sgp4 differential correction

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SGP4 Differential Correction

David A Vallado, Vince Coppola, Paul Crawford

Center for Space Standards and InnovationPresented at the AGI Users Exchange, Washington DC, August 28, 2007


Outline Motivation Estimation Theory

Technical Review Solution Alternatives

Standalone Approach Discussion Availability

STK Approach Nomenclature Some Examples

Conclusions


Motivation SGP4 routinely used for satellite analysis

Updated Code available Vallado et al (2006)

http://www.centerforspace.com/downloads/

TLE is convenient for Distributing compact state information Developing a quick ephemeris

Fast, modest accuracy

Usually sufficient to start a numerical DC technique Useful in antenna pointing


Motivation Availability of data from which to form a TLE

Ephemerides Observations

Several SGP4 DC programs are in use AFSPC

No release of code nor technical details Others

Various versions of SGP4 used

With the release of SGP4 code (2006) Intent to release a publicly available version of DC


Estimation Theory - Review Least Squares

Minimize sum square of residuals Normal Equation (linear form)

X = (ATWA)-1 ATWb State (in X ) is found directly

For the orbit problem, its non-linear dx = (ATWA)-1 ATWb Corrections to the state (dx ) found through iteration


Estimation Theory - Review Identification of symbols

X State, either vectors or orbital elements

A Partial derivative matrix

W Weighting matrix (1/sigma squared)

b Observation / Residual Matrix (observed calculated)

(ATWA)-1 Covariance matrix


Estimation Theory - Review

X State either vectors or orbital elements Position and velocity vectors Classical Orbital Elements

a, e, i, W, w, n

SGP4 Orbital Elements n, e, i, W, w, M, and B*

Kozai form, but based on Brouwer theory

Equinoctial Elements Recommended for near singular cases

Other


Estimation Theory Review Bstar

Equinoctial elements( )

( )

( )

( )W

==

W

==

W++==+W==

+W==

cos2

tan

sin2

tan

sin

cos

iq

ip

MLeha

neka

e

e

eg

ef

M

y

c

wlw

w

KRmAcB D 0

*

2r=


Estimation Theory - Review Formation of A

A

where H

Observation matrix

Error State Transition Matrix X, Xo

State either vectors or orbital elements at current time, and at epoch

oXX

=F

XobsH

=

F=

=

= H

XX

Xobs

XobsA

oo



W Weighting matrix (1/sigma squared) If range-only data,

W = [ 1/sr^2]

Difficult to form If using TLE generated ephemeris, no knowledge

AAS 07-127 (GPS paper using TLEs) If using obs, need sensor weights

If from numerical ephemeris, may be small



b Observation / Residual Matrix (observed calculated)



(ATWA)-1 Covariance matrix Not provided with existing TLE data

Observation data too sparse Modeling not complete Little historical emphasis on covariance

Never been trusted from SGP4 Very large values and unrealistic

Exists in a DC operation Potential to investigate its value


Estimation Theory - Review H matrix Observation matrix

Several types of observations Range, Azimuth, Elevation, other Position and velocity vectors from ephemeris

Can use just position vectors GPS Direction cosines Accelerometer Other

XobsH

=


Estimation Theory - Review Fit span is critical

Tradeoff between length of time (propagation) To obtain enough observations To not introduce too much error from the propagation

Kalman filters do not have this problem


Discussion of Alternatives Two major areas

1. Formation of the A matrix Analytical, numerical Finite differencing

2. Solution of the normal equations Gauss-Jordon Single Value Decomposition, SVD


Formation of the A Matrix Calculating the A matrix

Analytical Partial derivatives Can be time consuming depending on state vector elements Often used for H as limited number of observation types Error State Transition Matrix, , model dependant

Analytical Constant system

Calculate from F matrix Numerical integration of variational equations

Most complex

Finite differencing Accomplishes both tasks (H and ) in one step Uses exact propagation technique


Technical Details Finite Differencing

Ensures complete compatibility with the propagation routine Finite differencing vs Central Differencing

Lots of propagations! Simple to code, overall system independent of choice of model.

ddd

dd

2)()(

)()(

--+

=

-+=

XfXfXobsXfXf

Xobs

oo


Technical Details

Epoch(to)

Nominal Orbit

Propagation

True Orbit Modified Orbit(1 of 6 for each component

of the state vector)

Observation times(ti)

rnompert

vnom

Earth rnom

rnom

vnom

vnompert

vnompert

rnompert


Solution of the normal equations Matrix inversion

Gauss-Jordan elimination LU decomposition and back-substitution Solution of matrix inverse can be difficult

Potential near singular values

Single Value Decomposition (SVD) processing Returns a value even if A is singular Diagonal matrix w lets you find which elements are important to

solution


SVD Processing Normal Equation

ATWb = ATWA dx Sb = SAdx

SVD solves the weighted least-squares problem as dx = V (diag[1/ wj]) UT S b Matrices U, V, and vector w are the SVD decomposition

of the weighted Jacobian (design) matrix SA Covariance information in V and w

( ) xyUV dd = Tjwdiag ]/1[


DC Matrix Inverse Approach FOR i = 1 to the number of observations (N)

Propagate (SGP4) the nominal state to the time of the observation (TEME) Find the slant range vector, sensor to the propagated state in the topocentric (SEZ) coordinate

system TEME > ECEF > SEZ

Determine nominal observations from the SEZ vector Find the b matrix as observed nominal observations Form the A matrix

Finite (or central) differences Analytical partials

H, Partials depending on observation type , Partials for state transition matrix.

