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State of the art Mathematical model Conclusion
Equilibrium Conditions for Tethered
Nanosatellite Constellations
Denilson Paulo Souza dos Santos
denilson.santos@unesp.br
Engenharia Aeronáutica
17 de junho de 2024
Denilson Paulo Souza dos Santos UNESP-SJBV Space Tethers 17 de junho de 2024 1 / 14

Sumário
1 State of the art
Introduction
2 Mathematical model
3 Conclusion

State of the art
The idea of connecting Earth to Heaven already debated previously
and been described in Christian mythology. The Tower of Babel was
supposed to be the way to ascend to Heaven, coming once again
the story, the patriarch Jacob who described a vision where angels
descended and ascended from heaven to Earth via a stairs, are the
first descriptions of supposed space elevators.

Introduction
Figura: Artist Concept of a Space Elevator, from NASA Online Image Gallery

Introduction
Figura: The system composed by a massless tether and three nanosatellite.

Introduction
Figura: Space–Net System

Introduction
Figura: Tethered SPHERES nano-satellites developed by the MIT, ( Chung, Soon-Jo, Thesis, Nonlinear Control and Synchronization of
Multiple Lagrangian Systems with Application to Tethered Formation Flight Spacecraft)

Six-body system model in reference frame
Figura: Six-body system model in reference frame

The masses of the bodies (mi ) are the same, in the first hypotheses.
For the coordinates system the components of center of mass posi-
tion vector (x0, y0, z0) are:



















x0 = ρ cos(ν) y0 = ρ sin(ν) z0 = 0
x1 = x0 + l1 cos(ν + ϕ) cos(ψ) y1 = y0 + l1 sin(ν + ϕ) cos(ψ) z1 = l1 sin(ψ)
x2 = x0 − l2 cos(ν + ϕ) cos(ψ) y2 = y0 − l2 sin(ν + ϕ) cos(ψ) z2 = −l2 sin(ψ)
x3 = x0 − l3 sin(ν + ϕ) cos(ψ) y3 = y0 + l3 cos(ν + ϕ) cos(ψ) z3 = l3 sin(ψ)
x4 = x0 + l4 sin(ν + ϕ) cos(ψ) y4 = y0 − l4 cos(ν + ϕ) cos(ψ) z4 = −l4 sin(ψ)
x5 = x0 − l5 cos(ν + ϕ) sin(ψ) y5 = y0 − l5 sin(ν + ϕ) sin(ψ) z5 = l5 cos(ψ)
x6 = x0 + l6 cos(ν + ϕ) sin(ψ) y6 = y0 + l6 sin(ν + ϕ) sin(ψ) z6 = −l6 cos(ψ)




















The masses of the bodies (mi ) are the same, in the first hypotheses.
For the coordinates system the components of center of mass posi-
tion vector (x0, y0, z0) are:



















x0 = ρ cos(ν) y0 = ρ sin(ν) z0 = 0
x1 = x0 + l1 cos(ν + ϕ) cos(ψ) y1 = y0 + l1 sin(ν + ϕ) cos(ψ) z1 = l1 sin(ψ)
x2 = x0 − l2 cos(ν + ϕ) cos(ψ) y2 = y0 − l2 sin(ν + ϕ) cos(ψ) z2 = −l2 sin(ψ)
x3 = x0 − l3 sin(ν + ϕ) cos(ψ) y3 = y0 + l3 cos(ν + ϕ) cos(ψ) z3 = l3 sin(ψ)
x4 = x0 + l4 sin(ν + ϕ) cos(ψ) y4 = y0 − l4 cos(ν + ϕ) cos(ψ) z4 = −l4 sin(ψ)
x5 = x0 − l5 cos(ν + ϕ) sin(ψ) y5 = y0 − l5 sin(ν + ϕ) sin(ψ) z5 = l5 cos(ψ)
x6 = x0 + l6 cos(ν + ϕ) sin(ψ) y6 = y0 + l6 sin(ν + ϕ) sin(ψ) z6 = −l6 cos(ψ)
























l1 = m1lx
m1+m2
l2 = m2lx
m1+m2
l1 + l2 = lx
l3 = m3ly
m3+m4
l4 = m4ly
m3+m4
l3 + l4 = ly
l5 = m5lz
m5+m6
l6 = m6lz
m5+m6
l5 + l6 = lz






The equations of motion of the spacecraft
The Lagrange Equations of motion is ordinary differential equations,
which describe the motions mechanical systems under the action of
forces, can be obtained by L = T − V. For the following analysis, the
generalized coordinates are ϕ and ψ.










d
dt

∂L
∂ϕ̇

− dL
dϕ
= Qϕ
d
dt

∂L
∂ψ̇

− dL
dψ
= Qψ










d
dt

∂L
∂ϕ̇

− dL
dϕ
= Qϕ
d
dt

∂L
∂ψ̇

− dL
dψ
= Qψ
4(1+e cos(ν))

ϕ0 + 1

cos2(ψ)

lxlx0 + lyly0

+4lzlz0

ϕ0 + 1

sin2(ψ)(1+e cos(ν))+lz2

sin2(ψ)

2ϕ00(1 + e cos(ν)) − 4e sin(ν)+
3 sin(2ϕ) + 2ϕ0

ψ0 sin(2ψ)(1 + e cos(ν)) − 2e sin(ν) sin2(ψ)

+ 2ψ0 sin(2ψ)(1 + e cos(ν)) = cos(ψ)

2

lx2 + ly2

2

ϕ0 + 1

ψ0 sin(ψ)(1 + e cos(ν)) + e sin(ν) cos(ψ)

− ϕ00 cos(ψ)(1 + e cos(ν))

+ 3

ly2 − lx2

sin(2ϕ) cos(ψ)
2

lx2 + ly2 + lz2

(1 + e cos(ν))ψ00 − 2e sin(ν)ψ0

+

ϕ0 + 1
2
sin(2ψ)(1 + e cos(ν))

lx2 + ly2 − lz2

+ 4ψ0(1 +
e cos(ν))

lxlx0 + lyly0 + lzlz0

+ 3 sin(2ψ)

lx2 cos2(ϕ) + ly2 sin2(ϕ)

− 3lz2 cos2(ϕ) sin(2ψ) = 0

Equilibria
Considering the cable length in the direction constant z (lz = const)
and uniform rotations (ϕ = ων +ϕ0), with ψ = π
2 , the equation provides
solution, depending on the orbit eccentricity (Eq. 11), not dependent
on the cable length (lx, ly, lz).

Equilibria
Considering the cable length in the direction constant z (lz = const)
and uniform rotations (ϕ = ων +ϕ0), with ψ = π
2 , the equation provides
solution, depending on the orbit eccentricity (Eq. 11), not dependent
on the cable length (lx, ly, lz).
Mathematically, a point is in equilibrium when its speed and accele-
ration are equal to zero, then assume ϕ0
= ψ0
= l0
(lx, ly, lz) = 0 and
ϕ00
= ψ00
= l00
= lx00
= 0.
e =
3 csc(ν) sin(2ων)
4(ω + 1)

Equilibria
Figura: Tether length control for eccentricity (e), ψ = π
2
and ϕ0 = 0.

Conclusion
1 The mathematical model is based on a model where perturbations
of the motion are not included, and where the gravity does not
influence the orientation of the system.
2 Stability conditions give the values of e and ω where the system is
stable, Stability intervals were found for the system with numerical
integration for the monodromy matrix small perturbations won’t
change the behavior of the system in these intervals.

Obrigado pela Atenção!
Contato:
denilson.santos@unesp.br
Figura: https://orcid.org/0000-0003-2682-4043

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