Equilibrium Conditions for Tethered Nanosatellite Constellations

CNMAC

1. State of the artMathematical modelConclusion
Equilibrium Conditions for Tethered
Nanosatellite Constellations
Denilson Paulo Souza dos Santos
[email protected]
Engenharia Aeronáutica
17 de junho de 2024
Denilson Paulo Souza dos Santos UNESP-SJBV Space Tethers 17 de junho de 2024 1 / 14

2. State of the artMathematical modelConclusion
Sumário
1State of the art
Introduction
2Mathematical model
3Conclusion
Denilson Paulo Souza dos Santos UNESP-SJBV Space Tethers 17 de junho de 2024 2 / 14

3. State of the artMathematical modelConclusion
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
rst descriptions of supposed space elevators.
Denilson Paulo Souza dos Santos UNESP-SJBV Space Tethers 17 de junho de 2024 3 / 14

4. State of the artMathematical modelConclusion
Introduction
Figura:
Denilson Paulo Souza dos Santos UNESP-SJBV Space Tethers 17 de junho de 2024 4 / 14

5. State of the artMathematical modelConclusion
Introduction
Figura:
Denilson Paulo Souza dos Santos UNESP-SJBV Space Tethers 17 de junho de 2024 5 / 14

6. State of the artMathematical modelConclusion
Introduction
Figura:
Denilson Paulo Souza dos Santos UNESP-SJBV Space Tethers 17 de junho de 2024 6 / 14

7. State of the artMathematical modelConclusion
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)
Denilson Paulo Souza dos Santos UNESP-SJBV Space Tethers 17 de junho de 2024 7 / 14

8. State of the artMathematical modelConclusion
Six-body system model in reference frame
Figura:
Denilson Paulo Souza dos Santos UNESP-SJBV Space Tethers 17 de junho de 2024 8 / 14

9. State of the artMathematical modelConclusion
Six-body system model in reference frame
The masses of the bodies (mi) are the same, in the rst hypotheses.
For the coordinates system the components of center of mass posi-
tion vector (x0,y0,z0) are:
8
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
:
x0=cos() y0=sin() z0=0
x1=x0+l1cos(+') cos( )y1=y0+l1sin(+') cos( )z1=l1sin( )
x2=x0l2cos(+') cos( )y2=y0l2sin(+') cos( )z2=l2sin( )
x3=x0l3sin(+') cos( )y3=y0+l3cos(+') cos( )z3=l3sin( )
x4=x0+l4sin(+') cos( )y4=y0l4cos(+') cos( )z4=l4sin( )
x5=x0l5cos(+') sin( )y5=y0l5sin(+') sin( )z5=l5cos( )
x6=x0+l6cos(+') sin( )y6=y0+l6sin(+') sin( )z6=l6cos( )
9
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
;
8
>
<
>
:
l1=
m
1lx
m
1+m
2
l2=
m
2lx
m
1+m
2
l1+l2=lx
l3=
m
3ly
m
3+m
4
l4=
m
4ly
m
3+m
4
l3+l4=ly
l5=
m
5lz
m
5+m
6
l6=
m
6lz
m
5+m
6
l5+l6=lz
9
>
=
>
;
Denilson Paulo Souza dos Santos UNESP-SJBV Space Tethers 17 de junho de 2024 9 / 14

10. State of the artMathematical modelConclusion
Six-body system model in reference frame
The masses of the bodies (mi) are the same, in the rst hypotheses.
For the coordinates system the components of center of mass posi-
tion vector (x0,y0,z0) are:
8
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
:
x0=cos() y0=sin() z0=0
x1=x0+l1cos(+') cos( )y1=y0+l1sin(+') cos( )z1=l1sin( )
x2=x0l2cos(+') cos( )y2=y0l2sin(+') cos( )z2=l2sin( )
x3=x0l3sin(+') cos( )y3=y0+l3cos(+') cos( )z3=l3sin( )
x4=x0+l4sin(+') cos( )y4=y0l4cos(+') cos( )z4=l4sin( )
x5=x0l5cos(+') sin( )y5=y0l5sin(+') sin( )z5=l5cos( )
x6=x0+l6cos(+') sin( )y6=y0+l6sin(+') sin( )z6=l6cos( )
9
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
;
8
>
<
>
:
l1=
m
1lx
m
1+m
2
l2=
m
2lx
m
1+m
2
l1+l2=lx
l3=
m
3ly
m
3+m
4
l4=
m
4ly
m
3+m
4
l3+l4=ly
l5=
m
5lz
m
5+m
6
l6=
m
6lz
m
5+m
6
l5+l6=lz
9
>
=
>
;
Denilson Paulo Souza dos Santos UNESP-SJBV Space Tethers 17 de junho de 2024 9 / 14

11. State of the artMathematical modelConclusion
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 byL=TV. For the following analysis, the
generalized coordinates are'and .
8
>
>
>
<
>
>
>
:
d
dt

@L
@_'


dL
d'
=Q'
d
dt

@L
@
_



dL
d
=Q

4(1+ecos())

'
0
+1

cos
2
( )

lxlx
0
+lyly
0

+4lzlz
0

'
0
+1

sin
2
( )(1+ecos())+lz
2

sin
2
( )

2'
00
(1+ecos())4esin()+
3sin(2') +2'
0


0
sin(2 )(1+ecos())2esin() sin
2
( )

+2
0
sin(2 )(1+ecos()) = cos( )

