Fundamentals of Attitude
OPS-G Forum
19.05.2006
Uwe Feucht
JFK/RB, 2005-11-30

Background
This Presentation is compiled from:
Lecture on Satellite Technique, TU Umea/TU Lulea
Spacecraft Operations Course, DLR

Content
1. Introduction to Spacecraft Attitude
2. Parameterization of Attitude
3. Deterministic Attitude Determination
4. Attitude Control

1. Introduction to Spacecraft Attitude
What is 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:

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 matrix
It describes vectors in one system w.r.t. another system
E.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:
⎡1 0 0 ⎤
⎢0 cos ϕ sin ϕ ⎥
AX = ⎢ ⎥
⎢0 − sin ϕ cos ϕ⎥
⎣ ⎦

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), then
with θ around the new 2-axes (rotated y-axis), finally
with 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 the
quaternion
q = [q1, q2, q3, q4] with
q1 = e1 sin φ/2
q2 = e2 sin φ /2
q3 = e3 sin φ /2
q4 = 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
CoM
r F2
F1

4. Attitude Control – The Control Loop
or magnetic effects: Geographic
Geomagnetic North
North
β
S
and others like internal, aerodynamic or solar radiation disturbances

4. Attitude Control – The Control Loop
Thus there is the need for an automatic attitude control:
Comparator G1(s) G2(s) G3(s)
Desired attitude Attitude
Attitude Actuator
Actuator Spacecraft
Spacecraft Actual attitude
controller
controller dynamics
dynamics
φin + φout
¯ φerror
φout H(s)
Actual attitude feedback Attitude
Attitude
sensors
sensors

4. Attitude Control – The Control Loop
torque S/C attitude
dynamics
dynamic equ. of motion kinematic equ. of motion
r
dω r r r ⎡ 0 ω3 − ω2 ω1 ⎤
J = T − ω × Jω
T dt ω ⎢ ω2 ⎥ q
dq 1 ⎢− ω3 0 ω1 ⎥q
J K inertia tensor =
dt 2 ⎢ ω2 − ω1 0 ω3 ⎥
r ⎢ ⎥
T K torques ⎣ − ω1 − ω2 − ω3 0⎦

4. Attitude Control – Sensors
Sun Sensors:
φz
φx φy
I1
I2
I3
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
FOV
εφ {
εθ

4. Attitude Control – Sensors
Star Sensor (Sodern):

4. Attitude Control – Sensors
Mechanical Gyros: z
H = I ωg z
ζ
Gimbal
frame
Scale
ωb
Spring constant
k T
x y

4. Attitude Control – Sensors
Laser Gyros:
Phase meter
d1
d2
Quantum
R of light
ωb
∆φ ⎛ 2 ωb R2 ⎞
Laser emitter U out = U in cos = U in cos⎜ 4π N
⎜ ⎟
2 ⎝ λc ⎟
⎠

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
Sensor Type Accuracy Price
Sun Sensors 0.05….5 deg €…€€€
Earth Sensors 0.1….5 deg €…€€€
Star Sensors 5 arcsec €€€€
Mech. Gyros 0.01 deg €€€
Laser Gyros 0.005 deg €€€
Magnetometer 3 deg €
GPS 1 deg €€

4. Attitude Control – Actuators
Reaction Wheels:
(here: 1 spare wheel skewed)

4. Attitude Control – Actuators
Thrusters (cold or hot):
(also for wheel unloading)
ω max − ω 0
F (T =) I wheel = 2 FL (= M )
∆t
L
I wheel
∆t = (ωmax − ω0 )
2 FL
M
L
F

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):

4. Attitude Control – Results
Uncontrolled spacecraft:
Euler error angles [deg]
Interval: 1 orbital period
(i.e. 5,700 sec)

4. Attitude Control – Actuators
Attitude control by reaction wheels: