ops forum fundamentals of attitude 19.05.2006
DESCRIPTION
Uwe Feucht, a flight dynamics expert, will present fundamental concepts related to the mathematics and physics of attitude determination and discuss attitude control during flight.TRANSCRIPT
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: