SEMIANALYTICAL SATELLITE THEORY
AND
SEQUENTIAL ESTIMATION
by
Stephen Paul Taylor
B.S., University of Wisconsin - Madison
(1978)
Submitted in partial fulfillment of the
requirements for the degree of
MASTER OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 1982
©
Stephen Paul Taylor, 1982
Signature of Author
Departmnt
September
Approved
by
Paul J. Cefo aO,
Certified
sis Advisor
by
J.(aKrl Hedrick,
Accepted
Er
of Mecechaniccal
1981
Thesis Supervisor
by
Chairman
Wa-ar M. Rohow,
Department Graduate Committee
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SEmIANALYTICAL SATELLITE THEORY
AND
SEQUENTIAL ESTIMATION
by
Stephen Paul Taylor
2LT, U. S. Army Engineers
Mechanical
of
Deoartment
to
the
Submitted
oartial
in
1981
September
in
Engineering
fulfillment
of the
requirements
for the degree
of
Master of Science in Mechanical Engineering
ABSTRACT
a
with
combined
are
techniques
Filtering
Kalman
semianalytical orbit generator to develoo two sequential
orbit determination algorithms, analogous to the standard
Linearized Kalman Filter (LKF) and Extended Kalman Filter
The new .__algorithms are called the
(EKF), respectively.
Extended
and - the
(SKF)
Kalman. Filter
Semianalytical
design
The
( ESKF).
Filter
Kalman
Semianalytical
implications of the interaction between the filter and the
perturbations
theory
are
numerical tests conducted
presented.
discussed.
to verify design
The
results
assumptions
of
are
in the testbed
The new algorithms were imlemented
provided by the Research and Development version of the
Goddard Trajectory Determination System (Ri) GTDS). They are
and
investigated for comoutational efficiency, accuracy,
the special
with
by comoarison
radius of convergence
perturbations LKF and EKF previously implemented in the RD
A short-arc test case is used to
GTDS FILTER capability.
A many-orbit test case using
examine transient behavior.
ESKF and EKF steady
illustrates
data
real observational
state
erformance in the oresence of real-world observation
and force model errors. The ESKF is shown to meet or exceed
EKF performance for these tests.
An algorithm is derived that allows calculation of a
hysical
riori
suitable process noise strength based on a
This algorithm is verified using the real
considerations.
data test case.
2
The
real
data
test
case
led
to
the
discovery
of
an
a truncated 3x8 GEM 9
interesting force model anomaly:
erformance than the
gravity field gave better prediction
full 21x21 field. The satellite was in a low altitude orbit
geopotential
the 15th order
with
in sharp
resonance
The 16th
harmonics.
The atmosohere was relatively quiet.
order GEM 9 gravitational coefficients are shown to account
for the prediction performance degradation. Additional work
is required.
Thesis Supervisor:
Title:
J. K. edrick, Ph.D.
Associate Professor of Mechanical
Engineering
Thesis Suoervisor:
Title:
P. J. Cefola, Ph.D.
Section Chief
The Charles Stark Draner Laboratory
Lecturer
Deot. of Aeronautics and Astronautics
3
ACKNOWLEDGEMENTS
I wish to thank my advisors, Prof. Karl Hedrick (MIT),
and Dr. Paul Cefola (CSDL), for their continued patience and
guidance
throughout this thesis effort.
I owe Paul
especially a very sincere note of aooreciation for his role
in directly supervising my thesis research.
I owe Dr.
Richard Battin (MIT/CSDL) a warm note of thanks for my
initial elegant introduction to Astrodynamics and the orbit
determination problem in oarticular.
This thesis research has made extensive use of orevious
in Semianalytical Satellite Theory.
I wish to thank
work
Wayne
McClain,
Leo
Early,
Caot.
ndrew
J. Green
(UJS),
Dr.
Sean Collins, Cant. Jeffery Shaver (USAF), Dr. Mark Slutskv,
Dr.
Ronald
Proulx,
and Mr.
Aaron
Bobick
for the use of their
work and for numerous discussions, suggestions, and sunport. These men have contributed greatly to the richness of
my eductional experience during their tenure at CSDL.
During this thesis research, I have been privileged to
have the benefit of helo and advice from many members of the
technical community at large. I wish to thank in particular
Drs. Anne Long and Joan Dunham of the Computer Sciences
Corporation, Art Fuchs and Danny Mistretta of NASA-Goddard,
Jim Wright of General Electric Company, Ron Roehrich and
Paul Major of ADCOM, and Lt. Neil McCasland (USAF) for their
contributions to my thesis effort.
I wish to thank Bill Manlove, Kevin Daly, and Peter
Kampion, all of CSDL, for their very tangible suppoort and
guidance during this work.
Sandy Williams and Susan
Fennelly of MIT have similarly been of very great assistance
to me.
A final and very secial
Smith, who typed this thesis.
the
final
document
is due
note of thanks is due Karen
The highly olished nature of
solely
to
the
high
level
of
her
professional standards and a matching expertise.
This work has been oerformed while the author is on
educational leave from the U.S. Army. The Fannie and John
Hertz Foundation has been most generous in their suppoort
during that time.
4
I hereby assign my
Charles
Stark
Draner
Massachusetts.
copyright of
Laboratory,
this thesis to the
Inc.,
Cambridge,
Stegh n Paul TaVlor-- U
Permission is hereby granted by the Charles Stark
Draper Laboratory, Inc. to the Massachusetts Institute of
Technology
to reproduce
any or all of this
5
thesis.
TABLE OF CONTENTS
Section
1.
INTRODUCTION.................
1.1
1.2
1.3
2.
Page
................
*ee
.
eee....eeeee
.
..............................20
BACKGROUND
2.1 Semianalytical Satellite Theory ............... 20
A First Order Semianalytical
2.1.1
Satellite Theory ..................... 23
Fourier Series Development of
2.1.2
the Short Periodics .................. 30
General Notation..................... 33
2.1.3
The Variational Equations ............ 36
2.1.4
snects of
Computational
2.1.5
Semianalytical Satellite Theory ......41
2.1.5.1 Mean Element and Variational
Equation Computations....... ......... 42
2.1.5.2 Short Periodic Coefficient
Comoutation................. ......... 44
2.1.5.3 Position and Velocity
Interoolation............... ......... 46
Summary of Semianalytical
2.1.6
Satellite Theory............ ......... 43
2.2 Introduction to Sequential Estimation
...............................
Theory
2.2.1
2.2.2
2.2.3
2.2.3.1
2.2.3.2
2.2.3.3
2.2.4
3.
...... 12
Previous 'ork.......... * ...... * *....
. .. . ...... 12
Semianalytical Satellite Theory at CSDL. ......
...... 17
Overview of Thesis......
......... 43
Problem Formulation ....... ......... 49
Ontimal Linear Filtering.... ......... 53
Subootimal Nonlinear Filters ......... 56
The EKF ..................... ......... 59
The LKF ............................. 62
The Second Order Gaussian Filter.... 64
Summary of Estimation Theory .........56
SEMIANALYTICAL FILTER DESIGN...
3.1 Solve-Vector Choice .......
3.2
SKF Design................
on the Integration
Ooerations
3.2.1
Grid.............
3.2.2
Operations
67
68
71
........
71
on the Observation
Grid.............
3.3
........
........
........
.... .....
........ 73
ESKF Design...............
Integration ........76
3.3.1
Ooerations
on the
3.3.2
Grid............. ...
e
i..... ........
Observation
Ooerations on the
........30
Grid.............
.lleeeeeeeee
6
.
TABLE OF CONTENTS
(cont.)
Page
Section
3.4
3.5
4.
Verification of SF and ESKF Design
Assumptions........................
3.4.1
The Process Noise Test....
Evaluation of Dynamical
3.4.2
Nonlinearities............ ...........
Evaluation of Measurement
3.4.3
Nonlinearities............
........... 83
Summary ....
985
...........
90
........... 95
..
................
SIMULATION TEST CASES ................... . . . . . . ...96
4.1 Test Case Philosophy............... . . . . . . ...97
4.2 Test Case One: FILTEST 1.......... . . . . . . ..102
4.3
Test
Case
4.3.1
4.3.2
4.3.3
4.3.4
4.3.5
4.3.6
4.4
5.
83
rWNMTST ............. .
Test Case Formulation..... .
SKF Properties............ .
ESKF Properties........... .
LKF Prooerties............ .
EKF ProPerties............ .
Two:
.
.
.
.
.
.
. . .
. ..
.
.
.
.
.
.
.
.
.
. ..
. . .
. ..
..15
..105
. . 111
..126
.133
..142
Test Case Two Performance
...
....... 155
Summary..............
.
....... .......... .......... 158
Conclusions
THE REAL DATA TEST CASE .................
5.1 Test Case Formulation..............
Real Data Test Case Filter Results.
5.2
5.2.1
·
·
·
·
.....
·
.
.
·.
.....
150
150
..... 174
The Effects of Drag Coeffi cient
Errors
..............................
177
Preliminary Timing Results ......... 180
ESKF Integration Grid Length
Selection .
..........................
183
EKF and ESKF Steady State
5.2.4
Performance .........................139
5.2.5
Process Noise Model Verification .... 193
199
Semianalytical Modelling Errors .....
5.2.6
............ 200
Real Data Test Case Summary
5.2.2
5.2.3
5.3
6.
CONCLUSIONS AND FUTURE ORK .................... ... 204
...207
6.1 Conclusions............................
... 210
65.2 Future Work ...............................
LIST OF REFERENCES ............................
7
242
.
LIST OF APPENDICES
Page
Section
A.
PROCESS NOISE MODEL SELECTION .................. ... 213
A.1 Process Noise Analysis for Semianalytical
Satellite Theory .......................... ...213
A.1.1 Real Data Test Case Process Noise.. ... 218
A.1.2 Process Noise Modelling Extensions. ... 224
A.2 Process Noise Transformations ............. ...224
B.
SOFTWARE OVERVIEW ................................. 227
B.1 Software Descriotion ......................... 227
B.1.1 SKF and ESKF Subroutine
Descriptions .......................... 227
B.1.2 Test Case Support Subroutine
Descriptions .......................... 231
B.1.3 Software Bug Removal .................. 233
B.1.4 Interaction Diagrams .................. 233
B.2 Program Execution ............................ 233
8.2.1 JCL Setup ............................. 233
B.2.2 RD GTDS Control Cards ................. 237
B.2.3 Software Switches ..................... 239
8
.
LIST OF FIGURES
Figure
3-1
The
Page
Interaction
of
the Filter
(Ohs Grid)
and the Integrator (Integration Grid) when
the Integrator is Relinearized ............ ........ 69
4-1
4-2
4-3
4-4
4-5
4-6
4-7
4-8
4-9
4-10
4-11
4-12
4-13
4-14
4-15
4-16
4-17
4-18
4-19
4-20
4-21
Semianalytical Truth Radial Track Error....
112
Semianalytical Truth Cross Track Error.....
113
Semianalytical Truth Along Track Error..... .......
114
SKF Along Track Prediction Error, Run F.... .......
121
SKF Semimajor Axis Error History, Run F.... .......
122
SKF Semimajor Axis Error History, Run D.... .......
123
SKF Position Error History, Run F.......... .......
124
SKF Position Error History, Run D.......... .......
125
ESKF Along Track Prediction Error, Run A... .......
131
ESKF Along Track Prediction Error, Run E... .......
132
ESKF Along Track Prediction Error, Run G...
133
ESKF Position Error History, Run A......... .......
135
ESKF Position Error History, Run E......... .......
136
ESKF Position Error History, Run G......... .......
137
LKF Position Error History, Run D.......... .......
143
EKF Along Track Prediction Error, Run F.... .......
148
EKF Along Track Prediction Error, Run H ... ...... 149
EKF Semimajor Axis History, Run F.......... .......
151
EKF Semimajor Axis Error History, Run H.... . . . . . . 152
EKF Position Error History, Run F.......... .......
153
EKF Position Error History, Run H.......... ....... 154
EKF ALong Track Errors, Real Data Test Case .......
173
ESKF Run D Along Track Errors, Real Data
Test Case.................................. .......
179
5-3 ESKF Semimajor Axis Error History, Run A... .......
184
5-4 ESKF Semimajor Axis Error History, Run B... .......
185
5-5 ESKF Semimajor Axis Error History, Run C... .......
186
5-6 ESKF Semimajor Axis Error History, Run D... ... .... 187
5-7 EKF Semimajor Axis Error History
V .......... .......138
5-8 EKF Position Error History................ .......
191
5-9 ESKF Position Error History, Run D......... .......
192
5-10 EKF Osculating Keplerian Element Histories. .......
195
5-11 ESKF Mean Equinoctial Element
Error Histories, Test D.................... .......
197
5-12 EKF Predicted Inclination Error............ ......
201
.
.
.
5-1
5-2
202
5-13
ESKF Predicted
B-1
Software Interaction Diagrams .............. ....... 234
Inclination
9
Error,
Run D)....
.
LIST OF TABLES
Page
Table
2-1
2-2
Unified Equations for First Order
Semianalytical Satellite Theory............
Short-Periodic Coefficient Interpolator
......
. 34
......
Errors.......................................
2-3
45
Position and Velocity Interpolation
......
.
Errors......................................
47
3-1
3-2
3-3
Process Noise Covariance Test Results........ *......86
Results of the Dynamical Nonlinearity Test... ...... 39
Results of the Observation Nonlinearity Test. ...... 93
4-1
4-2
4-3
4-4
4-5
4-6
..103
FILTEST 1 Truth Model ....................
..104
FILTEST 1 SemianalVtical Truth Model.....
Test Case Two Truth Model ................ , .. .. .. .. 106
..103
Test Case Two Observation Data...........
Filter Test Case Two Semianalytical Truth ode l. . 110
Osculating and Mean Early Orbit Elements
.. 115
and Initial Perturbations ...............
for the S KF.......117
Issues and Options Investigated
..113
SKF Test Grid Runs and Performance.......
..128
Summary of ESKF Test Ootions .............
ESKF Parameter Test Results.............. trots··
. . . . . . . ..129
Final Keolerian Element ESKF Estimation E rrors.. ..134
LKF Parameter Test Results ............... .. .. .. ..141
EKF Parameter Test Results...............
........ : ..146
Final Keolerian Element EKF Estimation Er,rots.....150
..157
Test Case Two Performance Summary........
..157
Test Case Two Timing Estimates ...........
· · · · ·
·
e·····
e·®e··e·
· ·
4-7
4-3
4-9
4-10
4-11
4-12
4-13
4-14
4-15
4-15
l·
eeee·
· · · · ·
· · ·
ee
·
l·
leeoe··
· · · ·
·
·
·
·
·
· · ·
5-3
5-4
5-5
·..162
Osculating and Mean Filter Initial Conditions.
Solar Radiation Flux and Geomagnetic Index
History ............................. .............. 164
167
Real Data Observation History ...................
Real Data Test Case Truth Model ................... 16f
Real Data Test Case Semianalvtical Truth Model....169
5-5
Semianalytical
5-1
5-2
· s
Truthl Eohemeris
and Cowell
5-3
RMS Differences ................................... 171
GEM 9 Gravitational Coefficient Model
172
Anomaly Data .............................
Real Data Filter Performance Results .........
1..
176
5-9
EKF and
5-7
ESKF Element
Differences
..................
131
5-10 Real Data Test Case Timing Estimates .............. 131
10
LIST OF TABLES
(cont.)
Table
Page
A-1
A-2
A-3
Geopotential Coefficients and Errors .............. 220
Mean Element Rates...............................222
Mean Element Rate Probable Errors ................. 223
B-1
Print Flag Settings ...........................
11
241
1
Chapter
INTRODUCTION
Orbit determination processes require two capabilities:
the capability to accurately predict an orbit, given initial
estimation
indicating
how
and
observations
of the satellite orbit should be used in up-
dating
some
algorithm
conditions;
Advances in the technology of either
the ephemeris.
capability
imply corresponding advances
This
tion processes.
Semianalytical
thesis exploits
in orbit determinarecent
advances
in
Satellite Theory to develop and test two new
sequential orbit determination algorithms.
In their overview
of earth satellite orbit determina-
tion, Kolenkiewicz and Fuchs
[1] predict widespread use of
sequential estimation techniques by the mid 1990's.
Indeed,
orbit determination systems relying on sequential algorithms
are
already operational, motivated by increases in timeli-
ness, accuracy, or efficiency of the results.
search
A literature
turned up several examples that highlight
interest-
ingly the differences between the algorithms designed herein
and previous work.
1.1
Previous Work
Telesat
[2], a
satellite
communications
system,
has
been using a sequential algorithm to support station keeping
for several years now, with both improved
operations
accuracy and reduced costs.
This system uses an Extended
Kalman Filter (EKF) in conjunction with a special perturbaAny orbit generator that applies
tions ephemeris generator.
12
a high precision numerical integrator directly to a formulation
of
the
equations
of
motion
[e.g.,
Cowell,
is called a Special Perturbation Theory.
Encke,
VOP]
Special perturba-
tions theories have the advantage of being highly accurate,
but suffer from computational expense due to the small stepsizes
required
special
by
the
perturbations
Sciences
Corooration
study of autonomous
point
well.
CSC
high
precision
filter
(CSC)
develooed
in
navigation
recommends
integrators.
by
suooort
of
technology
use of
a
the
The
Computer
NASA/Goddard's
illustrates
10 or
15
second
this
step-
size in order to achieve the desired accuracy [3].
Alternate
Perturbations
satellite
theories,
theories
that
exist,
use
called
analytical
methods
improve comoutational
efficiency;
they require
tions
model
analytical
in
the
force
implicitly
causing losses
Air
Aerospace
Force
for
in accuracy.
Defense
Command
simplifica-
tractability,
(ADCOM)
approximate ephemerides
earth satellites.
Efficiency
task;
to
The United States
catalog containing
orbit determination
General
is mandatory
maintains
a
for all manmade
for this
DCOM uses general
large
erturbations
satellite theories and a recursive least squares estimation
algorithm to accomplish the routine uodating of the catalog
[4], [5].
However, a special perturbations
theory is still
required for high accuracy orbit determination.
The Global
orbit
Positioning System
determination
ture
[6].
tions
of
architec-
This system was designed to estimate
the oosi-
GPS
real time, while
satellites
with
with
an
uses a sequential
interesting
the
system
(PS)
a
1.5
meter
accuracyv
simultaneously having low operating
in
costs
for continuous operation. The expensive real time comoutations are minimized by using current observations to tune a
13
highly accurate
restored
nominal trajectory with a Kalman
Filter (LKF) linearized about that trajectory.
trajectory
and
all other
parameters
needed
The nominal
by the
filter
are
generated offline, once again with a soecial oerturbations
theory.
also
One of the algorithms
uses
a
investigated in this thesis
LKF to tune a orestored
nominal
trajectory.
Filter,
The last examole is the Eooch State Navigation
not
operationally
employed,
!rooosed
but
by
Battin,
to
Crooonik, and Lenox [71, and later studied in alication
autonomous
tion
of
by
4Menendez [31.
(VOP)
formulation
navigation
Parameters
notion is an essential similarity
lite
Theory,
special
although
the
The
of
use of a Varia-
the
equations
to Semianalytical
above
investigations
Satel-
emoloyed
erturbations methods thereafter.
Semianalytical
Satellite Theory
reoresents
a unifica-
tion of the strengths of soecial oerturbations
perturbations
methods.
Semianalytical
starts with the same force model
By using VP
formal
as soecial perturbations
The small magnitudes
in orinciole.
apolication
taken
in
of
asvototic
investigations
of
of these forces
mthods,
the
Navigation Filter. The asymtotic method is
the method of averaging, which was
Bogoliubov
Theory
equations of motion, only the perturbing forces
must be integrated.
beyond that
and general
Satellite
theories, so the same accuracy is achievable
allows
of
for analysis of nonlinear
a
step
Eooch State
atly
based
on
evelooed by Krylov and
oscillations.
Heuris-
tically, the method of averaging removes the oeriodic
content of the VOP equations of motion, allowing either
or
numerical
large
solution
steosizes.
It is the latter aonroach where the fundamental
14
integration
with
analytical
THIS PAGE INTENTIONALLY LEFT BLANK.
15
difference
with
analytically
general
intractable
perturbations
portions
of
methods
the
occurs:
force model
are
treated numerically, preserving real world accuracy.
1.2
Semianalytical Satellite Theory at CSDL
Since
the
sequential
discussed
in this
Satellite
Theory
orbit
thesis are
determination
based
currently being
on the
algorithms
Semianalytical
developed
at the Charles
Stark Draper Laboratory (CSDL), that theory is described
in
more detail below.
McClain
theory
in
(GMA).
[9] formulated the asymptotic expansion of this
terms
Use
development
arbitrary
of
the
of
the
of
formal
order.
GMA
Generalized
allows
the
rigorous
equations
Equinoctial
Method
for
of
Averaging
and
recursive
approximations
variables
were
of
selected
to
avoid artificial singularities in the VOP equations for near
circular
or equatorial
and Collins
equinoctial
third body
gravity
tions
recursion
generality
Gcreen [13]
and
element
[10], McClain
relations
flexibility
More
the
while
important
for
elements.
It
is
force
model
the
efficiency.
treatment
orbit
though was his development
osculating
func-
maximizing
of
determination
averaged out of
efficient
recovery
these short periodics that makes this Semianalytical
lite
Theory
as accurate
as special perturbations
while preserving the efficiency gains
analytical development.
Recent work
16
drag
of a Fourier series
expansion of the short periodic components
the
special
maintain
numerical
for
to central and
employing
to
[11]
expressions
rates due
perturbations,
formulated
perturbations.
processes,
Cefola
[12] have developed analytical
the averaged
and
orbits.
accruing
at CSDL
of
Satel-
theories
from the
has been
directed toward increasing the efficiency of short periodic
recovery
by further analytical
16] ,
[14], [15] ,
perturbations
[17] , [18], and the design
[19].
of appropriate interpolation algorithms
theory
can be
5 to 100
as efficient
times
of the gravity
development
The resulting
as special
pertur-
bations theories, depending on the application.
Satellite Theory has been investi-
The Semianalytical
gated
for
motivated
orbit
determination
by an interest
convergence
He used his results in a batch
properties
quite
finding
comparable
to
He also proposed a sequen-
high precision Cowell results.
tial algorithm
was
processes
(DC) estimation algorithm,
differential corrections
and
work
Green's
in orbit determination
for low altitude satellites.
accuracies
before.
that generated the interest motivating
this
work.
1.3
Overview of Thesis
This thesis reports on the application
tical
satellite
theory
to sequential
orbit
of semianalydetermination.
The sequential estimation techniques used are adaptations of
the Kalman Filter.
Although
the Kalman Filter is not the
optimal solution to the nonlinear estimation problem posed
by orbit
determination,
experience
[20] has shown that it
can be adapted to perform quite adequately.
The first
algorithm
design
is quite
used by the GPS system described above.
orbit generator
and
associated
accurate
values
is used to generate
short periodic
similar
The Semianalytical
an averaged ephemeris
interpolators.
for the osculating
to that
Thus
highly
satellite position
and
velocity can be recovered at observation times; the result-
17
ing
about
linearized
are
residuals
observation
the mean
used
trajectory
by
a
Kalman
filter
to improve the esti-
The resulting algorithm is called the Semianalytical
mate.
Kalman Filter (SKF).
orbit
Sequential
that
global
linearizations
designed
that employs
shown
than
oerformance
has much suoerior
the EKF generally
has
exnerience
determination
like the LKYF. Another algorithm was
ideas where
extended-tyoe
oossible.
This algorithm is called the Extended Semianalvtical
alman
Filter (ESKF).
Chaoter
2
semianalytical
Chaoter
mathematical
theory
satellite
for the SKF
background
These
oresents
and
introductions
theory
estimation
to
as
and ESKF designs.
the
3 discusses
of
designs
filters are based on the standard
SKF
the
and
ESKF.
Kalman algorithms
rather than on the square root or similar algorithms since
numerical
stability
is not
an
issue
in this
implementation.
The results of numerical tests which justify key assumptions
are also
resented.
Chapter 4
test
cases
cases serve
indications
resents the results of orbit determination
employing
simulated
to validate
observations.
the software
of the orooerties
and
These test
and give oreliminary
erformance
of the SKF and
ESKF.
Chapter 5 gives the results of EF
to real observational
data.
The results
18
and ESKF aoolication
are vervy
romising.
Conclusions and suggestions for future work are stated
in
Chapter
6.
The Appendices A and B discuss process noise modellinq
and the software implementation, respectively.
19
Chapter
2
BACKGROUND
This
chapter
presents
the
mathematical
background
required for the design of the Semianalytical Kalman Filters
(SKF
and
ESKF)
satellite
discussed
in
theory is described
3.
Semianalytical
first to define the notation
An introduction to filtering
its advantages.
and motivate
Chapter
theory surveys candidate estimation algorithms.
2.1
Semianalytical Satellite Theory
The accurate and efficient propagation of an ephemeris
requires both a precise model of the forces acting on the
satellite and an accurate and efficient means of integrating
the equations of motion.
The equations of motion are
iven
by Newton's Second Law as
d2 r
dt
2 +
%+
3r
(2-1)
r
The terms from left to right are the satellite's acceleration, the point-mass gravitational attraction, and all other
(disturbing)
due
accelerations,
bodies,
etc.
several
orders
The disturbing
of
magnitude
force.
20
to oblatness,
accelerations
smaller
than
drag,
are
the
third
typically
point
mass
Now any integrator
is most accurate and efficient
systems with only small nonlinearities
in the force model.
Historically,
models
integrated
which
and
ephemeris
retain
numerical
analytical
the
and
full force model
to
which use simplified
approximations
efficiently,
integrators
and low frequencies
this fact has motivated
tradeoffs between analytical methods,
force
obtain
for
to obtain
the
numerical
methods,
and use high
recision
the
integrated
ehemeris
quite accurately.
To increase the efficiency of an ehemeris
it
is necessary
to
nonlinearities
as well
force
The
model.
reduced by choice
Keplerian
decrease
as
both
the high
magnitude
of
the
of the orbital
and equinoctial elements
magnitude
frequency
the
generator,
content
nonlinearities
elements.
of
the
of
the
can
be
For examole,
incorporate the effects
of the point-mass acceleration, leaving only the disturbing
acceleration to be accounted for.
The transformation from
cartesian
osition
and
velocity
to
such
an
element
set
is
the basis of Gauss' VOP equations.
The VOP equations for a general satellite orbit can be
written
a
0=
=
ef(a,)
n(a)
+
(2-2)
eg(a,0)
21
Here the vector a represents
shape
and
plane
appropriate
high
of
the
set of
frequency
orbit,
hase
while
variables
variations
Examples of such
those elements describing
in
the
ing third body perturbations.
body
ohase variable.
magnitudes
of
ef(a, 8) and
8 is
accounts
satellite
an
for the
acceleration.
Sometimes the angle of the sun,
moon, or a planet is considered a
third
that
vector
hase variables are the satellite angle and
the Greenwich Hour Angle.
the
the
the
can
be
the
Alternatively,
modelled
The
svmbol
by
£
Just
taking
the motion of
tine
itself
formallv denotes
generalized
g(a, 8).
hase variable in examin-
disturbing
as a satellite's
as
a
the small
accelerations
mean
anomaly
is
analagous to time through Keoler's equation, the close relationshio between time and other
in
Equation
generalized
rate.
(2-2)
the
dominant
mean motion, n(a),
Since a satellite orbit
the generalized
are almost
variables
the
by
disturbing
hase variables is indicated
term
in
a nonzero,
8 being
almost
the
constant
is almost periodic in time,
accelerations
f(a, 8) and
g(a, 8)
eriodic in 8; this is consistent with the ohase
accounting
disturbing
for the high
accelerations.
frequency components
Semianalytical
of
Satellite
Theory averages the VP
equations (2-2) over the oeriods of
the
The method
hase
variables.
is illustrated
developing a first order theory (in
below
) with a single
in
hase
variable, taken to be the satellite angle.
See McClain [91,
Green
to higher
order,
angles,
resoec-
weak
[131,
and Collins
time dependence,
tively.
McClain's
[121 for extensions
and
work
multiple
hase
is fundamental
and
comprehensive,
while Green's and Collins' results show the oower and success of semianalytical satellite theory in two alications.
22
2.1.1
A First Order Semianalytical Satellite Theory
When the satellite mean longitude is taken as the only
phase variable, the VP
a
=
Ef(a, X)
A =
This
equations become
n(a)
develooment
+
)
cg(a,
(2-3)
uses equinoctial
variables, which
expressed in terms of classical Keolerian element as
a
=
can be
T
[a,h,k,,q,l
a = a, the semimajor
axis
h = e sin( w + I)
k = e
cos( w + I)
= tan
q = tan
= M +
I =
±,
I
I
(i/2)
(2-4)
sin
(i/2) cos
+ IQ
the retrograde
23
factor
Thus
n(a),
(2-3),
in the
expression
for
rate
of
X in Equation
is the mean motion
n(a)
=
n(a)
=
a
If
the
the
right
hand
sides
(2-5)
3
of
equations
(2-3)
depended
explicitly only on time, then periodicity would imply that a
and X could be expanded
in a
Fourier series whose
coeffi-
cients were linearly dependent on the small parameter
.
In
actuality, a Fourier series expansion in X is asymptotically
valid,
but
the
dependence
of
the
coefficients
on
is
complex, making expansions of various orders possible.
Generalized
Method
of
Averaging
(GMA)
The
developed
Mitropolsky provides
a rigorous mathematical
obtaining asymptotic
approximations of arbitrary order.
first
order,
averaging
the
the
of
the
GMA
are
formalism for
equivalent
To
to
the VOP differential equations over the period of
satellite
rates.
results
by
angle,
The
0
resulting
<
X <
mean
2r,
to
elements
constant term in the Fourier series.
tion is denoted by a super-bar,
are denoted
a
and
X.
obtain
mean
correspond
element
to
the
The averaging opera-
so that the mean elements
The relationship
between
the mean
elements and the unaveraged or osculating elements is stated
by the near-identity transformation
a
=
a + en(a,X)
A
=
X +
n 6 (a,X)
24
(2-6)
4
The
functions
n
periodic
functions
and
and
cn6
to
first
are
order
called
the
account
1--a1--
short
for
the
oeriodic terms in the Fourier series; thus they satisfy the
constraints
Sni ,(a'
+
nni (,
=
2r)
-A)
(2-7)
and
W
2 rT
I
ni (a,-)
(
=
d
0
(2-3)
0
for
i = 1,2,...,5.
Since the mean elements are obtained by integrating the
average of the equations of motion (2-3), the mean element
rates cannot depend on X or
elements
for the mean
a
=
=
.
Thus the equations of motion
can be written
A(a)
n +
A (a)
25
(2-9)
The mean element rates and short periodic functions are
completely
tions
determined
(2-3)
and
by asymptotic
(2-9),
applying
identity transformation
(2-7) and (2-8).
formal
Taylor
matching
as
between
constraints
Equa-
the
near
(2-6) and short periodic properties
The matching is most easily carried out by
series
expansion;
differentiate
(2-6)
to
obtain
aen
S
a
=
aen
*
a +
a +
aa
=
ax
A +
aa
+
Substitutinq for
(2-10)
a
aa
and X from
(2-9) and retaining terms to
first order in e for the asymptotic match
a
=
ives
nA(
A(a) +
n
3en
=
A 6 (a) +
n +
-
n
(2-11)
ai
Equation (2-3) asymptotically becomes
a
=
cf(a,j)
=
n(a)
+
) (a - a) +
aa
26
q(a,X)
(2-12)
since the residuals a - a and
By differentiating
-
are
first order in
the equation for the mean motion,
.
(2-5),
the final result is
=
f(a,3i)
X -n_- 3 -an
1
the
mean
Explicit
equations
for
periodic
functions
result
(2-11)
and
and (2-8).
(2-13)
and
(a,x)+
a
g(a,)
element
from
rates
comparison
application
of
the
(2-13)
and
of
short
equations
constraints
(2-7)
Comparison yields
3
E f(a,x)
E q(a,)
=
-
A(a) + -
3n
n(,
nl(a,X)
n
=
eA 6 (a) +
2a
n
ax
(2-14)
Integration
over
0 < X < 2
(2-8) by use of the
allows
identities
27
application
of
(2-7)
and
A
2i
aen (a, )
I
_
o
ax
n dA
n[cni(a,+2w)
=
-
ni(a,A)] =
(2-15)
and
2w
I0
2w
-
£nl;(a,A) dX
ia
=
a
J
s
(a,A)
0
d
=
o
(2-16)
Carrying out the averaging of equations (2-14) yields
2w
E A(a)
2
e f(
,)
d!
n)
2w
£
A 6 (a)
=
2
I
(2-17)
e q(a,X) di
0
With
the
coupled
Ai
now
partial
known,
equations
differential
(2-14)
equations.
become
The
satisfying the constraints is easily verified to be
28
a
set
of
solution
4
E n(a,)
£
=
1
=
n6(a,X)
[
I
n
j
f(a,S)
[
3
-
f
2a
0
- e A(a)]
gq(a,t])-
d
A6 (a)]
c nl(,S)
d
+
n(a,0)
dS
n(a,O)
+
(2-18)
Equations
(2-17) and
(2-18) provide a natural structure
ephemeris
generation
by
The
Averaged
elements
Orbit
by use of
Generator
(SPG)
Generator
(2-9) and
recovers
solution of (2-18).
results.
The
Satellite
Theory
Semianalytical
the
(AOG)
Satellite
computes
(2-17); the
short
peridic
Theory.
the
Short
to
mean
Periodic
variations
by
These equations are still purely formal
efficiency
depend
and
on
accuracy
of
maximizing
Semianalytical
the
analytical
development, with numerical methods used when further analysis is not possible.
bations
in
[111], while
(2-17)
Thus the averaginq of gravity perturis
done
analytically
[101,
the mean element rates due to drag and solar
radiation
are
Additional
formal development
tions
completely
computed
is possible.
The
by
numerical
29
[21].
of the short periodic
approach
described below.
quadrature
due
to Green
func-
[13]
is
2.1.2
Fourier Series Development of the Short Periodics
Accurate
tical
solution
satellite
periodic
of the VOP
theory
requires
But
variations.
equations
direct
recovery
by semianalyof
the
short
of equations
application
since integration of the osculating
(2-18) is undesirable,
force model would once again require small stepsizes.
equations
constraint
periodic
the
(2-8) imply that the short
(2-7) and
in a Fourier series in
functions can be developed
mean-mean
X.
longitude,
The
[13] developed
Green
this
expansion to achieve generality in the short-periodic model;
this
concept
also
leads
to
improved
computational
efficiency.
The
SPG
equations
(2-18)
imply
that
forces also have a Fourier series expansion.
the
disturbing
The derivation
starts with the assumption of such an expansion
E f(a,r)
=
C X 0 (a)
eX (a) cos al
+
a=l
+ cZ (a) sin aX
c g(a,X)
=
eX 6 0 (a) +
X 6 (a)
cos aX
a=l
+
(a) sin
Z
aX
(2-19)
30
The
Fourier
written
coefficients
of
the disturbing
forces
can be
as
s X (a) =
2~
2
-
E
f(a,X)
dX
-61 2w
e A(a)
=
2~
eX (a)
{
-
£
j
c X6,(_a) =
X
6,a
¢
f(a,X)
{
sin
2
(a)
-
I
e A 6 (a)
=
cos aX
2w
1
-
Z (a)
---
eX6
dX
q(a,X)
=0
1
f
E
0
cos
q(a,)
a
d
X
_}
d
sin aX
Z6,a
(2-20)
The Fourier coefficients
tions
can
disturbinq
be related
function
of the short periodic
to these
known
by substituting
coefficients
equations
of the
(2-19) into
(2-18) and carrying out the integration explicitly.
31
func-
Write the short periodic expansions as
£
n(a,X)
ECC sin aX -ao
a=l
D
-
cos aX
(2-21.)
6
c n 6 (a,)
EC 6
=
6,a
a=1
The
short
solved
periodic
Fourier
sin
a
coefficients
- ED
may
6,a
cos
aX
finally
be
for as
1
eC
-a (a)
1
eD
-a (a)
an
C6
6,a (a)
-
D
X
-a
an
=_1
(a) = 1
£
(a)
-
Z (a)
-a
X6,a (a)
-
Z
(a)
+
3
2aa
3
D1 a(a)
C1
(a)
2caa
(2-22)
32
Although
implicit integra-
this formulation still requires
tions of the disturbing force models as in equations
(2-20),
three factors contribute greatly to efficiency gains
1.
the
integrations
be done
can
analytically
for
all
gravity perturbations;
the Fourier coefficients depend only on the slowly
2.
and
varying mean elements,
selves,
extensive
allowing
so vary slowly
use of
them-
interpolators
[13];
3.
the
series
converge
quickly,
and
so
can
he
trun-
cated at low harmonics.
issues
Computational
Satellite
Theory
for the whole
are discussed
of
Semianalytical
in more detail below
after
discussing the variational equations.
2.1.3
General Notation
The development
so far has
different mathematical
and
the other
orbital
equations for each.
the essentially
emphasized
character of the satellite angle X,
elements,
a,
by deriving
separate
is necessary to
While this distinction
understand and implement Semianalytical Satellite Theory, it
is
less
important
for
the
application
Satellite Theory to orbit determination.
notational
conventions
similar
to
the
Delta, a common description of both the
tions can be developed.
of
Semianalytical
By using vector
scalar
Kronecker
and the a equa-
The unified equations are presented
in Table 2-1, referenced to the text above by "primed" equation numbers.
The
rest of the thesis will reference
equations in Table 2-1.
33
the
Table 2-1
Unified Equations for First Order
Semianalytical Satellite Theory
Indicial vector
eT
--
=
[o
**.
r)10
...
r ]T
1
t ith element,
out of six elements
Orbital Elements
a
T
=
(2-4)'
[a,h,k,o,q,
VOP Equations
a
=
+
ne
(2-3)'
f(a)
Near Identity Transformation
a
=
E n(a)
a +
(2-5)'
Mean Element Equations
a
=
ne
+
6
(2-9)'
Mean Element Rates
2r
1
2 r
(2-17)'
E f(a)
0
Short Periodic Functions
1
En(a)
-X
I
[Ef(a) -
A(a)
dA
n
3
2a
·P : *I
fn l(
(a)a
~-5
34
(2-18)'
Generalized Disturbing Acceleration Fourier Series
Coefficients
E a(a)
f( a)
=r
--a -
c-Za(-%)
a)
cos
2 r
1
al
} d
{
sin
0
(a>l)
(2-20)'
-
Short Periodic Function Fourier Series Coefficients
1
EC (A)
LX (a)
+
an
_.3
-26
2 oa
l,
(a
(2-22)'
(.a)
.
1
CD (a)
3
a(a)
--a
CZ
dn
2 cra
(a -)
e6
e-6 EC1,
~l, a
o(~)
Short Periodic Function Fourier Series
rn(a)
=
C
sin
a
a=1
35
-
E Da cos
(2-21)'
2.1.4
The Variational Equations
In
addition
satellite's
indicate
to
state
how
prescribing
over
to
the
time, most
compute
the
method
satellite
variational
of
computing
theories
equations,
a
also
or
partial derivatives of the satellite motion.
These partials
are required for orbit determination; the variational equations and all other partial derivatives required for orbit
determination
with
Semianalytical
Satellite Theory are
presented here, following Green's development
The variational
equations describe
[13].
to first order the
effects of small perturbations or variations in the initial
conditions on the satellite motion at later times.
The
equations of motion are linearized about the motion corresponding to the nominal initial conditions, resulting in a
linearized differential
of the perturbation.
differential
equation
derivatives
of
the
equation describing the propagation
The state transition matrix of this
is also called the matrix of partial
satellite
motion.
For
Semianalytical
Satellite Theory, the partial derivatives of the motion give
the dependence of the mean equinoctial elements at a time
tl
to
the
mean
elements
at
an
Denote the state transition matrix by
earlier
time
to.
(tl,t 0); then
a a(t )
0(tl,to )
(2-23)
a
a(t
36
0
)
is
a
notational
identity.
Letting
t1
be
an
arbitrary
time t and differentiating with respect to time gives
d
i(t,t0 )
respect
to
a(t)
a
a(t)
a
a(t
a
a(t
(2-24)
0 )
0 )
rule for the partial derivatives
Application of the chain
with
a
=
a(t 0)
and
substitution
of
equations
(2-9) for the mean element rates gives
a
(tt)
A(a)
[a s
2a
3n
T6
(t
t)
(2-25)
The effects of variations of dynamic parameters in the equations
of
tions.
motion
are
also described
by variational
equa-
Let c be the vector of dynamic parameters and define
aa(t)
*J(tt
)
=
a
(2-26)
Chain rule expansion of the derivative
gives
rise
to two
terms
of equation
in the variational
?(t,t 0)
37
(2-26)
equation
for
3a A(a)
~(t,t
=
)
ae A(a)
{
3n
a a(t)
e6el}
?(tt
+c
2a
(2-27)
The first term accounts
rates
due
to
the
for variations
implicit
dependence
in the mean element
of
a(t)
on
c;
the
second term describes the explicit dependence of the rates.
Use of Green's notation
A(t)
a
=
[13]
A(a)
3n
a a(t)
2a
e6e
T
as A(a)
D(t)
ac
(2-28)
B2(t)
=
(t,t 0 )
B3(t)
=
(t,t 0 )
gives his results for the Averaqed Partial Generator (APG)
B2
=
AB 2
(2-29)
3
AB
3
+ D
38
The
initial
(2-29)
c
conditions
state
for a fixed
the
for equations
independence
time
of the
(2-25) ,
(2-27) , and
a(t 0)
variables
and
to
=
o(t,t)
B 2 (t)
=
I
(2-30)
=
B 3 (t 0 )
the desired
output
T(tot
Since
0
)
=
0
of Semianalytical
Satellite
Theory is the high accuracy osculating position and velocity
at
a given
time
t, the partials
with
respect
pondinq mean elements are also needed.
from
mean
elements
to
osculating
to the
corres-
The transformation
position
and
velocity
occurs in two steps:
the near identity transformation
gives
equinoctial
the osculating
elements, which
(2-6)
are then
converted to position and velocity by a well-known transformation
[22].
The partial derivatives are developed by chain
rule as
a
ax(t)
ax(t)
[+
=3t
a(t)
aa(t)
Here
of
(2-31)
]
-
'+
x is the six component
substitution
en(a)
(2-6)
vector
results
factor.
39
in
of position
the
form
and velocity;
of
the
second
For
position
dynamical
special
perturbations
and velocity
do
not
theories,
have
any
the
dependence
c if the orbital elements
parameter
osculating
time are also taken as independent variables.
on
the
at the same
However, the
explicit dependence of the short periodics on the disturbing
forces, shown in (2-18), does result in nontrivial partials
for Semianalytical Satellite Theory.
when
(t)
ax(t)
ax(t)
ac
aa(t)
and
taken
c are
Specifically
ac-n(a)
ac
(2-32)
as the independent variables.
In
Green's notation
a sn
B1
(a)
=
a1
(2-33)
asen(a)
B
4
=
ac
giving
ax(t)
ax(t)
aa(t)
aa(t)
[I + B 1 ]
(2-34)
ax(t)
ax(t)
ac
=
aa(t)
' B4
40
The
of
partials
osculating
from
transformation
the
equinoctial elements to position and velocity are well-known
and
can
and
B4
be
computed
matrices
must
next.
discussed
(APG);
The
performed
is
matrices
the
B1
by
computed
B 1l
the
and
of
computation
the
by
B4
Averaged
matrices
are
B 2,
B 3,
Semianalytical
of these and
are
Theory
Satellite
Semianalytical
of
outputs
other
be
the
The details of computation
generator.
orbit
explicitly;
the
B2
and
B3
Generator
Partial
computed
by
the
Short Periodic Partial Generator (SPPG).
2.1.5
Aspects of
Computational
Satellite
Semianalytical
Theory
The central goal of Semianalytical Satellite
heory is
the development of the general formulation of a satellite's
equations
of motion
Full
has
use
been
and
computation,
for optimal
made
of
truncatable
computational
special
mean
rate
element
has
coefficient
or
a
been made
functions,
expansions
in the analytical
short
periodic
as efficient
Fourier
the
environment,
series
as possible
still maintaining the generality of the theory.
determination
recursive
Thus each evaluation of
development of the AOG and the SPG.
a
efficiency.
osculating
while
In an orbit
and
position
velocity must be computed at arbitrary times and arbitrarily
frequently, to allow utilization of the observations.
analytical
local
Theory relies
Satellite
internolation
heavily
Semiand
on global
strategies developed by L. Early
[19],
A. Bobick [231, and P. Cefola [19] to meet this requirement
efficiently.
that
It is the interaction with these interpolators
constrains
algorithm.
the
design
of
a
sequential
estimation
Their structure is discussed here in detail.
41
2.1.5.1
Mean Element and Variational Equation Comoutations
The
motion
fundamental
(2-9),
rooerty of the averaged equations of
(2-25),
frequency terms.
and
(2-27)
is
the
absence
of
high
Integration stensizes from a half day for
low altitude cases to several days for high altitude cases
easily
oreserve
Runge-Kutta
oroblems
computational
integrator
of any
arc
accuracy.
rovides
length.
the
The
A
self-starting
flexibility
mean element
to handle
rates
are
comouted by either analytical averages or numerical averaging quadratures.
Computation
of
the A and D matrices
(2-28)
for computing the rates in the variational equations (2-29)
can be done analytically for the oblateness perturbation, or
by central finite differences
rates
applied to the mean element
4 hermite interpolator
for all other perturbations.
recovers
the values
derivatives
times.
times
of the motion
The state
tl
of the mean elements
and
is
the
artial
(referenced to epoch) at output
transition
t2
and
matrix between
comouted
by
two arbitrary
the
semigroup
orooerty
~Dt=
and
=,t
(t 2)
(t,t)
(2-35)
its corollary
D(tot)
=
(tlt)
42
(2-36)
The same hermite interpolator used to compute the partials
*(t,t 0)
can
be
used
to
compute
its
matrix
inverse
in
(2-36) by using the rate equation
(to)
This
equation
- c- t,to)
is obtained
(t~tD) m (ttto)
by differentiating
(2-37)
the identity
-1
0(t,t 0)
A
(t,t 0)
= I.
similar development
partial
derivatives
of
can
the
be
used
motion
to
with
calculate
respect
to
the
the
dynamic parameters at arbitrary times, from epoch referenced
values.
The variational equation interpolator computes the
solution
?(t,t 0)
times.
The
to
equation
solution
(2-27)
to equation
at
(2-27)
required
can
output
be written
formally by variation of parameters as
t
?(t,t
0)
f
=
(t,T)
D(T)
(2-38)
dT
t0
The
partials
?(t
2 ,tl)
between
times
arbitrary
t1
and t 2 can be computed by expanding this integral
t2
f
to
t1
(t 2, )
D(
)
=
d
(t,t2
1
)
f
to
(tl,T)
D( r) dT
t2
(t2
+
t1
43
T) D()
dr
(2-39)
Appropriate
identification
of the terms
leads
to the desired
result
(t2'tl )
Use of (2-35) and
=
(t2',t0) -
(t2 ,tl)
(tl,t0)
(2-40)
(2-40) allows all needed mean quantities
to be computed quite efficiently.
The interpolator for the
transition
on
the
matrix inverse results in off-diagonal elements
order of 10-10, compared to 10-16 for the exact
inverse.
2.1.5.2
Short Periodic Coefficient Computation
As part of his study of the Fourier series expansion of
the short periodic functions, Green examined the computation
of the coefficients.
Since the coefficients depend only on
the slowly varying mean elements
longitude
, smooth
[13] verified
show
Early
interpolator
time histories
are expectable.
Green
this hypothesis; his plots of short periodic
coefficients
spans.
and not on the mean-mean
very
smooth
behavior
over
several-day
[19] implemented a variable order Lagrangian
for these coefficients with the same stepsize
as the Runge-Kutta interator.
He found accuracies depending
on the order and the stepsize as shown in Table 2-2.
the short periodic
once per step,
coefficients have
it is clear
to be evaluated only
that large gains
in efficiency
can be made even with low densities of output points.
44
Since
U)
40
4-_
I
1
I
I
I
.CN
Ccc)
i
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L
CeC
C
*
C03
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Cr
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iil
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r
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$4
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CU
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45
_
C-
U)
0~
OH
Position and Velocity Interpolation
2.1.5.3
Many orbit determination situations call for infrequent
with high data
station passes,
of observations
numbers
sensors taking
rate
the desired
to achieve
large
accuracy.
Now each call of the short periodic coefficient interpolator
described above requires the computation of six coefficients
for each frequency retained in the Fourier series expansion
(usually at least five).
Then
the Fourier series must be
and finally,
summed in order to obtain the short periodics,
the
resulting
equinoctial
osculating
transformed
to position
situations,
the computational
accomodate
elements
must
In high data
and velocity.
rate
load would grow rapidly.
such cases more efficiently,
be
To
L. Farly developed
low order, short arc interpolators for the osculating position and velocity;
polators
for
P. Cefola developed corresponding
the
partial
derivatives
of
position
velocity with respect to the epoch mean elements.
stepsizes
interpolator
lengths
up
polator
to 9 or
for
velocity;
are
and
used
a
a
or
two minutes,
Early
10 minutes.
position
Cefola
one
used
Lagrangian
Lagrangian
interand
Typical
with
a hermite
arc
inter-
interpolator
for
for
the
interpolator
partial derivatives.
The variation of the accuracy of the
position and velocity
interpolator with order and stepsize
is shown in Table
tors has
(DC)
orbit
been
2-3.
assessed
determination
analytical
perturbations
Satellite
The performance of both interpolain batch
differential
tests;
with
Theory
has
Cowell in efficiency.
46
their
corrections
use
exceeded
the
Semi-
special
Table
2-3
Position and Velocity Interpolation Errors
Points
Step
2
(sec)
Az(m)
z
(mm/sec)
.074
.392
60
.72
120
130
240
480
11.6
60
.002
.092
1. 03
.93
7.4
5.1
305
380
59
.8333
10 .6
135
2550
120
180
240
430
3
Av
640
.043
29
The reference position and velocity values were generated by
the Cowell high recision orbit generator for an AE-C
elliptical orbit over a ten-minute interval around erigee,
with a 4x4 Earth potential model, sun, moon, and
Harris-Priester atmosphere model.
The
step
size
is the
interval
between
interpolation points, in seconds.
47
successive
2.1.6
Summary of Semianalytical Satellite Theory
The
salient
features
Theory under development
of
the Semianalytical
Satellite
at CSDL have been discussed.
The
existing literature gives more details on analytical derivations
and
special
on
accuracy
perturbations
and
efficiency
Cowell.
The
comparisons
constraints
with
the theory
places on a sequential estimation algorithm have been introduced
and
will
become
clearer
after
filtering theory in the next section.
use
this
material
as
background
the
introduction
to
The next chapter will
for
the
design
of
a
Semianalytical Kalman Filter.
2.2
Introduction to Sequential Estimation Theory
The problem of orbit determination
is the accurate and
efficient estimation of a satellite ephemeris given observational data.
since
Sequential estimation algorithms are desirable
they make
real-time
immediate use of new observations,
availability
ephemeris.
Satellite
points
time;
in
discussed
estimate
above
from
estimate
be
observation
after
the
optimal
observations
the
will
filtering algorithm
new
of
are
Semianalytical
used
time
estimate
giving
of
the
taken at discrete
Satellite
to propagate
the
to observation
Theory
ephemeris
time.
A
is needed to specify how to determine a
receipt of another observation.
This
section presents results from sequential estimation theory,
as background
for the design of the complete orbit determi-
nation algorithms in the next chapter.
48
2.2.1
Problem Formulation
Modern estimation theory requires a
for the orbit
equations
determination
of motion
z(t)
(2-9)'
=
problem.
The
are modelled
f(z,t)
robabilistic model
as
z(t 0 )
+ w(t);
Semianalytical
=
(2-41)
The state vector z(t) is the estimated solve-for vector and
includes
the
mean
equinoctial
dynamical solve parameters.
elements
a(t)
and
other
Thus we can write
a(t)
z(t)
[
=
(2-42)
where c is the vector of dynamical solve
the coefficient of drag.
arameters, such as
The deterministic
force model
is
represented by the function f(z,t); equation (2-42) implies
that
f(z,t)
Notice
that
deoends
the
on
resence
the
of
mean
the
element
erturbation
not of direct significance in the estimation
vector
w(t)
random
inout
dynamics;
is
a
used
Gaussian
to
white
account
noise
for
rates
A(a).
arameter
e is
problem.
The
rocess.
model
£
errors
It
in
is
a
the
examples of such errors are given by atmospheric
density or solar radiation oressure model errors, geoooten-
49
tial
coefficient
tical
errors, and approximations
development
statistics
of
assumed
the
for
equations
of
orocess
noise
the
in the analymotion.
w(t)
The
and
the
initial condition zO are
Of
E{w(t)}
=
E w(t)
wT(
E Iz o
=
O
T)
}
function
(t- T)
(2-43)
-o
E{z
z)(z
--0 -z VI
The
(t)
=
- zT}
--
Po)
6( ) is the Dirac delta function and satisfies
6(t)
=
o
(t * 0)
(2-44)
and
I
f(t)
6(t)
=
f(0)
50
(2-45)
These properties of the delta function imply
6(0)
so
that
this
(2-46)
function
is not
well
defined
in
the usual
sense.
The
notation
E{.}
denotes
the
variable.
IFor a given
T
associated
[x, . . . ,nx] I
the
function is denoted by
random
PX(x)
PX1 ,...,X
_
random
value
variable
probability
n (Xl'i"
l·
expected
Oxn)
'n
of
a
x
=
density
(2-47)
Using this notation the expected values of x and a function
f(x) are computed by
x
=
E{x}
=
f
x Px(X)
-o
-
dx
(2-48)
and
f(
=
=
E{f(x)}
I
-aD
51
f(x)
PX (x) dx
-
(2-49)
The integral notation here is defined
f dx =
f dx
...
f dxn
(2-50)
Observations of the satellite ephemeris are modelled by
Yk
for
increasing
tion
of
the
times
mechanism
tk
for
all model
measurement
(k =
1,2,...).
the
obtaining
state,
z(tk).
The
Here
the
func-
deterministic
the
errors;
process.
(2-51)
+ Vk
tk)
represents
satellite
Vk describes
observation
The
of
measurement
there are many
statistics
k
model
A
the
noise
resent
in
of the vk are
=
E{vk
{vkv
for
h(Z(tk)
h(z(tk),tk)
current
any
=
=
The
k, =l1,2,...
k
(2-52)
kQ
symbol
6
kt
is
the
Kronecker
delta,
defined by
=
0
1
,
,
if k
if k =
52
(2-53)
I.,
with
unbiased
kth
imply
that
the
variance
R ,
and
statistics
These
Yk
measurement
the
that
is
for
errors
different observations are independent.
(2-41)
Equations
of
estimation
correlated
can
biased,
riate
is
noise
process
or
state-dependent,
all
be
of
augmentation
the
state,
functions f(z,t) and h(z,t)
above
use
noise
by
context
the
is
are
measurements
appropof
redefinition
z, and
[20].
requires
formulation
rigorous
the
or
in the
where
measurement
the
correlated,
treated
Cases
results.
theoretic
general
problem for aplica-
formulation of the orbit determination
tion
a
provide
(2-53)
through
the
Further extensions or a
the
of
calculus
or
estimate.
In
Ito
measure theory.
Optimal Linear Filtering
2.2.2
There are many definitions of an otimal
the linear case, most of them are equivalent and result in
the
Filter.
Kalman
The
fundamental
criterion
for
applications is the minimum mean square error
determination
With this criterion, the optimal estimate is the
criterion.
conditional mean of the state given the measurements.
Zk
be
at
time
Yi
the
orbit
value
t
in (2-51)
=
of
the
tk.
Let
up to and
Yn YZ=
=
process
stochastic
Y
be
including
{y
1 'Y 2 ' .
{Y'FY2''
53
'Y}
z
the
time
set
t
z(t)
of
in
Let
(2-41)
observations
z
(2-54)
Use
the
the
process
Ye.
When
when
k>z,
notation
at
time
k=X,
the
zk
tk
to
represent
based
this
on
estimate
estimate
is
the
is
the
set
the
a prediction,
of
estimate
observations
filter
the
estimate is a smoothed estimate.
Using this notation,
the
mean
estimate
error
J.
JK
Choosing
of the filtering
k )T (zk -
= E_(zk
E{(z
Zk
k
^k
zk to minimize
at time k is
k)}
)T
the mean
when
solution;
k<z,
square
and
of
(2-55)
square error yields
the
conditional mean estimate
k
-~!
kk
Ez
~_
k
(2-56)
- k IY k
When the dynamics and measurement models are linear, equations
(2-41)
and
(2-51)
z(t)
=
become
F(t)
z(t) + w(t)
; z(t 0)=
z0
(2-57)
and
Yk
=
Hk z(tk) + Vk
54
(2-58)
The
statistics
z 0,
of
w(t)
vk
and
are
given
equa-
by
tions (2-43) and (2-52), respectively.
The
Kalman
estimate of z(t).
filter gives an optimal
The filter has a natural division into two
tion algorithm and an udate
algorithm.
arts, a predic-
These equations are
well-known and are summarized below.
rediction
The prediction algorithm yields the otimal
^k-1
k-1
the
use in processing
Pklfor
its covariance
z k land
new observation
The state is
k.
redicted by
k-l =k-l
Z~k
(t't
=
(tk'tk-1)
k-)
=
( tk-l tk-l)
The covariance is
(t)
Zk-1
(2-59)
41(t~tk-1)
I
redicted by
k
(t tk-l
) k-1T (tktk1) + A(tktk-)
tk
A(tk,tk
1)
=
I
( tk, T
)
T)
(tk,T)
T
tk-l
(2-60)
55
Notice
that
filtered
the
predictions
estimates.
depend
only
The new filtered
on
the previous
estimates
depend only on the latest predictions.
similarly
The update algorithm
computes the new filtered estimate and is qiven by
K
=
k
P
k
k-l
T
k-i
T
H
k
k (Hk Pk
Hk + Rk)
k-l
+
k
(2-61)
k-1
Kk(Yk -k H
k
z
)
(2-62)
and
k
Pk
The
=
Kalman
practical
(I -
Filter
nonlinear
kk
Kk Hk)
(2-63)
equations
estimation
form
the
basis of many
alqorithms.
Three
such
algorithms are discussed next.
2.2.3
Suboptimal Nonlinear Filters
The
all
the
solution
general
above
random
to
section discusses
variables
the
linear
case, where
are
the Kalman Filter.
Gaussian,
filtering
either
the
it
problem.
is
the
In
initial condition,
When
optimal
the
more
process
noise or measurement noise is not Gaussian, the Kalman
Filter is optimal only among estimators with a linear form.
Most extensions
of the Kalman Filter to nonlinear estima-
56
tion problems make additional linearizing assumptions, based
on a perturbation
(2-41)
and
series expansion of the system equations
(2-51).
The method
and
its justification
are
illustrated by the following scalar example.
Let
random
h(x)
be
variable
a
given
x.
Let
differentiable
x have mean
function
of
x and variance
the
a
2
The Taylor series expansion of h(x) around x
h(x)
=
h(x)
+ h'(x)
- 1
(x - x) +
h"(x)
- 2
(x - x) +
(2-64)
can
be
h(x).
and
used
to evaluate
the
expected
value
of
the
function
Substitution of (2-64) into the expectation operator
use of its linearity
E{h(x)}
=
qives
h(x)
+
An additional assumption
development
(2-65) .
of
+
1
h"(x)
a
2
3 } + ...
h"'
'(x)E{(x- x)
(2-65)
that x is Gaussian allows further
All odd order
terms must
vanish,
while even order terms can be calculated explicitly in terms
of
2
a .
The
4
h(4)(x)a
,
(x-x) ,
(2-64)
fourth
as
which
of
h(x).
order
term
compared
is
the
term
In general,
57
[in
with
in
the
the
nth
(2-65)]
is
1/24
h(4)(x)
original
order
term
1/8
expansion
in
(2-64)
will
become
computed.
nth
order
in a when
its
expected
value
is
Thus the convergence properties of the two series
will be similar, with
versa when
Ix-xl>>a.
(2-64) faster when Ix-x <<a and vice
Depending on the magnitude of a, equa-
tion (2-65) can be truncated at first or second order with
little accuracy loss.
Suboptimal filters for the nonlinear estimation problem
posed
in equations
applying
the
(2-41) through
above
expansion
(2-53) can be derived by
method
f(z,t) and observation model h(z,t).
series
are
Kalman
Filter
result.
truncated
the
force
model
When the perturbation
first order,
either
the
Extended
(EKF) or the Linearized Kalman Filter
(LKF)
When terms through second order are retained, the
result
is called
filters
require
The
at
to
difference
a Gaussian
the
Second
estimation
between
the
Order
Filter.
errors
filter
to
estimate
be
and
These
small.
the
true
state will grow unstably due to neglected nonlinearities
the estimation
also result
model.
errors ever get too large.
from errors
Divergence
if
can
in the force model or observation
Jazwinski has designed two filters to control filter
divergence.
One filter
[24] adaptively adjusts the process
noise covariance to provide feedback on filter gains magnitude;
the other
subtracts
divergence
their
[25] actively
effects
is not
applications,
these
a
from the
problem
two
estimates
errors
filter estimate.
in most
filters
model
are
orbit
not
and
Since
determination
presented
here.
Only the extended, linearized, and second order filters are
presented here.
58
6
2.2.3.1
The EKF
The Extended Kalman Filter results from direct application of the method presented
in equations
(2-64) and
(2-65).
Perturbation
series truncated at first order are
used for both the dynamics and observation models.
The
expansions are based on the last filter estimate,
k-l;
it is tacitly assumed that "k-1 is in fact the mean of
the true state.
equation
For the state prediction the perturbation
is
z(t)
=
The assumed mean
z(t)
z(t)
+ Az(t)
(2-66)
(t) obeys
=
f(z(t),t)
(2-67)
z(tk-l)
k-l
=
k-l
1
=k-
Differentiation of (2-66) gives
z(t)
=
z(t)
+ A(t)
59
(2-68)
while expansion of f(z,t) in (2-41) yields
z(t)
=
F(t)
=
+ F(t)
f(z,t)
Az(t)
+ w(t)
(2-69)
where
af( z,t)
aaz
(2-70)
z(t)
Thus the perturbation Az(t) obeys
A_(t)
=
F(t)
Az(t)
+ w(t)
(2-71)
The solve vector prediction is the conditional mean; thus it
is predicted by
k-1
lk
The covariance
prediction
for
=
(2-72)
z(tk)
is predicted by application of the usual KF
equations
to
(2-71).
the EKF are
60
The prediction
equations
zkk -
=
z(t k
z(t)
=
f(z(t),t)
)
(2-73)
.k-1
=
(tk-l)
k--
Zk-l
(
k
tk)
O(tk'tk-1)
$(t,tkl
1)
=
F(t)
)
=
k-l
P-
(ttk-
T(tk
1
'tk-1
) + A(tktk_
1)
(2-74)
0( tk-ltk
1
I
tk
A(tktk
1)
=
I
(tk, T) Q( T)
T( tk,
T)
dT
tk-l
where
F(t) is defined
in equation
(2-70).
The EKF update
equations also use the oerturbation equation
(2-66).
The
measurement equation (2-51) is expanded to yield
k
=
Yk
=
h (zk
h(k-l
,tk)
+ k AZ(tk) + Vk
where
61
(2-75)
ah(Zk'tk)
a=
Hk
Ak-1
zk
Since
is
assumed
predicted observation
to
to
zk
date
is
;
the
mean
be
estimating
this
equations
1
of
the
process,
the
k-l
=Yk h(z k
EKF
k
is
k-1
The
zk
(2-76)
-Zk
z =
k
the
estimate
as
tk)
the
(2-77)
correction
Az(tk)
is accomplished
Kalman
Filter
to
by
be
the
algorithm
added
same
up-
(2-61)
to
(2-63).
2.2.3.2
The LKF
The
Extended
Kalman
trajectory
integration
Filter
in
continually
Equation
used
(2-67),
for
the
the
(2-70) and
dynamics
and observation
(2-76).
The trajectory update is based on the latest state
estimate;
thus
the
model
updates
linearizations,
computations
in
(2-67),
(2-70),
and
(2-76) must all be done in real time. The Linearized Kalman
Filter allows more efficient computation by assuming a
global
nominal
trajectory
linearizations;
correction
the
for
filter
the
integration
estimates
to this trajectory
by
the
the usual
(2-67) and
nonzero
Kalman
mean
Filter
equations.
This filter is more attractive for use with
Semianalytical Satellite Theory due to the latter's ability
to generate
long time trajectories
very efficiently.
The derivation
(e.g., 1 day
of the LKF is straight-
forward; only new equations are presented here.
62
or more)
A nominal trajectory for the LKF is generated exactly
as
it is for the
EKF, by
AN(t)
=
f(zN t)
(2-78)
=
z-N(tO)
The perturbed system becomes
Az(t)
=
FN(t) Az(t) + w(t)
(2-79)
AYk
=
Az(tk) + vk
HN
k
where
Ayk
is
computed
from
the
real
observation
Yk
by
AYk
The
linear
=
Yk - h(N(tk
coefficients
(2-S0)
) 'tk)
FN(t)
and
HNk
in
(2-79)
are
computed by linearization about the nominal trajectory
af(z,t)
FN(t)
=
a
=
-
K
(2-81)
ah(Zktk)
HNk
aZk
k =
63
N(tk)
The Kalman Filter equations are now aolied
system (2-79) to produce
the estimate.
directly to the
Since the nominal
trajectory is not locally undated, the estimated corrections
^k
Sk
can become quite large; the LKF tends to produce
less accurate estimates and to diverge sooner than the
One means of correcting this
roblem
KF.
is to use the Second
Order Gaussian Filter discussed next.
2.2.3.3
The Second Order Gaussian Filter
The Second Order Gaussian Filter retains terms through
second
order
in
contributions
the
perturbation
expansion
the filter name.
robability distribution, giving rise to
Second order filters tend to have better
accuracy and convergence characteristics
EKF,
order.
[26]
and
since
nonlinearities
Second order
Athans,
are
than either the LKF
accounted
for
et al.
[29] and Jazwinski
surveys
this paoer,
for
the
[27];
second
order
[201 contain
of nonlinear
complete
analysis
of
second
the
effects
of
Gelb
as oart
techniques.
In
Filter is used only
nonlinearities
SKF; thus only the new equations are
of non-
[28].
derivations
estimation
the Second Order Gaussian
analysis
to
filters have been derived by ¶Widnall
linearities was first done by Denham and Pines
of their
The
of the new second order terms are analyzed by
assuming a Gaussian
or
(2-64).
for
the
resented.
Gelb defines a linear operator (eq. 6.1-25 of Ref 29)
a 2 (g,3)
-
= trace {[2x
D
64
q
]
}
(2-82)
The
is
are
arguments
(nxl),
B
Using
(nxn)] .
is
function
a scalar
this
g(x)
and
operator,
a matrix
a
B
[x
dynamical
bias correction term is computed, using B = P, the estimate
covariance,
g
and
force model f(z,t).
=
ith
component
of
the
The ith component of the bias is
= a (fi
bi
the
fi(z,t),
P)
(2-83)
The EKF prediction equation (2-73) becomes
z(t)
=
f(z,t)
+
b
(2-84)
while the LKF prediction equation for Az(t) becomes
=
Az
FN(t)
AZ +
b
(2-85)
The bias due to an observation nonlinearity is
c
=
a2 (h,P)
(2-86)
The predicted observations are corrected
65
k-l
Yk
for the
EKF
k-1
=
h(Zk
[see
(2-77)],
k
Yk
[see
(2-80)].
1
'tk) +
c
(2-87)
and
1
for the
LKF
-
h(zN(tk)'t t k )
c
(2-8
There is also a second order correction to the predicted measurement covariance used in the Kalman ain computation; this correction effectively augments the measurement
noise
Rk.
the analysis
2.2.4
The
covariance
correction
of nonlinearities
is
and so is not
not
needed
included
for
here.
Summary of Estimation Theory
Nonlinear
estimation
theory
has
been
three estimation algorithms were presented.
to have better computational
form better
in the presence
introduced
and
The LKF appears
form, but the EKF should perof large nonlinearities.
The
Second Order Gaussian Filter uses bias correction terms to
reduce nonlinear effects; these terms can be used to assess
the impact of nonlinearities.
The next section uses the
discussions of Semianalytical Satellite Theory and Filtering
Theory to design a Semianalytical Kalman Filter.
66
Chapter
3
SEMIANALYTICAL FILTER DESIGN
chapter
This
two sequential
for use with Semianalytical Satellite
estimation algorithms
They are analogous to the Linearized Kalman Filter
Theory.
and
(LKF)
of
the design
discusses
the
Kalman
Extended
Filter
(EKF)
discussed
previously; they are called the Semianalytical Kalman Filter
Filter (ESKF)
(SKF) and the Extended Semianalytical Kalman
respectively.
When
orbit
two
role,
determination
Theory
Satellite
Semianalytical
time
is cast
in an
definitions
frame
are
important:
1.
the integration grid is the time frame used by the
integrator
and
coefficient
interpolators
short-periodic
associated
in
the
software
implementation;
2.
the observation
grid specifies the arrival times
of observations and consequently the output times
for the satellite position and velocity generated
by the integrator.
The
Semianalytical
makes
discussed
previously
operation
of Semianalytical
relinearization
of
integration grid.
boundary
of an
self-startinq,
the
it
clear
Satellite
equations
implementation
that
the
cannot
Theory
of
efficient
motion
allow
within
the
Relinearization should occur only at the
integration grid;
it
Theory
Satellite
can
be
since
relinearized
67
the
at
integrator
the
cost
of
is
one
additional
equation
the
evaluation
of the mean element
force models.
filter
and
the
The
resulting
integrator
on
and variational
interaction
their
respective
frames is shown in Figure 3-1.
The relationshios
Figure
and the ESKF.
3-1 apply
It
is
to both
clear
straightforward
the
SKF
apolication
Semianalytical
approximations
that
the SKF
can
of
integrator.
be
ideas
ESKF
by
with
requires
before EKF concepts are used.
time
shown in
designed
LKF
The
between
the
the
some
Before these
designs are presented, the issue of the filter solve vector
is discussed.
3.1
Solve-Vector Choice
The
choice
trivial.
elements
For
would
of
the filter
examole,
solve-vector
for
be a much
tao-body
more
is not always
dynamics
natural
solve
Kenlerian
vector
than
nosition and velocity; this example is striking in that the
element choice makes the equations of motion linear.
For
more
nonlinear
general
regardless
force
models,
of element
the
choice.
dynamics
are
The equations of
motion and the observation model must then be linearized for
application
chain
rule
follows
of
the
LKF
and
the
linearity
immediately
mathematically
correction,
updated.
as
that
irrelevant
long
or
as
EKF
equations.
of
the
the
filter
solve
nominal
of
the
equations,
it
vector
to the computation
the
By
use
choice
is
of the filter
trajectory
is
not
When the nominal trajectory is updated, second and
higher order terms in the element set transformation cause
differences in the updated trajectories.
68
4-J
U)
Cd
H
r-'
0
-H
--I
· .,
n'~
a)
0
4J
4-'
4-)
k
)N
0)
(N
44-
4J O
H )
N
a) -H
C
4-
O)
rd -H-
*
Cd (
H
.
O
-
0
rd0
(N
'~
H
11
-
k
4-'
O
4d
rl H
4'
4-)
a)¢a)
J)
0)
t -
U
4-3
U)-H
-,
4
rn
4-3
U
-
)
O -H
O-'0 0 )
4J () E "r
4- O
h
0
0-H 0
w
4 4-IJ
O
S400
n 0 -H--
* H
u4
(d3
H4-)
n
HE
a aa)
kOrdt00 *H
O
O0i
- H
En U)
_r
_~
tr
-~
.u
Cd
4-'
1H
O
r.
69
h4
0)
>Cd:>
:-
r
-H
0)
S4
4-
O0
Z
r·
W
·r
,1
The
Semianalytical
Differential
Corrections
(SDC)
software at CSDL can update both the epoch mean position and
velocity
and the mean equinoctial elements.
Generally,
no
distinct trends have been observed in either SDC accuracy or
convergence rate with the choice of update elements.
The
for
equations
include
bias
partials
of
the
terms
correction
the
force
Second
model
Order
Gaussian
Filter
on
second
depending
and
the
model
observation
with
These second partials do not
respect to the solve vector.
transform linearly; application of the chain rule yields a
term
the
containing
partial
of
the
element
For example, the equation
transformation.
a2 h
X
2
( ax
aa
-
ah
Da
describes
second
x
ax
(3-1)
x)(h 2)
aa +
Ba2
the transformation of the second
partial of
the
model with respect to the element sets a and x.
observation
Clearly solve vector choice will affect filter performance
here, but in a complex and unpredictable way.
The
solve
natural
vector
uses the mean equinoctial
Since
integrator.
equations
and
infrequently,
accuracy.
the
this
The
the
Semianalytical
elements,
SKF
trajectory
choice
solve
for
and
will
should
vector
may
filters
(t), produced
by the
linear
filter
ESKF
be
have
use
updated
minimal
also
relatively
impact
include
on
dynamic
parameters, c, when they are estimated.
70
a
3.2
SKF Design
The SKF algorithm is a direct result of the application
of
the
LKF
algorithm
to
the
Semianalytical
Satellite
Although conceptually simple, the implementation is
Theory.
more complex due to the interaction of the observation grid
and integration grid, as shown in Figure 3-1.
The algorithm
is detailed explicitly below to emphasize this interaction.
The
algorithm
is
statement
broken
down
by
operations
performed on the integration grid and those performed on the
observation
grid.
The
integration
usually executed much less frequently
grid
operations,
due
to
the
long
grid
are
than the observation
integration
allowed by Semianalytical Satellite Theory.
stepsizes
Due to the use
it suffices to consider only
integrator,
of a Runge-Kutta
operations
one integration step; all others are processed identically.
Operations on the Integration Grid
3.2.1
1.
At
the
time
t
new
=
t
update
integration
the
grid
nominal
with
initial
the
filter
state
for
correction
from the previous grid
ZN
+= Az
where
z
=
N0
Update
the initial
Initialize
the
[-]
c
covariance
filter
matrices
71
P0 = P0
0
0
correction
and
transition
--0
0
O(t0 ,t)
=
I
T(tot
=
0, save
0)
-l(t0 ,t)
and
compute
force
00
=
I,
in
save
evaluations
s
in
s
5~~~
for
the
equations
of
motion and variational equations
aN(t0), i(t,t0),
2.
Do the
evaluate
averaged
integration
(t,t ), and - (tt
until
time
obtain aN(t),
(t,to
0 ),
invert
to get
(t,t 0)
the corresponding
rates
mean interpolators for
aN, I,
72
',
t = to +
0)
At
P(t,to) and
-(t,t
to allow
0)
set u
of the
3.
Compute short periodics EC (aN), ED (aN
at
time
t0
coefficient
3.2.2
and
t
for
set
up
of
)
the
short
periodic
interpolators
Operations on the Observation Grid
The
SKF
executes
the
following
steps
when
a
new
observation is received.
1.
Obtain
2.
Interpolate for aN(ti), '(ti,t 0 ),
the new observation,
y(t i ), at time
we already have - l(ti_l,t0)
3.
(tit) 0
s
Interpolate for short periodic coefficients
EC (aN(ti
4.
in
t = ti
D
N(ti))
Construct the osculating elements
N
aN(t
i )
=
-N(ti) +
c
EC
sin aX - ED
a=transform
to
artesian
coordinates ti
transform to cartesian coordinates N(ti)
73.
cos
X
5.
Compute the nominal observation
YN(ti) = h(XN(ti),ti)
=
h(N(ti),ti)
and the observation residual
Ay(ti)
=
y(t i) - YN(ti)
compute the observation partials
4
Hi =
(z
B = aCnl(a
1
[I +BlB4]
=
N)
a-N
B
6.
aenl ( a N )
4
-
a0
Dc
Compute the transition matrix and variational partials
0(titi-1) = (tito)
is
=
=-(ti'ti)
?(ti't0)
- (titi-1) s
using
7.
s = -(ti_l,t
0
), and
s
=
(ti-l't0)
Obtain predicted solve vector and covariance
Aa
.
i =
1)
74
_a(titi
ai_+ Y(titi
)
A
i-l
i-i
-i-i
--1
(t
pi-=
i
[
ti
_
i
i-l
i-1i " ti t i-l ) T(tiIFti' )
Tti Ft
Ii-lI
)
0i-i
0
+ A(ti t i _l )
A(tit_
) =
lete
thei-hase
8.
Comnvoete the
·(tof -the
tii-l1)
update
hase
of the
filter.
pi-i q T
Calculate
the gain
1
K.1
1
i-l
(H.i
P
= Azi
uodate the state Az
update
9.
the covariance
1
Pi
H
T
+R)
+ K. [(t)
(I - KiHi)
Interpolate for the transition matrix
-
.iAz7, ]
Pi
inverse and save
for next observation
The
SKF
s=
7l(ti ,t)
Ts=
(ti,t0)
continues
with
and
step
one
until
the
integration
grid
boundary is crossed; then the integration grid algorithm is
reoeated.
When
t
t.1
onlyv steos
75
1,
5,
7
and
8
must be executed.
Since the SKF does not update the nominal
trajectory
the
within
corrections
to cause
This
integration
grid,
the
solve
vector
estimated by the filter can become large enough
filter
tendency
divergence
is reduced
due
to
model
nonlinearities.
for the ESKF algorithm presented
next.
3.3
ESKF Design
The
impact
of
ESKF
is motivated
observation
approximately
by
the desire
nonlinearities,
and
to reduce
is allowed
by
the
the
linear nature of the Semianalytical dynamics.
The mean elements obey the equation of motion (2-9)'
a
=
n e-6 +
A(a)
; a(t)
a(3-2)
(3-2)
Thus, to zeroth order
a(t)
=
n e 6 At + a 0
(3-3)
The zeroth order state partials are
aa(t)
I + e
e
aao
-
a
76
At
(3-4)
The
mean
motion,
n,
is small
for most
for
small
time
At,
differences
the
equinoctial elements
a(t) exhibit linear behavior.
equation
linear,
(3-2)
propagated
is
optimally
then
by either
The
the second term, so,
semimajor axis, a, further attenuates
especially
satellites.
the
estimate
Now if
a(tk)
is
KF prediction
the original
That is, optimal
equations or the LKF prediction equations.
prediction, analogous to EKF prediction,
mean
is accomplished by
the LKF prediction equations.
Let
t. 1
after
This argument
can
be made more
zN(til)
the
a
and
a
be
let
i-1
Az
measurement
priori
be
at
the
nominal
Kalman
t
time
explicit
An
as follows.
at
state
Filter
EKF
time
correction
would
use
the
new state
z(ti)
= -N(
z (ti-l)) + Az-i-1
i-l)
(3-5)
as the initial condition for the state propagation equations
and
a
corresponding
equations.
When
relinearization
of
the
filter
the dynamics are linear, this new initial
condition results in a predicted state at time ti
i
z.
-1
=
-
z(ti)
=
(titi_)
i-l
77
z(til)
i-l
(3-6)
Substitution of (3-5) and use of the notational definitions
gives
^i-1
+ Az i
= ZN(ti)
Equation
optimal
(3-7)
states
EKF prediction
LKF-predicted
filter
the
(3-7)
desired
result
can be accomplished
correction
to the
explicitly:
by adding
nominal
the
trajectory
when the dynamics are linear.
The
ESKF
dynamics
design
(3-2) are linear.
in Section 3.4 below.
only
application
of
is
computed
predicted
the
lies
The measurement
observation,
mean
the
semianalytical
This assumption is investigated
EKF concepts
based
predicted by (3-7).
that
The above arqument indicates that the
update computations.
HK
assumes
on
the
however,
in the measurement
linearization
nominal
is
based
matrix
trajectory;
on
the
the
state
Computation of an observation based on
equinoctial
elements
generated
by
the
Semianalytical integrator requires three transformations:
1.
computation
of
short
periodic
functions
and
position
and
osculating equinoctial elements;
2.
computation
of
the
osculating
velocity; and
3.
computation of the resulting observation.
78
The
prediction
state
equinoctial
mean
in Equation
(3-7)
The
elements.
is implemented
in
observation
resulting
computation can be written
Yi
=
+
h[x(aN(ti) + Aai
exact
Clearly
concepts
requires
periodic
functions,
efficient
EKF
precluding
the
(3-8)
predicted
observation
of
of
use
and velocity
implementations
ai)
recomputation
explicit
or position
coefficient
of
application
+
n(aN(ti)
of
the
short
the
either
More
interpolators.
(3-8) allowing
use
of
these
interpolators can be obtained by successive linearization of
the arguments
For use with
of h(x).
(
interpolator,
coefficient
the
just the short
periodic
computations
following
are
optimal for accuracy and efficiency.
Recall
partial
(i)
ai = aN (t.) i1
+ (I + B1)
--
(ii)
xi
=
(iii)
yi
= h(xi)
B1
matrix
that
of
the
the
short
-i-)
i
-a
+ ena
---(ti))
(a i
x(ai)
(3-9)
is
periodics
defined
with
in
(2-33)
respect to
as
the
the mean
short arc
When the position and velocity
elements.
interpolator is used, the following computations implement
the ESKF predicted observation calculation.
79
Green
the
(i)
aN(ti)
= aN(ti) + -n(aN(ti))
:IN 1
-N
(ii)
x(ti)
= x(aN(ti))
(iii)
xi = xN(ti)
(iv)
Yi
=
state
[I aaN
+
+
(3-9) (i) and
history
output,
computing the predicted observation.
estimation
algorithmic
algorithm
flow
for
(1]1
i
h(xi)
[13] proposed using
filter's
(3-10)
ax(a -i-1
is
the
quite
ESKF
(ii) for generating
but
not
for
use
in
Green's semianalytical
similar
to
employing
the
SKF.
equations
An
(3-9)
follows; an algorithm using (3-10) is quite similar.
3.3.1
Operations on the Integration Grid
These
due
to the
operations
use of
are
identical to those for the SKF,
the assumed
linearity
of
the
Semianalytic
dynamics.
3.3.2
Operations on the Observation Grid
The ESKF performs the following steps in processing a
new observation.
1.
Obtain
the new observation,
80
y(ti),
at time t = t i
2.
Interpolate
for aN(ti),
we already have
3.
D(ti,t0),
-1(til,to)
(tit)
0
in s
Interpolate for short periodic coefficients
C (aN(ti ) )t
e
i )u(anN(t
compute the short periodic functions
N
cn (
4.
)
N
I)
=
i
til
)
=
( iti-)(ti,
(ti t
0 )
(ti
1 )-
cos
S
(t i, ti
1)
S
Predict the filter corrections
i 1 = (t ,t
Ea
A
6.
-
Comoute the transition matrices
O(t
5.
sin
EC
a=1
i -1
=
ii
(ti ti ) --i-l
i-l
i-
Compute the predicted osculating elements
a(t
)=aN(ti)
+ i
transfor
to cartesian
eleents
(ti
transform to cartesian elements x(t i )
81'
i-l
7.
ComPute the oredicted observation
y(ti ) = h(x(ti) t i )
and the observation residual
by(ti) = (t i) - Y(ti )
compute the observation
a-ZN(Z-Nti
H
1
partials
- ----N [I + BiS3
4]
I
1
a-N
a4l(a=)
B
8.
=
4
--
the filter covariance
Predict
i-l [ ti t i-)
Pi
_ T
(ti
0
t i-1 1Pi1
I
[
i-1
(tti i_
)
0
+ A(ti,ti_l)
A(t i ,ti_ 1 )
9.
Comlete
=
the update
(t
1 - tii-1 )
hase
Calculate the gain K.1
of the filter.
pi-i H T
P
H
----
i
(H.
11p
82
H.+R)
1
(ti ti- 1) T
I
update
Azi =
the state
A
+ K
uodate the covariance P i
(I - Ki.i) Pii-i
i
1
10.
Interpolate
save
for
the
(ti)
transition
matrix
and
inverse
and
for next observation
s
=
t )
- l(t
0
(ti'
s= (tito)
The ESKF continues with step 1 until all observations
have
been
crossed.
processed
or
integration
grid
boundary
is
When the boundary is crossed, processing continues
as indicated previously.
same
the
time,
t
=
ti
1
When two observations come at the
then
only
steps
1,
7, and
9
must be executed for the second observation.
3.4
Verification of SKF and ESKF Design Assumptions
Several assumptions have been made in the design of the
SKF and ESKF
the
algorithms.
computation
of
predicted covariance
the
One
not commented
process
noise
on
reviously
contribution
to
is
the
(see SKF, step 7, and ESKF, step 8).
The process noise term is assumed to grow linearly in time.
This assumption and the nonlinearities in the observation
and dynamics models are tested below.
3.4.1
The Process Noise Test
The SKF and ESKF model the orocess noise contribution
to the
follows
redicted covariance as being linear in time.
the
assumption
used
83
in
the
Goddard
This
Trajectory
Determination
GTDS) software
System, Research
version
(RD
[31, which served as the testbed for SKF and
ESKF development.
prediction
and Development
The equation for Kalman Filter covariance
(2-60) derives the process noise term as
t
A(t,t 0)
f
=
T
D(t,T)
Q(T)
(3-11)
(t,rT) dT
to
where
Q(t)
is the process
E{w(t)
and
w
(T)}
noise
=
strength
Q(t)
at time
t
(3-12)
6(t - T)
(t,T) is the system state transition matrix.
is a constant
equation
matrix
(t,T)
is the
identity
matrix,
then
(3-11) reduces to the GTDS assumption
A(t,t
Shaver
and
If Q(t)
0)
[30] computed
equinoctial elements
oblateness effects.
=
Q
(t - t0 )
the state transition matrix
(3-13)
for mean
explicitly, including two body and
His state transition matrix has the
form
D(t,t0)
- A(t,t
0) + B(t,t 0 )
84
(3-14)
The matrix A has order 1 constant and sinusoidal elements;
the matrix EB has a secular growth in (t-t 0).
Equation
(3-14) leads to a process noise covariance form
- Cl(t - t0) + C2(t - t)2 + ...
A(t,t)
This
result
(t-t 0).
for
Equation
the
(3-13) was
GTDS
aroach
verified
by
for
small
numerical
test
large time intervals using the low altitude satellite
described below.
and drag.
The state
rule,
exactly computed.
as
transition matrix
included J
2
The exact equation (3-11) was integrated by the
trapezoidal
3-1
validates
(3-15)
is
the main
the
so
terms
in
A(t,t)
were
is shown in Table
The value used for
value
diagonal
quadratic
of
A(t,t 0 )
terms behave
after
16
linearly
hours.
Clearly
in time;
notice,
however,
that the model (3-13) does not account for cross
correlations that develop in the exact equation (3-11). In
actual
filter
adequately.
choosing
and
tests,
the
model
(3-13)
has
Observe that the same methodologies
the
Performed
aoly
rocess noise strength, Q, under both
for
(3-11)
(3-13).
3.4.2
Evalution of Dynamical Nonlinearities
The
correction
The filter
Second
terms
Order
Gaussian
for the dynamics
rediction equation is
85
Equations
specify
and measurement
bias
models.
Table
3-1
Process Noise Covariance Test Results
satellite!
WNMTST (see Table 4-3)
orocess noise strength
=
-1
diag [10-10 ,1 -l
'
i
,,
-17,,10,
1017
,
-1 3
,10- 3
orocess noise covariance contribution
at time
t = 16 hours
0.580-05
0.130-10
-0.810-10
0.14D-10
-0.140D-10
-0.38D-07
A(t,0)
0.0
= 57500.
0.130-10
0.50-11
0.13D-14
-0.410-15
-0.220-14
-0.5,0013
-0.810-10
0.130-14
0.570-11
-0.60D-14
0.580D-14
0.31D-12
0.0
0.0
seconds
-0.38D0-07
-0.580-13
0.31D-12
-0.15D-12
0.910-13
0.610-08
0.140-10 -0.140-10
-0.410-15 -0.220-14
0.5&D-14
-0.600-14
0.60D-12 -0.820-15
0.55D-12
-0.82D-15
0.91D-13
-0.15D-12
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.580-03
K
?Note:
value
below.
z
of
=
[a
cD
is
Notice
So that
similar
the
The
rag orocess noise is included.
to
strong
those
emoloved
correlation
oredictable from equation (3-4).
86
in test
between
a
cases
and
X,
2
Az(t)
=
F(t)
Az(t)
(3-16)
+ b
where the vector b accounts for the dynamical nonlinearities
causing a bias in the filter estimate; the ith component of
b is given
by
bi
=
1
2
a2f
tr{
1
(3-17)
P}
2
az-N
The other terms in equations
(3-16) and (3-17) are defined
by
Az
= estimated filter solve vector
f.i
= ith component
of
f(z,t)
f(z,t) = system dynamics force model
zN
= nominal system state
F(t)
P
=
P
= covariance
af(z,t)
=
covariance matrix= of
87
matrix
of Az
The
bias
correction
vector
provides
an
increment
to the
filter estimate Az given by
t
6z(t)
=
f
$(t,T)
b(T)
dT
(3-18)
to
The
matrix
Fquation
hours;
is
'
the
(3-18) was
the
oblateness
state
integrated
variational
by
equation
effects.
analytically.
computed
system
The
Second
matrix
partials
derivatives
was
matrix.
Euler's method
force
of
by finite differencing
numerical
transition
model
F(t)
the
for
included
was
force
by
the
the
computed
model
F(t); the accuracy
verified
11
were
of the
equality
of
off-diagonal terms, i.e.
a 2 f.
az
indicating
partials
_
n
that
az
the
n
1z
(3-19)
m
analytical
Since magnitude
reaching
finite
differenced
The results are shown
in
the
value
shown
after
11 hours.
of the 6z correction is much smaller than
filtering
dynamical
and
The perturbation 6z had fairly regular growth in
finally
typical
a1z
had the same accuracy.
Table 3-2.
time,
m
a2 f
corrections,
nonlinearities
have
it
is concluded
negligible
that
impact.
conclusion supports the design assumptions of the ESKF.
88
the
This
Table
3-2
Results of the Dynamical Nonlinearitv Test
satellite:
WNMTST (see Table 4-3)
state covariance:
diag[10-6 10 -12, 0-12
0
10-12, 1012,
10
0
]
Nonlinearity bias correction:
at
[
z (t)
Note:
t =
time
P
11 hours
-0.126D-0
0.134D-16, -0.167D-14 -0.75D-15
0.772D-15
imrl ies
-- 0
AZ
clearly
Azi >>
P , implying
--o
6z.i
validity
[10-3
.0-I10-6
These
results
10-5
are linear
for all scalings
89
of P
--o
-6 -5 ]T
in
0.258D-1
]
3.4.3
Evaluation of Measurement Nonlinearities
The
analysis
of
observation
model
essential to the design of the ESKF.
nonlinarities
is
Any nonlinearities
in
the observation model result directly in filter biases. The
Second Order Gaussian Filter update equations approximate
this bias as
2
c
=
tr{i-
1
y
P
(3-20)
azN
-N
where
c
is
the
bias,
y
is
the
observation,
ZN
is
nominal filter state, and P is the filter covariance.
bias
can
be
expanded
explicitly
in
terms
of
the
This
the
computational method used by Semianalytical Satellite Theory
for obtaining predicted observations.
Semianalytical Satellite theory computes an observation
by the following sequence of calculations
(i)
a
(ii)
x
(i)
xLT
=
=
a + en(a)
T(a)
;
D x
R;
x =
xLT
[]
= [LTv
-LT
90
]
(iv)
The
=
y
in
transformation
transformation
to
earth-fixed
for
accounts
(iii)
from inertial oosition
coordinates),
1950
h(xLT)
x
and velocity,
coordinates,
the
and
(in
from
there to the local tangent frame at the observation station
location,
R.
(i) and
The transformations
(ii) have been
on the
discussed above; the observation model (iv) deends
current observation type.
First and second
be
by
comouted
of
application
of the bias
for analysis
the
chain
rule
to
The second order oartials are
(i) - (iv).
transformations
desired
artials of the observation model can
correction
term
(3-20).
Three terms will arise naturally in the chain rule expansion
of the second order partials;
second
(ii),
or
the nonlinear
of one of
partials
(iv).
The
transformation
1
=
tr {(
will
contain
(i),
transformations
(iii)
is
linear,
the
so
has
Writing
vanishing second partials.
c
the terms
+ B + C)P}
(3-21)
the three terms A, 3, and C become
ax
[D(-)
+(I
+
+
]T
]T
aa2 y2
[D
ax
a
) (I + B 1)]
(3-22)
(LT
82x
=
-D
-LT
(I +
aa
91
1)
: (I +
1)
(3-23)
aB
x
=
C
aY
ax
D
.
aa
LT
The
quantity
two
multiplications
(3-24)
-
aa
a2x/Sa2
is
by
a
(I
third
+
order
B 1)
tensor;
the
from
the
result
application
of the chain rule in each of the partials with
respect
a.
to
The
quantity
aB1 /aa
is
similarly
third
order.
The
relative
magnitudes
of
the
three
terms
A, B, and
C
were evaluated in a station pass assuming range observations
were
taken.
Analytical
the Semianalytical
RD
GTDS
for
matrix,
the
the
range
Satellite Theory in the CSDL version of
the
equinoctial-to-cartesian
J 2 -short
obs
implementations already existed in
periodic
partials.
were implemented.
in B and
C remained
to be
partials
Analytical
partials
partials,
(B1
the
matrix),
second
order
D
and
range
Only the second order partials
implemented;
they were
implemented
by finite differences operating on the first order analytic
partials.
Accuracy
off-diagonal terms.
B, and
the
C matrices
start
printouts
of
the
was
verified
by
the
B
and
their
pass,
the
sum
T; the
second
first
is at the
so
can
should
observation
type
transformation
periodics.
printout
is at
middle.
These
show that A and B usually have the same order of
and
algorithm
of
Table 3-3 gives two printouts of the A,
magnitude; C is several orders of magnitude
or
equality
be
neglected.
Thus
include the nonlinear
and
smaller than A
an
effects due
the equinoctial-to-cartesian
but can neglect nonlinearities
One
will
extended-type
expect
92
element
in the short
an ESKF algorithm
(3-9) to perform better than an algorithm using
to the
employing
(3-10).
Table 3-3
Results of the Observation Nonlinearity Test
satellite:
TqNMTST (see Table
4-3)
station: MALABQ
height
-4.0 (km)
latitude
23n
longitude
279 °0
station
1' 12"
ass interval
observation tyoe:
------
-----
-
18'
50.4"
03:45:00.0 to 03:55:00.0
range
-
DATE JUNE 8, 1979
3HRS
46tlIlS
ELAPSEDTIXE FROl EPOCH=
0 DAYS
FLIGHT CENTRALBODY IS EARTH
I I
1 2
1 3
1 4
I 5
1 6
2 1
2 2
2 3
2 4
2 5
2 6
3 1
3 2
3 3
3 4
3 5
3 6
4 1
4 2
4 3
4 4
4 5
4 6
5 1
5 2
5.3
5 4
S 5
5 6
6 1
6 2
6 3
6 4
6 5
6 6
A.
A,
A,
A,
A,
A.
A,
A,
A.
A,
A,
A,
A.
A,
A,
A,
A,
A,
A,
A,
A,
A,
A,
A,
A.
A,
A.
A,
A,
A,
A,
A,
A,
A,
A,
A,
B,
B.
B.
B.
B,
B,
B,
8,
B,
B,
B,
8,
B,
B,
B,
B,
B.
B,
B.
B,
B.
B,
8.
B,
S,
5B
B.
B.
B.
8,
B,
B,
B,
B.
B.
S,
C,
C.
C,
C,
C,
C,
C,
C,
C,
C,
C.
C,
C,
C,
C,
C,
C,
C,
C,
C,
C,
C,
C,
C,
C,
C.
C,
C,
C,
C,
C,
C,
C,
C,
C,
C,
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
'
=
=
=
=
=
=
=
0.296860-03
0.274340+01
-0.27774D00
0.43501000
-0.353140D00
0.85239000
0.27434D+01
0.30409D+05
=
-0.937710+04
=
0.1180D9005
=
-0.121910t05
=
0.120340+05
= -0.277740+00
=
-0.937710+04
= 0.943140.04
=
-0.108850+05
=
0.123530405
=
-0.639570+04
=
0.435010+00
=
0.11809D005
=
-0.108850'05
=
0.126090f05
=
-0.142520D05
=
0.764700+04
=
-0.353140+00
=
-0.11910D+05
=
0.12353005
=
-0.1425:D+05
=
0.161790+05
=
-0.835220D04
=
0.85239D000
0.120340+05
=
-0.639570+04
0.76470D+04
=
-0.83522D+04
=
0.536450+04
6.379SECS (JULIAN DATE 2444032.6570)
0 HRS
2 IHNS
0.000 SECS
0.900760-07
-0.563270+00
0.155150+01
-0.352490+00
0.286740D00
-0.787160+00
-0.563270+00
0.458150+03
0.20354004
-0.687290+04
0.248950+04
0.264530 04
0.15515001
0.203540D04
-0.65575D+04
0.52039004
0.297920-07
0.214170-02
-0.582080-03
-0.179510-03
-0.463050-04
-0.13383D-02
0.21417D-02
0.0
.0.0
-0.11615D+02
-0.316710+01
-0.224280+01
-0.582080-03
0.0
0.0
0.153310+02
0.391680+01
0.355630+02
-0.475.90+02
0.57419D004
-0.352490+00-
-0.179510-03
-0.687290D04
-0.116150+02
0.520390.04
-0. 780830+04
0.332030,04
-0.398440D+04
0.286740D+00
0.248950+04
-0.475290D+02
0.296980-03
0.218220+01
0.12731D001
0.833330-01
-0.664530-01
0.638840-01
0.21822D+01
0.308670+05
-0.73416004
0.49246D+04
-0.97044D+04
0.146770+05
0.127310+01
-0.734160+04
0.2873;D+04
-0.566570.04
0.123090+05
-0.61822D+03
0.83333D-01
0.492460D04
0.153310D02
-0.128430+02
-0.56657D+04
-0.162510D01
-0.129110+02
-0.109330.05
-0.463050-04
-0.316710401
0.3916CD001
-0.664530-01
0.478760+04
0.364970D04
-0.970440D+04
0.123090+05
-0.16251001
-0.109330+05
0.136030D+04
0.68918D+03
-0. 787160+00
0.264530D04
0.57410D+04
-0.473950+00
0.3S8400D01
-0.133830-02
-0.224286001
0.175390D05
-0.765910D04
0.355630+02
-0.618220+03
-0.3?844D#04
-0.129110.02
0.689180D03
0.38840D+01
0.364970D04
-0.76591D+04
0.33:030+04
-0.313010+04
93
-0.227520+02
0.638840-01
0.14677D+05
0.271170+04
DATE JL"4E 8, 1979 i 3HRS
511ItNS
6.37)SECS (JULIAN DATE = 244,032.6605)
ELAPSED T!E F.:1 EPCCH
0 DAYS
0 HRS
7 INS
0.000 SECS
FLIGHT CENTRALBODY IS EARTH
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
C
3
4
5
6
1
2
3
4
5
6
A
A,
A,
A,
A,
A,
A,
A,
A,
A.
A,
A,
A,
A
A.
A,
A,
At
A,
A.
A,
A.
A,
A,
A,
A,
A,
A,
A,
A,
A,
A,
A,
A,
A,
A,
B, C,
C,
B. C.
B, C,
B, C,
B, C.
B, C,
B. C,
B, C,
B, C,
B, Co
B, C,
B. C,
BD C.
B, C,
B. C,
8, C.
B, C,
B, C,
B, C,
B, C,
B, C,
B, C.
B, C,
B, C,
B, C,
B. C,
Bi C,
B, C,
B, C,
B, C,
B, C,
, C
B, C,
B. C,
D, C,
T
T
T =
T =
=
T
T
T
T
T =
T T =
T =
T =
T =
T =
T =
T =
T =
T =
T =
T =
T =
T =
T =
T =
T =
T =
T .
T
T
T =
T =
T
T
T
0.47344D-03
0.33641D+01
0.33.40D+00
-0.34720CD00
0.14505D001
0.265130+00
0.336410+01
0.36618D+05
-0.35523D+05
0.131550405
-0.71647004
0.217970D05
0.332400+00
-0.35823D+05
0.114920+06
-0.471650D05
0.534940+05
-0.596220D05
-0.34,28D00
0.131550D05
-0.471650D05
0.194520+05
-0.225330D05
0.242750405
0.14505D001
-0.716470D+04
0.534940D05
-0.22533005
0.28454D+05
-0.26553D+05
0.26513D00
0.21797D*0
-0.562D.05
0.24275D+05
-0.26553D#05
0.3133.+05
-0.511380-07
0. 415020+00
0.514360+00
0.176880-06
0.212310-03
-0.54619D+00
-0.17730D-02
-0.37617D-03
0.6864680-03
-0.193530+00
-0. 12860D-02
0.41502D+00
-0.38230D+03
0.174020+04
-0.38050:D04
-0.29369D404
0.515280D02
0.51436D*00
0.174020.04
-0.727600+04
0.143030D05
-0.31851D+04
0.212310-03
0.0
0.0
0.13163D+00
0.280240D
+01
-0.14689D0+01
0.689820+01
-0.177380-02
0.0
0.0
0.40269D+04
0.532730D+01
-0.19475D+01
0.833060D+00
0.131630D00
-0.3805D2004
0.14303D005
-0.37617D-03
0.28024D+01
0.53273D001
-0.523793004
-0.95117D+01
0.32C260D04
-0.36945D01
-0.73330D004
-0.893100D01
-0.54619D+00
0.68648D0-03
-0.C9365D+04
-0.31861D+04
-0.14630SD01
-0.194750D+01
0.322260+04
-0.56649D+04
0.11571D+04
-0.369450+01
-0. 2800D000
-0.10353D000
0 .51528002
-0.128600-02
-0.13156D+01
-0.733300+04
0.689820D+01
0.833060D00
-0.8931CD+01
0.11571D04
-0.13156D+01
0.40690D+04
-0.382260+04
94
-0.119400D+02
0.473560-03
0.37794D+01
0.844990D00
-0.21603D+00
0.904980D+00
0.703150-01
0.37794D+01
0.36236D+05
-0.340820D05
0.935230D04
-0.101030D+05
0.218550+05
0.8449?0+00
-0.34082D+05
0.107640D06
-0.32856D+05
0.503060+05
-0.555940+05
-0.216030+00
0.93523D+04
-0.328560D+05
0.141630D05
-0.193140D05
0.1698D*005
0.904980D+00
-0.101030+05
0.50306D+05
-0.19314D+05
0.22789D+05
-0.25397D+05
0.70315D-01
0.218550+05
-0.5559T",05
0.169:8D+05
-0.25397D05
0.275030+05
3.5
Summary
This
chapter
resented
the design
semianalytical orbit determination
the eSKF.
Preliminary
assumptions made.
of
two sequential
algorithms,
numerical
tests
The next chaoter presents
simulation test cases.
95
the SKF and
verified
the
results from
Chapter
4
SIMULATION TEST CASES
This chapter discusses the results from two simulateddata test cases.
The Semianalytical Kalman Filter (SKF) and
Kalman
Semianalytical
Extended
Filter
(ESKF) designed
in
this thesis are compared with the Linearized Kalman Filter
(LKF) and Extended Kalman Filter (EKF) previously implemented
in
the
and
Research
System (RD GTDS).
Determination
Trajectory
of
version
Development
Goddard
the
All filters are
by the
compared against the performance baselines provided
application
of
to
algorithms
estimation
batch
the
same
observational data.
The question of test uniformity is fundamental to performance
dity of performance
The vali-
between different filters.
comparisons
can
comparisons
on
be questioned
two
bases:
1.
The
estimation
lated,
meaning
distributed;
problem
that
is stochastically
all
Monte Carlo
are
results
testing
formu-
randomly
is usually re-
quired to achieve a high degree of confidence
in
the results.
2.
The
particular
intrinsically
compared
filters
different
estimate
here
quantities;
the
SKF and
ESKF estimate mean equinoctial elements, while the
LKF and EKF estimate osculating position and velocity.
Thus
different
input parameters
quired by the different filter types.
of
any
resulting changes
carefully addressed.
96
are re-
The impact
in performance
must be
The
section
on test case philosophy
addresses
below
this
The two test cases are then discussed,
question in detail.
with important points summarized at the end of the chapter.
4.1
Test Case Philosophy
The RD GTDS software package provides a natural struc-
ture for
the
implementing
cases
test
simulated-data
dis-
This package has five basic capabi-
cussed in this chapter.
lities important for this discussion.
1.
EPHEM:
The EPHEM program allows
the propagation
of an ephemeris from a given set of initial conditions, using one of a large variety of satellite
theories and force models.
high
precision
The capabilities
Cowell numerical
integration
for
and
semianalytical ephemeris propagation are important
here.
2.
The DATASIM program can simulate a wide
DATASIM:
variety
of
observation
types
range,
observations
range-rate,
a
specified
The capability to simu-
tracking station network.
late
from
azimuth,
and
elevation
from a C-band tracking station net-
work, with random errors included, was used here.
3.
EARLYORB:
The EARLYORB program provides
estimates of a satellite orbit using
observation sets.
initial
just a
few
The algorithms used are similar
to Gauss' method; typical errors are on the order
of
50
kilometers
in
velocity.
97
the
initial
position
and
4.
DC
and
FILTER:
differential
sequential
RD
GTDS
corrections
filtering
capabilities
implements both
(DC)
batch
estimators
algorithms
(FILTER).
have been extended
and
Both
to allow the use
of Semianalytical
Satellite Theory as the epheme-
ris
The
generator.
called
rithm
the
Semianalytical
employing
precision
resulting
the
(SDC).
special
numerical
Cowell DC (CDC).
DC
DC algorithm
The
DC algo-
perturbations
integrator
is
is
hiqh
called
the
The FILTER algorithm abbrevia-
tions are defined above.
5.
COMPARE:
The
COMPARE program allows
the point-
by-point comparison of the time histories of two
ephemerides.
ephemeris
The
with
comparison
the
of
simulation
an
estimated
truth
ephemeris
provides a measure of estimator accuracy.
Each simulated-data test case required defining a truth
ephemeris,
simulating
the observations
C-band
observations,
with the various filters.
and
processing
The first test
case was directed primarily toward software verification and
included only these steps.
The second test case investi-
gated input parameter selection and performance for a difficult
filter convergence
problem;
the EARLYORB program was
used to provide the initial orbital estimate, while DC runs
and the COMPARE program gave performance baselines and performance measures, respectively.
Cowell
The EPHEM program with the
integrator was used to generate the truth ephemeris
-- high precision
Cowell integration provides
a generally
accepted high accuracy standard for ephemeris prediction.
98
Valid performance comparisons can be achieved either by
comparing
estimator
performances
for
parameter
choices, or by comparing the optimal performance
of each algorithm over all possible
parameters.
chapter.
corresponding
choices
input
of the
input
Both of these approaches are used within this
The
input
parameters
to
be
selected
are
the
initial orbital elements, the a priori covariance of these
elements, the process noise model, and the force model used
in the equations of motion and the variational equations.
The
CDC,
LKF,
and EKF all estimate
truth
posi-
These estimators can be initialized with
tion and velocity.
the Cowell
the osculating
initial elements
for software
validation
tests, or with the EARLYORB elements when initial errors are
desired.
The SDC, SKF, and ESKF, on the other hand, all
estimate mean equinoctial
elements
elements.
The mean equinoctial
to a given osculating
corresponding
initial posi-
tion and velocity can be obtained in two ways.
1.
A Precise Conversion of Elements
(PCE) procedure
consists of solving for epoch mean elements with a
SDC, using exact osculating position and velocity
measurements
taken at a uniformly high data rate
as the input observations.
high
precision
generate
Cowell
the position
An EPHEM run using the
integrator
and velocity
is required
to
measurements.
This initialization procedure has been used before
with excellent
results
[12],
[13],
[18].
It gives
highly accurate mean elements, especially appropriate
for
ephemeris
generating
corresponding
truth ephemeris.
99
a
Semianalytical
truth
to the simulation Cowell
2.
An Epoch Point Conversion
the
near
the
mean
Walter
identity
and
transformation
osculating
[31] proved
formation
(EPC) procedure inverts
(2-6) relating
equinoctial
elements.
that the near identity trans-
is a contraction mappinq, ensuring
convergence
of
the
successive
the
substitutions
iteration
ak+
This
a-
iteration
n(a k)
has
(4-1)
been
programmed
usinq
Zeis'
[16] explicit J 2 -short periodic expressions, which
are
zeroth
order
in
the
converged mean elements
The
eccentricity.
cannot be expected to be
highly accurate, due to the short periodic model
truncation.
conversion
They are adequate, however,
of
osculating
EARLYORB
for the
elements
to
corresponding mean initial elements.
When
corresponding
position and velocity
usual
result
transformation
mean
equinoctial
initial covariances
equation
and
osculating
are desired,
for covariance applies.
the
The
is
P (t0 )
=
[
ax(t o )
]Pa(to)
aa(t o )
100
o)
ax(t
[ 1 ]
aa(t o)
T
(4-2)
Here
P
-x
and
P
-a
are
the
initial
osculating
position
and velocity covariance matrix and mean equinoctial covariance matrix, respectively.
The required partial derivatives
are discussed in Section 2.4.
covariances
can
be
Corresponding
obtained
in
a
process noise
similar
fashion,
as
discussed in Appendix A.
Highly accurate orbit determination usually requires as
complete
a
force
model
for
possible.
Semianalytical
flexibility
in
force
the
equations
Satellite
model
of
Theory
truncations
motion
allows
as
great
without
accuracy
considerations.
Trunca-
tion decisions are typically based on assumptions
like the
loss, based on order-of-maqnitude
small eccentricity approximation, fourier series coefficient
attenuation, and perturbation magnitude.
mance
the
impact of force model
technical
Consult
description
the references
clarification.
tion
the
tional
costly.
force
[15],
accuracy
models
[18],
necessary
Any variational equation
achievable
is assessed, only
is given.
[19] for further
May [32] investigated the variational equa-
force model
ence.
truncations
of
[13],
When the perfor-
is highly desirable,
derivatives
(36)
for good
force model
DC converg-
simplification
since the number of varia-
makes
their
integration
quite
She found that inclusion of the J 2 perturbation was
usually sufficient
for good convergence and accuracy.
The
filter tests here use only the J 2 and drag perturbations
in
the variational
is
equations;
the validity
of
this model
confirmed by the overall filter performance.
Relative
ways.
filter
performance
was
assessed
in
three
Efficiency was measured by the CPU time required for
the observation processing.
Predictive orbit determination
101
accuracy, which measures the accuracy of the filter estimate
without input observations,
hour prediction
orbit
by comparing a 24
of the final filter estimate with the simu-
lation truth ephemeris.
tive
was computed
The filter convergence and defini-
determination
accuracy
were
measured
by
the
history of the magnitude of the position error during the
observation span.
The
single
results
Monte
presented
Carlo
in
trial.
this
Since
chapter
complete
represent
Monte
a
Carlo
a large number of trials to achieve high
testing requires
confidence in the results, such testing was ruled out by the
consequent
cost.
Rather, an EARLYORB initial estimate was
used to give realistic
initial errors.
The consequence of
this economical approach is the need for a word of caution;
the results are promising,
but further testing and experi-
ence are required.
4.2
Test Case One:
FILTEST 1
This test case was used to verify the software implementations
of the SKF,
is
altitude
a
low
force model.
EKF, and
satellte;
LKF.
only
The satellite
J2
was
observed
included
in
the
C-hand range and range rate observations were
taken; no random errors were added to the observations.
The
LKF and
conditions
and
and model
are
EKF
both
used
the
truth
initial
dynamical
model.
These
initial conditions
presented
in
4-1.
Table
Table
4-2
shows
the
corresponding
initial conditions and force model for the semianalytical
truth ephemeris, also used by the SKF.
This truth was
generated
by
an
SDC
run;
the
errors
from
the
Cowell
truth
averaged about 10 centimeters in position error and 5 millimeters per second in velocity error.
102
Table
4-1
FILTEST 1 Truth Model
Initial Conditions:
epoch = March 21, 1979
a
= 6673.0 km
=
0.01
65.0 deg
i
=
Q
(
=
0.0
=
0.0 deg
0.0
.= deg
M
deg
Dynamical model:
J2 only in the equations of motion and variational
equations
Observations:
range and range rate (C-band), no errors
1 day san
Truth Integrator:
12th order Cowell/Adams Predictor-Corrector
Steo Size:
50.0 sec.
EKF/LKF Integrator:
4th order Runge-Kutta Fehlberg
Steo Size:
10.0 sec.
103
Table
4-2
FILTEST 1 Semianalytical Truth Model
Initial Conditions.
epoch = March 21, 1979
a
= 6664.673
e
=
i
Q
=
=
64.9335 deg
0.0 deg
o
=
360.9999 deg
it
=
km
0.0093
0.0001
Integrator Step Size:
deg
43200.0 sec.
Force Model:
AOG:
J 2, J 22
e0
SPG:
J
e
APG:
J2
SPPG:
J 2 e
2,
J
2
104
The final estimation errors for all filters from their
were less than 10 centimeters
respective truth ephemerides
The SKF tests all used the
maximum in total position error.
short periodic coefficient interpolator and the state transition
All
tested
the
tests
were
These results verify the software implementations
positive.
of
others
relinearization.
grid
end-of-integration
and
interpolator
velocity
and
position
run also used the local
One
interpolator.
matrix
the SKF,
EKF, and
LKF.
ESKF was developed
The
later;
its
is
its
implementation was verified by similar tests.
4.3
Test Case Two:
A
WNMTST
test
critical
of
a
performance
filter's
The transient
transient response to the initial conditions.
response
tion
is especially
because
problem
important for the nonlinear estimafilter
if
diverge
estimates
the
This test case is posed
required corrections are too large.
as a rigorous test of the model accuracy of the Semianalytical
Theory
Satellite
atmospheric drag in the force model.
the
complete
of
presence
large
by including a (21x21) gravity
errors
condition
in the
problem
each
of
statement.
the
SKF,
initial
field and
The next section gives
Then
ESKF,
the
LKF,
performance
and
EKF
are
properties
of
described,
following with a section comparing the filters'
performance with the SDC and CDC baselines.
4.3.1
Test Case Formulation
This test case was formulated along the lines discussed
in Section 4.1.
is given
in
The complete description of the truth model
Table
4-3.
The
truth
105
trajectory
in osculating
Table 4-3
Test Case Two Truth Model
satellite:
WNMTST
June
epoch:
8, 1979,
reference frame:
3 hrs,
44 min,
6.4 sec
1950
a = 7016.7363 (km)
coordinates:
e = 0.0535734
i = 97.8520
= 224.4888
w = 127.3311
(deg)
(deg)
(deg)
M = 237.6709
(deg)
drag coefficient:
area:
A
= 1.0
m
mass:
m =
217.
density model:
CD =2.0
(m
2)
(kg)
1964 Harris-Priester atmosphere
(F10.7
1 0 7=150)
gravity
field:
integrator:
steosize:
21 x 21 (GEM-9)
12th order Cowell/Adams oredictor-corrector
30.0
(sec)
106
position
and velocity
was generated
by the high precision
Cowell integrator implemented in the GTDS testbed.
The GTDS
DATASIM program was used to generate simulated observations;
observation types and statistics, and the resulting observation history are summarized in Table 4-4.
There are several
interesting facets of this orbit determination problem.
First, the observations are extremely accurate;
and Pines
Denham
[28] argue that the effects of observation model
nonlinearities
become
observations.
Thus the situation is favorable for extended
more
significant
for
more
accurate
filters; linearized filters might be expected to diverge.
Second, both
latitude;
stations
since the orbit
but at different
problem
in
eccentricity
and
the
the
same
this means
that
the same relative
times.
observation
at approximately
is nearly polar
both stations will observe
orbit
are
in the
This creates an interesting
geometry
argument
point
of
for
the
perigee
estimator;
may
be hard
the
to
estimate.
Finally,
minutes
only
333
total observation
observations
(approximately
20
time) are taken of the satellite
during its passage of two stations.
The resulting estimate
accuracies will be an interesting indication of the rate of
filter convergence.
A Semianalytical truth ephemeris was generated from the
Cowell truth ephemeris usinq a PCE.
their generating
osculating
The PCE elements
Semianalytical model corresponding
to the
model of Table 4-3 are shown in Table 4-5.
resulting mean-plus-short
periodics
107
trajectory
and
The
is compared
Table
4-4
Test Case Two Observation Data
Station
1:
AtIOSq
TRANSMITTER :
TYPE-C-BAND
FREQUENCY-2200.000 tH.
HEIGHT
3.048 KM
COORDINATES
ERROR
0.0
LATITUDE
20 42 0.0
0 0 0.0
n,
MINIMUM ELEVATION ANGLE IS
observation
(DDD tM SS.SSS)
(
Ml SS.SSS)
A
5.000DEGREES
type (C band)
Statistics (standard dev)
0.1
Range
Azimuth
Elevation
Station
LONGITUDE
203 42 0.0
0 0 0.0
(DOD MM SS.SSS)
(
HM SS.SSS)
0.003
(m)
(deg)
0.003
(deg)
2:
MALABq
TRANSMITTER:
COORDINATES
ERROR
TYPE-C-BAND
FREQUENCY-2200.000 MH.
HEIGHT
-0.004 KM
LATITUDE
28 1 12.000 (ODD MM SS.SSS)
(
Mr!SS.SSS)
0 0 0.0
0.0
MIINItUMELEVATION ANGLE IS
LONGITUDE'
279 18 50.400
0 0 0.0
(0DD MM SS.SSS)
(
MT SS.SSS)
5.000DEGREES
observation type (C band)
Range
Azimuth
Elevation
Statistics (standard dev)
2.0
(m)
0.000556
(deg)
0. 000556
(deg)
A
108
Station Pass History
REV NUM I 0
1
2
3
;START 17906081790608179060817906081
I ;:,,6371 3i4061 5S1'61 659051
0S I
I
ol LO I
sI
I
QIELItAXI
I AOSLI
I
I
I
I
I
I
I
IlIAOS I
Al
I
LI LOS I
Al
I
BIELWAXI
QI AOSLI
I
I
17906081
1 36001
7900o
35501o
I
31
17906031
123151
AI
I
I
II
I
I
1
I
I
I
I
17906o31
701101
o7906081
1 709301
I
151
17906081
12035581
I1
I
iS I
I
I
I
I
I
I
I
ELMAX = MAXIMUM
ELEVATION ANGLETHIS PASS
AOSL = LOCAL TIME OF AOS
AOS = ACQUISITION OF SIGNAL
LOS = LOSS OF SIGNAL
Total observation time:
3 hrs, 23 min from the start of obs
by ALABQ until the comoletion of
obs by AMOSQ
109
Table 4-5
Filter Test Case Two Semianalytical Truth Model
satellite:
epoch:
JTNMTST
June
8, 1979,
reference frame:
3 hrs.,
44 min.,
6.4 sec.
1950
coordinates:
Mean Equinoctial
Mean
a = 7003.0073 km
a = 7003.0073 km
h = -0.0073252
k = 0.0533125
e = 0.05331336
i = 97.353567 deg
= -. 03041503
drag
eolerian
= 224.43964
deg
q = -0.8136052
w=
A = 229.43174 deg
M = 237.30524 deg
= C
coefficient
area:
a = 1.0
mass:
m = 217.
density model:
127.63585
deg
= 2.0
(m2)
(kg)
1964 Harris-Priester atmosohere
(F 1 0 .7=150)
gravity fieldAOG
SPG
integrator:
steosize:
21x21 (Gem-9) averaged otential
resonance (15,15) - (21,15) shallow
second order
2 and Draq and couolinq
zonal short eriodics (21x9)
sixth order e
m-dailv short eriodics (21x21)
sixth order e
tesseral short eriodics (21x19)
sixth order e
4th order Runge-Kutta
43200.
(sec) = 1/2 dav
110
to the Cowell truth in Figures 4-1 through 4-3.
The quad-
ratic growth in the errors is due to slight initial condition errors.
The excellence of the Semianaytical model
shown by the small magnitude of the errors during
soan from 0 to 5 hours.
fit
san
is 5.6 meters,
millimeters
The position error RMS during the
while
the
velocity
error
these elements
SKF
RMS
program was used to generate
orbital elements for filter initialization.
for
the fit
is 4.7
er second.
The EARLYORB
used
is
and
ESKF
directly;
The LKF and EKF
corresponding
initialization
were
aooroximate
mean
elements
generated
by
PCE
applied to the osculating EARLYORB elements and by EPC using
the iteration (4-1).
The EPC converged
to elements consistent
errors
from
the
PCE
in five iterations
to 10 decimal olaces.
elements
are
small.
element sets are shown in Table 4-5.
The
11
are also given.
osculating
elements
kilometers.
These
and
The
the
initial error
mean
elements
initial errors
of
the
these
erturbations of
the osculating and PCE mean EARLYORB elements
elements
Clearly
rovide
from the true
in both
is
about
a good
the
50
initial
error for testing filter convergence.
4.3.2
SKF Prooerties
This section presents results from an investigation of
the
effects
of
four variables
on
SKF
erformance.
The
variables that were investigated are:
1.
force
model
selection
for
the
and the variational equations;
111
equations
of
motion
*.4.
,
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a.4
4041,MI
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iI
=
J
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d
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=
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a.
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Table 4-6
Osculating and Mean Early Orbit Elements and
Initial Perturbations
Early Orbit Keplerian Elements
Osculating
PCE Mean
EPC Mean
a
7004.285
7005.566
7005.173
e
0.054123
0.054359
0.054302
i
97.9699
97.9765
97.9645
a
224.848
224.849
224.347
M
127.380
127.730
127.824
M
237. 566
237.207
237.114
Osculating and Mean Perturbations
Mean Equinoctial
Osculating Cartesian
32.86 (km)
Aa
2.44
33.71 (km)
$4h
0.3033E-3
AVz
12.36 (km)
X
0.5920E-3
AVx
23.26
AVy
0.47 (m/s)
aq
0.337E-2
AVz
3.34 (m/s)
AX
0.529E-2
I I.Ar
48.67 (km)
IrI
LI
48.55 (mn)
I
22.21
Ax
II
A*
(m/s)
lAX 23.50 (m/s)
115
tfl
(km)
0. 63E-2
(/s)
selection of the process noise covariance and the
2.
initial state covariance;
3.
4.
trajectory
update considerations
trajectory
used
in the SKF;
for the nominal
and
mean state initialization by PCE and EPC.
The filtering issues (1) to (4) above provide a natural
test grid
for exploring
the properties
of
the
SF.
Two
options are presented for each issue or consideration.
With
a total of four questions to be examined, the resulting grid
has
sixteen points.
diagonalized
tion
of
tests
reported
have essentially
the grid, providing insight into the interac-
each
summarized
The
in
of
the
Table
issues.
4-7
in
The
the
options
context
available
of
the
list
are
of
issues.
The force model issue explores the effect of one force
model
truncation on filtering accuracy.
options
test
strength.
the sensitivity
The process noise
of the SKF to the process
noise
The relinearization of the SKF nominal trajectory
after
a station pass causes
EKF.
The relinearization option was implemented by changing
the integration grid length.
approximate
EPC
measurements
in actual runs.
Seven
options.
SKF
procedure
runs
tested
the SKF to look more like an
Additional confirmation of the
(4-1)
is
various
given
by
performance
combinations
of these
The options used for each run and the resulting
performance are given in Table 4-8. The final estimate of
each SKF run was used as the initial condition for a 24-hour
predicted ephemeris.
The RMS values given in Table 4-8 are
e) 4'
1u C)
I
um.. C
OD,
IC.
IC
)
cI
]a0)U
·
-,-
U
a .,~
C' r3 t E4·
CC
C4 1D
xc
c
r,,
i,
0!
C)
X< C
*~'
EnU)
all
4-)
E C
C C
C
.
Q
44
I
E
rt 4
C
E:
.
4-)
c .....
C
..}
0)
4
-
N
Cr
(u
0)
,
C
24
0X
0
0)
- .,-H
0)
C
IN
U?
,C
-'.
,,
(N
0¢U)
v
..4,
'4-) O
C
0
0-
o9r0_ RII
C
.jH
OJ
R
X
I
CC
.
'-H
_ (N
-U)_
-
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-
1
ICrlr-H
CC
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-00--_D
0). .,C,_.r
,:
,. e~ *^ - -.-E
nsI nI CI .
O
I
(2 C't
(
a
>-
RI
(P
t
-a! IH(tlU)
U a
u
C-H
H::)l
O
fr
c)t
CT
]
.
(C)
-H
C
CV
4(N
,.NR
N
tt4-4
a(c' c,,i
(N)
cX
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X
H
-
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C)
(.4s
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o
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C- _ 0
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,tn
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C4 - I aH,
a-
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4-U
Na!]
ir -:
IC
l- I 4n
R
OC
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( iU
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- r
C
II 4~o, -H.I-3
40,;)0
c-
C:
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-rmC-117)
r, --
r C4
Table
4-3
SKF Test Grid Runs and Performance
TEST
RUNS
1
Force
Model
2
Process
3
Trajectory
4
Early
Noise
Upd.
Orbit
A
1B*
2
C*
D
E
F
G
1
1
2
2
2
1
1
2
2
2
1
2
2
1
1
1
1
2
2
1
1
1
1
1
1
1
2
2.5
.25
.3
.24
2.7
.19
.3
PERFORMANCE (km)
Radial
rMS
Cross
MS
3.6
.21
.3
.22
3.2
.37
.3
Along
RMS
21.6
7.2
7.3
7.0
19.2
1.14
9.0
Total
RMS
22.0
7.2
7.3
7.0
19.7
1.22
9.0
* Run B used the coefficient
t'he position and velocity
interpolator
interpolator
well as the coefficient interpolator.
118
only; Run C used
(steo = 100 sec) as
the
root
mean
ephemerides
squared
residuals
of
these
from the Cowell Truth ephemeris
predicted
of Table 4-3.
The results shown support the following statements:
1.
use of the position and velocity interpolator does
not seriously impact SKF performance
(the observed
RMS increase is probably accentuated by the highly
2.
accurate observations)
[B, C];
the
model
truncated
impact
3.
SKF
force
SKF performance
accuracy
is very
does
not
seriously
[B, D];
sensitive
to
the
process
noise strength choice [A, D and E, F];
4.
more frequent trajectory updates in a convergence
test
do
improve
5.
letters
supporting
necessarily
speed
[A, E and
D, F];
accuracy
convergence
SKF performance
in brackets
the
statement.
