Attitude Determination. - Using GPS

Transcription

1 Attitude Determination - Using GPS

2 Table of Contents Definition of Attitude Attitude and GPS Attitude Representations Least Squares Filter Kalman Filter Other Filters The AAU Testbed Results Conclusion 20/ (MJ) Danish GPS Center 2

3 What is Attitude? Z V U Orientation of a coordinate system (u,v,w) with respect to some reference system (x,y,z) W Y X 20/ (MJ) Danish GPS Center 3

4 When is Attitude information needed? Controlling an Aircraft, Boat or Automobile Onboard Satellites Pointing of Instruments Pointing of Weapons Entertainment industri (VR) Etc... 20/ (MJ) Danish GPS Center 4

5 Attitude sensors Currently used sensors include: Gyroscopes Rate gyros (+integration) Star trackers Sun sensors Magnetometers GPS 20/ (MJ) Danish GPS Center 5

6 Advantages of GPS Adding new functionality to existing equipment No cost increase No weight increase No moving parts (solid-state) Measures the absolute attitude Disadvantages Mediocre accuracy (0.1-1º RMS error) Low bandwidth (5-10 Hz maximum) Requires direct view of satellites 20/ (MJ) Danish GPS Center 6

7 Interferometric Principle Carrier Wave (from GPS satellite) Integer Component Master Antenna k Fractional Component, ϕ Baseline, b(3x1) Slave Antenna e(3x1) Unit Vector to GPS Satellite 20/ (MJ) Danish GPS Center 7

8 Interferometric Principle Measurement equation: The full phase difference is the projection of the baseline vector onto the LOS vector: 20/ (MJ) Danish GPS Center 8

9 Attitude Matrix Z V 9 parameters needed: U W Y X When (x,y,z) is a reference system: 20/ (MJ) Danish GPS Center 9

10 Properties of A A rotates a vector from the reference system to the body system The transpose of A rotates in the opposite direction (back again) 20/ (MJ) Danish GPS Center 10

11 Properties of A Rotation does not change the size of the vectors: Every rotation has a rotation-axis (and a rotationangle) The rotation-angle is the eigenvalue of A 20/ (MJ) Danish GPS Center 11

12 Euler sequences A sequence of rotations by the angles (φ,θ,ψ) about the coordinate axes of the reference system Single axis: Multiple axes: 20/ (MJ) Danish GPS Center 12

13 Quaternions A quaternion consists of four composants Where i,j and k are hyperimaginary numbers 20/ (MJ) Danish GPS Center 13

14 Quaternions A quaternion can be thought of as a 4 dimensional vector with unit length: 20/ (MJ) Danish GPS Center 14

15 Quaternions Quaternions represent attitude as a rotation-axis and a rotation-angle 20/ (MJ) Danish GPS Center 15

16 Quaternions Quaternions can be multiplied using the special operator defined as: 20/ (MJ) Danish GPS Center 16

17 Quaternions The attitude matrix can be formed from the quaternion as: Where 20/ (MJ) Danish GPS Center 17

18 Least Squares Solution Including attitude information into the measurement equation Linearization of the attitude matrix 20/ (MJ) Danish GPS Center 18

19 Least Squares Solution Forming the phase residual 20/ (MJ) Danish GPS Center 19

20 Least Squares Solution 20/ (MJ) Danish GPS Center 20

21 Least Squares Solution Estimate update 20/ (MJ) Danish GPS Center 21

22 Extended Kalman Filter A Kalman filter consists of a model equation and a measurent equation 20/ (MJ) Danish GPS Center 22

23 Extended Kalman Filter And their linearized counterparts. And 20/ (MJ) Danish GPS Center 23

24 Extended Kalman Filter Algorithm z(t k ) x(t k- ), P(t k- ) Estimate of x(t k ), P(t k ) Estimate Update x(t k+ ), P(t k+ ) k = k + 1 Time Propagation x(t k+1- ), P(t k+1- ) 20/ (MJ) Danish GPS Center 24

25 Extended Kalman Filter Tuning of the filter Noise variance determined experimentally 20/ (MJ) Danish GPS Center 25

26 Extended Kalman Filter Tuning of the filter Noise variance determined by trial-and-error 20/ (MJ) Danish GPS Center 26

27 Extended Kalman Filter Determining the system model N Dynamic ω Kinematic q model model 20/ (MJ) Danish GPS Center 27

28 Other Filters Raw phase measurements Non-iterative Iterative Integer resolution Extended phase measurements Line bias calibration Corrected phase measurements Vector Determination Vector estimate Attitude Determination Kalman filter ALLEGRO Cohen Least Squares Bar-Itzhack New Algorithm Euler-q Geometric Descent Bar-Itzhack 2 Attitude estimate 20/ (MJ) Danish GPS Center 28

29 Testbed LON network Controller box Labtop PC RS232 Motor commands Encoder readings GPS receiver Antenna array Surveyor platform 20/ (MJ) Danish GPS Center 29

30 Software Start Load simulated attitude angles.txt Start User Interface + Initialize Testbed Proces 1 Store phases and LOS vectors from receiver in file Proces 2 Store angle-values from encoders in file Proces 3 Send angle commands to LON nodes GPS receiver phase.out los.out angles.out LON nodes Stop 20/ (MJ) Danish GPS Center 30

31 Motor Control 20/ (MJ) Danish GPS Center 31

32 Motor Angles 200 Angle of Antenna Array Angle [degrees] Time [seconds] 20/ (MJ) Danish GPS Center 32

33 Local Horizontal System 20/ (MJ) Danish GPS Center 33

34 Results Based on actual and simulated data, the following performance parameters were evaluated Accuracy Computational efficiency Ability to converge 20/ (MJ) Danish GPS Center 34

35 Accuracy 20/ (MJ) Danish GPS Center 35

36 Speed Number of Floating Point Operations 6 x 104 Speed Analysis with three baselines Kalman filter w. bias est. 5 Kalman filter Geometric Descent 4 Bar Itzhack x ALLEGRO New Algorithm + Euler-q New Algorithm + SVD Single-point Cohen Bar Itzhack + SVD Number of satellites 20/ (MJ) Danish GPS Center 36

37 Convergence Number of Samples Kalman Filter Number of baselines = 3 Convergence = 100% Average = 5.75 Number of Samples Single-point Algorithm Number of baselines = 3 Convergence = 100% Average = Convergence Time (Samples) Convergence Time (Samples) Number of Samples ALLEGRO Algorithm Number of baselines = 3 Convergence = 100% Average = 4.56 Number of Samples Geometric Descent Algorithm Number of baselines = 3 Convergence = 100% Average = Convergence Time (Samples) Convergence Time (Samples) 20/ (MJ) Danish GPS Center 37

38 Convergence Number of Samples Kalman Filter Number of baselines = 2 Convergence = 96.61% Average = 7.87 Number of Samples Single-point Algorithm Number of baselines = 2 Convergence = 69.34% Average = Convergence Time (Samples) Convergence Time (Samples) Number of Samples ALLEGRO Algorithm Number of baselines = 2 Convergence = 69.75% Average = Convergence Time (Samples) 20/ (MJ) Danish GPS Center 38

39 Conclusion Kalman filter is by far most accurate, but also computationally very heavy Single-point (LSQ) offers good accuracy + high speed Vector matching algorithms has the lowest accuracy but does not suffer from convergence problems Performance depend on satellite constellation Results were affected by mechanical problems with levelling of the testbed 20/ (MJ) Danish GPS Center 39