JFK/RB, 2005-11-30

Fundamentals of Attitude

OPS-G Forum19.05.2006

Uwe Feucht

BackgroundBackground

This Presentation is compiled from:

Lecture on Satellite Technique, TU Umea/TU Lulea

Spacecraft Operations Course, DLR

ContentContent

1. Introduction to Spacecraft Attitude2. Parameterization of Attitude3. Deterministic Attitude Determination4. Attitude Control

1. Introduction to Spacecraft Attitude

Mathematically attitude is a coordinate transformation

In space attitude is the orientation of the spacecraft main axes w.r.t. a reference system

An example for a spacecraft coordinate system:

What is Attitude ?

1. Introduction to Spacecraft Attitude

Example for a body coordinate system:

! Not valid for all Satellites !

2. Parameterization of Attitude

There are 3 common ways of describing attitude:

1) Direction Cosine Matrix

The DCM is a 3x3 rotation matrixIt describes vectors in one system w.r.t. another systemE.g. multiplication of a vector in body coordinates with the DCM can transform its coordinates into the reference system

E.g. a rotation with φ around the x-axis:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ϕϕ−ϕϕ=

cossin0sincos0

001AX

2. Parameterization of Attitude

2) Euler Angles

3 angles describe 3 successive rotations around 3 body axes.

Numbers 1,2,3 describe the type of body axes and the rotation order.

E.g. an Euler 1-2-3 rotation stands for the following rotation sequence:

with φ around the 1-axes (x-axis), thenwith θ around the new 2-axes (rotated y-axis), finallywith ψ around the new 3-axes (rotated z-axis)

2. Parameterization of Attitude

2. Parameterization of Attitude

3) Quaternions

Quaternions are hypercomplex numbers with 1 real and 3 imaginary components.

A rotation with φ around an axis [e1, e2, e3] can be expressed by thequaternion

q = [q1, q2, q3, q4] with

q1 = e1 sin φ/2q2 = e2 sin φ /2q3 = e3 sin φ /2q4 = cos φ /2

3. Deterministic Attitude Determination

Attitude is described by 3 parameters, thus in terms of vectors:

At least 2 vectors in both body- and reference system are needed,e.g. sun- and earth-vector or 2 star-vectors, or….

With these u and v in both systems an orthogonal frame is set up:

with q = u, r = u x v and s = q x r

and the body and reference matrices MB = [qB rB sB ], MR = [qR rR sR ]

yields the attitude matrix

A = MB MRT

4. Attitude Control – Why ?

Basically a satellite remains intertially fixed in space:

4. Attitude Control – Why ?

But there are disturbances, e.g. the gravity gradient:

M

CoMr F2

F1

4. Attitude Control – The Control Loop

or magnetic effects: GeographicGeomagnetic North

Northβ

S

and others like internal, aerodynamic or solar radiation disturbances

4. Attitude Control – The Control Loop

Comparator G1(s) G2(s) G3(s)

Desired attitude Actual attitude

φin + φout¯ φerrorφout H(s)

Actual attitude feedback

AttitudecontrollerAttitude

controllerSpacecraftdynamics

Spacecraftdynamics

AttitudesensorsAttitudesensors

ActuatorActuator

Thus there is the need for an automatic attitude control:

4. Attitude Control – The Control Loop

torquesT

tensorinertiaJ

JTdtdJ

KrK

rrrr

ωωω×−=

attitudeS/Cdynamics

torque

dynamic equ. of motion kinematic equ. of motion

T ω qq

dtdq

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−−

−−

=

00

00

21

321

312

213

123

ωωωωωωωωωωωω

4. Attitude Control – Sensors

Sun Sensors:

φz

φx φy

I1

I2I3

I4

4. Attitude Control – Sensors

Earth Sensors:

I1

I2 I3

I4

And: Combined earth- and sun sensor (CESS) based on thermistors

4. Attitude Control – Sensors

Star Sensors:

εψ

εφ {εθ

FOV

4. Attitude Control – Sensors

Star Sensor (Sodern):

4. Attitude Control – Sensors

Mechanical Gyros: z

H = I ωg z

ζ

Gimbalframe

Scale

ωbSpring constant

k Tx y

4. Attitude Control – Sensors

Phase meter

d1d2

QuantumR of light

ωb

Laser emitter⎟⎟⎠

⎞⎜⎜⎝

⎛=

∆=

cR

NUUU bininout λ

ωπφ 2

24cos2

cos

Laser Gyros:

4. Attitude Control – Sensors

and:

Magnetometers – measuring the direction of the earth magnetic field

GPS – using interferometry of the carrier signal

4. Attitude Control – Sensor Accuracies

€…€€€€€€€€€€€€€€€€

0.1….5 deg5 arcsec0.01 deg0.005 deg3 deg1 deg

Earth SensorsStar SensorsMech. GyrosLaser GyrosMagnetometerGPS

€…€€€0.05….5 degSun Sensors

PriceAccuracySensor Type

4. Attitude Control – Actuators

Reaction Wheels:(here: 1 spare wheel skewed)

4. Attitude Control – Actuators

Thrusters (cold or hot):(also for wheel unloading)

FL

M

L

F

)(2)( 0max MFLt

IT wheel ==∆−

=ωω

)(2 0max ωω −=∆

FLIt wheel

4. Attitude Control – Actuators

Wheel unloading (momentun dumping):

ωmax

ω = 0

ωmin

Day1 2 3 4 5 6 7 8 9 10 11

4. Attitude Control – Actuators

Magnetic Torquers (interacting with the earth magnetic field):

Euler error angles [deg]

4. Attitude Control – Results

Uncontrolled spacecraft:

Interval: 1 orbital period(i.e. 5,700 sec)

4. Attitude Control – Actuators

Attitude control by reaction wheels: