1. Spacecraft Attitude Determination Using GPS Signals C1C Andrea Johnson United States Air Force Academy

2. Outline Concept review/ Prior work Goals Receiver arrangement Integer resolution Assumptions/ Coordinate Frames Minimizing the loss function Results Conclusions Recommendations

3. Concept Review Two receivers detect the same GPS satellite signal Phase differences can be used to determine the angle of the line defined by the 2 receivers

4. Determine matrix, A, that transforms baseline vector from body frame to LO Issues Find n Accurate loss function minimization Concept Review Cont.

5. Prior Work Minimizing the loss function Linear least squares ALLEGRO (Attitude-Lean-Loping-Estimator using GPS Recursive Operations)

6. Linear least squares with motion-based integer resolution: Non-linear, predictive filter assuming n has already been resolved: Prior Work Cont.

7. Project Goals Integer resolution algorithm Non-IC dependent minimization technique incorporating integer phase difference measurements Design computer code to perform attitude determination

8. Receiver Arrangement 2 master antennas, 2 slaves, 4 intermediate Non-military frequency: 1575.42 MHz, λ = 0.1903 m 12.5 0.5 λ 5λ5λ Master antenna Intermediate antenna Slave antenna 12.5 0.5 λ

9. Integer Resolution Intermediate receivers Variation of integer search Unique solution to 2 phase difference measurements if baselines not multiples of each other Third provides check Accurate even for large baselines 2λ2λ 3λ3λ Φ1Φ1 Φ3Φ3 Φ2Φ2

10. Assumptions/ Coordinate Frames Algorithm uses single set of 3 receivers Same 2 GPS satellites always in view No masking or multipathing “Inertial” reference frame: local orbital Body frame = LO when roll, pitch, and yaw = 0 x lo y lo zl o

11. Assumptions/ Coordinate Frames Cont.

12. Minimizing the Loss Function Linear Diverges for poor initial guesses Motion-based integer resolution ALLEGRO Does not account for n in algorithm Separate motion-based integer resolution Gauss-Newton Not sensitive to initial conditions Always converges Designed for minimization of squared functions

13. Minimizing the Loss Function Cont. Generating Test Data 3 orbit propagators 1 for spacecraft, 2 for GPS satellites 2-body EOM, no perturbations Ode5/Dormand-Prince numerical integration Fixed time-step: 1 sec 1 hour simulation

14. Minimizing the Loss Function, Cont. 1 attitude propagator Euler moment, no disturbance torques Initialization program generates actual fractional phase differences and quaternions Noise added with

15. Minimizing the Loss Function, Cont. Gauss-Newton/ Gauss-Newton-Levenberg-Marquardt Receiver locations written in body frame coordinates, units of wavelengths

16. Minimizing the Loss Function, Cont. Unknown value is the A-matrix, must be converted to a vector for GN/GNLM

17. Minimizing the Loss Function, Cont. Minimization equation requires solving for state using Gaussian elimination or decomposition This is GN method

18. Sometimes a singularity occurs: To counter this, an additional term is needed: If the singularity still occurs, multiply λ by 10 and recalculate Minimizing the Loss Function, Cont.

19. Defining variables:

20. Minimizing the Loss Function, Cont. Jacobian matrix:

21. Determining attitude from the transformation matrix: Minimizing the Loss Function, Cont.

22. Minimizing the Loss Function Cont. S/C actual quaternion GPS 1, GPS 2, & S/C IJK vectors Orbit Propagators (3) Attitude Propagator Initialization Program Integer Resolution Program GN/ GNLM Program Transformation matrix/ quaternions 3 integer phase differences 3 noisy Phase measurements

23. Results Initial Guess# IterationsMethod% Error Identity matrix100GNLM94.34 Identity matrix100GN217.88 Actual100GNLM468.47 Actual100GN26.15 Actual10GN243.82

24. Conclusions Significant errors caused by several factors GN/GNLM intended for vectors of parameters, not vectorized matrix Use of constant to prevent singularities Linear receiver arrangement Only 2 sightlines used (minimum of 4 available) GN/GNLM sensitive to measurement errors

25. Conclusions, Cont. ALLEGRO remains most accurate GN/GNLM with modifications may or may not perform better

26. Recommendations Use matrix for singularity avoidance Determine better method for comparing results of matrix calculations (compare entire matrix, elements thereof, or a combination of both) Integrate integer resolution algorithm into GN/GNLM algorithm If cannot use GN/GNLM, incorporate integer resolution algorithm into ALLEGRO algorithm

27. Questions?