (1 − ν)(1 −1 + x

ρ31

) − x(1 −ν

ρ3) =

ǫ20

2

ν − 1

ρ51

[

3(sin φ2 + n) sin φ2 − (1 + x)S2(n1

ρ1

)

]

−ν

ρ5

[

3n sin φ2 − xS2(n

ρ)

]

y + 2x + y(1 − ν)

ρ31

− y(1 −ν

ρ3) =

ǫ20

2

1 − ν

ρ51

[

3(sin φ2 + n) cos φ2 sin φ1 + yS2(n1

ρ1

)

]

+ν

ρ5

[

3n cos φ2 sin φ1 + yS2(n

ρ)

]

z + zν

ρ3+ z

(1 − ν)

ρ31

=ǫ20

2

ν − 1

ρ51

[

3(sin φ2 + n) cos φ2 cos φ1 − zS2(n1

ρ1

)

]

−ν

ρ5

[

3n cos φ2 cos φ1 − zS2(n

ρ)

]

dφ1

dτ=

(

1 +cos φ1 cos φ2

2Ω⊥

)

sin φ2 cos φ1

cos φ2

+3

2

(1 − ν)

Ω⊥

(sin φ2 + n)(cos φ2 + b)

ρ51 cos φ2

+3

2

ν

Ω⊥

n b

ρ5 cos φ2

dφ2

dτ= −

(

1 +cos φ1 cos φ2

2Ω⊥

)

sin φ1 + (y cos φ1 + z sin φ1)

3

2

(1 − ν)

Ω⊥

(sin φ2 + n)

ρ51

+3

2

ν

Ω⊥

n

ρ5

dΩ⊥

dτ= sin φ1 sin φ2 cos φ1

where the quantities(n, b) and the fast variableφ3 are given by

n = x sin φ2 − (y sin φ1 − z cos φ1) cos φ2

b = x cos φ2 + (y sin φ1 − z cos φ1) sin φ2

dφ3

dτ= Ω⊥ −

(

1 +cos φ1 cos φ2

2Ω⊥

)

sin2 φ2 cos φ1

cos φ2

−3

2

(1 − ν)

Ω⊥

(sin φ2 + n)(cos φ2 + b)

ρ51

tan φ2 −3

2

ν

Ω⊥

n b

ρ5tan φ2

Remember: for a FRT the non-dimensional variableΩ⊥ is a large number, that is,Ω⊥ ≫ 1.


T

DG The Hill approach for fast rotating tethers

• Frequently, the parameterν is small. The Hill approximation give us the right order of

magnitude of distance to the small primary. We introduce this approximation by means of the

change of variables:

x = ℓ ν1/3 ξ, y = ℓ ν1/3 η, z = ℓ ν1/3 ζ, ρ = ℓ ν1/3 ρ

• For a FRT the parameterΩ⊥ is large (Ω⊥ ≫ 1). For example, taking Jupiter and Io as

primaries,Tp ≈ 1.769 days. IfTFRT ≈ 25 minutes thenΩ⊥ > 100. It is reasonable to take

the limit Ω⊥ → ∞ in the governing equations.

• The tether’s characteristic length,λ =a2

2

(

L

ν1/3 ℓ

)2

appears in a natural way in the

problem (hereν = µ/(µ1 + µ) is the reduced mass of the small primary or central body).


T


ξ − 2η = (3 − 1

ρ3)ξ − λ

ρ5

3N sin φ2 − ξS2(N

ρ)

η + 2ξ = − η

ρ3+

λ

ρ5

3N cos φ2 sin φ1 + ηS2(N

ρ)

ζ = −ζ(1 +1

ρ3) − λ

ρ5

3N cos φ2 cos φ1 − ζS2(N

ρ)

dφ1

dτ= cos φ1 tan φ2

dφ2

dτ= − sin φ1

N = ξ sin φ2 − (η sin φ1 − ζ cos φ1) cos φ2

These equations should be integrated from the initial conditions:

at τ = 0 : ξ = ξ0, η = η0, ζ = ζ0, ξ = ξ0, η = η0, ζ = ζ0, φ1 = φ10, φ2 = φ20

If the initial conditions areφ10 = φ20 = 0 ⇒ φ1(τ) = φ2(τ) ≡ 0.


T


TETHER ROTATION PARALLEL TO THE PRIMARIES PLANE :

ξ′′ − 2 η′ = 3 ξ − ξ

ρ3− 3

2λ

ξ

ρ5

(

1 − 5ζ2

ρ2

)

η′′ + 2 ξ′ = − η

ρ3− 3

2λ

η

ρ5

(

1 − 5ζ2

ρ2

)

ζ′′ = −ζ − ζ

ρ3− 3

2λ

ζ

ρ5

(

3 − 5ζ2

ρ2

)

Formally equal to a Hill-J2 problem:

• J2 helps to stabilize high inclination orbits

• we have control overL and, therefore, overλ!

• λ might take “high” values with feasible tether lengths• λ = 0.01 ⇒ Metis: L ≈ 7.2 km (R⊕ ≈ 50 km)• . . . when consistent with ourO(L/r)3 approximation


T

DG Hill+oblateness long-term dynamics

• Studied since the dawn of the space era (Lidov, Kozai)

• β: ratioJ2 to third body perturbations (β2 ∝ a−5)

0.0 0.5 1.0 1.5 2.0 2.5 3.090

100

110

120

130

140

β

I(deg)

T

DG ARIADNA PROGRAM

The END of thisGentil Introduction


T

DG ARIADNA PROGRAM

Inert Tethersand Periodic Motions


T

DG Inert tethers and periodic motions

In the CRTBP for a mass particle there are

three well known families of periodic mo-

tion originated from the collinear points:

1) Lyapunov periodic orbits, starting from

highly unstable small ellipses with retro-

grade motion around the collinear points

3) Eight-shaped orbits

2) Halo orbits bifurcated from the Lya-

punov family

Figure is an sketch of three starting orbits

Vertical oscillations

Halo orbits

Lyapunov orbits

We carried out numerical explorations of the tethered-satellite problem. More precisely, we discuss how

the known periodic solutions of the Hill problem are modifiedin the case of tethered satellites. The most

favorable configuration is found for tethers rotating parallel to the plane of the primaries, a case in which

the attitude of the tethered-satellite remains constant onaverage. In this case, the effect produced by a

non-negligible tether’s length is equivalent to introducing aJ2 perturbation on the primary at the origin, or

intensifying it if the primary at the origin is an oblate body, and it is shown that either lengthening or

shortening the tether may lead to orbit stability. Promising results are found for eight-shaped orbits, but

regions of stability are also found for Halo orbits.


T

DG Summary

• Lyapunov orbits:there is no beneficial effects on orbit stability by using tethered satellites

• Eight-shaped orbits:the presence of the tether in general, reduces instability.Stability

regions are found for certain lengths of the tether when the Jacobi constant remains below

roughlyC = 3.

• Halo orbits:stability regions for the tethered satellites are generally found, even when

starting from highly unstable orbits of the Hill problem. The general effect of lengthening the

tether is to narrow and twist the Halo, sometimes convertingthe Halo in a thin, eight-shaped

orbit. However, depending on the Jacobi constant value of the starting Halo orbit, the required

length of the tether to stabilize the orbit may be small and the stabilized orbit may retain most

of the Haloing characteristics.

Two different motivations to use a tether: 1) to stabilize unstable Halo orbits and 2) to take

advantage of the tether; it enjoys the stability propertiesof some stable Halo orbits (for values

of C close to the lower limit).

Therefore,at first sight, the most promising regions are close to the Halo family’s stabilityregion of the Hill problem , where tethers of few tenths of kilometers may stabilize Halo

orbits of the simplified model. For specific applications oneneeds to check that the tether

length required for stabilization is feasible, say of few tenths of km, and that the corresponding

Halo orbit does not impact the central body. Dynamics and Stability of Tethered Satellites at Lagrangian Points– p. 35/100

T

DG Summary

k1 stability curve

k2 stability curve

Minimum distance curve

Maximum distance curve

1.5 2.0 2.5 3.0 3.5 4.0-4

0

4

28

215

1622

0.01

0.2

0.39

0.58

0.78

0.97

Jacobi constant

Sta

bilit

yin

dice

s

Dis

tanc

eHH

illun

itsL

k1 stability curve

k2 stability curve

Jupiter

The Moon

Europa

IoEnceladus

Deimos

Phobos

0.0 0.1 0.2 0.3 0.4 0.5

-4

0

4

28

215

1622

Minimum radiusHHill problem unitsL

Sta

bilit

yin

dice

s

• Left: Stability diagram of the Halo family of the Hill problem (no tether) showing the

maximum and minimum distance to the origin.

Right: Stability diagram of the Halo family of the Hill problem (no tether) as a function of the

minimum distance to the origin.


T

DG Summary

k1 stability curve

k2 stability curve

Eros

MercuryThe Moon

EuropaIo

Enceladus

0.00 0.05 0.10 0.15 0.20-4

-2

0

2

4

Minimum radiusHHill problem unitsL

Sta

bilit

yin

dice

s

Right: Stability diagram of the Halo family of the Hill problem (no tether) as a function of the

minimum distance to the origin.


T

DG Summary

0.0

0.2

0.4

-0.4

-0.2

0.0

0.2

0.4

-0.5

0.0

0.00.1

0.2

0.3

0.4

-0.4

-0.2

0.0

0.2

0.4

-0.5

0.0

• Left: Halo orbit close to Io, stabilized with a tether lengthof ∼ 70 km. The minimum

distance to Io is about one half of it radius.

Right: Halo orbit about Eros, stabilized with a tether length of ∼ 110 km.


T

DG ARIADNA PROGRAM

The END of Inert Tethersand Periodic Motions


T

DG ARIADNA PROGRAM

Io Exploration

with Electrodynamic Tethers


T

DG POWER AND M ASSES OF SOMERTG’ S

Mission Thermal W Electrical W Mass (kg)

Pionner 10 2250 150 54.4

Voyager 1 y 2 7200 470 117

Ulysses 4400 290 55.5

Galileo 8800 570 111

Cassini 13200 800 168

PROBLEMS OF THE RTG’ S

• Very high cost: from $40,000 to$400,000 per W

• High potential risk

• Limited power (mass grows strongly)


T

DG DEORBITING SATELLITES WITH ELECTRODYNAMIC TETHERS

ANY SATELLITE CAN BE DEORBITED WITH AN ELECTRODYNAMIC TETHERWORKING IN THE

GENERATOR REGIME

∆h = ai − af

∆h

~vi

~vf

~fe

~fe

Deorbiting a satellite

The mechanical energy lost in the deorbiting process is

∆E =mµ

2afai∆h ≈ mµ

2a2f

∆h(1 − ∆h

af)

and it is invested in several task. Basically:

• to attract the electrons from the infinity to the

tether

• ohmics loss in the tether when the current is

flowing

• some useful work that we can obtain in the form of

electrical energy (charging batteries, for example)

We need:magnetic fieldandplasma environment. Both are

present in Jupiter.

