OPS Forum Fundamentals of Attitude 19.05.2006
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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.

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 OPS Forum Fundamentals of Attitude 19.05.2006 Presentation Transcript

  • 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: