SPACECRAFT DYNAMIC AND CONTROLSByZuliana Ismail

INTRODUCTIONModes of OperationReference FrameSatellite Attitude RepresentationOrbital ElementsExternal DisturbancesDynamicsKinematicsSatellite Attitude Control

INTRODUCTIONPurpose of Attitude Control Systems:• To stabilize the spacecraft and orients it in desired directions

during the mission despite the external disturbance torques acting on it.

TYPICAL MODES

Attitude Acquisition:Acquire definite attitude (e.g. sun pointing, earth pointing) from arbitrary

initial dynamic condition (Attitude, angular rate)Safe attitude:Sun pointing with S/C z-axis, slow rotation around z-axis to assure safe power

and thermal conditionsNominal Attitude:Steady state earth pointing (roll/pitch bias capability) supplying the mission objectives (e.g. telecommunication)

REFERENCE FRAME Require to describe the motion of a satellite• Inertial Earth (IE) coordinate system, • Satellite’s Body (B) coordinate system • Local-Vertical-Local-Horizontal (LVLH) coordinate system (assigned for nadir

pointing)XLVLH

YLVLH

ZB

YB

ZLVLH

ZIE

XIE

YIE

earth

XB

ATTITUDE REPRESENTATION• Attitude• Orientation with respect to (w.r.t.) a given reference frame• Satellite’s attitudes (roll, pitch, yaw) are defined with respect to

the LVLH coordinate system.

Attitude representation• Orientation of a body fixed axis w.r.t. reference frame• Three techniques to represent the satellite’s attitude: Euler

Angles, Direct Cosine Matrix and Quaternion.

QUATERNIONS

EULER ANGLE

EULER ANGLE

Example: 3-2-1 (z-y-x) rotation.

xx1

z

y

z1

y1

z2 z1

y2

y1

x1

x2

x2

x3

y2

y3

z2

z3

zyx

AAAzyx

ZYX

3

3

3

QUATERNION

Quaternion: More computational efficient.Euler’s Theorem : any finite rotation of a rigid body can be represented

by the rotation through a definite angle (Euler-angle, ) around a definite axis (Euler-axis, e)

the simplest way to describe the quaternion is using the Euler axis e and Euler angle Φ

YB

ZB

XB

e

XLVLH

YLVLH

ZLVLH

QUATERNIONS (SYMMETRIC EULER PARAMETERS)• Representation of Euler-Axis and Euler-Angle by a 4-dimensional vector• Interpretation of this vector as ‘Quaternion’ (=hypercomplex number)• Quaternion algebra is applicable for attitude kinematics computations

2cos

2sin

2sin

2sin

4

33

22

11

q

eq

eq

eq

Tqqqq 4321q==>

QUATERNION ALGEBRA

• If Frame AB = 0

• Determination of Euler axis/angle from a quaternion qq4 0;

0<<180 deg (direction of rotation included in e)

q4 = 0; = 180

4

23

22

21arctan2

qqqq

1000

q

32

22

21

43

42

41 1

qqqqsignqqsignqqsignq

eee

e

z

y

x

Tqqqe 321

QUATERNION ALGEBRA

• For transformation of vectors between two coordinate systems such as the LVLH and satellite’s body coordinate systems, the equation can be related as

LVLH BqLVLH/B

LVLHB

B LVLH / B LVLH

B LVLH

XXY A q YZ Z

2 2 2 21 2 3 4 1 2 3 4 1 3 2 4

2 2 2 2LVLH / B 1 2 3 4 1 2 3 4 2 3 1 4

2 2 2 21 3 2 4 2 3 1 4 1 2 3 4

q q q q 2(q q q q ) 2(q q q q )

A(q ) 2(q q q q ) q q q q 2(q q q q )

2(q q q q ) 2(q q q q ) q q q q

Direction Cosine Matrix computed from a quaternion

QUATERNION SUCCESSIVE ROTATIONSBy multiplying two known values of attitude quaternions, the desired unknown attitude quaternion can be found.

• Quaternion multiplication

1

LVLH / B IE / LVLH IE / B

4x14x1

q q q

1LVLH / B IE / B IE / LVLHq S(q ) q

4 3 2 1

3 4 1 2IE / B

2 1 4 3

1 2 3 4

q q q qq q q q

S qq q q qq q q q

QUATERNION SUCCESSIVE ROTATIONS

Inertial Earth LVLH

Satellite’s Body

XLVLH

YLVLH

ZLVLH

ZIE

XIEYIE

XB

ZB

YB

IE / B LVLH / B IE / LVLHq S(q ) q

1IE / LVLH LVLH / B IE / Bq S(q ) q

1LVLH / B IE / B IE / LVLHq S(q ) q

1IE / LVLHq

1IE / Bq 1

LVLH / Bq

QUATERNION TO EULER ANGLESTransformation from quaternions to Euler angles

1 4 2 32 2

1 2

4 2 3 1

4 3 1 22 2

2 3

2(q q q q )arctan

1 2(q q )

arcsin 2(q q q q )

2(q q q q )arctan

1 2(q q )

ORBITAL ELEMENTS

Ω

ωi

XIE

ZIE

YIE

Ascending Node

Descending Node

Vernal Equinox Direction γ

θ

Equatorial plane

Perigee

Orbital plane

Earth Pointing Satellite

XLVLH

YLVLHZLVLH

ORBITAL PARAMETERSFor circular orbit• orbital period

• Using the define value of RAAN and inclination, the initial reference quaternion can be known

3eo

h RT 2

oo

2T

IE / LVLH

i isin cos sin cos cos sin2 2 2 2 2 2i isin cos sin cos cos sin2 2 2 2 2 2i isin cos sin cos cos sin2 2 2 2 2 2isin cos sin cos2 2 2

q

i cos sin2 2 2

o otinstantaneous angle of satellite position

Earth’s orbital frequency

EXTERNAL DISTURBANCES

• The major source of external disturbance torques:• Gravity Gradient Torque, T gg

Exist form the variation of the Earth’s gravitational force over the asymmetric body that orbiting the Earth

• Aerodynamic Torque, T Aero

Caused by the interaction between the upper atmosphere with the satellite surface

• Magnetic Torque, T Mag

Caused by the interaction between the satellite’s residual magnetic field and the geomagnetic field

• Solar Radiation Torque, T Solar

Exist from the solar radiation particle that hit the satellite’s surface

EXTERNAL DISTURBANCES

Axis Disturbance Torques

Roll (solar)

Pitch(aero+solar)

Yaw(aero)

5dx oT 8 10 sin t Nm

6 5 5dy o oT 8 10 5 10 cos t 8 10 sin t Nm

6 5dz oT 8 10 5 10 cos t Nm

0 1 2 3 4 5-1.5

-1

-0.5

0

0.5

1

1.5x 10

-4

Time[Orbits]

Td[N

m]

Tdx

Tdy

Tdz

Worst Case Torque Condition:

Solar radiation torques act along the roll and pitch axis.(Solar torque parallel to yaw axis)

Aeodynamic torques act along the pitch and yaw axis.

DYNAMICS EQUATIONS OF MOTION

• Angular momentum at Body Coordinate System

Th I

………(20)

………(17)

zxyyxz

yxzzxy

xyzzyx

Thhh

Thhh

Thhh

hhh BI ………(18)

ThhB ………(19)Euler’s Moment Equation

zxyyxzz

yzxzxyy

xyzyzxx

TIII

TIII

TIII

………(21)

DYNAMICS EQUATIONS OF MOTION

IE / B IE / B IE / B Iω T ω Iω

B w h h h

w w h T h ω Iω h

s b s w w d= - ×( + ) - h ω h h h T

hs : Satellite’s angular momentumωb: Satellite’s body Angular velocity

w.r.t Inertial Earthhw : Wheel’s angular momentum.Td : External disturbances torques

x

y

z

I 0 00 I 00 0 I

I

With reaction wheels:

x x x y z z y wz y wy z

y y y z x x z wx z wz x

z z z x y y x wy x wx y

I T I I h h

I T I I h h

I T I I h h

LINEARIZED EQUATIONS OF MOTION

Angular velocity vector of a rotating vector

LVLH/B Angular velocity vector of the body frame w.r.t LVLH frameI/LVLH angular velocity vector of the LVLH frame w.r.t Inertial frameI/LVLH/B I/LVLH w.r.t body frameI/B angular velocity vector of the body frame w.r.t Inertial frame

BLVLHIBLVLHBI //// ………(2)

zxyyxzz

yzxzxyy

xyzyzxx

TIII

TIII

TIII

Euler’s Moment Equation

………(1)

LINEARIZED EQUATIONS OF MOTION

• For small Euler Angle

• The angular velocity of LVLH coordinate system w.r.t Inertial coordinate system

11

1

/

BLVLHA

0

0

0/ LVLHI

0

0

0

0

////

0

0

11

1

LVLHIBLVLHBLVLHI A

BLVLH /

Because of small Euler Angles

………(3)

…(5)

…(4)

LINEARIZED EQUATIONS OF MOTION

• Insert equation (4) and (5) into equation (2)

0

0

0

/

BI

00

0

0

/

BI

………(6)

………(7)

LINEARIZED EQUATIONS OF MOTION

• Insert equation (6) and (7) into equation (1)

zzyxxyz

yy

xzyxzyx

TIIIIII

TI

TIIIIII

020

020

SATELLITE ATTITUDE KINEMATICS

LVLH / B1 ( )2

q ω q

z y x

z x yLVLH / B

y x z

x y z

00

( )0

0

ω

Since quaternion is used for attitude representation, the derivatives of the Euler parametes can be updated using the kinematics equation as follows:

differential equation, 1st order, dimension 4ADVANTAGE: no trigonometric functions

SATELLITE ATTITUDE CONTROL

ATTITUDE PERFORMANCES

simple, cheap

cheap, slow, lightweightLEO only

inertially oriented

RWs: Expensive, precise, faster slew, Momentum Unloading

CMG: Expensive, heavy, quick, for fast slew, 3-axesThrusters: Expensive, quick response, consumables

GG: Long booms-Restricted maneuverability

GRAVITY GRADIENT

• An elongated object in a gravity field tends to align its longitudinal axis to the Eart’s center.

• Earth oriented• Requires stable inertia – limited accuracy• No Yaw stability (can add momentum wheel)• Only effective in LEO – because gravity varies with the square of the distance.

Gravity-Gradient

X

Y

Z

Gravity-gradient satellite with momentum wheel

-Momentum wheel for yaw stability -Satellite body rotates along Y-axis -at one revolution per orbit

GRAVITY GRADIENTExample - UoSAT

Satellite mass : 70 kgSatellite moment of inertia : (120, 120, 1) kgm2

Satellite body : 40 x 40 x 60 cmBoom : 8 m

SINGLE SPIN STABILIZED SATELLITE

• Make use of physical principles/elements for s/c attitude control. • Entire s/c rotates so that its angular momentum vector remains fixed in

inertial space.• An advantage of this technique is the capacity achieve a relatively long

operational life. The typical disadvantages are the poor attitude accuracy and the dependence of the environmental elements

• Because single spin stabilized satellites have a fixed pointing w.r.t inertial space, they are not a good choice for Earth-pointing missions.

H

H

H

DUAL SPIN STABILIZED SATELLITE

Stowed (during launch)

In orbit

• One way to avoid Earth-pointing limitations of spin stabilization is to use a

dual-spin system. These systems consists of an inner cylinder called the ‘de-spun’ section, surrounded by an outer cylinder that is spinning at a high rate.

de-spun section : stays pointed at the Earth

spun section : provides stiffness

DUAL SPIN STABILIZEDExampleTACSAT 1• Launched in 1969 and was the dual spin stabilized

satellite.• The antenna is the platform, and is intended to

point continuously at the Earth, spinning at one revolution per orbit.

• The cylindrical body is the rotor, providing gyroscopic stability through its 60 RPM spin.

H

THREE AXIS CONTROL TECHNIQUE

Actuators – require continuous feedback and adjustment:• Thrusters, • Magnetic Torquers• Momentum-control devices

• Biased momentum systems• Zero-bias systems• Control-moment gyroscopes

• Fast; continuous feedback control• Relatively high power, weight and cost

Active Control Systems directly sense spacecraft attitudeand supply a torque command to alter it as required. This technique require energy consuming attitude actuators.

Good attitude accuracies can be achieved

ADCS BLOCK DIAGRAM

SpacecraftActuatorsController

Sensors

Physical output: the current attitude, real

commands

Difference: error signal, error

System Input: desired attitude, desired

+

Measured Output: the measured attitude, measured

-

++

Disturbance Torques

-Gyros & Accelerometer -Sun Sensors -Star Sensors -Horizon Sensors -Magnetometer

-Thrusters -Reaction Wheels -Momentum Wheels -Control Moment Gyros -Magnetic Torquers

ACTUATORS : MOMENTUM-CONTROL DEVICES

Biased momentum system“momentum wheel” with a large fixed momentum to provide gyroscopic stiffness. The wheel’s speed gradually increases to absorb disturbance torques

Zero-bias system“reaction wheel” with little or no initial momentum. Each wheel spins independently to rotate the spacecraft and absorb disturbance torques

Control-moment gyroscope“wheel” with a large fixed momentum. The wheel is mounted on gimbals, rotating the wheels about their gimbals changes the satellite orientation

MOMENTUM BIASED PRINCIPLE

• The same concept used by spin-stabilized spacecraft. Only in this case, instead of spinning the whole spacecraft, only a small wheel (momentum wheel) inside the spacecraft is spinning providing a gyroscopic stiffness.

• Momentum vector (momentum wheel) perpendicular to orbit plane (parallel to satellite pitch axis)

• Pitch Axis : continuous control through change of wheel speed• Roll/Yaw Axis : improved passive

stabilization due to increased momentum stiffness through pitch bias momentum

X

Y

Z

h

ACTUATORS : MAGNETIC TORQUERS

The interaction between the Earth’s geomagnetic field and magnetic dipole moment within the satellite that normally comes from electrical equipments

onboard will generate a magnetic disturbance torque. Fortunately, this torque can be used for controlling purposes when it is generated in desirable amount and

direction. This is done by generating a controllable value of magnetic dipole moment within the satellite using an electromagnetic based device called

magnetic torquer.

MBT

-Often used for LEO satellites-Useful for initial acquisition maneuvers- Also commonly use for momentum desaturation - (“dumping”) in wheel-based system

3-AXES CONTROL VIA REACTION WHEELS• The reaction wheel

concept relies on the principle angular momentum conservation.

• When a satellite rotates one way due to the disturbance torque, the reaction wheel will be counter rotated to produce a same magnitude reaction torque in order to correct the attitude

3-AXES CONTROL VIA REACTION WHEELS

BASIC CONTROL LAWS

dzEpzcz

dyEpycy

dxEpxcx

KKT

KKT

KKT

Control command for Euler Angle Errors

E

E

E

E

E

qqqq

q

4

3

2

1

rKqqKT

qKqqKT

pKqqKT

dzEEpycz

dyEEpycy

dxEEpxcx

43

42

41

2

2

2

Control command for Quaternion Error Vector

MOMENTUM DUMPING• By controlling the satellite’s attitude using the reaction wheels,

the change in the angular momentum of the satellite will be transferred to the wheels and vice versa in order to compensate for the external disturbance torques.

• The constant disturbance torques can cause the reaction wheel angular momentum to constantly increase or decrease, hence induces a build-up of the angular momentums.

• Since the reaction wheels lack of the ability to remove the excess angular momentums and that the wheels have a limited capacity to store angular momentum.

• The angular momentum of the wheels will be accumulated and saturated over time thus preventing the application of any further wheel control torques.

MOMENTUM DUMPING

MOMENTUM DUMPING

2

kB m = B h

m BT = m×B

sin coscos

2 sin sinLVLH

x o

y o

oz

B B iB B i

B iB

B

Δh : excess momentum to be removed

k : unloading control gain. (PI Controller)

Magnetic Control Equation

Wheel Unloading law Simple Dipole Model

c k T h

k h m×B

MOMENTUM DUMPING

m

dT

wh

w2

kB m = B h

m bT m×B

Magnetic Dipole Moment

B

Magnetic Control Torquers

Dipole Saturation Limit

Disturbance Torques

Simplified Magnetic Model

Reaction Wheels

SatelliteDynamics