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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


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


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)


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)


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)


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)


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”


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


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


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


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


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


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)


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)


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


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

～


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


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


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)


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)


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


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)


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


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


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


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


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


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


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)


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)


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)


30

5.3 Control of Spin Spacecraft

Spin Spacecraft motion in inertial coordinate system

X

ZY

Spin Axis

Ascending

Node


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 ,,


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)


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)


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


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)