[B, G].
reference
the
The results
runs
in
in Table
4-8
Table 4-8
are
consistent with filter histories of the various runs.
means
or
The EPC mean EARLYORB procedure does not seriously
impact
The
not
that the filter runs have converged,
This
so the results
shown are not especially dependent on the particular filter
output estimate used to generate the predicted ephemeris.
The changes in performance in the sequence of runs A,
D, and F is remarkable.
The performance in each case is
dominated by the along track error.
The performance
improvements from run A to D to F correspond directly to
119
decreases in the error of the final semimajor axis estimate
from 130 meters
to 70 meters
to 7 meters.
Figures 4-4
through 4-8 show plots comparing runs D and F.
shows
the
contains
along
a
Figures
4-5
and
and
is
the
4-6
error
amplitude
This error
axis
histories
prediction
.5 kilometer
frequency.
semimajor
track
due
error
show
the
Figure 4-4
for
error
run
at
to
coupling
in
the
the
for runs F and D, respectively;
axis
it
orbital
between
satellite
semimajor
F;
the
angle.
filtering
the variable
DA
plots the error while PA is a 3a standard deviation bound.
Figure 4-6 graphically shows the bias in the final semimajor
axis estimate of run
similar
behavior.
D.
Both figures otherwise
Figures
4-7
and
4-8
show
show very
the
filter
histories of the total estimate position error DR, with the
3a bound PR also plotted.
Once again, the SKF runs D and F
show very similar behavior, including similar final position
errors of about 300 meters.
Both show final standard devia-
tions of about
It is interesting
10 meters.
error transient in Figure 4-8 for run D:
to note the
at 11800 seconds a
spike occurs indicating a position error of about 1.1 kilometers.
The
station
pass,
transient
after
transient
for
deviation
history.
consistent
run
a
F has
occurs
3 hour
been
at
the
of
the new
Any
such
similar
outage.
overwritten
The presence
of
by
the
such a
3a standard
transient
is
with the apparent divergence of these SKF runs;
the actual errors are 30 times their
Note that this divergence
tional observations
should
standard deviations.
is probably only apparent; addiserve
to make
their standard deviations consistent.
are drawn:
1.
start
the
errors
and
Two final conclusions
An adequate process noise model for the linearization errors due to initial condition errors would
120
I Y IY I~Y
· YY.YYYIY.YYYY
· YYYY
YLY
0
YYY -YY
-
S
U _YY
0
iI..
S
O
aO
~~~~~~~~~~~
ld*m
9
0.
r~~~~
3
S
**,
. Sw
|~~
3
~~
*
~
3
s
I
Ia
Im
a
.4
a
I
I
~t
O^
,
·
I
·
w
·
3
·
·
I
§
I~~~~~ ,l
', 3
*
3
i
*
,i
I
·
3
I-
IEU U
~~~~~
·.
II
IJ
ltd
*
3
^
~~~~i
.Y
·
·
*
s
s
O *
I ;
:
3~l
I~~~~~~~~~~~~mI
S'
3
II
:
me
I
1.4
I
Ia.
* a.
:j . rl
~~~~I.
m·
D-
a
P.
io
2
7
S3
a3
I
II
m~~~~~~~~~am
o o
td
~r ; r
o
e
o
*ur~~~~~
0
.4 0
.4
Ill
~
S -. ~~~~r
~ ~o ~~ I ~~~~~~~o
a
S
.4
.4
0
III
·
~P
I~O~oyn~O~u~uD1
c~tu u Pwurwou~oun
121
TEST CRSE 2: WNMTST ERROR HISTORY
ELEMENT ERROR HISTORY
o DR
PA
En
a)
a)
.0
-4
.1
oo0
T
Figure 4-5.
seconds
SKF Semimajor Axis Error History, Run F
122
TEST CRSE 2:
WNMTST ERROR HISTORY
ELEMENT ERRORHISTORY
DR
PR
O
,
a)
-l.ri
000
T
seconds
Figure 4-6.
SKF Semimajor Axis Error History, Run D
123
TEST CRSE 2:
NMTST ERROR HISTORY
a
ELEMENT ERROR HI STORT
o ODR
+ PR
0
OC
C
a
a0
0
C
C
C
Cv
a
-,c)
C
g0
g
0
1r4
6
aC
C
a
U0
Isoo
3D00
S0ooD
D
750
9000soo
:SOU
i 2000
5SD0o
seconds
Figure 4-7.
SKF Position Error History, Run F
124
15030
TEST CRSE 2: WNMTST ERROR HISTORY
ELEMENT ERROR HISTORY
o
DR
+ PR
-,En
'p
0
rri
10r
ooo
T
seconds
Figure 4-8.
SKF Position Error History, Run D
125
help
prevent
discussion
such
aparent
in Aooendix
Jazwinski's
[241
A
divergence.
provides
kdaotive
one
Noise
The
aooroach.
filter
gives
another oossibility.
2.
In the same vein, filters using global linearizations without further compensation are likely to
suffer convergence problems.
This can be taken as
directly motivating the dvelopment
4.3.3
of the ESKF.
ESKF Properties
This section describes a series of tests investigating
the
performance
the
?arameter s were aeld
tion above.
impact
of
five
input
arameters.
Three
of
in common with the SKF investiga-
Process noise sensitivity
not investigated;
%was
-11 ESKF runs used process noise option 2 from Table 4-7.
The
two
new
input
arameters
and
the
motivation
for
their
consideration are:
(i)
ESKF interpolator implementation:
sion of the
presented
SKF design,
for
structures
use
two
im'3lementations
according
being
In the discus-
to
the
accomodated.
were
interoolator
Equation
(3-9)
applies to the case where only the short periodic
coefficient interpolator is used; equation (3-10)
is employed
osculating
used.
(ii)
31
These
matrix
hen the short-arc
position and
are otions
and
(3-10)
assume
the
short
periodic
126
velocity
1 and
comiutation:
3oth
that
interpolator for
the
function
is also being
2, respectively.
equations
31
matrix
state
(3-9)
(i.e.,
oartials
matrix)
is being computed.
This corresponds
to
the linearization of the short periodic functions
about
the nominal
with
the
not
SDC
important
convergence.
the
B1
has
B1
shown
to
Experience
that
SDC
the
B1
estimation
[19]
matrix
is
accuracy
or
The SKF tests above did not employ
matrix.
matrix
state
trajectory.
for
A
the
ESKF
not
simple:
the
by
the
option
Option
2
1
includes
perturbation,
J2
large
if
the
significant estimate bias will be introduced
tion;
becomes
truncate
a
neglect.
Aa
is
to
enough,
its
correction
decision
neglects
the
matrix
due
to
analytically
to
B1
computed
computa-
B1
zeroth order in the eccentricity.
Table 4-9 summarizes the input parameters
for the ESKF.
investigated
Table 4-10 presents the test results.
All of
ESKF test runs show a marked improvement over correspondinq
SKF runs.
1.
This data supports the following statements:
The approximate
EPC procedure does not seriously
impact ESKF performance
2.
The
truncated
impact
3.
More
ESKF
relinearization
4.
The
model
performance
frequent
performance
force
[A, B];
(significantly)
not
trajectory
does not necessarily
of
updating
and
improve ESKF
F, G];
the
B1
ESKF accuracy
127
seriously
[A, C];
nominal
[C, D and
computation
does
matrix
can
improve
[C, F] ([D, G]);
Table
4-9
Summary of ESKF Test Otions
Input
Parameter
Ootion 1
Otion
1. Force Model
full
truncated
2
Comments
see Table
4-7
2. Trajectory
Uvdates
un1ate after
uodate at the
each
end of the
station
pass
3. Mean
EARLYORB
init.
4. ESKF*
mode
see Table
4-7
observation
span
PCE
EPC
PV intero.
off
see Table
4-7
PV intero.
on
nv off:
(3-9)
or) on:
(3-10))
5. Short
oeriodic
oartials
comoutation
neglect
B
the
matrix
comoute
the
B 1 matrix
B1 is
comnutel
analvtically
incluiq
J2 at e
*
hen used, the
osition and velocity
steo size of 120 seconds for a total
123
interoolator
had a
san
of 4 minuts.
A
Table 4-10
ESKF Parameter Test Results
Input
Parameter
Test Runs*
A
B
C
D
E
F
G
1
1
2
2
2
2
2
1
1
1
2
1
1
2
1
2
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
1
2
2
.197
.213
.133
.196
.231
.166
.174
Cross RMS
.020
.024
.019
.013
.026
.016
.019
Along
.930
.336
1.153
1.345
.559
1.185
.500
1.
force
2.
nodel
trajectory
undates
3.
mean
init.
4.
ESKF
5.
31
mode
matrix
Performance**
Radial
* all
R~MS
RMS
test
runs used
8
diag[l.0,10
P
=
Q
= diag[0l
10
covariance
.10
,10,
12
8
,10-
10-
12
6
arameters
6 lo6]
,10
,10-
14
,10-
14
,10
12
** oerformance is measured by computing the root mean
squared residuals of the differences between the Cowell
truth ehemeris of Table 4-3 and a 24 hour predicted
eohemeris based on the final filter estimate.
129
5.
The
ESKF
using
the
position
and
velocity
interpolator can produce satisfactory predictions.
The
data
in
Table
4-10
is limited
in depth;
quite
clearly only very conservative conclusions can be drawn from
these results.
And yet, even in this light, statement
above is startlingly conservative.
(5)
The reason for this lies
in the limitations of the prediction error RMS performance
measure used in Table 4-10.
Figures
4-9, 4-10,
and
4-11 show
the dominant
along
track prediction errors for runs A, E, and G, respectively.
Each of the ESKF runs uses the same strategy to reduce the
prediction
the
RMS
RMS:
the along track error is biased such that
is reduced
by
the distribution
of
initial
errors
the mean semimajor axis and the mean-mean longitude.
cases,
both
the
semimajor
axis
longitude estimate are too large.
axis causes a decrease
growth
in
longitude
the
will
causing the reduction
summarizes
the
mean
the
mean
The increased semimajor
cross
longitude.
zero within
Thus
errors
the mean
the predict
in the along track RMS.
final estimation
Keplerian elements
and
In all
in the mean motion, implying slower
too-large
error
estimate
in
span,
Table 4-11
in the osculating
for each of the runs A, E, and G.
Note
the generally superior accuracy of run G.
The inadequacy of the prediction error RMS performance
measure alone
is indicated
Figures 4-12 to 4-14.
tion
error
Notice
that
history
by the filter history plots in
These figures show plots of the posifor
runs
run G produces
followed by run A and run E.
A,
E,
and
the smallest
G
respectively.
position
errors,
The histories for runs A and G
are essentially the same; both ESKF runs produce meter-level
130
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133
ws-Wwv
Table 4-11
Final Keolerian Element ESKF Estimation Errors
Run
Element
A
a*
9.0
e
3.6
i**
2.2
3.4
2.9
a**
2.7
3.5
1.9
-54.0
+220.0
-33.0
-12.0
-290.0
+20.0
0.3
1.5
c**~
M**
0.5
x 10-5
8.0
*
E
4.2
units are meters
** units are microradians
134
2.7
x 10-5
3.3 x 10a'
TEST CRSE 2: NNMTST ERROR-HISTORY
ELEMENT ERROR HISTORT
O OR
+ PR
En
..Ia)
J
Q
0
-I
.,-I
T
seconds
Figure 4-12.
ESKF Position Error History, Run A
135
TEST CRSE TWO: STELLITE
WNMTST
ELE,1ENTERROR HISTORY
Y DOR
Z PR
Co
I
o
I
I
I
1
1
1
1
I
I
I
I
I
o
ID
i1
o
oo
o
td.
o
W o
,fq
0
N
150
i5
0
;~U 7 0
60
00
100
k
§
13 0
15000
0
o
o
o
7
O0
1500
3000
4500
6000
T
7500
9000
10500
6
0
13500
seconds
Figure 4-13.
ESKF Position Error History, Run E
136
15000
TEST
CRSE 2:
NNMTST ERROR HISTORY
ELEMENT ERBOR HISTORY
o OR
+ PR
0
U)
,-I
o
Ix
o000
seconds
Figure 4-14. ESKF Position Error History, Run G
137
accuracy
history
in
of
their
run
transients
E
is
estimates.
The
fundamentally
position
different.
error
Fstimate
of several kilometers occur during both tracking
station passes
within
final
200
and the final estimate
meters.
This
results from the additional
degraded
is only accurate
performance
linearization
to
probably
of the two body
element transform from equinoctial variables to position and
velocity.
Certainly the results of the observation
model
nonlinearity test of Section 3.4.3 indicate that the element
transformation nonlinearities are often of the same order of
magnitude
care
must
as
the
be
observation
exercised
in
model
the
use
nonlinearities.
of
the
position
Thus
and
velocity interpolator with the ESKF.
The significant results of this section are:
1.
The
in greatly improved performance
ESKF results
over comparable SKF runs, both in the filter error
history
ESKF
and in the prediction RMS; the fact that
performance
does
not
change
greatly
with
parameter variations is also important;
2.
Calculation
the
of
B1
matrix
can
be
important
ESKF
algorithm
for ESKF accuracy;
3.
The
coefficient-interpolator-only
gives
much better
performance
than the position
and velocity interpolator version.
4.3.4
LKF Properties
The LKF runs discussed here required making choices for
three input parameters.
tion parameters.
The first two are typical estima-
They are
138
All
(1)
covariance parameter selection, and
(2)
coefficient of drag estimation.
LKF
runs
used
the
same a
priori
covariance;
it was
selected to be consistent with the initial condition errors
and has value
6
= diaq[100.,100.,100.,10-
P
-6
10
6
,10
]
Many values for the process noise strength were tested in a
trial and error search for the best value.
All choices were
diagonal and used the same variance for all position coordiA process
nates and all velocity coordinates respectively.
noise choice
diaq[l0 - r
diaq[
=
is
denoted
by
Q
=
below.
If r=O then
replaced
by zero.
Several
[r,s]
-r
,10
in
the
-s
,lO
tested
-s
lO
-s
,lO
presentation
the corresponding
of the runs
drag estimation.
-
,Q
of
elements
the impact
results
in
of coefficient
are
of
While accurate draq coefficient estimation
cannot usually be achieved over observation spans as short
as that of this test case, the presence of the additional
estimation
variable
does
sometimes
improvements.
139
allow
performance
The
emulate
last
SKF
input
parameter
tested
end-of-integration
allows
qrid
trajectory
The two options here allow relinearization
pass
and
global
linearization
the
over
the
LKF
to
updating.
after a station
observation
span,
respectively.
The LKF test run parameter choices and the corresponderrors are shown
ing RMS prediction
in Table 4-12.
nhese
results support the following statements.
1.
Coefficient
of
Drag
estimation
for
this
short-arc
problem does not significantly help or hurt performance
2.
The
[A, B and
dependence
D,
of
E].
LKF
noise
strength selected
ing.
Runs
noise
should be modelled
velocity,
unmodelled
this;
[H, I] and
contrary
on
performance
process
is complex and interest-
[D, F] indicate
to
that process
for position as well as
its
acceleration.
the interesting
the
interpretation
Runs
as
an
[A, C] contradict
thing about run A is that
the large value of position process noise caused
such
an increase
in the semimajor
axis variance
during the data outage between station passes that
the
LKF thought
at the
start of the second pass.
The changes in perform-
ance
E
noise
from
tion
run
scalinq
performance
3.
the orbit was hyperbolic
J
to
H
changes
to
are
to
B
as
also
the
process
interesting:
is not a monotonic nor a convex func-
of the scaling.
Short-arc
or
station-pass
relinearization
improve the performance of the LKF.
140
may
.~~~~~~~~~~~~~~~~,i,
|
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cr
m
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(U
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ge
-
141
C
0H-
-
E,
Figure
4-15
shows
typical LKF run (D).
magnitude
each
are
initial condition
results
osition
error
history
of
a
Two transients of about 50 kilometers
shown.
The
first
corresponds
error; the second results
outage between station
The
the
of
to
the
from the data
asses.
testing
the
of
the
LKF
are
best
described by two additional conclusions.
The global linearization of the satellite dynamics
1.
used by the
gence
when conver-
in the presence of large initial errors is
required.
preference
2.
LKF is not appropriate
This
is consistent
for the EKF over
Imorovements
must
be
with
the
common
orocess
noise
the LKF.
made
in
modelling for convergence situations.
this
is advocating
the use
of
the
Essentially
Gaussian
Second
Order Filter when there are large initial errors.
This
filter
covariance
augments
the
redicted
with a orocess noise-like
measurement
term.
This
correction term depends on the estimate covariance
and measures the
4.3.5
EKF Properties
This
of
robable linearization error.
section presents
the effects
the results of an investigation
of the a oriori
covariance
strength selection on EKF performance.
drag
was
estimated
in
all
experience.
142
the
and
process
noise
The coefficient of
tests,
based
on
LKF
NMTST ERROR HISTORY
TEST CRSE 2:
ELEMENT ERFOR HISTORY
Z PR
DOR
C
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,
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000
Sa
f'"~
z
.
~,
.
I,
seconds
Figure 4-15.
LKF Position Error History, Run D
143
.
11500
iOseconds
I, ,,
.
s5000
initial EKF runs showed the same poor
The
RMS prediction performance as corresponding
to select a process noise
inability
for good performance
error
and
the
LKF runs.
to the development
process
covariance
noise
transformation equations discussed in Appendix A.
routine
to
covariance
a
transforming
for
mean
A utility
a
equinoctial
position
corresponding
a
This
strength by trial and
led directly
of
implementation
(or worse)
priori
velocity
and
covariance was also implemented.
A
total
of
conducted.
The
a
the same as for the LKF
used was either
priori covariance
(Option
tests were
EKF
eight
1),
P1
6,10 6,10 - 6
0.,100. ,10
diaq[100.
or the transform of the SKF/ESKF a priori covariance (Option
2), given in equinoctial coordinates as
P2 = diag[l.0,10-
The
process
diagonal
noise
trials
used
was
10
-8
,1n - 6 ,0
either
one
66
10
of
a
]
series
of
the
second SKF
As in the LKF discussion
above, the
or else
process noise option.
8
the
transform
of
symbol [r,s] is used to denote a diagonal process noise
Q1 = diag [l0 r,1
0
-
144
r 0- r l-s
l0-s]
-
Recall that the process noise strength used as Option 2 of
the SKF tests
is
Q2
=
[
-12 '10-12
10
-
10-14 10-12]
Note that this value of process noise was also used in all
of the ESKF runs.
The options used for each of the eight EKF tests and
the resulting performances are presented in Table 4-13.
These results support the following statements:
1.
The
of
dependence
EKF
performance
on
the
proces
noise strength used is complex and unpredictable.
This is probably due to the time varying nature of
the linearization error being poorly modelled by a
process
constant
Second
Order
noise
Filter
A
strength.
would
probably
Gaussian
reduce
this
problem.
2.
A
process
noise
strength
giving
acceptable
RMS
prediction performance was selected [run F].
3.
The transformation of the a priori covariance and
the process noise strength from equinoctial coordinates to position and velocity coordinates gives
Both
performance.
RMS prediction
acceptable
transformations
performance
appear
[runs G, HI.
145
to be
required
for
best
Table 4-13
EKF Parameter Test Results
Test Runs
Input
A
Parameter
A
B
1
1
1
off
[138,22] [0,13]
Priori
E
D
C
F
G
H
1
1
1
1
2
[0,141
[10,14]
[0,10]
SKF
SKF
Covariance
Process
Noise
Performance
Radial RMS
2.2
5.3
2.2
10.2
11.5
0.31
3.0 0.99
Cross
2.5
2.6
2.5
3.3
7.7
0.03
0.4
58.6 132.4
53.1
263.1
243.9
9.47
RMS
AlongRMS
Notes: (i)
the coefficient
C
D0
2.0
(true
Drag was
value)
(2 =
84.3 23.4
in all
estimated
runs;
o-6
10)
c
riori Covariance:
(ii) A
Option
1 = same as LKF
Ootion
2 = transform
SKF means otion
(iii) Q
Q
=
of
0.02
[r,s] means
unless
2,
Q = diag [
r = 0, which
of SKF
the transformed orocess noise
- r1
means
- s ,1-S
Q = diag [o,,0O,o10-S,10
14 6
10 ,0
-r
,
1
s01
0
-
]
The
Since
EKF achieved
best
erformance
these runs used very different
in runs
methods
F and
for a
H.
riori
covariance
initialization and process noise calculation, a
comparison of their Performance is of interest.
detailed
Figures
4-15
and
4-17
prediction errors.
run
H
final
is much
semi
respectively.
elements
than
axis
The
for
runs
their
dominant
along
track
The growth of the along-track error for
larger
major
show
F
for
errors
run
of
F.
295
final
errors
and H
are
in
shown
This
is due
meters
all
and
of
95
the
in Table
to
the
meters,
Keolerian
4-14.
These
errors are consistent with the element errors for both runs
over the last fifty observations; they are insensitive to
the
final
filter
The
oosition
outout
filtering
errors
for
through 4-21.
is striking.
time.
histories
runs
F
for
and H
the
are
semimajor
shown
in
axis
and
Figures
4-13
The difference between the respective plots
The
lots
for run
F
show
aooarent
divergence
and large errors for much of the observation span, with good
convergence only in the latter oart of the second station
pass.
The olots for run H show good convergence throughout
the observation span, although the semimajor axis plot shows
explicitly the final bias reoorted in Table 4-14.
