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Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
1
Chapter 5 Attitude Stabilization with Spin
5.1 Equations of Motion with Spin
5.1.1 Fundamentals of Spin Motion
Angular moment of momentum
Spin Stabilized
Spacecraft
Tzzyyxx III H
Euler’s equation for the moment of momentum
zxyyxz
yzxxzy
xyzzyx
MHHH
MHHH
MHHH
zyxxyzz
yxzzxyy
xzyyzxx
MIII
MIII
MIII
Equations of Motion in terms of angular velocities
(5.1-1)
zI
Bx
By
Bz
xIyI
z
x
y
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
2
T
s
ss
T
Ts
I
I
I
II
1
zI
Bx
By
Bz
xIyI
z
x
y
Tyx III
sz II
Axially symmetrical body spinning around axis
No external torque is applied
Bz
0M
0
0
0
zs
xzsTyT
zyTsxT
I
III
III
Equations of motion Eq.(5.1-1) becomes
(5.1-2)
Constant angular velocity of spinsz
0
0
xy
yx
Equations of motion Eq.(5.1-2) is reduced to
MOIR: Moment of
Inertia Ratio
(5.1-3)
where
:transverse
:spinning
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
3
State equation of motion derived from Eq. (5.1-3)
(5.1-6)
TTT ωAω
y
x
T
ω
where
0
0
TA
Laplace transformation of state equation
y
x
T
y
xLL
s
s
ωΩ
Characteristic equation
0det 22 jsjssss TT AIAI
Eigen values of motion for the spinning spacecraft
js 2,1
(5.1-4)
where
0Ω
s
s
(5.1-5)
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
4
jsjsjsjs
j
jsjs
j
jsjs
s
s
jsjss
ss T
11
2
111
2
11
2
11
2
1
11
1AI
Let’s
Transition matrix can be derived using inverse Laplace transformation
tjtjtjtj
tjtjtjtj
TT
eeeej
eej
ee
sLt
2
1
2
22
1
11AIφ
(5.1-7)
(5.1-8)
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
5
0
00
y
x
T
ω
Assuming the following initial condition
tt
tt
eeeej
eej
ee
tt
yx
yx
y
x
tjtjtjtj
tjtjtjtj
TTT
cossin
sincos
2
1
2
22
1
0
00
00
0
0
ωφω
The free response of angular velocities are derived as
Sinusoidal oscillations with natural frequency of in and
axes, which has 90 degrees phase difference between themBx
By
(5.1-9)
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
6
2
0
2
0
2
00
2
00 cossinsincos
yx
yxyxTT tttt
ω
Total angular velocity amplitude remains constant
TTT xω
Resultant angular vector rotates around axis with natural
frequency of and with constant amplitude of Tω Bz
T
BzTTBzByBx zxzyxω
Total angular velocity vector
(5.1-12)
Moment of momentum around spin axiszsz IH
TTT IH Moment of momentum around axis
Bz
Tx
Total moment of momentum
BzTTBzsByBxT HHII zxzyxH
(5.1-10)
(5.1-11)
(5.1-13)
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
7
Total angular velocity vectors
and total moment of momentum
vector lie in the same plane,
where angular vector and spin
axis locate
ω
H
Tx
Bz
Important findings on “Nutation”
Total moment of momentum
vector is constant and fixed
in the inertial coordinate systemH
Total angular velocity vector
has constant amplitude
determined by initial condition
ω and angular
velocity oscillation vectorx y
Nutation
sNutation Angle
Spin Axis
Total Moment
of Momentum
ZH
Bz
H
TH
TTxBx
By0
H
dt
d
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
8
How “Nutation” occurs
1. Caused by moment of momentum normal to spin axis
2. Total moment of momentum vector does not have the same
direction of spin axis
Body Cone: Caused by angular vector around spin axis
Space Cone: Caused by angular vector around total moment of
momentum vector
ωω
H
Bz
Body Cone
Space Cone
zH
Bz
H
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
Body ConeBody Cone
Space ConeSpace Cone
Bz
H
s
Bz
H
s
9
1T
s
I
I 1
T
s
I
I:Disk Body :Rod Body
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
10
axis of inertia coordinate
system has the
same direction of total moment of
momentum vector
5.1.2 Spin Motion in Inertial Coordinate System
