Spacecraft Dynamic Modeling

Spaacecraft Dynamic
D
M
Modelingg
Introd
duction:
t formulatiion and goverrning assump
ptions for obttaining the
This document covers the
down dynamics of spacecraft similar to that of Hayabusa. To devvelop the
overaall and touchd
equattions of motio
ons (EOM’s), Newton’s seccond law wass implementeed as well as EEuler’s
rotational equations of motion..
Backgground:
busa spacecraaft was design
ned to touchddown and collect a materiaal sample
The Hayab
ere would be three samplees that it wou
uld collected. While
from the asteroid Itokawa. The
mode
eling the spacecraft is difficcult, the deve
elopers createed a model th
hat assumed tthe solar
panels would reactt as if there were
w
a mass-sspring-dampeer mechanical setup and co
ould only
move
e in the verticaal direction. This model iss depicted bellow.
t
from “TTouchdown Dynamics
D
for Sample Colleection in Hayaabusa
This illlustration is taken
Missio
on.”
The figure above iss a simplified three-dimensional model of the Hayab
busa spacecraaft.
Two Dimensional
Dynamic Model:
D
To staart out the un
nderstanding of how the th
hree-dimensi onal hayabussa mission sattellite body
was fo
ormulated, a similar two-d
dimensional case
c
can be sttudied. This m
model consisted of a box
with two
t masses attached on either side by a spring-dam
mper system. The box was not allowed
to rottate, which would give only vertical motion.
After using Newton’s second law
w, the follow
wing EOM’s caan be develop
ped:
Mass 1:
=
−
+
−
−
−
+
−
−
=0
Mass 2:
=
=0
Main Body:
=
+
−
−
−
−
−
−
−
−
=0
Variab
bles:
,
,
: Th
he masses of the main bod
dy as well as tthe two massses
,
,
: The vertical displacements of the main boddy and the tw
wo masses
,
,
: The horizontal dissplacements of the main bbody and the two masses
: Spring constant and
d damping con
nstant
: Input th
hruster forces
: Graavity
The next step in fin
nding the equ
uations of mo
otion were to introduce tw
wo-dimension
nal motion as
a rotation of the body. Byy doing this, the
t non-linea r equations ccan be found. Once these
well as
are fo
ound, the equ
uations can be
e validated byy setting the aangular displaacement to zzero as well ass
all of its time derivvatives. The image below depicts the m
model.
+
y
2
x
+
The fo
ollowing syste
em has two masses
m
attach
hed to either eend of a rigid
d bar or box, w
which is donee
to inccorporate the dynamical re
eactions of so
olar panels. TTo develop the EOM’s, a seeparate
coord
dinate system
m had to be created for eacch mass. Thiss is done becaause of the m
masses being
able to
t move. Oncce these coorrdinates are im
mplemented,, the following EOM’s are obtained:
Main body:
=−
=
sin
cos
+
−
sin
cos
+
−
Mass 1:
=−
sin
−
siin
sin
cos
−
+
sin
n
−
ssin
−
siin
cos
+
co
os
+
co
os
−
=
cos
+
cos
sin
+
sin
−
Mass 2:
=
=−
cos
−
cos
−
Rotational Element:
=
=
−
2
ℎ
−
−
2 −
+
−
+
2
+
+
+
2
+
Accelerations:
= +
+
= +
2
= +
2
= +
sin
cos
+
sin
+
sin
2
+
−
sin
2
cos
−
cos
2
cos
2
+
−
2
−
2
sin
+
cos
cos
−2
sin
−2
−
sin
−2
sin
−
+
−
sin
−2
sin
−
+
cos
sin
cos
sin
−
−
−
−
+
+
+
+
cos
cos
cos
cos
Variables:
,
,
: The masses of the main body as well as the two masses
,
,
: The vertical displacements of the main body and the two masses
,
,
: The horizontal displacements of the main body and the two masses
: The extension of each spring
: Rotation angle
,
,
,
: Spring constant and damping constant
: Input thruster forces
, ,
: The length of the main body and respective original lengths of the springs
: Gravity
Since the non-linear EOM’s have been found, the validity of said equations must be shown. Like
stated previously, assuming that the satellite only moves vertically and does not rotate, it can be
compared to the vertical moving case.
Assumptions:
=
=
= 0 : For all relative coordinates as well
=
=
=0
=
−
=
−
=
−
=
−
=
=
=
=
After replacing the above assumptions, the equations do in fact reduce down to the first twodimension case. The work can be seen in the appendix.
Conclusion:
The two-dimensional non-linear model is correct. After the implementation of the assumptions,
the EOM’s reduce to the equations of just the vertical case. Further work will need to be done
in simulating the results as well as formulating the three-dimensional equations of motion.
References
[1] Takashi Kubota, Masatsugu Otsuki, Tatsuaki Hashimoto, Member, IEEE. “Touchdown
Dynamics for Sample Collection in Hayabusa Mission.” 2008 IEEE International
Conference on Robotics and Automation, 2008.
Ap
ppendix
Verticcal Model:
Mode
el with Rotatio
on:
Validaation of Non-Linear Modell: