An Inverse Dynamics Attitude Control System with Autonomous Calibration. Sanny Omar Dr. David Beale Dr. JM Wersinger

Transcription

1 An Inverse Dynamics Attitude Control System with Autonomous Calibration Sanny Omar Dr. David Beale Dr. JM Wersinger

2 Outline Attitude Determination and Control Systems (ADACS) Overview Coordinate Frames used in ADACS Algorithms Attitude Determination Attitude Control Legacy algorithms Inverse dynamics PD controller Improved inverse dynamics PD controller Improvements since paper was written Active Momentum Dumping Autonomous Calibration

3 3 Attitude Determination and Control Systems (ADACS) Responsible for determining satellite orientation and pointing satellite correctly Necessary to properly orient science, power, propulsion, and communication devices

4 4 ADACS Sensors and Actuators Reaction Wheels (3) Magnetorquers (3) Magnetometer (3-axis) Rate Gyros (3-axis) Sun Sensors Position Sensitive Detectors

5 Coordinate Frames in ADACS Algorithms Earth Fixed Inertial Frame Origin at Earth center, x-axis toward vernal equinox, z-axis through North Pole Orbital Frame Origin at satellite cm, x-axis aligned with velocity, z-axis toward Earth Satellite Body Frame Origin at satellite cm and moves with the satellite Maneuvering Frame Aligned with target body orientation 5

6 6 Coordinate Frame Transformations Direction Cosine Matrix rx a11 a1 a13 rx1 r = a a a r y 1 3 y1 r z a31 a3 a 33 r z1 Quaternion Rotation vector (nn) plus rotation about vector (θθ) q q 1 x = q ny sin( θ ) q 3 cos( θ ) n sin( θ ) n z sin( θ )

7 Attitude Determination Vectors Sun vector in body frame from sun sensors Magnetic field vector in body frame from magnetometer Magnetic field vector in orbital frame from IGRF model Sun vector in orbital frame from heliocentric orbit propagator Angular velocity vector (in body frame) from rate gyros 7

8 8 q Attitude Determination Defined as direction cosine matrix between orbital frame and body frame Triad algorithm gives DCM based on two vectors known in both frames R1 R R1 R = [ A] r1 r r1 r DCM to quaternion conversion ( ) ( ) tr( A) + 1 a + a + a + 1 = = 4 4 a3 a3 q1 = 4q q q 3 = = a a a 4q a 4q 1 1

9 9 Error Quaternion Relates current orientation (q m ) to desired orientation (q d ) Used in control algorithm Quaternions are specified rotating the orbital frame to the body frame q = q q 1 e d *( m) q d * q 1 m q q q q q 1 3 d d d d m q1 q d q d 3 q d q d 1m = q q q q q d 3d d 1d m q q q q q 3d d 1d d 3m

10 1 De-Tumble Algorithm B-Dot de-tumble algorithm Minimize angular velocity by torqueing opposite to direction of angular velocity vector BB Satellite Angular Velocity Over Time 6 = BB ωω μμ = kkbb 4 (kk < ) TT = μμ BB Satellite angular velocity (rpm) - wx wy wz x 1 4.W Max Continuous Power (36g magnetorquers, 3U sat)

11 11 Legacy PD Control Algorithm Torques calculated directly from controller Tx kpxq e1 kdxωx T y kpyq e kdyω = + y T z k m pzq e3 kdzω z Controller does not take wheel angular momentum into account Can result in instability Controller gains dependent on satellite inertia properties Separate gain value needed for each parameter Instability of Direct PD Controller when Angular Momentum is Large

12 1 Inverse Dynamics PD Controller Takes satellite dynamics into account Ensures stability in all conditions PD law determines desired angular accelerations Equations of motion used to calculate required torques Gains independent of inertia properties Exhibits some steady state oscillations αx q e1 ωx α y kp qe k d ω = + y α z q ω e z 3

13 13 Improved Inverse Dynamics PD Controller Use the quaternion derivative for the derivative term instead of angular velocity 1 q m = * qm where q m is orbital to body ω body Mitigates steady state oscillations α x α α M q q 1 1 y p e d m z sc e m 3 3 dh = = dt Tx α1 I11 ωx ω1 ωx Ixx αx T y I w α I ω y I w ω ω y Iyy α = = + y T α I ω ω ω I α e z m 3 w 33 sc z sc 3 w z sc zz sc z sc m = k q + k q q q

14 14 Mathematical Considerations Quaternions must be defined as rotations from the orbital frame to the body frame Define maneuvering frame aligned with the target orientation All control relative to this frame Desired quaternion always [1,,, ] Prevents undesired qq sign changes Controller suboptimal for large ωω De-tumble beforehand

15 15 Analytical Calculation of Controller Gains Controller is quasi-linear, so block diagram can be drawn and used to calculate gains for linearized controller α x q e q 1 m1 α y = kp qe k + d q m α q q z e m 3 3 θdes θ kp + θ kd = θ T() s = s k p sk k d p + Critically Damped: Underdamped: p Overdamped: k k > p k k < d k k = p d 8 8 d 8

16 16 Simulation Results Controller Simulation in MATLAB System behavior independent of inertia properties.333 Isat =.4 kg * m.67 6 I = 5.3*1 kg * m wheel kd =.1, kp =.15 ω = (.1,.8,.1) rad/s Error Quaternion Error Quaternion Over Time err quat 1 err quat err quat Satellite angular velocity (rpm) Satellite Angular Velocity Over Time wx wy wz Reaction wheel angular velocity (rpm) x 14 Reaction Wheel Angular Velocity Over Time x wheel y wheel z wheel

17 Underdamped Case ωω = kk dd =.1 kk pp =.5 Error Quaternion Error Quaternion Over Time err quat 1 err quat err quat 3 Satellite angular velocity (rpm) Satellite Angular Velocity Over Time wx wy wz Critically Damped Case ωω = kk dd =.1 kk pp =.15 Error Quaternion Error Quaternion Over Time err quat 1 err quat err quat 3 Satellite angular velocity (rpm) Satellite Angular Velocity Over Time wx wy wz Overdamped Case ωω = kk dd =.1 kk pp =.1 Error Quaternion Error Quaternion Over Time err quat 1 err quat err quat 3 Satellite angular velocity (rpm) Satellite Angular Velocity Over Time wx wy wz

18 18 Active Momentum Dumping Magnetorquers constantly work to reduce wheel angular momentum (H w ) ω1 Hw = I w ω ω 3 µ ( ) id = k* H w B ( ) T = µ id B = k* H w B B Error Quaternion Reaction wheel angular velocity (rpm) Reaction Wheel Angular Velocity Over Time 8 6 x wheel y wheel 4 z wheel Error Quaternion Over Time err quat 1 err quat err quat

19 19 Autonomous In-Flight Calibration Estimate moments of inertia and angular velocity based on kinematic response M Calibrate to correct bias error in rate gyros. I = + ω sc α dh dt = I H α s s w w

20 Conclusions 3-axis attitude determination possible using inexpensive sensors B-Dot controller effectively de-tumbles satellite Inverse dynamics controller provides consistent, reliable performance in all scenarios Performance independent of satellite inertia properties and stored angular momentum Gains can be analytically calculated Possible to autonomously dump momentum On-orbit system calibration possible Calculate II and ωω

21 1 Questions? Contact: Sanny Omar AubieSat Team