0000 ,, zyx
0z
H
Coordinate system rotation
from inertial coordinate system
to body (moving)
coordinate system BBBB zyx ,,
0000 ,, zyx
Spin Axis
Total Moment of
Momentum Vector H
Bz0z
By
y
0y
x Bx
0x
Nutation Angle
x
: Nutation Angular Velocity
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
11
1. Rotate around axis until
the plane where nutation angle
is defined
2. Rotate nutation angle
around axis until axis
meet with axis of body
coordinate system
3. Rotate around axis until
axis meet with and axis of
body coordinate system
0z
y z
Bz
BBBB zyx ,,
z
Bx
Coordinate System Rotation
By
Spin Axis
Total Moment of
Momentum Vector H
Bz0z
By
y
0y
x Bx
0x
x
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
12
0
0
0
323
z
y
x
CCC
z
y
x
B
B
B
Transformation of coordinate
system rotation from inertial
coordinate system
to body (moving) coordinate
system BBBB zyx ,,
0000 ,, zyx
0
0
0
0
0
0323 CCC
z
y
x
review (2.3-1)
Spin Axis
Total Moment of
Momentum Vector H
Bz0z
By
y
0y
x Bx
0x
x
review (2.1-7)
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
13
cos0sin
sinsincossincos
cossinsincoscos
cos0sin
010
sin0cos
100
0cossin
0sincos23 CC
Calculating
Knowing nutation angle is constant:
0
Definition of nutation angle
z
Tz
H
H
H
H tan,cos
zs
Tzs
I
H
H
I
11 tancos
(5.1-14)
(5.1-15)
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
14
cos
sinsin
cossin
0
0
0
023 CC
z
y
x
1st and 2nd equations of Eq.(5.1-16) result in nutation angular velocity
Nutation angular velocity
is proved constant in the
inertia coordinate system
Multiplying transvers moment of inertia
TI
HHHH
III
yx
yTxT
T
sin
sin
sin
sincos
sin
sincos
(5.1-16)
(5.1-17)
sin
sincos yx
Body coordinate system rotates at the constant angular velocity of
around the fixed moment of momentum vector
H
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
15
coscoscos
coscos
T
ssszs
I
HII
IIH
Eq.(5.1-15) and Eq.(5.1-16) result in an equation of total moment of
momentum
ssT
s
Ts IH
II
IH
I
H
I
H
cos1
cos1cos
(5.1-18)
Eq.(5.1-17) and Eq.(5.1-18) result in nutation
angular velocity calculation
cos11
1cos
coscos1cos
coscos
coscoscoscos
T
s
T
z
T
zs
T
I
I
I
H
I
I
I
H
(5.1-19)
Spin velocity
is proved constant
in the inertia
coordinate system
~
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
16
Definition of nutation angle and inclination angle between total
angular velocity and spin angular velocity
s
T
zs
TT
z
T
I
I
H
H
tan,tan
ω sω
(5.1-20)
Thus
tan
1tan
zs
TT
I
I(5.1-21)
Important findings on “Nutation”
1. Unless outer torque is applied, nutation angle and its angular
velocity remain constant
2. Nutation angular velocity is proportional to spin angular
velocity and MOIR (Moment of Inertia Ratio)
3. In case of , nutation angular velocity is larger than spin
angular velocity * Eq.(5.1-19)
z
1
z
Inclination angle is constant
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
17
5.2 Stability of Spin Spacecraft
5.2.1 Case of No Energy Dissipation
Spin Stabilized
Spacecraft
zI
Bx
By
Bz
xIyI
z
x
y
zyxxyzz
yxzzxyy
xzyyzxx
MIII
MIII
MIII
Euler equations of motion in terms of angular
velocities
(5.1-1)
Spacecraft has its initial angular velocity
condition
T000 ω
Tzyx 0ω
Applying external torque, angular velocities
become
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
18
0
0
0
0
0
yxxyzz
xzzxyy
zyyzxx
III
III
III
Small perturbed equations of motion in terms of angular velocities
(5.2-1)
Eliminating second order of small perturbed terms
0
0
0
0
0
zz
xzxyy
yyzxx
I
III
III
Laplace transformation of Eq. (5.2-2) is described as
0
sE
sE
sI
II
I
IIs
y
x
y
zx
x
yz
0
(5.2-2)
(5.2-3)
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
19
Characteristic Equation becomes
0det 2
0
20
yx
zxyz
y
zx
x
yz
II
IIIIs
sI
II
I
IIs
D
The stability condition is
02
0
yx
zxyz
II
IIII 02
0
yx
zyzx
II
IIII
zyzx
zyzx
IIII
IIII
,
,“When the spin axis has the maximum,
or the minimum moment of inertia
compared with the transverse moment of
inertia, the spacecraft becomes stable”
(5.2-4)
(5.2-5)
(5.2-6)
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
20
5.2.2 Case of Energy Dissipation
Actual spin spacecraft has
Nutation Damper
Energy Dissipation Elements such as
Fuel Sloshing
Friction of Driving Elements
Elastic Structure Damping
222222
ssyxT IIH
Total moment of momentum
Spin Stabilized
Spacecraft
zI
Bx
By
Bz
xIyI
z
x
y
H
Total energy of motion
222
2
1ssyxT IIT
wheresI
TI
s
Moment of inertia of spin axis
Moment of inertia normal to spin axis
Bz
Bz
Spin angular velocity around axisBz
T
(5.2-7)
(5.2-8)
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
21
Eliminate using Eq. (5.2-7) and Eq. (5.2-8)22
yx
2222 12 s
s
TssTssT
I
IIIIITIH
H
I ss 1cos
Using nutation angle
2222 cos12
T
s
s
TsTssT
I
I
I
IHIIITIH
cossin12
sI
HT
Eq. (5.2-9) results in
Differentiating Eq. (5.2-11) and knowing , equation of energy
dissipation becomes
(5.2-9)
(5.2-10)
(5.2-11)
(5.2-12)
0H
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
22
For the case of energy dissipation
0T
1
1
(Disk type)
(Rod type)
0
0
: Stable
(nutation angle decreases)
: Unstable(nutation angle increases)
Turn style antenna
Nutation damper for spacecraft with
Viscosity of fluid
Eddy current
1
(5.2-13a)
(5.2-13b)
T
s
I
I
MOIR: Moment of
Inertia Ratio
Spacecraft of Explore 1st
Stability Failure (Flat Spin)
1
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
23
5.2.3 Stability of Dual Spin Satellite
Spin stability spacecraft
Advantage : simple control
Disadvantage : difficult to conduct pointing mission
S Band Antenna
VISSR(Visible and Infrared
Spin Scan Radiometer)
UHF Antenna
VHF Antenna
Sun Sensor
Earth Sensor
Solar Cell
Panel
Dual Spin Spacecraft
Turn style
antenna
Rotor(Spinning)
Platform(Anti-spinning:
relatively stable in inertia
coordinate system)
H
r
s
p
s
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
24
2222222 p
s
p
s
r
s
r
sTTssyxT IIIIIH
222
2
1 p
s
p
s
r
s
r
sTT IIIT
Total moment of momentum H
Total energy of motion T
where suffix : rotor
: platform
: tangential to spin axis
rp
(5.2-14)
(5.2-15)
Differentiae Eq.(5.2-14) in terms of time
02 p
s
p
s
r
s
r
s
p
s
p
s
r
s
r
sTTT IIIII
Differentiae Eq.(5.2-15) in terms of time
p
s
p
s
p
s
r
s
r
s
r
sTTT IIIT
(5.2-16)
(5.2-17)
T
Rotor
Platform
H
r
s
p
s
p
sI
r
sI
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
Eliminating and using Eq.(5.2-16) , Eq. (5.2-17) becomes
where : nutation angular velocity of rotor
: nutation angular velocity of platform 25
T
p
s
p
s
T
p
s
p
s
r
s
r
sp
s
r
s
r
s
T
p
s
p
s
r
s
r
sr
s
p
s
p
s
p
s
r
s
r
s
r
s
p
s
p
s
r
s
r
s
p
s
p
s
r
s
r
s
T
I
III
I
III
IIIIIII
T
1
Total nutation angular velocity is derived by Eq. (5.1-17)
T
p
s
p
s
r
s
r
s
I
II
0
The final equation of energy dissipation is
p
sp
p
s
r
sr
r
s
p
s
p
s
p
s
r
s
r
s
r
s
II
IIT
00
r
sr 0
p
sp 0
TI
H (5.1-17)
(5.2-18)
(5.2-19)
T
0
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
26
Energy dissipation of rotor and platform occur independently
pr TTT
where
p
pp
s
p
s
r
rr
s
r
s
TI
TI
,
p
p
sp
r
r
srTTT TTI
11
p
p
r
r
Tp
p
p
r
rr
T
TT
TTTTI
000 11
1
Thus Eq. (5.2-17), Eq. (5.2-21) and Eq.(5.2-21) becomes
Finally the angular acceleration of normal axis to spin axis
(5.2-20)
(5.2-21)
(5.2-22)
(5.2-23)
p
s
p
pr
s
r
rpr
p
s
p
s
p
s
r
s
r
s
r
sTTT
TTTTIITI
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
27
Stability condition of dual spin spacecraft means not to
diverge the nutation angle (transverse angular velocity)
0T
0,0 pr TT
Assuming energy dissipation of rotor and platform
the combination of nutation velocity for rotor and platform results in
the stability condition based on Eq. (5.2-23)
(a)
(b) and
(c) and
0,0 pr
0,0 pr p
p
r
rTT
0,0 pr p
p
r
rTT
(5.2-24a)
(5.2-24b)
(5.2-24c)
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
28
0
tan
T
p
s
p
s
r
s
r
s
TT
II
I
Knowing that the nutation angle of dual spin spacecraft is
Finding of nutation characteristics for dual spin space craft
minus transverse angular velocity (normal angular velocity to
spin axis) by energy dissipation results in stabilization of
nutation angle
If the platform does not rotate or almost stable in the inertia
coordinate system:
p
s
r
s
Defining the MOIR (Moment of Inertia Ratio) as
T
r
s
I
I
(5.2-25)
(5.2-26)
(5.2-27)
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
29
Nutation angular velocities can be simplified
r
s 0
r
s
r
sr 10
r
s
p
sp 0
Stability condition of dual spin spacecraft (non spinning platform)
(Disk type)
(Rod type)
1
1p
p
r
rTT
Platform energy dissipation
must be larger than rotor
energy dissipation
Advantage of dual spin spacecraft
The stability condition of dual spin spacecraft is more relaxed than
that of single spin spacecraft
(5.2-28)
(5.2-29)
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
30
5.3 Control of Spin Spacecraft
Spin Spacecraft motion in inertial coordinate system
X
ZY
Spin Axis
Ascending
Node
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
31
Spin Axis
Ascending
Node
Angle transformation from inertial coordinate system
(orbital coordinate system) to body coordinate system
1. Orbital coordinate system
is employed as the inertial
coordinate system
2. Rotate angle around axis and
angle around axis to meet spin
coordinate system
that decline from normal direction
to the orbital plane
3. Finally rotate around
axis to meet body coordinate
system
0000 ,, zyx
0z
ssss zyx ,, sy
sz
BBBB zyx ,,
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
32
External torque defined in the body coordinate system
TBzByBx TTTT
BBBB zyx ,,
Euler equations of motion Eq. (5.1-2) become
Bzzs
ByxzsTyT
BxzyTsxT
TI
TIII
TIII
where : spin axis
: moment of inertia around spin axis
: moment of inertia around normal to spin axis
sB zz
sI
TI
0
0
0
323
z
y
x
CCC
z
y
x
B
B
B
Transformation from orbital
coordinate to body coordinate
system
(5.2-30)
(5.2-31)
(5.2-32)
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
33
Transformation of coordinate angular velocity to angular velocity in
body coordinate system
cos
cossinsin
sincossin
0
0
0
0
0
0
0
0
0
0
0
0
323
323323
CCC
CCCCCC
z
y
x
sincos
sincoscossinsincossinsin
cossinsinsincoscoscossin
z
y
x
Derivative of Eq.(5.2-33) in terms of time(5.2-33)
(5.2-34)
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
34
szBzs
syByBxTssT
sxByBxsTT
TTdt
dI
TTTIIII
TTTIII
cos
cossinsincos
sincoscoscos2sin
Put Eq.(5.33) Eq.(5.34) to Eq.(5.31) and arrange
the equations using
Assuming no torque is applied to spin axis , the 3rd equation
of Eq.(5.35) becomes0szT
.cos constsz
syTss
sxssT
TII
TII
sin
sin
Eliminating 2nd order small perturbed terms except
of 1st and 2nd equations of Eq.(5.35) becomessz
(5.2-35)
(5.2-36)
(5.2-37)
cos,sin
Advanced Course of Aerospace Guidance and Control Chapter II-5 Attitude Stabilization with Spin
35
Introducing new parameter
sin1
1
Eq.(5.2-37) is reduced to the following equations
syTss
sxssT
TII
TII
Eq.(5.2-38) is equivalent to the equation of motion for spacecraft with
bias-momentum control system
zBz
xBx
MhI
MhI
(5.2-38)
(5.2-39)