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

2

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

3

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

4

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

5

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

6

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

7

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

8

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

9

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

10

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

12

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

14

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

15

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

17

Controlled Bias-momentum Satellite