The Dynamics of Tethered
Debris With Flexible
Appendages and Residual Fuel
(current status of the research)
Vladimir
S.
Aslanov
and
Vadim
V.
Yudintsev
3rd
European
Workshop
of
Space
Debris
Modelling
and
Remedia;on
-­‐
CNES
HQ,
June
16-­‐18
2014
Samara
State
Aerospace
University,
Russia
1

Purpose
Development
of
mathema;cal
model
for
analysis
of
an
aLtude
mo;on
of
large
debris
using
tethered
space
tug
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2

Large
Space
Debris:
Types
Satellite
as
a
rigid
body
Satellite
with
flexible
appendages
Upper
stages
and
saQellte
with
fuel
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June
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3

Three
transportaEon
schemes
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4
1.
Space
tug
2.
Space
tug
+
Balloon
3.
Balloon

MathemaEcal
model
structure
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1.
Equa;ons
of
the
center
of
mass
2.
Equa;ons
of
the
rela;ve
mo;on
of
the
bodies
3.
Equa;ons
for
addi;onal
elements
(fuel,
flexible
appendages)
5

• Tug’s
thrust
• Tether
tension
(depend
on
Young's
modulus,
cross
sec;on
area,
damping)
• Aerodynamic
forces
• Gravita;onal
forces
and
torques
• Magne;c
forces
and
torques
• Solar
pressure
• Control
forces
and
torques
Forces
and
Torques
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6

ax
,
ay
,
az
are
disturbance
accelera;ons,
that
depend
on
tug’s
thrust,
drag,
gravity
gradient,
…
EquaEons
of
the
mass
center
in
osculaEng
elements
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02 y
p
a
dp
t
r
d µ
= 0 0
sin 1 cosx y
er rp
a a
p p
de
dt
θ θ
µ
⎛ ⎞
= ⎜ ⎟
⎝ ⎠
⎡ ⎤⎛ ⎞
+ + +⎢ ⎥⎜ ⎟
⎢ ⎥⎝ ⎠⎣ ⎦
0
0
1
cos 1 sinx y
p rp
a a
d
dt r e p
µθ
θ θ
µ
⎛ ⎞⎛ ⎞
− − + +⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠
=
0
cos( )z
di
d
r
a
pt
θ ω
µ
+=
0 01
cos 1 sin cot sin( )x y z
d
dt
r rp
a a a e i
e p p
θ θ θ ω
µ
ω ⎡ ⎤⎛ ⎞
− + + − +⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦
=0 sin( )
sin
z
d
dt
r
a
ip
θ ω
µ
Ω
=
+
7

The
aLtude
mo;on
of
the
space
debris
can
be
described
by
well
known
equa;ons
where
and
Ao2
is
a
matrix
that
transforms
coordinates
from
the
space
debris
principal
frame
to
the
orbital
frame
is
an
angular
velocity
tensor
associated
to
the
angular
velocity
of
the
debris
rela;ve
to
the
orbital
frame
is
an
angular
velocity
vector
of
the
orbital
frame
rela;ve
to
an
iner;al
frame
AOtude
moEon
of
space
debris
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8
2 2 2 2 2 2ω ω ω+ × =J J M
dAo2
dt
= − !Ω2
Ao2
!Ω2
ω2
= Ω2
+ωo
ωo
System’s
center
of
mass
viscous-­‐elas;c
tether

Space debris as a rigid body
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MoEon
of
the
rigid
space
debris
• The
space
debris
is
considered
as
a
rigid
body.
• The
space
tag
is
a
mass
point.
• The
space
tug
equipped
with
a
rocket
thruster
and
connected
to
the
passive
spacecra_
by
the
viscous-­‐elas;c
tether.
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10
System’s
center
of
mass

Influence
of
several
factors
to
the
moEon
of
the
space
debris
• Tether
proper;es
• Gravita;onal
torque
• …
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11
System’s
center
of
mass

The
influence
of
the
tether
damping
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• Tether
is
slack:
L0=27
<
L
=
30
m
• Tether
Young's
modulus:
60
Gpa
• Tether
diameter:
2
mm
• Tether
damping:
dT
=
16
Ns/m
• Tug’s
force:
20
N
• ϑ0
=
0.6
rad
• The
effect
of
the
tether
damping
on
the
aLtude
oscilla;on
of
the
space
debris
rela;ve
to
the
tether
is
insignificant

The
influence
of
gravitaEonal
torque
• If
tug’s
thrust
is
low
high
amplitude
oscilla;on
of
the
tether
can
occur
rela;ve
to
the
ro
direc;on
due
to
the
ac;on
of
the
gravita;onal
torque.
• The
space
tug
control
system
should
compensate
these
high
oscilla;ons
of
the
angle
a
by
changing
the
direc;on
of
the
thruster
force
vector
F.
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13

