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Spacecraft Attitude Determination Using GPS Signals: 

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

Outline: 

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

Concept Review: 

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

Concept Review Cont.: 

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

Prior Work: 

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

Prior Work Cont.: 

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

Project Goals: 

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

Receiver Arrangement: 

Receiver Arrangement 2 master antennas, 2 slaves, 4 intermediate Non-military frequency: 1575.42 MHz, λ = 0.1903 m 12.50.5λ 5λ 12.50.5λ

Integer Resolution: 

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

Assumptions/ Coordinate Frames: 

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

Assumptions/ Coordinate Frames Cont.: 

Assumptions/ Coordinate Frames Cont.

Minimizing the Loss Function: 

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

Minimizing the Loss Function Cont.: 

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

Minimizing the Loss Function, Cont.: 

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

Minimizing the Loss Function, Cont.: 

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

Minimizing the Loss Function, Cont.: 

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

Minimizing the Loss Function, Cont.: 

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

Minimizing the Loss Function, Cont.: 

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.

Minimizing the Loss Function, Cont.: 

Minimizing the Loss Function, Cont. Defining variables:

Minimizing the Loss Function, Cont.: 

Minimizing the Loss Function, Cont. Jacobian matrix:

Minimizing the Loss Function, Cont.: 

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

Minimizing the Loss Function Cont.: 

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

Results: 

Results

Conclusions: 

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

Conclusions, Cont.: 

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

Recommendations: 

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

Questions?: 

Questions?