Spacecraft Attitude Determination Using GPS

Signals

C1C Andrea JohnsonUnited States Air Force Academy

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

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

)(cos nb

Determine matrix, A, that transforms

baseline vector from body frame to LO

Issues Find n Accurate loss

function minimization

Concept Review Cont.

ns

vsAbn T ˆ

Prior Work

Minimizing the loss function Linear least squares ALLEGRO (Attitude-Lean-Loping-

Estimator using GPS Recursive Operations)

m

i

n

jj

Tiijijij sAbnAJ

1 1

22 ˆ

Linear least squares with motion-based integer resolution:

Non-linear, predictive filter assuming n has already been resolved:

Prior Work Cont.

N

l

m

i

o

jjll

Tiij

lijN sAbnnAAAJ

1 1 1

2

21 ˆ,...,

kkkkT

kk tWdtdtytyRtytyJ2

1ˆ~ˆ~

2

111

111

Project Goals

Integer resolution algorithm Non-IC dependent minimization

technique incorporating integer phase difference measurements

Design computer code to perform attitude determination

Receiver Arrangement

2 master antennas,

2 slaves, 4 intermediate Non-military

frequency: 1575.42 MHz, λ = 0.1903 m

12.50.5λ

5λ

Master antenna

Intermediate antenna

Slave antenna

12.50.5λ

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

2λ

3λ

Φ1

Φ3Φ2

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

xlo

ylo

zlo

Assumptions/ Coordinate Frames Cont.

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.

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.

1 attitude propagator Euler moment, no disturbance torques

Initialization program generates actual fractional phase differences and quaternions

Noise added with 026.0

Minimizing the Loss Function, Cont.

5.2

7678.1

7678.1

1rec

5.2

0858.1

0858.1

2rec

5.2

7678.1

7678.1

3rec

5.0

7678.1

7678.1

4rec

5.2

7678.1

7678.1

5rec

5.2

0858.1

0858.1

6rec

5.2

7678.1

7678.1

7rec

5.0

7678.1

7678.1

8rec

Gauss-Newton/ Gauss-Newton-Levenberg-Marquardt

Receiver locations written in body frame coordinates, units of wavelengths

Minimizing the Loss Function, Cont.

)3,3(

)2,3(

)1,3(

)3,2(

)2,2(

)1,2(

)3,1(

)2,1(

)1,1(

ij

ij

ij

ij

ij

ij

ij

ij

ij

ij

A

A

A

A

A

A

A

A

A

p

M

mm pfpE

1

2

obsm

cmmm HpHWpf

Unknown value is the A-matrix, must be converted to a vector for GN/GNLM

Minimizing the Loss Function, Cont.

Minimization equation requires solving for state using Gaussian elimination or decomposition

This is GN method

n

nnn

n

n

p

f

p

f

p

fp

f

p

f

p

fp

f

p

f

p

f

A

21

2

2

2

1

2

1

2

1

1

1

kTkkk

Tk fApAA

kkk ppp 1

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.

kk pEpE 1

kTkkk

Tk fApIAA

Minimizing the Loss Function, Cont.

)3(

)2(

)1(

i

i

i

i

b

b

b

b

)3(

)2(

)1(

ˆ

j

j

j

j

s

s

s

s

ij

ijij

ij

jiijiijiij

ij

jiijiijiij

ij

jiijiijiijij

nsbAbAbA

sbAbAbA

sbAbAbAf

)2(ˆ)2()2,2()3()2,3()1()2,1(

)3(ˆ)2()3,2()3()3,3()1()3,1(

)1(ˆ)2()1,2()3()1,3()1()1,1(

Defining variables:

Minimizing the Loss Function, Cont.

2,3

23

2,3

23

2,3

23

2,3

23

2,3

23

2,3

23

2,3

23

2,3

23

2,3

23

1,3

13

1,3

13

1,3

13

1,3

13

1,3

13

1,3

13

1,3

13

1,3

13

1,3

13

2,2

22

2,2

22

2,2

22

2,2

22

2,2

22

2,2

22

2,2

22

2,2

22

2,2

22

1,2

12

1,2

12

1,2

12

1,2

12

1,2

12

1,2

12

1,2

12

1,2

12

1,2

12

2,1

21

2,1

21

2,1

21

2,1

21

2,1

21

2,1

21

2,1

21

2,1

21

2,1

21

1,1

11

1,1

11

1,1

11

1,1

11

1,1

11

1,1

11

1,1

11

1,1

11

1,1

11

)3(ˆ)3()2(ˆ)3()1(ˆ)3()3(ˆ)2()2(ˆ)2()1(ˆ)2()3(ˆ)1()2(ˆ)1()1(ˆ)1(

)3(ˆ)3()2(ˆ)3()1(ˆ)3()3(ˆ)2()2(ˆ)2()1(ˆ)2()3(ˆ)1()2(ˆ)1()1(ˆ)1(

)3(ˆ)3()2(ˆ)3()1(ˆ)3()3(ˆ)2()2(ˆ)2()1(ˆ)2()3(ˆ)1()2(ˆ)1()1(ˆ)1(

)3(ˆ)3()2(ˆ)3()1(ˆ)3()3(ˆ)2()2(ˆ)2()1(ˆ)2()3(ˆ)1()2(ˆ)1()1(ˆ)1(

)3(ˆ)3()2(ˆ)3()1(ˆ)3()3(ˆ)2()2(ˆ)2()1(ˆ)2()3(ˆ)1()2(ˆ)1()1(ˆ)1(

)3(ˆ)3()2(ˆ)3()1(ˆ)3()3(ˆ)2()2(ˆ)2()1(ˆ)2()3(ˆ)1()2(ˆ)1()1(ˆ)1(

sbsbsbsbsbsbsbsbsb

sbsbsbsbsbsbsbsbsb

sbsbsbsbsbsbsbsbsb

sbsbsbsbsbsbsbsbsb

sbsbsbsbsbsbsbsbsb

sbsbsbsbsbsbsbsbsb

A

Jacobian matrix:

Determining attitude from the transformation matrix:

Minimizing the Loss Function, Cont.

5.04 )3,3()2,2()1,1(1

2

1ijijij AAAq

)2,3()3,2(4

1

41 ijij AA

qq

)3,1()1,3(4

1

42 ijij AA

qq )1,2()2,1(

4

1

43 ijij AA

qq

Minimizing the Loss Function Cont.

S/C actual quaternion

GPS 1, GPS 2, & S/C IJK vectorsOrbit

Propagators (3)

Attitude Propagator

InitializationProgram

IntegerResolutionProgram

GN/GNLM

Program

Transformationmatrix/ quaternions

3 integerphase differences

3 noisyPhase

measurements

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 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.

ALLEGRO remains most accurate GN/GNLM with modifications may

or may not perform better

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?