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# OPS Forum Fundamentals of Attitude 19.05.2006

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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.

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### OPS Forum Fundamentals of Attitude 19.05.2006

1. 1. Fundamentals of Attitude OPS-G Forum 19.05.2006 Uwe Feucht JFK/RB, 2005-11-30
2. 2. Background This Presentation is compiled from: Lecture on Satellite Technique, TU Umea/TU Lulea Spacecraft Operations Course, DLR
3. 3. Content 1. Introduction to Spacecraft Attitude 2. Parameterization of Attitude 3. Deterministic Attitude Determination 4. Attitude Control
4. 4. 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:
5. 5. 1. Introduction to Spacecraft Attitude Example for a body coordinate system: ! Not valid for all Satellites !
6. 6. 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⎥ ⎣ ⎦
7. 7. 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)
8. 8. 2. Parameterization of Attitude
9. 9. 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
10. 10. 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
11. 11. 4. Attitude Control – Why ? Basically a satellite remains intertially fixed in space:
12. 12. 4. Attitude Control – Why ? But there are disturbances, e.g. the gravity gradient: M CoM r F2 F1
13. 13. 4. Attitude Control – The Control Loop or magnetic effects: Geographic Geomagnetic North North β S and others like internal, aerodynamic or solar radiation disturbances
14. 14. 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
15. 15. 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⎦
16. 16. 4. Attitude Control – Sensors Sun Sensors: φz φx φy I1 I2 I3 I4
17. 17. 4. Attitude Control – Sensors Earth Sensors: I1 I2 I3 I4 And: Combined earth- and sun sensor (CESS) based on thermistors
18. 18. 4. Attitude Control – Sensors Star Sensors: εψ FOV εφ { εθ
19. 19. 4. Attitude Control – Sensors Star Sensor (Sodern):
20. 20. 4. Attitude Control – Sensors Mechanical Gyros: z H = I ωg z ζ Gimbal frame Scale ωb Spring constant k T x y
21. 21. 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 ⎟ ⎠
22. 22. 4. Attitude Control – Sensors and: Magnetometers – measuring the direction of the earth magnetic field GPS – using interferometry of the carrier signal
23. 23. 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 €€
24. 24. 4. Attitude Control – Actuators Reaction Wheels: (here: 1 spare wheel skewed)
25. 25. 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
26. 26. 4. Attitude Control – Actuators Wheel unloading (momentun dumping): ωmax ω=0 ωmin Day1 2 3 4 5 6 7 8 9 10 11
27. 27. 4. Attitude Control – Actuators Magnetic Torquers (interacting with the earth magnetic field):
28. 28. 4. Attitude Control – Results Uncontrolled spacecraft: Euler error angles [deg] Interval: 1 orbital period (i.e. 5,700 sec)
29. 29. 4. Attitude Control – Actuators Attitude control by reaction wheels: