Spacecraft Attitude Determination Using GPS Signals C 1

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

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

Concept Review n n 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

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

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

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

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

Receiver Arrangement 12. 50. 5λ n n 2 master antennas, 12. 50. 5λ 2 slaves, 4 intermediate Non-military frequency: 1575. 42 MHz, λ = 0. 1903 m 5λ Master antenna Slave antenna Intermediate antenna

Integer Resolution n Intermediate receivers n n 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 2λ 3λ Φ 1 Φ 2 Φ 3

Assumptions/ Coordinate Frames n n n 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 zlo xlo ylo

Assumptions/ Coordinate Frames Cont.

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

Minimizing the Loss Function Cont. n Generating Test Data n 3 orbit propagators n n n 1 for spacecraft, 2 for GPS satellites 2 -body EOM, no perturbations Ode 5/Dormand-Prince numerical integration Fixed time-step: 1 sec 1 hour simulation

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

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

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

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

Minimizing the Loss Function, Cont. n n n 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. n Defining variables:

Minimizing the Loss Function, Cont. n Jacobian matrix:

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

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

Results Initial Guess # Iterations Method % Error Identity matrix 100 GNLM 94. 34 Identity matrix 100 GN 217. 88 Actual 100 GNLM 468. 47 Actual 100 GN 26. 15 Actual 10 GN 243. 82

Conclusions n Significant errors caused by several factors n n n 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

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

Recommendations n n 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

Questions?