Chapter 4 Attitude Control with Reaction Wheels (2/2)...Pitch Axis Momentum Wheel with Large...
Transcript of Chapter 4 Attitude Control with Reaction Wheels (2/2)...Pitch Axis Momentum Wheel with Large...
Chapter II-4 Attitude Control with Reaction Wheels (Part 2)飛行力学
where : external torque 1
4.2 Bias-Momentum Attitude Control
Chapter 4 Attitude Control with Reaction Wheels (2/2)
Bias-momentum attitude
control system has a pitch axis
momentum wheel with large
momentum.
4.2.1 Equations of Motion
Angular momentum of reaction
wheels:
(4.2-1)
Total angular momentum of spacecraft:
wwT HIωHHH
MHωH
T
B
T
dt
d
Euler’s equations:
MHHωHH ww
TBBw h 00 H
M
Pitch Axis Momentum Wheel with
Large Momentum
Unloading Thrusters
Yaw Axis
Earth Sensor
Pitch Axis
Roll Axis
Z
X
Y
Bh
(4.2-2)
(4.2-3)
Chapter II-4 Attitude Control with Reaction Wheels (Part 2)飛行力学
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Thus
For small Euler angles
Bz
y
x
ω
If the body coordinate
system agrees with
the principle axes of
inertia
zz
yy
xx
z
y
x
z
y
x
z
y
x
I
I
I
I
I
I
H
H
H
00
00
00
IωH
M
kji
HH
zByx
zyxw
HhHH
zxyByxz
yzxxzBy
xByzzyx
MHhHH
MHHhH
MhHHH
*see Capt. I-2
*see Capt. I-2 Eq.(2.4-4)
(4.2-4)
Chapter II-4 Attitude Control with Reaction Wheels (Part 2)飛行力学
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Assuming 0
Euler’s equations becomes
The following equations are obtained:
Spacecraft Dynamics Reaction Wheel
Control Dynamics
Disturbance
Torque
zxByz
yzxBy
xByzx
MIhII
MIIhI
MhIII
zByxzz
yBxzy
xBzyxx
MhIIII
MhIII
MhIIII
2
2
2
(4.2-5)
(4.2-6)
(4.2-7)
Chapter II-4 Attitude Control with Reaction Wheels (Part 2)飛行力学
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Neglecting the second order small terms of Euler angle
Assuming
zByxzz
yBy
xBzyxx
MhIIII
MhI
MhIIII
zyxB IIIh ,,max
zBBz
yBy
xBBx
MhhI
MhI
MhhI
(4.2-10)
Euler’s equations of biased-
momentum system becomes
Pitch Axis Momentum Wheel with
Large Momentum
Unloading Thrusters
Yaw Axis
Earth Sensor
Pitch Axis
Roll Axis
Z
X
Y
Bh
(4.2-8)
(4.2-9)
Chapter II-4 Attitude Control with Reaction Wheels (Part 2)飛行力学
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4.2.2 Transfer Function and Time Response of Roll and Yaw Attitude
in Terms of External Torque
Pitch axis (independent motion)
Roll and yaw axes (coupled motion of roll and yaw axes)
yBy MhI
zBBz
xBBx
MhhI
MhhI
Laplace transformation results in
sM
sM
s
s
hsIsh
shhsI
z
x
BzB
BBx
2
2
zBzB
xBBx
MhIh
MhhI
(4.2-12)
(4.2-13)
(4.2-11)
Chapter II-4 Attitude Control with Reaction Wheels (Part 2)飛行力学
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Mathematical approximation 222
22
zxBB
BBBzx
IIhh
hhhII
2222
22224
2222
shsII
hshhIIsII
shhsIhsI
Bzx
BBBzxzx
BBzBx
2222
2
2222
2
shhsIhsI
hsI
sM
s
shhsIhsI
hsI
sM
s
BBzBx
Bx
z
BBzBx
Bz
x
Transfer functions of
roll and yaw attitudes
in terms of external
torque
Denominator of transfer function becomes
(4.2-14)
(4.2-15)
(4.2-16)
Chapter II-4 Attitude Control with Reaction Wheels (Part 2)飛行力学
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Mathematical approximation
Numerator of transfer function becomes
ZBB
xBB
Ihh
Ihh
B
BZxBz
B
zBBxBz
B
BBBzBz
B
BZxBx
B
xBBzBx
B
BBBxBx
h
hsIIshI
h
IhhsIhI
h
hhshIhsI
h
hsIIshI
h
IhhsIhI
h
hhshIhsI
22222
22
22222
22
(4.2-17)
(4.2-18a)
(4.2-18b)
Chapter II-4 Attitude Control with Reaction Wheels (Part 2)飛行力学
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Final approximated transfer functions of roll and yaw attitude in terms
of external torque
2222
2222
1
1
shsII
hI
hsM
s
shsII
hI
hsM
s
Bzx
Bx
Bz
Bzx
Bz
Bx (4.2-19)
Step external torque of roll and yaw axes
s
MsM
s
MsM
zz
xx
(4.2-20)
Chapter II-4 Attitude Control with Reaction Wheels (Part 2)飛行力学
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Transfer functions of roll and yaw attitude in terms of step external
torque
(4.2-21a)
22
2
22
2
2
2222
1
ssh
M
ssh
IM
shsII
hI
hs
Ms
B
x
p
p
B
zx
Bzx
Bz
B
x
<roll axis>
<yaw axis>
22
2
22
2
2
2222
1
ssh
M
ssh
IM
shsII
hI
hs
Ms
B
z
p
p
B
xz
Bzx
Bx
B
z
(4.2-21b)
where
zx
Bp
II
h22
Chapter II-4 Attitude Control with Reaction Wheels (Part 2)飛行力学
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Time response of roll and yaw attitude in terms of step external
torque (Inverse Laplace transformation)
th
Mt
h
IMt
B
xp
B
zx
cos1cos12
th
Mt
h
IMt
B
zp
B
xz
cos1cos12
(4.2-22a)
(4.2-22b)
Short Period Motion:
Nutation of Spin Axis
(Pitch Axis)
Long Period Motion:
Precession of Spin
Axis (Pitch Axis)
External torque is suppressed in inverse-proportion with angular
momentum of pitch reaction
Pitch angular momentum is called “Gyro Stiffness”
Chapter II-4 Attitude Control with Reaction Wheels (Part 2)飛行力学
Pitch Axis Momentum Wheel
with Large Momentum
Unloading Thrusters
Yaw Axis
Earth Sensor
Pitch Axis
Roll Axis
Yaw Reaction Wheel
Z
X
Y
Bh
where : external torque 11
Angular momentum of reaction
wheels:
(4.2-23)
Total angular momentum of spacecraft:
wwT HIωHHH
MHωH
T
B
T
dt
d
Euler’s equations:
MHHωHH ww
M
4.2.3 Active Control of Roll and Yaw Attitude
Controlled bias-momentum
attitude control system has
a bias-momentum attitude
control system with a
reaction wheel at roll and
yaw axis.
TBzBw hh 0H
(4.2-24)
(4.2-25)
Chapter II-4 Attitude Control with Reaction Wheels (Part 2)飛行力学
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Assuming
and the momentum of yaw axis reaction wheel is small
As the same procedure of bias momentum system, Euler’s equations
becomes
zBzyxzz
yzBxzy
xzBzyxx
MhhIIII
MhhIII
MhhIIII
2
2
2
Neglecting the second order small terms of Euler angle
zBzyxzz
yzBy
xzBzyxx
MhhIIII
MhhI
MhhIIII
zyxB IIIh ,,max
(4.2-26)
(4.2-27)
(4.2-28)
Chapter II-4 Attitude Control with Reaction Wheels (Part 2)飛行力学
Pitch Axis Momentum Wheel with
Large Momentum
Unloading Thrusters
Yaw Axis
Earth Sensor
Pitch Axis
Roll Axis
Yaw Reaction Wheel
Z
X
Y
Bh
13
zzBBz
yBy
xzBBx
MhhhI
MhI
MhhhI
The Euler’s equations of controlled bias-
momentum attitude control system are as follow:
zzBz
xBx
MhhI
MhI
Assuming the terms with orbital
angular velocity are small
(4.2-30)
Detecting roll angle error and
performing PD (Proportional and
Derivative) feedback control of yaw
reaction wheel
DPz KKh (4.2-31)
(4.2-29)
Chapter II-4 Attitude Control with Reaction Wheels (Part 2)飛行力学
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zDPBz
xBx
MKKhI
MhI
Closed Euler’s equations with PD feedback
zPBDz
xBx
MKhKI
MhI
Laplace transformation results in
sM
sM
s
s
sIKshK
shsI
z
x
zPBD
Bx
2
2
Transfer functions of roll and yaw angle in terms of yaw axis external
torque
(4.2-33)
sKhshKhsII
sI
sM
s
sKhshKhsII
sh
sM
s
PBBDBzx
x
z
PBBDBzx
B
z
24
2
24
(4.2-32)
(4.2-34)
Chapter II-4 Attitude Control with Reaction Wheels (Part 2)飛行力学
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Assuming the step external force of yaw axis
s
MsM z
z
Terminal response can be calculated by “Final Value Theorem”
P
zz
PBBDBzx
B
s
st
K
M
s
M
sKhshKhsII
shs
sst
240
0
lim
lim<roll angle>
<yaw angle>
0lim
lim
24
2
0
0
s
M
sKhshKhsII
sIs
sst
z
PBBDBzx
x
s
st
The “gyro stiffness” contribute to stabilize roll angle, whereas the yaw
angle augmentation is performed by the roll error feedback to yaw
reaction wheel.
(4.2-35)
(4.2-36)
(4.2-36)
Chapter II-4 Attitude Control with Reaction Wheels (Part 2)飛行力学
16Bias-momentum Satellite
Momentum Wheel
Chapter II-4 Attitude Control with Reaction Wheels (Part 2)飛行力学
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Controlled Bias-momentum Satellite