THE ELECTRODYNAMIC TETHER IS ABLE TO RECOVER A SIGNIFICANT FR ACTION OF THE

MECHANICAL ENERGY LOST DURING THE DEORBITING PROCESS


T

DG SCENE OF THE MISSION

Jupiter inner moonlets

rAdrastea = 128971 km, rMetis = 127969 km

rAmalthea = 181300 km, rThebe = 221895 km

The plasma environment is co-rotating

with Jupiter (as a rigid body); the an-

gular velocity is

ωJ ≈ 1.7579956104 × 10−4s−1

The orbital velocity in a circular or-

bit of radius r is vc =

√

µ

r(µ =

1.26686536 × 1017m3/s2). It exists

an orbital radius for which the orbital

period coincides with the sideral pe-

riod of Jupiter (about 9.925 hours); it

is the stationary radius (rs) given by

rs = (T√

µ

2π)23 ≈ 160009.4329 km


T

DG FUNDAMENTALS OF THE MISSION

~vo

~f e

~f e

− ~f e

Amalthea:m = 2.09 × 1018 kg, af = 181300 km

The cable can be removed by using the gravitational

attraction of Amalthea to link the tether system and

the small Jupiter moon. The tether system moves

in an equilibrium position relative to the primaries:

Jupiter + Amalthea

We tray to deorbit one of the Jupiter moon-

lets. We joint the tether and the moonlet

with a cable. ~F e is the electrodynamic

drag acting on the tether. This force is trans-

mitted thought the cable to the small Jupiter

moon that will be deorbited.

∆h =2∆Ea2

f

mµ

To deorbit Amaltea 1mm (∆h = 1mm) im-

plies a loss of mechanical energy about

∆E ≈ 4 × 1015J≈ 1.1 × 109Kwh

From a practical point of view the reserve of

energy is unbounded


T

DG Equilibrium Positions at the Synodic Frame

JIoL1

L2

Thrust

Io atraction

• A permanent tethered observatory at Jupiter. Dynamical analysis, by J. Pelaez and D. J. Scheeres, Proceedings of the

17th AAS/AIAA Space Flight Mechanics Meeting Sedona, Arizona, Vol. 127 of Advances in the Astronautical Sciences,

2007, pp. 1307–1330

• On the control of a permanent tethered observatory at Jupiter, by J. Pelaez and D. J. Scheeres, Paper AAS07-369 of

the 2007 AAS/AIAA Astrodynamics Specialist Conference, Mackinac Island, Michigan, August 19-23, 2007


T

DG EQUILIBRIUM POSITIONS

0

0.2

0.4

0.6

0.8

1

1.2

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

χ =(

34

) 23

χmax χmax

ξe

ζe

ROOT 1ROOT 2

L1L2

χ = 0 χ = 0

Equilibrium Positions on the plane(ξe, ζe)


T

DG EQUILIBRIUM POSITIONS (AMALTEA )

0

0.2

0.4

0.6

0.8

1

1.2

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

χ = 0 χ = 0

χmax χmax

χ =(

34

) 23

ξe

ζe

ROOT 1ROOT 2

L2L1

In [1]

Comparison with the results summarized in[1]

[1] A permanent tethered observatory at Jupiter. Dynamical analysis, by J. PELÁEZ & D. J. SCHEERES,

(Paper AAS07-190) ofThe 2007 AAS/AIAA Space Flight Mechanics Meeting, Sedona, AZ, January 2007.


T

DG TETHER DESIGN IN M ETIS . NUMERICAL ANALYSIS

0.1

1

10

10 100

mT (in kg)

Wu (kw)

L = ctedw = cte

L = 50 km

L = 5 km

dw = 50 mm

dw = 5 mm

35 49 76 140

10 km

20 km

30 km

40 km

DIFFERENT OPTIMUM CONFIGURATIONS AT METIS ( h = 0.1 MM )


T

DG TETHER DESIGN IN M ETIS . THEORETICAL ANALYSIS

0.1

1

10

10 100

mT (in kg)

Wu (kw)

L = ctedw = cte

L = 50 km

L = 5 km

dw = 50 mm

dw = 5 mm

35 49 76 140

10 km

20 km

30 km

40 km

DIFFERENT OPTIMUM CONFIGURATIONS AT METIS ( h = 0.1 MM )


T

DG TETHER DESIGN IN I O. NUMERICAL ANALYSIS

0.1

1

10

10 100 200 300

2

3

4

170

mT (kg)

Wu (kw)

L = ctedw = cte

L = 8 km

L = 10 km

L = 15 km

L = 25 km

L = 35 km

dw = 50 mmdw = 5 mm

dw = 10 mm

dw = 36 mm

DIFFERENT OPTIMIZED CONFIGURATIONS IN I O (h = 0.05 MM )


T

DG THREE DIFFERENT CONFIGURATION

A B C

L (km) 25 35 35

dw (mm) 50 25 36

mT (kg) 169 117 170

ZC (Oh) 640 1265 886

Wu (w) 2100 2100 3000

Isc (A) 10.94 5.53 7.89

Iav (A) 1.43 1.04 1.49

φ∗ (deg) 39.7 40.06 40.06

χ · 103 5.79 5.91 8.43

de (km) 200,052 198,119 165,803

T (mN) 5.28 7.4 7.4

Three possible designs

• Aluminum tape 0.05 mm thick

• Self-balanced

• Mass of the S/C 500 kg

• Tether mass≤ 170 kg

• χ ∈ [0, 0.115] STABLE

• Power from 2 – 3 kw

• distancede large

• Tether tension small≈ mN

• Appropriate for the scientific explo-

ration of the Io plasma torus


T

DG ROTATING ELECTRODYNAMIC TETHER ORBITING I O

Toward Jupiter

Eπ

ϕ

vIo

B x

y

z

ω t

O

We take an inertial frameOxyz (origin Io)

Orbital velocity of Io:vIo ≈ 17.33 km/s

Plasma velocity at Io orbit≈ 74.17 km/s

Orbital velocity of S/C (around Io)≈ 1 km/s

Magnetic fieldB ≈ 2 · 10−6 T

E ≈ (vIo − vpl) × B

E = Eπ(cos ωt, sin ωt, 0),

Eπ ≈ 0.12 V/m


T

DG POWER GENERATION I O

A C

II

hB

IC

A C

V

Vp

Vt

EM

L

CH

I

hB

EE

ZT

II

Power generation: assumptions

• small values ofn∞ no Ohmic effects

• negligible ion losses at the cathodic segment

• control impedanceZC at the cathodic end

• OML regime

• control parameterζ =z∗

L= 1 −

ZC IC

EmL

Results of the analysis

• Iav = 1L

∫ L

0I(z)dz = I0 (1 − 2

5ζ)ζ3/2

• W = ZCI2C = I0EmL(1 − ζ)ζ3/2

I0 =4dw

3πen∞

√

2eEmL3

me


T

DG OPTIMUM POWER GENERATION

0

0.05

0.1

0.15

0.2

0 0.2 0.4 0.6 0.8 1

ζ

WI0EmL

ζopt =3

5, Zopt

C =3π

√2

8· 1 − ζopt

ζ3/2opt

· 1

dwn∞

√

meEm

e3L=

π√

30

12· 1

dwn∞

√

meEm

e3L


T

DG ROTATING ELECTRODYNAMIC TETHER ORBITING I O

Wav =1

π

∫ π/2

−π/2cos3/2 ϕ dϕ · 4dw

3πn∞

√

2E3πL5e3

me

[

(1 − ζ) ζ3/2]

0 10 20 30 40 50 600

2

4

6

8

10

12

14

P (kw)

Tether length (km)

First order approximation of averaged generated power

for a 5 cm wide tape tether of different lengths in Io

orbit.

Power generated with a 25 km long 5 cm wide tape

tether along circular retrograde orbits of different radii:

r = 1.1 rIo (grey solid line),r = 2.0 rIo (dark

solid line), andr = 3.0 rIo (dark dotted line).


T

DG POWER GENERATION IN I ONIAN ORBIT

0.1

1

10

10 100 200 300

2

3

4

170

L=8

L=10

L=15

d=50d=5

d=10

L=25

L=35

d=36

mT (kg)

Wu (kw)

L = ctedw = cte

DIFFERENT OPTIMIZED CONFIGURATIONS IN I O (h = 0.05 MM )


T

DG ORBIT STABILITY

• Io is the Galilean moon more affected by the gravitation of Jupiter

• Lara & Russela shows thatretrograde equatorial orbits around Europe are stable when eccentricity and

semimajor axis are smaller than critical values.

• Their results have been numerically extended to Io: if the apocenter is less than 3.5rIo the equatorial

retrograde orbit is stable for at least a year

Fav =1

π

∫ π/2

−π/2L Iav (u× B)dϕ =

1

πL

[

∫ π/2

−π/2Iav udϕ

]

× B

When the load impedance is controlled for maximum power generationζ =cte:∫ π/2

−π/2Iav udϕ =

1

Eπ

∫ π/2

−π/2Iav cos ϕEπ dϕ =

Eπ

Eπ·∫ π/2

−π/2Iav cos ϕ dϕ

This way we have

Fav =k′

√Eπ

(Eπ × B), k′ ≈ 0.0667 dwn∞

√

L5e3

me(5 − 2ζ)ζ3/2

The averaged force has a constant value, in a first approximation, on the synodic frame.

aOn the design of a science orbit about Europa, Paper AAS 06-168, 16th AAS/AIAA Space Flight Mechanics Conference. Tampa, Florida,USA, January 2006


T

DG ORBIT STABILITY

F ≈ F (− sin ω t, cos ω t, 0) with F = 0.0667 dwn∞B

√

EπL5e3

me· (5 − 2ζ) ζ3/2

0 5 10 15 20 25 301

1.5

2

2.5

3

3.5

rrIo

time (days)

Evolution of orbital radius for circular retrograde orbitsof different radii (1.2, 2.0, 2.5 and 3 Io radii) under

the effect of the Lorentz force of a 25 km long 5 cm wide tape tether with power-optimized impedance

control.


T

DG ORBIT CONTROL

Eπ

F ∝ Eπ × B

IovIO

Drag arc

Thrust arc

Schematic of electrodynamic thrust and drag arc

for a generic Ionian orbit. We neglect the orbital

velocity around Io in the determination of mo-

tional electric field

The main contribution to the motional electric

field comes from Io orbital velocity rather than

the spacecraft orbital velocity around Io. The

Lorentz force is thrusting the S/C in part of the

orbit (thrust arc) and braking it (drag arc) in the

rest of the Ionian orbit.

Preliminary test with a simple control strategy:

the control parameterζ is switched to 1 (maxi-

mum Lorentz force) on thrust arcs and switched

to 0 (no force) on drag arcs. A numerical sim-

ulation has been conducted assuming a 500 kg

spacecraft equipped with a 25 km long and

5 cm wide electrodynamic tether starting from

a low altitude retrograde circular orbit (r = 1.2

Io radii).

B⊗


T

DG ORBIT CONTROL

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

xrIo

yrIo

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 1500

1

2

3

4

5

6

7

8

9

10

rrIo

time (days)

Spiral-out trajectory (left) and orbital radius evolution(right) for a 500 kg spacecraft propelled by a

25 km long 5 cm wide tape tether with simple current control strategy. Escape from Io gravitational field

require less than 5 months, in this case.


T

DG ARIADNA PROGRAM

The END of Io Exploration

with Electrodynamic Tethers


T

DG ARIADNA PROGRAM

Stability of tethered satellites

at collinear lagrangian points


T

DG PREVIOUS ANALYSIS

• The Stabilization of an Artificial Satellite at the InferiorConjunction Point of the Earth-Moon System,

G. Colombo, Smithsonian Astrophysical Observatory Special Report No.80, November 1961

• The Control and Use of Libration-Point Satellites, R. W. Farquhar , NASA TR R-346, September 1970.

pp. 89-102

• Tether Stabilization at a Collinear Libration Point, R. W. Farquhar , The Journal of the Astronautical

Sciences, Vol. 49, No. 1, January-March 2001, pp. 91-106.

• Dynamics of a Tethered System near the Earth-Moon Lagrangian Points, A. K. Misra, J. Bellerose, andV. J. Modi , Proceedings of the 2001 AAS/AIAA Astrodynamics SpecialistConference, Quebec City,

Canada, Vol. 109 of Advances in the Astronautical Sciences,2002, pp. 415–435.

• Dynamics of a multi-tethered system near the Sun-Earth Lagrangian point, B. Wong and A. K. Misra,

13th AAS/AIAA Space Flight Mechanics Meeting, Ponce, Puerto Rico, February 2003, Paper No.

AAS-03-218.

• Dynamics of a Libration Point Multi-Tethered System, B. Wong and A. K. Misra, Proceedings of 2004

International Astronautical Congress, Paper No. IAC-04-A.5.09.


T

DG EQUILIBRIUM POSITIONS AT THE SYNODIC FRAME

L1 L2L3

L4

L5

E1 E2

Equilibrium position

with an inert tether


T

DG EXTENDED DUMBBELL M ODEL . H ILL APPROACH

VARYING LENGTH INERT TETHER

ξ − 2η − (3 − 1

ρ3)ξ =

λ

ρ5

3N cos ϕ cos θ − ξS2(N

ρ)

η + 2ξ +η

ρ3=

λ

ρ5

3N cos ϕ sin θ − ηS2(N

ρ)

ζ + ζ(1 +1

ρ3) =

λ

ρ5

3N sin ϕ − ζS2(N

ρ)

θ + (1 + θ)

[

Is

Is− 2ϕ tan ϕ

]

+ 3 cos θ sin θ =3N

ρ5

(−ξ sin θ + η cos θ)

cos ϕ

ϕ +Is

Isϕ + sin ϕ cos ϕ

[

(1 + θ)2 + 3 cos2 θ]

=3N

ρ5(− sin ϕ[ξ cos θ + η sin θ] + ζ cos ϕ)

whereρ =√

ξ2 + η2 + ζ2 and the quantityN and the functionS2(x) are given by

N = ξ cos ϕ cos θ + η cos ϕ sin θ + ζ sin ϕ, S2(x) =3

2(5x2 − 1)


T


λ =

[

Ld

ℓ

]2

· a2

ν2/3, a2 =

Is

m L2d

∈ [1

12,1

4]

where,ℓ is the distance between primaries,ν is the reduced mass of the small primary (usuallyν ≪ 1),

m = m1 + m2 + mT is the total mass of the system andIs is the moment of inertia about a line normal

to the tether by the center of massG of the system;a2 is of order unity and takes its maximum value

a2 = 1/4 for a massless tether with equal end masses (m1 = m2). For a tether of varying length the

parameterλ is a function of time since the deployed tether massmd and the deployed tether lengthLd(t)

are changing. Moreover, some terms of these governing equations involve the ratio:

Is

Is= 2

Ld

LdJg, Jg = 1 − Λd (1 + 3 cos 2φ)

(

3 sin2 2φ − 2Λd

) , Λd =md

m, cos2 φ =

(

m1

m+

Λd

2

)

This formulation includes the mass of the tether through theparameterΛd and the mass angleφ. In order

to neglect the tether mass, we only have to introduce the condition Λd = 0 in the above expressions. For a

tether of constant length the parameterλ is also constant and the quotientIs/Is vanishes, that is,

Is/Is = 0. For the sake of simplicity we will assume a massless tether in what follows.


T


In the search of equilibrium position we will consider the tether tension given by

T0 ≈ m1 m2

m1 + m2ω2Ld

ϕ2 + cos2 ϕ(1 + θ)2 + 3 cos2 θ − 1 +1

ρ33(

N

ρ

)2

− 1 − Ld

Ld

This expression has been derived under the following assumptions: 1) inert tether, 2) massless tether, and

3) the Hill approach has been performed. At any equilibrium position,tether tension must be positivebecause a cable does not support compression stress and the above expression takes the form

T0 ≈ Tc

cos2 ϕ1 + 3 cos2 θ − 1 +1

ρ33(

N

ρ

)2

− 1

, Tc =m1 m2

m1 + m2ω2Ld

We will use this expression for checking the tether tension in a given steady solutions of the equations of

motion; if the tension is positive the equilibrium positionwill exist; if negative, the equilibrium position

will not exist.


T

DG VALUES OF λ FOR SUN AND DIFFERENT PLANETS

1e-016

1e-014

1e-012

1e-010

1e-008

1e-006

0.0001

0 1000 2000 3000 4000 5000

MercuryVenusEarthMarsJupiterSaturnUranusNeptune

Tether length (km)

λ


T

DG VALUES OF λ FOR BINARY SYSTEMS IN THE SOLAR SYSTEM

1e-014

1e-012

1e-010

1e-008

1e-006

0.0001

0.01

1

100

10000

0 200 400 600 800 1000

Earth-Moon

Mars-PhobosMars-DeimosSaturn-MimasSaturn-EnceladusSaturn-RheaSaturn-Titan

Neptun-Triton

Tether length (km)

λ


T

DG VALUES OF λ FOR BINARY SYSTEMS IN THE JOVIAN WORLD

1e-014

1e-012

1e-010

1e-008

1e-006

0.0001

0.01

1

100

0 20 40 60 80 100

λ

MetisAdrasteaAmaltheaThebeIoEuropaGanymedeCallisto

T

DG NON ROT. TETHERS. CONST. L ENGTH . EQUILIBRIUM POSITIONS

Small Primary

L2 E2

Center of massG

Sketch of the equilibrium positionE2 in the neighborhood ofL2.

There is another one, similar to this,on the leftof L1.

ξe = ±ρe, ηe = ζe = ϕe = 0, θe = 0, π, λ =ρ2

e

3

(

3 ρ3e − 1

)

, ρ3e > 1/3

This expression can be expanded whenλ ≪ 1 is small and provide the asymptotic solution

ξe ≈ (1

3)13 + 3

13 λ − 9 λ2 +O(λ3)

The convergence of this serie is poor but, for really small values ofλ the two first terms give a useful

approximation.


T

DG NON ROT. TETHERS. CONST. L ENGTH . STABILITY ANALYSIS

A linear stability analysis shows that the equilibrium positions of that family are unstable for

any value ofλ.

10−6

10−4

10−2

2.5

2.55

2.6

2.65

2.7

2.75

2.8

2.85

R(s5)

λReal part of the unstable eigenvalue as a function ofλ


T

DG NON ROTATING TETHERS. VARIABLE L ENGTH

The idea original of Colombo and Farquhar is as follows: let us assume that the tether length isL and for

that length we have the equilibrium position labeled with(1) in this figure. On the right of such a position,

the system center of massG is acted by a force that impulses it toward the right. By increasing the tether

length up toL(1) = L + ∆(1)L we move the equilibrium position up to the point labeled with(2) in

figure; now the force acting onG impulses it toward the left. Then we decrease the tether length,

L(2) = L(1) + ∆(2)L in order to move the equilibrium position on the left side ofG . . . Thus, by

changing the tether length in an appropriate way the center of massG can bestabilizedand kept in the

neighborhood of the collinear lagrangian pointL2.

Dim.

N. Dim. ξL2 = ( 13)13

xL2 = ( 13)13 Lc

313 λ

313 λ Lc

∆xL2

(1) (2)O

G

Distances from the small primary and from the collinear point L2 whenλ ≪ 1 (Lc = ℓν1/3)


T

DG NON ROTATING TETHERS. VARIABLE L ENGTH

We carried out two different analysis:

• Linear approximation for small values ofλ

• Full problem. Proportional control

We finish the analysis of non rotating tethers pointing to some drawbacks associated with this

kind of control

• Control drawbacks


T

DG L INEAR APPROXIMATION FOR SMALL VALUES OF λ

ξ = ξL + 31/3 λ0(1 + u(τ)) , η = 31/3 λ0 v(τ) , λ = λ0(1 + s(τ))

d2 u

d τ2− 2

d v

d τ− 9 u(τ) +

9

2

(

3 cos2 θ cos2 ϕ − 1)

s(τ) +27

2

(

cos2 θ cos2 ϕ − 1)

= 0

d2 v

d τ2+ 2

d u

d τ+ 3 v(τ) − 9 cos2 ϕ cos θ sin θ (1 + s(τ)) = 0

d2 w

d τ2+ 4 w(τ) − 9 cos ϕ cos θ sin ϕ (1 + s(τ)) = 0

(1 + s(τ))d2 θ

d τ2+

d s

d τ(1 +

d θ

d τ) − 2 tan ϕ

d ϕ

d τ

(

1 +d θ

d τ

) (

1 +d s

d τ

)

+ 12 cos θ sin θ (1 + s(τ)) = 0

(1 + s(τ))d2 ϕ

d τ2+

d s

d τ

d ϕ

d τ+ (1 + s(τ)) sin ϕ cos ϕ

[

(

1 +d θ

d τ

)2

+ 12 cos2 θ

]

= 0

s(τ) = e−β τ (A cosΩτ + B sinΩτ)


T

DG ONE DIMENSIONAL MOTION

3.6

3.8

4

4.2

4.4

4.6

4.8

5

5.2

0 200 400 600 800 1000

t (hours)

x(km)

940

960

980

1000

1020

1040

1060

1080

1100

1120

1140

1160

0 200 400 600 800 1000

t (hours)

x(km)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 200 400 600 800 1000

t (hours)

L(m/s)

0.0145

0.015

0.0155

0.016

0.0165

0.017

0.0175

0.018

0.0185

0.019

0 200 400 600 800 1000

t (hours)

T (N)

Massless tether with two equal masses (500 kg) at both ends inthe Earth-Moon system. The selected parameters are

β = 0.5, Ω = 1.5, for the initial conditionsu0 = 0.15 andu0 = 0. The nominal tether length isL = 1000 km


T

DG BI -DIMENSIONAL MOTION

3

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5

5.2

0 100 200 300 400 500 600 700 800 900 1000

t (hours)

x(km)

940

960

980

1000

1020

1040

1060

1080

1100

1120

1140

0 100 200 300 400 500 600 700 800 900 1000

t (hours)

L(km)

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 600 700 800 900 1000

t (hours)

L(m/s)

0.007

0.0075

0.008

0.0085

0.009

0.0095

0.01

0 100 200 300 400 500 600 700 800 900 1000

t (hours)

T (N)

T

DG FULL PROBLEM . PROPORTIONAL CONTROL

The length variationλ(τ) is governed by a proportional control law:

λ = λe +∑

Ki (xi − xe,i)

whereKi are gains andxi stands for the variablesξ, η, θ.

d y

d t= My

whereyT =

δξ, δη, δθ, δξ, δη, δθ

.

The detailed stability analysis of this equation is cumbersome. Nevertheless, we firstly consider the simpler

case in which only one gainKξ is different from zero. The characteristic polynomial takes the form:

s3 +

[

3 Kξ

ξ4e

−(

2 − 4

ξ3e

)]

s2 +

[(

6 Jg + 27

ξ4e

+6

ξ7e

)

Kξ −(

105 − 6

ξ3e

+4

ξ6e

)]

s+

+6

[(

9

ξ4e

− 3

ξ7e

)

Kξ −(

45 +9

ξ3e

− 2

ξ6e

)]

= 0


T

DG FULL PROBLEM . PROPORTIONAL CONTROL

The Descartes rule of signs providesa sufficient condition for stability to be fulfilled byKξ . Such a

condition is drawn in figure as a function of the equilibrium valueλe for a massless tether (Jg = 1).

10−6

10−4

10−2

2

2.5

3

3.5

4

4.5

5

K∗ξ

λe

Sufficient condition forKξ to stabilize the system as a function ofλe. Values over the curve (Kξ > K∗ξ )

provide stability.


T

DG CONTROL DRAWBACKS . TETHER LENGTH

∆L

∆xe=

1

2 313√

a2 λ

10−6

10−4

10−2

100

100

101

102

103

λ

d Ld xe

Non linearLinear

Ratio between variation of tether length and deviations

of the center of mass vs.λ.

10−2

100

102

104

102

104

106

Sun-EarthEarth-MoonMars-PhobosJupiter-AmalteaJupiter-IoSaturn-Enceladus

L (km)

d Ld xe

Ratio between variation of tether length and deviations

of the center of mass vs. the tether length (km)


T

DG CONTROL DRAWBACKS . TETHER TENSION

10−5

10−4

10−3

−5

0

5

10

15

20

TTc

∆ξ

Maximum and minimum tether tension vs. the

initial perturbation∆ξ for λe = 10−2.

10−4

10−3

10−6

10−5

10−4

λe

∆ξ

Zero tether tension∆ξ vs. λe (blue line).

Zero tether length∆ξ vs. λe (red line).

To avoid the zero tension problem (slack tether) is the most strong requirement of this control strategy


T

DG ROTATING TETHERS . EQUILIBRIUM POSITIONS

ξe = ±ρe, ηe = ζe = 0, λe =4ρ5

e

3

(

3 − 1

ρ3e

)

, ρ3e > 1/3

For small values ofλ the above solution provides the asymptotic solution

ξe ≈ (1

3)13 + 3

13

λ

4− 9 (

λ

4)2 +O(λ3)

Comparing with the same results for a non-rotating tether

ξe = ±ρe, ηe = ζe = 0, λ =ρ2

e

3

(

3 ρ3e − 1

)

, ρ3e > 1/3

ξe ≈ (1

3)13 + 3

13 λ − 9 λ2 +O(λ3)


T

DG ROTATING TETHERS . PROPORTIONAL CONTROL

λ = λe + Kξδξ + Kξδξ + Kηδη + Kηδη

Through the Routh-Hurwitz theorem, it has been found thatasymptotic stability is guaranteed if the gains

of the control law satisfy the relations

Kξ ≥ 4

3ρe(

15ρ3e − 2

)

Kξ > 0 Kη = 0 Kη = 0

We carried out two simulations of a rotating tether with different values ofKξ andKξ in order to see the

qualitative behavior of the system. The characteristics ofthe simulations are:

λe = 0.0401 (ξe = 0.707); ξ0 − ξe = 10−3, ηe = 10−3; Ω⊥ = 50.


T

DG SIMULATIONS

0.7055 0.706 0.7065 0.707 0.7075 0.708 0.7085 0.709−3

−2

−1

0

1

2

3x 10

−3

δξ

δξ

Kξ = 4

0.7065 0.707 0.7075 0.708−15

−10

−5

0

5x 10

−4

δξ

δξ

Kξ

= 3

Kξ

= 0.5

−1.5 −1 −0.5 0 0.5 1 1.5

x 10−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

−3

δη

δη

Kξ = 4

−5 0 5 10

x 10−4

−1.5

−1

−0.5

0

0.5

1x 10

−3

δη

δη

Kξ

= 3

Kξ

= 0.5

λe = 0.0401 (ξe = 0.707); ξ0 − ξe = 10−3, ηe = 10−3; Ω⊥ = 50


T

DG SIMULATIONS

0 5 10 15 20 25 3048

50

52

54

56

58

60

62

64

66

τ

Ω⊥

Kξ = 4

0 5 10 15 20 25 3050

51

52

53

54

55

56

57

58

τ

Ω⊥

Kξ

= 3

Kξ

= 0.5

0 5 10 15 20 25 300.185

0.19

0.195

0.2

0.205

0.21

0.215

√λ

τ

Kξ = 4

0 5 10 15 20 25 300.19

0.195

0.2

0.205

0.21

0.215

√λ

τ

Kξ

= 3

Kξ

= 0.5


T

DG ARIADNA PROGRAM

The END ofStability of tethered satellites

at collinear lagrangian points


T

DG Lyapunov orbits

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-3

-2

-1

0

1

2

3

x

y

Horizontal stability curve

Vertical stability curve

Orbit period curve

-15 -10 -5 0

-38

-9

-2

0

2

9

38

161

682

2889

3.4

4.

4.5

5.1

5.7

6.3

6.9

7.4

8.

8.6

Jacobi constant

Sta

bilit

yin

dice

s

Per

iod

• Left: sample orbits of the family of Lyapunov orbits aroundL1 (dashed) andL2 (full line) for, from

larger to smaller,C = −1, 0, 1, 2, 3, 4. Right: stability-period diagram of the family of Lyapunovorbits

of the Hill problem. Note thearcsinh scale used for the stability curves.(C is the Jacobi constant)


T

DG Tethered family of Lyapunov orbits: unstable



Orbit period curve

Critical points

0 1 2 3-4

0

6

64

663

6870

6.2

6.4

6.5

6.6

6.8

6.9

Tether's characteristic length

Stab

ility

inde

x

Peri

od

-0.5 0.0 0.5 1.0 1.5

-2

-1

0

1

2

Ξ

Η

• Left: stability-period diagram of the family of Lyapunov orbits with Jacobi constantC = −0.408295

for tether’s length variations; the horizontal gray lines correspond to the critical valuesk = ±2 (in the

arcsinh scale). Rigth: Hill’s problem Lyapunov orbit (λ = 0, full line) and an orbit with a tehter’s

characteristic lenghtλ = 1 (dotted).(C is the Jacobi constant)


T

DG Tethered family of Lyapunov orbits: unstable



Orbit period curve

Critical orbits

0 1 2 3-2

1

16

186

2164

25 188

3.

3.2

3.3

3.5

3.6

3.7


Stab

ility

indi

ces

Peri

od

0.66 0.7 0.74 0.78

-0.2

-0.1

0.0

0.1

0.2

Ξ

Η

• Left: stability-period diagram of a family of Lyapunov orbits close toL2 for tether’s length variations;

the horizontal gray lines correspond to the critical valuesk = ±2 (in thearcsinh scale). Rigth: starting

orbit (λ = 0, full line) and an orbit withλ = 0.1 (dotted).(C is the Jacobi constant)


T

DG Family of eight-shaped orbits

0.00.2

0.40.6

Ξ

-0.20

0.2Η

-1

0

1

Ζ

k1 stability curve

k2 stability curve

Orbit period curve

Critical point

-8 -6 -4 -2 0 2 4

2

7

31

130

549

2327

3.1

3.7

4.4

5.

5.6

6.2

Jacobi constant

Stab

ility

indi

ces

Peri

od

• Left: sample eight-shaped orbits forC = 2 (red),1 (magenta), and0 (blue).Right: stability-period

diagram of the family of eight-shaped orbits of the Hill problem. (C is the Jacobi constant)


T

DG Tethered family of eight-shaped orbits

k1 stability curve

k2 stability curve

Orbit period curve

1: start

2

3

45

6: end

0.00 0.05 0.10 0.15 0.20-2

0

3

17

90

475

2501

2.79

2.93

3.07

3.21

3.34

3.48

3.62


Stab

ility

indi

ces

Peri

od

0.00.2

0.40.6Ξ

-0.070.07

-0.5

0.0

0.5

Ζ

• Left: stability-period diagram of a family of eight-shaped, periodic orbits with constantC = 2 for

tether’s length variations; the horizontal gray lines correspond to the critical valuesk = ±2. Rigth: stable

orbits forλ = 0.2 (left) λ = 0.215 (center), and unstable orbit of the Hill problem (λ = 0, right). (C is

the Jacobi constant)


T


k1 stability curve

k2 stability curve

Orbit period curve

1: start

2

34

5

6: end

0.00 0.05 0.10 0.15 0.20 0.25 0.30-2

1

10

87

788

7122

64 375

2.12

2.42

2.72

3.02

3.32

3.61

3.91


Stab

ility

indi

ces

Peri

od

0.00.2

0.40.6Ξ

-0.060.06

-0.5

0.0

0.5

Ζ

• Left: stability-period diagram of a family of eight-shaped, periodic orbits with constantC = 2.7 for

tether’s length variations; the horizontal gray lines correspond to the critical valuesk = ±2. Right: stable

orbits forλ = 0.17 (left, black)λ = 0.184 (center, blue), and unstable orbit of the Hill problem (λ = 0,

right). (C is the Jacobi constant)


T


k1 stability curve

k2 stability curve

Orbit period curve

Bifurcation orbits

0 1 2 3 4 5-2

0

2

9

38

161

682

2.4

2.6

2.7

2.8

2.9

3.


Stab

ility

indi

ces

Peri

od

• Stability-period diagram of a family of eight-shaped, periodic orbits with constantC = 3 for tether’s

length variations; the horizontal gray line corresponds tothe critical valuesk = +2. (C is the Jacobi

constant)


T

DG Family of Halo orbits

k1 stability curve

k2 stability curve

Orbit period curve

Reflection

1.5 2.0 2.5 3.0 3.5 4.0

-2.

0.

2.

8.

30.

100.

400.

1600.

1.5

1.8

2.

2.2

2.4

2.6

2.8

3.1

Jacobi constant

Stab

ility

indi

ces

Peri

od

0.00.2

0.40.6Ξ

-0.4

-0.2

0.00.2

0.4Η

-0.5

0.0

Ζ

• Left: stability-period diagram of the family of Halo orbitsof the Hill problem. Right: sample stable orbit

for C = 1.08 (the blue and red dots linked by a gray line are the origin andL2 point, respectively).(C is

the Jacobi constant)


T

DG Tethered family of Halo orbits

k1 stability curve

k2 stability curve

Orbit period curve

0 0.00001 0.00003 0.00005 0.00007

-1

0

1

2

2.26

2.27

2.28

2.29

2.3

2.31

2.32


Stab

ility

indi

ces

Peri

od

0.00.2

0.40.6Ξ

-0.5

0.0

0.5

Η

-0.5

0.0

Ζ

• Left: Stability-period diagram of the family of Halo orbitswith C = 1.07 for tether’s length variations.

Right: stable (full line) and unstable (dashed) Halo orbitsof the Hill problem.(C is the Jacobi constant)


T


k1 stability curve

k2 stability curve

Orbit period curve

1: start

2: reflection

3

4: end

0.000 0.001 0.002 0.003 0.004 0.005 0.006-3

-1

0

2

5

14

2.

2.1

2.2

2.3

2.4

2.5


Stab

ility

indi

ces

Peri

od

0.00.2

0.40.6Ξ

-0.5

0.0

0.5

Η

-0.5

0.0

Ζ

• Stability-period diagram of the family of Halo orbit withC = 1.15 for tether’s length variations. Right:

unstable Halo orbits of the Hill problem (red and magenta) and stable (blue) Halo orbits with a tethers

characteristic lengthλ = 0.0052. (C is the Jacobi constant)


T


k1 stability curve

k2 stability curve

Orbit period curve

1: start2: reflection

3

4: end

0.000 0.002 0.004 0.006 0.008 0.010-4

-1

0

2

6

19

1.9

2.1

2.2

2.3

2.5

2.6


Stab

ility

indi

ces

Peri

od

0.00.2

0.40.6Ξ

-0.5

0.0

0.5

Η

-0.5

0.0

Ζ

• Stability-period diagram of the family of Halo orbit withC = 1.2 for tether’s length variations. Right:

unstable Halo orbits of the Hill problem (red and magenta) and stable (blue) Halo orbits with a tethers

characteristic lengthλ = 0.009. (C is the Jacobi constant)


T


k1 stability curve

k2 stability curve

Orbit period curve1: start

2: reflection

3

45

6

7: end

0.00 0.02 0.04 0.06 0.08-51

-3

1

23

383

6391

2.1

2.3

2.4

2.6

2.8

2.9


Stab

ility

indi

ces

Peri

od

• Stability-period diagram of the family of Halo orbit withC = 2 for tether’s length variations. The

horizontal, gray lines mark the critical valuesk = ±2 in thearcsinh scale.(C is the Jacobi constant)


Documents

Dynamics and Stability of Tethered Satellites at Lagrangian Points