2

lx
2
+ly
2


2

'
0
+1


0
sin( )(1+ecos()) +esin() cos( )

'
00
cos( )(1+ecos())

+3

ly
2
lx
2

sin(2') cos( )
2

lx
2
+ly
2
+lz
2

(1+ecos())
00
2esin()
0

+

'
0
+1

2
sin(2 )(1+ecos())

lx
2
+ly
2
lz
2

+4
0
(1+
ecos())

lxlx
0
+lyly
0
+lzlz
0

+3sin(2 )

lx
2
cos
2
(') +ly
2
sin
2
(')

3lz
2
cos
2
(') sin(2 ) =0
Denilson Paulo Souza dos Santos UNESP-SJBV Space Tethers 17 de junho de 2024 10 / 14

12. State of the artMathematical modelConclusion
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 byL=TV. For the following analysis, the
generalized coordinates are'and .
8
>
>
>
<
>
>
>
:
d
dt

@L
@_'


dL
d'
=Q'
d
dt

@L
@
_



dL
d
=Q

4(1+ecos())

'
0
+1

cos
2
( )

lxlx
0
+lyly
0

+4lzlz
0

'
0
+1

sin
2
( )(1+ecos())+lz
2

sin
2
( )

2'
00
(1+ecos())4esin()+
3sin(2') +2'
0


0
sin(2 )(1+ecos())2esin() sin
2
( )

+2
0
sin(2 )(1+ecos()) = cos( )

2

lx
2
+ly
2


2

'
0
+1


0
sin( )(1+ecos()) +esin() cos( )

'
00
cos( )(1+ecos())

+3

ly
2
lx
2

sin(2') cos( )
2

lx
2
+ly
2
+lz
2

(1+ecos())
00
2esin()
0

+

'
0
+1

2
sin(2 )(1+ecos())

lx
2
+ly
2
lz
2

+4
0
(1+
ecos())

lxlx
0
+lyly
0
+lzlz
0

+3sin(2 )

lx
2
cos
2
(') +ly
2
sin
2
(')

3lz
2
cos
2
(') sin(2 ) =0
Denilson Paulo Souza dos Santos UNESP-SJBV Space Tethers 17 de junho de 2024 10 / 14

13. State of the artMathematical modelConclusion
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 byL=TV. For the following analysis, the
generalized coordinates are'and .
8
>
>
>
<
>
>
>
:
d
dt

@L
@_'


dL
d'
=Q'
d
dt

@L
@
_



dL
d
=Q

4(1+ecos())

'
0
+1

cos
2
( )

lxlx
0
+lyly
0

+4lzlz
0

'
0
+1

sin
2
( )(1+ecos())+lz
2

sin
2
( )

2'
00
(1+ecos())4esin()+
3sin(2') +2'
0


0
sin(2 )(1+ecos())2esin() sin
2
( )

+2
0
sin(2 )(1+ecos()) = cos( )

2

lx
2
+ly
2


2

'
0
+1


0
sin( )(1+ecos()) +esin() cos( )

'
00
cos( )(1+ecos())

+3

ly
2
lx
2

sin(2') cos( )
2

lx
2
+ly
2
+lz
2

(1+ecos())
00
2esin()
0

+

'
0
+1

2
sin(2 )(1+ecos())

lx
2
+ly
2
lz
2

+4
0
(1+
ecos())

lxlx
0
+lyly
0
+lzlz
0

+3sin(2 )

lx
2
cos
2
(') +ly
2
sin
2
(')

3lz
2
cos
2
(') sin(2 ) =0
Denilson Paulo Souza dos Santos UNESP-SJBV Space Tethers 17 de junho de 2024 10 / 14

14. State of the artMathematical modelConclusion
Equilibria
Considering the cable length in the direction constant z (lz = const)
and uniform rotations ('=!+'0 ), with =

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'
0
=
0
= l
0
(
lx;ly;lz) =0 and
'
00
=
00
=l
00
=lx
00
=0.
e=
3csc() sin(2!)
4(!+1)
Denilson Paulo Souza dos Santos UNESP-SJBV Space Tethers 17 de junho de 2024 11 / 14

15. State of the artMathematical modelConclusion
Equilibria
Considering the cable length in the direction constant z (lz = const)
and uniform rotations ('=!+'0 ), with =

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'
0
=
0
= l
0
(
lx;ly;lz) =0 and
'
00
=
00
=l
00
=lx
00
=0.
e=
3csc() sin(2!)
4(!+1)
Denilson Paulo Souza dos Santos UNESP-SJBV Space Tethers 17 de junho de 2024 11 / 14

16. State of the artMathematical modelConclusion
Equilibria
Figura: =

2
and'
0=0.
Denilson Paulo Souza dos Santos UNESP-SJBV Space Tethers 17 de junho de 2024 12 / 14

17. State of the artMathematical modelConclusion
Conclusion
1
The mathematical model is based on a model where perturbations
of the motion are not included, and where the gravity does not
inuence the orientation of the system.
2
Stability conditions give the values ofeand!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.
Denilson Paulo Souza dos Santos UNESP-SJBV Space Tethers 17 de junho de 2024 13 / 14

18. State of the artMathematical modelConclusion
Obrigado pela Atenção!
Contato:
[email protected]
Figura:
Denilson Paulo Souza dos Santos UNESP-SJBV Space Tethers 17 de junho de 2024 14 / 14