Feasibility of Demonstrating PPT’s on FalconSAT-3

C1C Andrea JohnsonUnited States

Air Force Academy

Outline

Problems encountered with PPT’s Methods of demonstrating use

Spiral Transfer Attitude

Model Experimental Results Recommendations

Problems Encountered Low Thrust

160e-6 N maximum thrust 15e-6 second pulse, 2 Hz => 4.8e-9 N

average thrust Updated data indicates possibly higher

average thrust (50 μN-s) Power requirements Inaccuracy of original model

Uncoupled equations of motion Inaccurate disturbance torque models

Methods of Demonstrating

Spiral Transfer One PPT yields 1.6 cm change in

semimajor axis with no disturbance torques

No GPS receiver

Methods of Demonstrating Cont.

Attitude Control Z-axis only possibility for control

because of small moment of inertia (1.31 versus 67.4 kg-m2)

Model

Assumptions Equations of motion PPT modeling Disturbance torques Validation

Assumptions

Simplified satellite model Small center of pressure - center of

mass offset No products of inertia Constant, known PPT decay rate Negligible orbital perturbations

Assumptions Cont.

31.100

069.30

0064.3

31.100

045.670

0040.67

Body Mass: 35.5 kg

Boom Mass (without tip mass): 3.15 kg

Tip Mass: 7.45 kg

Total Mass: 46.1 kg

Inertia Tensor (Stowed Boom):

kg-m2

Inertia Tensor (Deployed Boom):

kg-m2

Coefficient of Drag (Cd): 2.6

Spacecraft Dipole: 0.05 A-m2

Orbit: Altitude = 560 kmSemimajor axis = 6938.137 kmInclination = 35.4o

Eccentricity = 0Right Ascension = 0o

Equations of Motion

1322311 )( TIII

2313122 )( TIII

3211233 )( TIII

PPT Modeling

t

15 usec 15 usec4.8 nNs

160 μN

t4.8 nN

Actual

Simulation1 sec

Disturbance Torques

Gravity Gradient Magnetic Drag Solar Pressure

Gravity Gradient

3323

2 )(3 bylobyloyzoGGx TTIIN

1333

2 )(3 bylobylozxoGGy TTIIN

2313

2 )(3 bylobyloxyoGGz TTIIN

Magnetic

13th degree, 13th order IGRF 10th generation model with

secular terms up to 8th degree and 8th order

Magnetic Cont.

k

n

n

m

mnmnmnn

r Pmhmgnr

a

r

VB

1 0

,,,2

sincos1

k

n

n

m

mnmnmn

nP

mhmgr

aV

rB

1 0

,,,

2

sincos1

k

n

n

m

mnmnmnn

Pmhmgmr

aV

rB

1 0

,,,2

cossinsin

1

sin

1

Magnetic cont.

sincoscossin BBBB rx

cossincossin BBBB ry sincos BBB rz

x

y

z

θ

r

φ

Magnetic Cont.

ECF to ECI coordinate frame conversion

Precession Nutation Sidereal time Polar motion

Drag

VVNAVCF dplanedragˆˆˆ

2

1 2,

VVZDLVCF dcylinderdragˆˆˆ1

2

1 22,

n

iiidrag FRN

1

Solar Pressure

NCCSCAPF dsunsssunsunplanesolar

ˆ3

1cos2ˆ1cos,

NCCECAP dearthssearthearth

ˆ3

1cos2ˆ1cos

ZACCSACCPF sundsunsdssunsuncylindersolar

ˆcos6

sin3

4ˆ63

11sin 1,

ZACCEACCP earthdearthsdsearthearth

ˆcos6

sin3

4ˆ63

11sin 1

Validation Integrator: Attitude and orbital

energy and momentum should be constant

Gravity gradient: Should match C program data

Magnetic field: Should match C program data

Drag and solar pressure validated using hand calculations

Integrator Energy and momentum constant if no

external torques Attitude

Orbit

Normalized error

I Ih T

2

1

R

V 2

2

VRh

o

o t

o

o

h

thh

Integrator: Attitude

Energy Momentum

Maximum error: 3e-14 Maximum error: 1.5e-14

Integrator: Orbit

Energy Momentum

Maximum error: 2.5e-14 Maximum error: 7.5e-15

Gravity Gradient Validation

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

1 JanJan 70

2 Fri

UoSat: Attitude Log File

Time

Roll Angle Pitch Angle Yaw Angle

Gravity Gradient Validation Cont.

Magnetic Field Validation

Magnetic field in ECF matched C program numerical output

8th degree, 8th order With secular terms

ECF to ECI conversion output matched C program

Estimation Theory

Kalman filter Truncate results Statistical mean smoother

Batch estimator Data used by filters comes from

attitude determination Kalman filter

Estimation Theory Cont.

xyyxbylobyloxyozz BMBMTTIII 2313

23

gkNNNII PPTo

solarz

dragzyxxy z

yxbylobylooxyzz TTIIIz 2313

23ˆ

gkNNNBMBMz PPTo

solarz

dragzxyyx z

gkNN PPTo

PPTz z

Estimation Theory Cont.

y

x

PPTo

M

M

N

Xz

yx M

z

M

z

N

zH

xy BBH 1

Batch Filter Algorithm

k

kTkT

R

HHWHH

k

kkTkT

R

zzHCOWH

ˆ

xy BBH 1

Batch Filter Algorithm Cont.

COWHWHHX TT 1

XXX

If X (user defined), then exit the loop. If not,

Experimental Results

No Noise Actual

PPT torque 0.0000

Dipole (x) 0.0000

Dipole (y) 0.0500

Percent Error Kalman w/o Smoothing

Percent Error Kalman w/ Smoothing

Percent Error Batch

N/A N/A N/A

N/A N/A N/A

3.4001E-10 8.6001E-10 0.0000E+00

Experimental Results Cont.

0.3E-6 on B field Actual

PPT torque 0.0000

Dipole (x) 0.0000

Dipole (y) 0.0500

Percent Error Kalman w/o Smoothing

Percent Error Kalman w/ Smoothing Percent Error Batch

N/A N/A N/A

N/A N/A N/A

2.0709 2.2331 0.0318

Experimental Results Cont.

No PPT's   With PPT's  

Noise Percent error Noise Percent error

0.3E-3 on w 0.379488 0.3E-3 on w 10.07

1.33E-6 on wdot 0.379488

1.33E-6 on wdot 10.07  

Experimental Results Cont. Batch filter is more accurate with and

without noise for longer firing times Kalman filter converges faster for short

firing times, but has comparatively poor accuracy

Recommendations 24 hour firing Magnetorquers and non-essential

systems off Magnetometer readings are taken or

IGRF data provided Attitude data for the entire firing period

is taken Initialize attitude determination Kalman

filter at the start of firing and provide batch filter data only after convergence

Questions?