• Space
debris
with
the
long
tether
(l0=150
m)
and
a
small
value
of
the
space
tug
force
F=0.2
N
is
considered.
• Figure
shows
oscilla;ons
of
the
angles
α
between
the
tether
and
the
axis
Oyo
of
the
orbital
rota;ng
frame
with
undesirably
high
amplitudes
while
θ
is
small.
• Marked
oscilla;ons
caused
by
the
gravita;onal
torque
that
is
created
by
the
difference
between
the
gravity
forces
act
on
the
space
tug
and
on
the
orbital
debris.
The
influence
of
gravitaEonal
torque:
example
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14

• Maximum
tether
length
as
a
func;on
of
the
space
tug
force
(F)
and
orbit
height
(h)
is
expressed
as
• If
l>lmax
high
amplitude
oscilla;ons
of
α
occur
Maximum
tether
length
as
a
funcEon
of
tug’s
thrust
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( )
3
max
13 m
eF R h
l
µ
+
=
15
lmax(F,h)

For
a
drag
augmenta;on
device
(e.g.
balloon)
maximum
tether
length
is
a
func;on
of
the
radius
of
the
balloon
(Rs)
and
orbit
height
(h)
Maximum
tether
length
as
a
funcEon
of
the
balloon’s
radius
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lmax
= Cx
πRs
2
ρV 2
Re
+ h( )
3
6µ m1
16
lmax(Rs,h)

Space debris with flexible
appendages
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• Space
tug
• Mass:
m1
• Thrust
force
F
• Orbital
debris
• Mass:
m2
• Moments
of
iner;a:
J2x,
J2y,
J2z
Debris
with
flexible
appendages
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18

• Flexible
appendages
are
considered
as
in-­‐plane
bending
homogeneous
beams,
characterized
by
• Bending
s;ffness:
EJi
• Length:
li
• Mass
per
unit
length:
μi
• To
describe
the
mo;on
of
flexible
appendages
the
normal
mode
expansion
technique
is
used:
Debris
with
flexible
appendages
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1
( ) ( )i j i ij
j
q tη ξ
∞
=
= Φ∑
19

• The
natural
frequency
of
the
tether
is
higher
than
the
frequency
of
the
flexible
appendages.
• The
vibra;ons
of
the
flexible
appendages
haven't
a
significant
influence
on
the
tether
vibra;ons
and
on
the
aLtude
mo;on
of
the
debris.
Case
1
–
Far
spaced
frequencies
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20
λt
>> λf

• The
amplitude
of
the
tether
vibra;ons
is
influenced
by
the
vibra;ons
of
the
solar
panels
and
and
vice
versa.
• At
t=15
the
deforma;on
of
the
panel
2
reach
the
breaking
strain
(doQed
red
lines)
causing
structure
failure.
Case
2
–
Closely
spaced
frequencies
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t fλ λ≈
21

DeterminaEon
of
the
parameters
for
safe
transportaEon
Let
us
consider
simplified
1D
equa;ons
for
and
where
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!!q = cqq
q + cqε
ε
!!ε = cεq
q + cεε
ε + aε
22
aε
=
F
l0
m1
cqq
=
EJla
m2
+ 2ma( )Ι4
µ 2ma
Ι1
2
− m2
+ 2ma( )la
Ι2
⎡
⎣
⎤
⎦
, cqε
=
2ct
l0
Ι1
2ma
Ι1
2
− m2
+ 2ma( )la
Ι2
,
cεq
=
EJla
ma
Ι1
Ι4
µl0
2ma
Ι1
2
− m2
+ 2ma( )la
Ι2
⎡
⎣
⎤
⎦
, cεε
=
ct
la
MΙ2
− 2µΙ1
2
( )
m1
2ma
Ι1
2
− m2
+ 2ma( )la
Ι2
⎡
⎣
⎤
⎦
,
( )1 2 / 2q q q= + ( )0 0/l l lε = −

Frequencies
esEmaEon
• Solu;ons
the
characteris;c
equa;on
where
D
is
a
discriminant
where
• D
characterizes
the
closeness
of
the
frequencies
of
the
tether
and
flexible
appendages.
• D
should
be
maximized
to
avoid
mutual
influence
between
tether
oscilla;ons
and
the
oscilla;ons
of
flexible
appendages
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( ) ( )
( )
22 4 4 2 2
1 2 1 1 4 1 2 1 1 4
22 2 2
1 1 1 2
4 4
4 2
t t t tc c M EJm M m c c M EJm
D
M m
M
m M m
µ µ µ µ
µ µ
Ι + Ι − − Ι + Ι − Ι + + Ι⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦
− Ι + − Ι
=
⎡ ⎤⎣ ⎦
1,2
2
b
Dλ = ±
23
Ι1
=
0
la
∫Φ1
η( )dη, Ι2
=
0
la
∫ Φ1
η( )⎡
⎣
⎤
⎦
2
dη, Ι3
=
0
la
∫ηΦ1
η( )dη, Ι4
=
0
la
∫ Φ1
′′ η( )⎡
⎣⎢
⎤
⎦⎥
2
dη
l1
= l2
= la
, M = m1
+ m2
+ 2µla
, EJ1
= EJ2
= EJ

0.00035
0.0005
0.0005
0.002
0.0020.01
0.01
0.05
0.1
0 50 100 150 200 250 300
100
200
300
400
500
600
700
800
ct, N m
m1,kg
O
A
B
C D
Amplitude
of
the
oscillaEons
as
a
funcEon
of
the
tether
sEffness
&
tug’s
mass
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24
A:
max|q1|=0.035;
B:
max|q1|=0.005;
C:
max|q1|=0.030;
D:
max|q1|=0.015
D(m,ct
)

The motion of upper stages
with fuel
(preliminary research)
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Upper
stage
with
fuel
1. Space
debris
is
a
orbital
stage
with
par;ally
filled
tanks.
2. Thruster
burn
phase
is
considered
3. Tether
is
weightless
and
viscoelas;c
4. F
=
const
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Simple
fuel
slosh
model
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• The
space
debris
is
considered
as
a
rigid
body
(m,
J).
• The
fuel
slosh
is
modeled
as
a
pendulum
(mf,
Jf,
lf).
• The
space
tag
is
a
mass
point
(m1)
• The
space
tug
equipped
with
a
rocket
thruster
and
connected
to
the
passive
spacecra_
by
the
viscous-­‐
elas;c
tether
(l,
ct,
E,
d)
27

Parameter
Value
Parameter
Value
Tug
mass
100
kg
Thrust
force
20
N
Debris
mass
(dry)
500
kg
Tether
length
50
m
Fuel
mass
200
kg
Tether
sEffness
80
MPa
[2.0;
0.1]
COF
2
m
Simple
example
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ρA

Case
1
–
fuel
is
“frozen”
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Ini;al
condi;ons
Oscilla;ons
of
α
and
θ
caused
by
the
shi_ed
aQachment
point:
ρA=[2;
0.1]
29
α0
= 0, θ0
= 0

Ini;al
condi;ons
The
influence
of
the
considered
fuel
slosh
to
the
mo;on
of
the
debris
is
insignificant
Case
2
–
Orbital
debris
with
fuel
slosh
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30
α0
= 0, θ0
= 0, ϕ0
= 0.2

What
has
been
done?
• We
have
been
learned
to
simulate
aLtude
mo;on
of
the
system
(debris+tether+tug)
• Some
simple
examples
are
shown
• We
found
that
characteris;cs
of
a
tether
and
a
space
tug
affect
on
the
level
of
vibra;ons
of
flexible
elements
• Wrong
choice
of
the
characteris;cs
can
lead
to
the
destruc;on
of
debris
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What
we
will
intend
to
do?
• To
study
the
capture
dynamics
of
debris
(using
harpoon,
net,
lasso
...)
• To
examine
a
stabiliza;on
phase
a_er
capture
debris
and
find
a
law
of
tether
control
• To
consider
a
par;cular
type
of
debris
at
all
removal
stages:
from
the
capture
to
the
atmospheric
stage
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PublicaEons
Published
• V.S.
Aslanov,
V.
V.
Yudintsev,
Dynamics
of
Large
Debris
Connected
to
Space
Tug
by
a
Tether,
J.
Guid.
Control.
Dyn.
36
(2013)
1654–1660.
• V.
Aslanov,
V.
Yudintsev,
Dynamics
of
large
space
debris
removal
using
tethered
space
tug,
Acta
Astronaut.
91
(2013)
149–156.
• V.
Aslanov,
A.
Ledkov,
Dynamics
of
towed
large
space
debris
taking
into
account
atmospheric
disturbance,
Acta
Mech.
(2014)
1–13.
SubmiQed
• V.
S.
Aslanov,
V.
V.
Yudintsev,
Behaviour
of
Tethered
Debris
With
Flexible
Appendages
(Acta
Astronau;ca)
• V.
S.
Aslanov,
V.
V.
Yudintsev,
Dynamics,
AnalyEcal
SoluEons
and
Parameters
EsEmaEon
for
Towed
Space
Debris
with
Flexible
Appendages
(Advances
in
Space
Research)
In
work
• V.
S.
Aslanov,
V.
V.
Yudintsev,
Modeling
of
Tethered
Space
Debris
with
Fuel
(Acta
Astronau;ca)
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HQ,
June
16-­‐18
2014
33

Thank
You
3rd
European
Workshop
of
Space
Debris
Modelling
and
Remedia;on
-­‐
CNES
HQ,
June
16-­‐18
2014
34

Authors
• Vladimir
S.
Aslanov
Prof.,
Head
of
the
Theore;cal
Mechanics
Department,
Samara
State
Aerospace
University
aslanov_vs@mail.ru
hQp://aslanov.ssau.ru
• Vadim
V.
Yudintsev
Associate
Prof.,
Theore;cal
Mechanics
Department,
Samara
State
Aerospace
University
yudintsev@classmech.ru
hQp://www.classmech.ru
3rd
European
Workshop
of
Space
Debris
Modelling
and
Remedia;on
-­‐
CNES
HQ,
June
16-­‐18
2014
35