Accumulate ATWA and ATWb

END FOR

Find P = (ATWA)-1 using Gauss-Jordan elimination (LU decomposition and back-substitution)

Solve dx = P ATWb

Check RMS for convergence

Update state X = X + dx

Repeat if not converged using updated state


DC SVD Approach FOR i = 1 to the number of observations (N)

Propagate (SGP4) the nominal state to the time of the observation (TEME) Find the slant range vector, sensor to propagated state in topocentric (SEZ) coordinate system

TEME > ECEF > SEZ Determine nominal observations from the SEZ vector Find the b matrix as observed nominal observations Form the A matrix

Finite (or central) differences Analytical partials

H, Partials depending on observation type , Partials for state transition matrix.

Accumulate ATWA and ATWb

END FOR

Use SVD to solve [S A], as U, V, and w, where (S = ATW)

Inspect wj, set any very small wj terms to zero rather than any 1/ wj term near-infinite

Solve dx = V (diag[1/ wj]) UT S b

Check RMS for convergence

Update state X = X + dx

Repeat if not converged using updated state


Misc Issues to consider Time, coordinate systems, Constants

UTC TEME WGS-72, WGS-84

Output Cartesian state vector Orbital elements

Kozai, Brouwer?

B* Solved for?


TEME details Frame in between TOD and PEF

TEME of Date Calculate nutation parameters at each propagation time We assume as the default

TEME of Epoch Calculate nutation parameters at epoch Use this value for all propagation times

Example Difference 24m in 3 days for a 7100 km altitude orbit

TEMEGMSTPEF

PEFGMSTTEME

GMSTGASTPEFGASTTOD

rROTrrROTr

EqeandrROTr

vv

vv

vv

)(3)(3

)(3

82

82

82828282

qq

qqq

=-=

+=-=


True Equator of Date (t1)

Mean Equator of Date (t1)

Dy

IPEF^e = e

- + DeITOD^

qGAST

z

Ecliptic

GCRS Equator

IMOD^



IITRF^

xp

IGCRF^

qGMSTITEME^EqEquinox


Constants WGS-72 stated

We use as default

WGS-84 possibleSymbol Calculation Value

m 398,600.4418 km3/s2 RK 6378.137 km J2 C2,0 = 0.000 484 166 850 00 0.001 082 629 989 05 J3 C3,0 = 0.000 000 957 063 90 0.000 002 532 153 06 J4 C4,0 = 0.000 000 536 995 87 0.000 001 610 987 61

XKE 60/sqrt(RK3/m) 0.074 366 853 168 71 /min

TUMin sqrt(RK3 /m)/60 13.446 851 082 044 98 min

Symbol Calculation Value m 398,600.8 km3/s2

RK 6378.135 km J2 0.001 082 616 J3 0.000 002 538 81 J4 0.000 001 655 97

XKE 60/sqrt(RK3/m) 0.074 366 916 133 17 /min

TUMin sqrt(RK3 /m)/60 13.446 839 696 959 31 min


Standalone Approach


Standalone Processing Fit observations or ephemeris Computing the Jacobian by finite difference

Each state element Xj perturbed and moved to observation time. Requires N more propagations than computing observations alone

Normal equation solution SVD processing

Scale and Weighting factors Used to mitigate the numerical magnitude differences in state

parameters. No effect on perturbations, just the processing. Weights for S based on circular orbit possible when no sensor

accuracy known.


Standalone Processing Finite differencing

Scaling

aGMvcir =

nGMa

vascir

13===

( )

W

D+

=

0

0

0

0

0

23016

91

aM

a

a

ai

ae

ta

GMa

w

x


Standalone Resolutions Constants

Use WGS-72

Coordinate System Use TEME Recommend conversion to PEF and then the standard

IAU coordinate conversions Recommend ephemerides be in J2000 (IAU-76/FK5) or

ITRF


Standalone Availability Beta code developed Public code

In development and test Designed to use AIAA 2006-6753 SGP4 code Release expected in a few months Check

http://www.centerforspace.com/downloads/


STK Approach


STK Approach Fits a TLE to an ephemeris Uses an iterative DC approach

Finite differences to compute state transition matrix Gauss-Jordan elimination to solve the Normal eqns

Convergence metric: RMS of residuals Current_RMS < Last_RMS (Last_RMS - Current_RMS) / Last_RMS < tolerance


STK Specifics Getting TLE data

Select Satellite/Generate TLE From an ephemeris

Trajectory sampling method Satellite Tools/Generate TLE BLS to minimize residuals

Options to solve for drag (Bstar)


STK Specifics Getting TLE data

Select Satellite/Generate TLE From an ephemeris

Single Point Conversion Fast Not as accurate


Generate TLE Tool in STK Observations

Cartesian position of the ephemeris in TEME Weight matrix currently is Identity, I

Results TLE Solution summary

Convergence performance

Residuals Graph, Report


User Controls Fit Method: Time Span or 1 Point in Time

Span to fit Time step for samples Epoch

B* Solves for B* User sets B* value

Convergence Tolerance Maximum Iterations


State elements Previous to STK 8.0 (Fall 2006)

State was Cartesian position and velocity Conversion to SGP4 Mean elements

Now STK uses mean equinoctial elements Improved performance Similar to Keplerian elements No problems with singularities near e ~ 0.0, i ~ 0.0, p

Reduces some problems associated with orbital regimes


State transition matrix Change in pos/vel from a change in state elements Forward difference Perturbation size chosen per element

Need a small perturbation large enough to make a difference larger than numerical noise

Reduced perturbation size near gravity resonance One rev per day, two revs per day


SGP4 Structural Organization (Revised)SGP4init

INITLif

method

if


Issue 1. Routine has several different code paths

Based on detected situations during processing Use deep space code or not Gravity resonance or not

Different code different code paths Code does not smoothly transition across paths Jumps in computed position and velocity Jumps lead to poor partials

Reference and perturbed state use different code paths Finite difference poorly represents derivative May cause divergence

Solution Reduce perturbation sizes

Tries to avoid different code paths We allow several times until a min size is reached


Issue (contd) 2. SGP4 may fail

A concern for e ~ 0.0 and e ~ 1.0 Cant create a valid partial

Solution Re-start procedure using a near-by solution


Issue (contd) 3. B* issues

Sensitivity to B* for low orbits Very small changes to B* may have large effect far from epoch May lead to divergence

B* may be unobservable (GEOs) Large changes in B* have almost no effect Can lead to very large B* values

Solution Set B* and do not solve for it Re-start


Issue (contd) 4. General divergence

RMS gets worse, not better Original ephemeris not well-modeled by SGP4

Maneuvers, Gravity resonance when m m_WGS72 Near circular near equatorial GEOs Re-entry, Launch

Solution Scale the state update by a < 1.0

We allow 5 such re-scalings before giving up Reduce the time span

We try to compute a fit for 1 rev Re-start using this value as a seed

Reset user controls Allow more iterations Loosen the convergence tolerance Reduce the time span


STK Specifics Output of data

Be careful to select/compare the proper system TEME of Date

Default TEME of Epoch

Possible AFSPC implementation Other coordinate system

J2000, MOD, etc

STK reports do proper conversions for each


Conversions Setup

Determines how STK interprets incoming data Reports are consistent

You can set STK up differently Locate the file

C:\Program Files\AGI\STK 8\STKData\Defaults\_Default.ap

Set the following default as necessary BEGIN SGP4FrameDefinition

ReferenceFrame TEMEOfDateEND SGP4FrameDefinition


Test Case Overview Verification test cases

Test each path through the code High eccentricity Data formats Other

Challenging cases GEO

Low inclination

HEO


Conclusions Two approaches

SGP4 differential correction with revised SGP4 code STK differential correction

Future release of code Standalone SGP4DC Documented

Technical equations in the literature Test cases and results

Widely available Softcopy: http://www.centerforspace.com/downloads/


Conclusions SGP4 Differential correction

TLEs are generated using a differential correction process and the SGP4 propagator

Proper selection of intermediary elements is critical for algorithmic robustness

Equinoctial appear to be best

Discontinuities due to conditional computation paths in SGP4 generate nontraditional challenges

Results reflect development work on STK implementation and a collaborative standalone effort


QUESTIONS???


So how do we know its right? POE comparison

Historical TLE comparison


Program Code Structure START

TwoLine2RVSGP4

Days2DMYHMS

SGP4Loop

Loop to read

input file of TLE data

SGP4init

Loop to propagate

each tle

Loop

JDay

Function Locations

if

if

DSPACE

DPPER

GETGRAVCONST

INITL GETGRAVCONST

GSTIME

if

if

DPPER

DSINIT

if DSCOM

GETGRAVCONST

SGP4GETGRAVCONST

DSPACEDPPER

SGP4Ext

SGP4IO

SGP4UnitOutput

sgp4 differential correction

Documents

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theory review x state

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n sgp4 orbital elements