3oth runs
have similar position errors for the last fifty observations
of
for
the
second
run
YH
semimajor
the
ass:
It
axis
is
and
corresponding
about
10 meters
interesting
osition
to
error
histories
for
for
run
observe
how
5 meters
similar
the
for run H are to
histories
the
F and
SKF
and
ESKF;
this
provides a good verification of the transformation method.
There
filtering
are
two
histories
explanations
for
runs
F
for
the
and
H.
restatement of statement (1) above:
147
difference
The
first
in
the
is
a
a time varying process
0
*
0*
0
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14
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14
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lor
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Table 4-14
Final Keplerian Element EKF Estimation Errors
Run
Element
F
H
a*
96.0
-295.0
e
-7.SE-6
-2.4E-5
i**
5.4
a2l**
k3.3
*
M**
-2.7
-2.0
-250.0
+240.0
* units are meters
** units are microradians
150
720.
-557.
TEST CRSE 2: WNMTST ERROR HISTORY
ELEMENT ERROR H!STORT
o
DR
PR
w
U4
rr4
0.
r-i
.1-
U
1aUU
JUUU
%UU
biUU
(UU
aUUU
aULUU
1eUUU
seconds
Figure
4-18. EKF Semimajor Axis History, Run F
151
I3zJJ
UY
&5UI
TEST CRSE 2:
NMTST ERROR HISTORY
ELEMENT ERROR HISTORY
DR
PI
0
En
a)
4-,
a)
E
0
r-
.14
14
o0
seconds
Figure 4-19.
EKF Semimajor Axis Error History, Run H
152
TEST CRSE 2:
WNMTST
ERROR HISTORY
EL EMENT ERROR HISTORY
Y QR
Z PR
0
a1
0
H
r-1
·cA
I
seconds
Figure 4-20.
EKF Position Error History, Run F
153
TEST CRSE 2: WNMTST ERROR HISTORY
ELEMENT ERROR HISTORT
Y DR
Z PR
o
C)
-4
01)
0
v-I
.r4
T
seconds
Figure 4-21.
EKF Position Error History, Run H
154
noise
The
a
between
of
noise
process
error.
to account the EKF linearization
is required
value
F
run
the
of
representative
a
strikes
probably
balance
convergence
requirement of the first station pass and the steady state
requirement of the latter part of the second station pass.
of the
The second explanation has to do with the geometry
noise
process
the
variables:
of
geometry
equinoctial
variables is uniformly valid throughout a satellite's orbit,
while
geometry
the
inertial
of
and
position
velocity
coordinates changes with the time varying constraint imposed
by the current satellite location within the orbit.
explanations
are
needed
to
for
account
Both
differences
the
between runs F and H.
Test Case Two Performance Summary
4.3.6
This
section
summarizes
the
results
of
the previous
four sections and provides efficiency estimates and comparisons with CDC and SDC baselines.
The DC runs were subjected to a slightly more strenuous
test case.
In addition
to
the
initial
and CDC were required to estimate
an initial
with
truth value of 2.0.
errors
EARLYORB elements, the SDC
given by use of the appropriate
starting
condition
estimate
the coefficient of drag,
of
2.1 as
compared
to the
Both runs converged to a final estimate
of 2.08, illustrating the difficulty in short arc estimation
of the drag coefficient.
these
DC results with
estimates
Truth.
give
And
the
respectable
Two factors allow
filter
tests.
predictions
second, the drag
comparison of
First,
DC
the
Cowell
estimate
varies
against
coefficient
both
over a range well including the true value over the course
155
for convergence.
of the five iterations required
final drag
Thus the
gives the best
coefficient
estimate
apparently
4-15 lists
the results of the best
fit.
Table
from each
of the
sections
above
for
other filters and with CDC and SDC.
comparison
with
the
The runs are identified
two runs are
letter from each section;
by the appropriate
filter run
listed for the EKF, reflecting the two essentially different
methods
for process
noise and a priori
covariance
choice.
These results indicate that the SKF and ESKF have converged
faster than the LKF and EKF for this test case.
in Table 4-16.
The results of timing tests are given
The times recorded are the CPU times required for processing
from the two station passes.
the observations
These timing
estimates should be conservative for two reasons:
1.
The
span
observation
was
very
short,
hours, compared with typically allowable
tion
grid
lengths
of
up
to
1 day;
only
3.5
integra-
since
much
of
the cost of semianalytical ephemeris generation is
due
short
to
integration
periodic
grid
force
coefficient
evaluations
computations,
and
any
increase in grid length will further increase the
timing advantage of the SKF and ESKF.
2.
The complexity
tion
allows
of a semianalytical
more
room
for
force evalua-
efficiency
gains
by
model truncation or code optimization
than does a
comparable
Most of the
Cwell
present code
force evaluation.
for semianalytical
in the RD GTDS testbed was
156
satellite
theory
implemented primarily
Table 4-15
Test Case Two Performance Summary
IK-------------Performance
CDC
SDC
Test Runs
SKF
(F)
ESKF
LKF
EKF
EKF
(G)
(J)
(F)
(~)
Radial RMS
.030
.051
.185
.174
2.79
.313
. 986
1.986
Cross
RiMS
.005
.007
.371
.019
.885
.033
.015
Along
RMS
.725
.367
1.145
. 500
31.9
9.47
28.4
Table
4-16
Test Case Two Timing Estimates
Filter Run
CPU Execution Time*
SKF B
0:30.97
SKF C
0:29.33
SKF D
0:15.49
ESKF G
0:17.58
LKF H
0:38.69
EKF F
0:40.90
* units are minutes:seconds
157
to verify accuracy, and not to achieve operational
system
this
efficiency.
is
tested
given
for
position
B1
the
by
A preliminary
the
SKF.
The
and velocity
matrix
two
force
timing
interpolator
calculation
indication
for
model
of
options
impact of
the
for the SKF and
the
ESKF
are
also
given.
The
results
of this test case
ive
indication of SKF and ESKF performance.
a very promising
Further testing is
required.
4.4
Conclusions
This chapter accomplished several important tasks:
1.
A survey of the problems involved in SKF and ESKF
testing and performance evaluation was presented,
and a resulting test methodology was detailed;
2.
Results
indicating successful software validation
were presented;
3.
The
transformation
equations
covariance
and
result
correspondnqg
in
process
noise
for
the
were
filter
a
priori
verified
histories
to
when
employed;
4.
The ESKF was shown to achieve position estimates
with accuracy equal to that of the conventional
EKF for the test case considered.
158
The
principal
simulated
data
test
case
discussed
in
this chapter was a short-arc case measuring the convergence
properties
of
the. SKF
and
Of
ESKF.
greater
importance
in
many applications is the steady state filtering accuracy in
the presence of real world errors.
of
the
next
chapter
provides
issue.
159
The real data test case
an excellent
test
of
this
Chapter
5
THE REAL DATA TEST CASE
This chapter oresents the results from the alication
of
the
EKF
and
low altitude
previously
the
ESKF
earth
to
the real
observational
data
of a
satellite.
The data had been obtained
from the Aerospace
Defense Command (ADCOM) for
use in orbit determination
is used to discuss
studies at CSDL.
the effects of model
This test case
errors on steady
state filter performance.
The
employed
described
organization
in Chapter
first,
of
4.
this chapter
The
formulation
are
to that
of
case
the
test
by GTDS.
The
second section
the actual filter results; only the EKF and
examined,
is
including some interesting results on the
GEM 9 gravity field employed
discusses
is similar
based
on
the
results
from Chaoter
4.
SKF
The
final section summarizes the important results.
5.1
Test Case Formulation
Nine days of
DCOM tracking data provided the basis for
the filter tests presented in this chapter.
sents the tracking history for Sace
Vehicle 10299 (SV10299)
in the ADCOM catalog over the time period
1977 to September
observational
7, 1977.
data, a
tests.
batch estimation
CDC performance.
from August 30,
provided
CSDL with the
tracking network description,
history of geomagnetic
determination
ADCOM
The data reore-
and a
and solar activity for use in orbit
P. Cefola
[191 conducted a series of
tests sing this data, comparing SDC and
The filter test case formulated here was
based on his experience.
160
The data employed did not include a satellite descriotion
or
satellite
addressing
initial
two
discusses
the
section
these
conditions.
issues,
this
modelling
test
of
tional forces, the observational
In
the
addition
case
drag
to
formulation
and
gravita-
data, and the question of
filter performance measurement.
Some
knowledge
estimate
its aerodynamic
force modelling.
fied
as
of the satellite
the
characteristics
in order
for correct
The satellite can be tentatively
COSMOS
characteristics
is required
947
[191.
satellite,
based
Confirmation
on
is offered
drag
identi-
its
by
to
orbital
the
fact
that reasonable drag coefficient estimates result when the
COSMOS 947 mass and area data [40] are used.
The results from Chapter 4 indicate that the RD GTDS
EARLYOR3
EKF
an
orogram gives
tests,
and that
adequate
set
initialization.
satisfactory initial conditions
the simole
of
EPC
corresponding
The osculating
iteration
mean
(4-1)
for
orovides
elements
for
E.ARLYORB elements
'ESKF
and
the
corresponding mean EPC elements used for this test case are
shown in Table 5-1.
Once again the EPC procedure converged
very quickly.
This table also oresents the estimated errors
in the Early
Orbit
elements,
commuted
as the
difference
between the EARLYORB-based elements and the best CDC and SC
estimates generated during Cefola's work.
Notice that both
the initial position errors and the semimajor axis error are
quite
large,
interesting
so that this test case will
convergence
the Early Orbit
choice
the
of
a mean
ESKF
tests.
oroblem.
mean equinoctial
equinoctial
The
The
covariance
161
estimated
elements
a oriori
orovide
another
errors
resulted
covariance
transformation
in
in the
for
use
in
equation
Osculating
Table 5-1
and Mean Filter Initial Conditions
Early Orbit Keplerian Elements
a
e
i
C4
W
My
Osculating
EPC Mean
6630.316
0.008686
72.826
126.216
71.332
96.966
5622.453 (km)
0.008572
72.337 (deg)
126.21.1 (deg)
57.393 (deg)
100.432 (deg)
Truth Minus Early Orbit Perturbations
Mean
Osculating
15.33 (km)
39.70 (km)
38.76 (km)
43.6 (m/s)
4.5 (m/s)
2.2 (m/s)
57.5 (km)
AVX
AVy
AVz
11
r_1H
Mean Equinoctial
PO
-0
=
13.36 (km)
Aa
Ah
I I
A
0. 0002
ak
0. 0013
0.0048
0.0035
Axrl
0.0032
57.4
11j
(rad)
(kin)
Priori Covariance
diag [100,10-
7 - ,107
-
0-6]
Position and Velocity Transformed Covariance
· 11 12 13
14 15 16
22 23 24
0.577872937D+02
-0.9107241940-02
0.935358494D+02
25 26 33
-0.2668856910-01
34 35 36
0.694130954D-01
44 45 46
55 56 66
0.1110074680-03
0.100737766D-03
-0.2478117620D+02
-0.1135780110-01
0.2333669910D02
-0.4293646320-01
-0.926367328D-01
-0.1406315620-04
-0.3830309730-05
162
0.2710832.380+02
0.3301723950-01
0.3619211290-01
0.1289029320D+03
0.3595369220-01
0.2584691780-04
0.495533978D-04
(4-2) was used to comoute the corresponding osculating Oosiand velocity
tion
a oriori
for use
covariance
with
the
KF.
These covariances are listed in Table 5-1.
The many filter tests resented in Chaster 4 indicate
that the correct choice of a orocess noise model is essential
for
filter
oerformance.
The
desire
orocess noise model of this test case to the
to
relate
the
robable real-
world force model errors led to the derivation and alicaresented in Aooendix A.
tion of the algorithms
The calcu-
lations detailed there led to a orocess noise strength of
= diag [2E-3,2E-16,2E-16,2E-17,2E-17,3E-16]
for use with the
was obtained
SKF.
by use of
The orocess noise used with the EKF
the transformation
equations
RD GTOS contains two density model ootions:
Jacchia
1971
Density
Model
or
the
1964
Atmosphere Density Model can be used.
considerations
[41],
the
simpler
Harris-Priester
Density
(-13).
either the
Harris-Priester
Based on oerational
Harris-Priester
Density
Model was selected.
The
Model
uses
different
density tables according to the current value of the mean
solar
radiation
flux,
F 1 0 .7.
Table
5-2
resents
values
for the solar radiation flux and the daily average value of
the
geomagnetic
30,
1977
and
index,
Seotember
Ap,
4,
small variations in either
for
1977.
each
day
Evidently
between
there
August
were
only
arameter, so only small changes
163
Table
5-2
Solar Radiation Flux and Geomagnetic Index [History
F10.
A
Aug 30
35.5
4.9
Aug 31
34.7
6.0
Sept 1
33.1
4.1
Sept 2
84.2
4.1
Seot 3
37.2
3. 4
Seot 4
34.2
8.0
Day in
1977
164
in
the
atmosoheric
during
that
density
rofile
the tracking period of
the
average
interest.
geomagnetic
second three day san,
should
index
occurred
Notice,
however,
does
increase
in the
reflecting a small geomagnetic
and
an accompanying
increase
in the
The
Harris-Priester
density
table
used.
have
The difference
between
the
atmospheric
for
Fl
storm
density.
7=10
tabular value
was
for the
solar radiation flux and the actual values was one motivation for estimating
the coefficient of drag; uncertainties
in the satellite's mass and aerodynamic area rovided another.
The filter tests used an initial value of 2.0 for the
coefficient
reflecting
of
the
drag
and
an a
aoproximately
priori
variance
20% uncertainty
of
0.333,
in the drag
force model.
All
of the filter tests used the GEM 9 Gravity Model
[36], which is the most recent gravity model available in
the CSDL version of RD GTDS.
A truncated version of this
gravity
model
was
used,
degree and order retained.
with
only
terms
through
eighth
The gravity model was truncated
for two reasons:
1.
Most of the tracking
data was not of very high
accuracy, and so did not warrant using a very high
precision force model; and
2.
Batch estimation tests [191 indicated that use of
the truncated field gave better
redictions
than
when the full (21x21) field was used.
The filter test force models
gravitational
radiation
also included the third body
forces due to the moon
ressure
was
neglected
the satellite.
165
due
and the sun.
to the
Solar
low altitude
of
Each filter test processed the observations recorded on
August
30,
satellite
muth,
The
1977.
A total
of
ten
that day, taking
elevation,
and
observations
Typical
standard
between
30. meters
tracked
deviations
not
of very
for
the
were
high
taken.
quality.
observations
1.5 kilometers
for
the
Range, azi-
rate observations
usually
and
stations
576 observations.
range
were
radar
were:
range, between
0.01 degrees and 0.04 degrees for azimuth and elevation, and
er second and 10.0 meters ner second for
between 1.0 meters
the range-rate
observations.
5-3 gives
Table
an explicit
account of each satellite station nass, listing the orbit it
in, the average satellite
occured
the
true anomaly during
ass, and the
oass, the elansed time since the last station
number of observations taken during the current
ass.
This
data will be important for analyzing the filter oerformance
results presented below.
Chapter
In
comparing
a
ephemeris.
filter
filtered
was
performance
ephemeris
with
the
history,
is the actual satellite
which
is unknown.
A
measured
simulation
In a real data orbit determination
truth ephemeris
city
4,
by
truth
problem, the
osition and velo-
truth
ephemeris
was
defined for this real data test case by using the converged
ephemeris estimated by a long arc CDC.
for
this
ephemeris
was
converged ehemeris.
tracking
1977.
data
Table
orovided
by
The CDC and SC
from
August
30,
A consistency check
the
SDC
used the three days of
1977 through
September
1,
5-4 summarizes the truth model used by the CDC
for generating
the real data truth eshemeris.
ponding SDC truth model is given in Table
analytical
corresponding
truth
ephemeris
agrees
166
5-5.
excellently
The corresThe Semiwith
the
0o
a
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N
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CN
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CX
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167
C
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4
E
Table
5-4
Real Data Test Case Truth Model
satellite:
eooch:
SV10299
August
30,
reference frame:
1977,
0 hrs.,
0 min.,
0 sec.
Mean of 1950
coordinates:
Position and Velocity
Keplerian
x = 3544.2402
y = -5517.4040
z = 1140.6632
a = 6644.2294
(km)
e = 0.009311916
i = 72.932366
(deg)
(km)
(km)
(km)
Vx = 2.655753 (km/sec)
Vy = 0.115046 (km/sec)
S= 125.7742 (deg)
= 68.576733 (deg)
Vz = -7.260209
M = 99.994253
drag coefficient:
area:
= 6.1
mass:
m = 5700.
(km/sec)
CD = 1.3421073
(m2)
(kg)
density model:
1964 Harris-Priester Atmosphere
( 10.7 = 100)
gravity field:
8x3 (GEM-9)
third bodies:
integrator:
steo size:
(deg)
Moon, Sun
12th order Cowell/Adams predictor corrector
45.0 seconds
168
4
Table
5-5
Real Data Test Case Semianalytical Truth Model
satellite:
epoch:
SV10299
August
reference frame:
0 min.,
1977, 0 hrs.,
30,
0 sec.
Mean of 1950
coordinates:
Mean Keplerian
Mean Equinoctial
a = 6635.3109
(km)
a = 6635.3109
e = 0.00979471
k = -0.00961044
= 0.60004557
(deg)
i = 72.969833
(deg)
= 125.770814
q = -0.43230204
X = 294.359281
mass:
m = 5700.
(kg)
1964 Harris-Priester atmosphere
(F10.7
size:
= 100)
3x3 (GEM-9)
Moon, Sun
third bodies:
integrator:
(deg)
2
A = 6.1
gravity field:
M = 103.22775
(m )
area:
density model:
w = 65.360707 (deg)
(deg)
CD = 1.8408701
drag coefficient:
step
(km)
h = -0.00189093
4th order Runge-Kutta
43400.
sec
= 1/2 day
AOG
8xO averaged otential
second order J 2 e and Drag-J2 couoling
lunar-solar single averaged (arallax=8,4),
SPG
zonals (3x0), e
(8x3),
m-dailies
tesserals (x3),
2
2
e
e
drag 8 frequencies
second order J 2 e
second order J2-m-dail
169
coupling (8x3), e
e
Cowell
truth
Table
5-6,
differences
fit san
ephemeris;
this
which
gives
between
the
the
is shown
RMS
quantitatively
position
two ephemerides
and
in
velocity
for the three day
and for the three day predict span.
The agreement
between the two ephemerides is quite good.
Examination
of
the truth models
Tables
5-4
and
5-5
shows
that
both
of
also use the truncated 8x8 gravity model
instead of the full 21x21 field.
for the filter tests:
diction performance.
The reason is the same as
the truncated field gives better
ore-
Table 5-7 presents the results of five
tests conducted in the study of this gravity model anomaly.
The
setup
for each
test
was
the same
as for
truth models, with only
the gravity model
coefficient
estimated
of drag was
the CDC
and SDC
changing.
in all tests.
The
The RMS
values shown are the weighted RMS observation residuals for
the given three day span; the predict span is from Seotember
2, 1977 through Seotember 4, 1977.
mance
degradation
can
The prediction
erfor-
be seen by comparing
the first and
third tests or the second and fourth tests.
The parallel
CDC and SDC tests show that the
theory dependent:
lite SV10299
was
henomenon is not satellite
it is a force model anomaly.
in a 201x331 kilometer
89.93 minute period;
orbit
The sateland had an
it made sixteen revolutions
per day.
Calculations indicated that the resulting resonance with the
sixteenth order geopotential harmonics was very share:
periodic
motions
period were
5-7
at
approximately
introduced.
indicates
that the
960
times
the
The last test presented
resonant
geoootential
long
orbital
in Table
coefficients
(the coefficients of sixteenth order and degree varying from
sixteen through twenty-one) account for the degradation of
the prediction performance.
170
5-5
Table
Semianalytical and Cowell Truth Ephemeris
RMS Differences
Fit Span Differences
August
30, 1977 to
September
POSITION RMS
1,
VELOCITY RMS
(KM)
RADIAL
CROSS TRACK
ALONGTRACK
TOTAL
1977
(KM/SEC)
0.446040-02
0.113520-04
0.193940-01
0.226290-04
0.512490-05
0.25830D-0'&
0.11655D-01
0. 30620-01
Predict Span Differences
September
2, 1977 to Setember
1977
VELOCITY RMS
POSITION RMS
(KM/SEC)
(KM)
RADIAL
CROSSTRACK
ALONGTRACK
TOTAL
4,
0.435560-02
0.310770-01
0.118310-03
0.362220-04
0.530050-05
0.101660+00
0.106390+00
0.123850-03
171
Table
5-7
GEM 9 Gravitational Coefficient Model Anomaly Data
Run
Type
Gravity
Field
Fit Span
RMS
Predict Span
RMS
CDC
3x3
1.63
13.41
1.842
SDC
3x8
1.46
19.06
1.841
CDC
21x21
1.61
57.40
1.891
SDC
21x21
1.34
53.45
1.339
SDC
plus (16,16)
-(21,16)
1.36
55.75
1.883
Drag
Coefficient
8x3
Notes: 1.
GEM 9 Gravity coefficients were used.
2.
The drag coefficient was estimated in all runs.
3.
All runs included drag effects and lunar-solar
third body erturbations.
4-
O
172
There
results:
1.
are
The
several
possible
sixteenth
explanations
order
9
GEM
these
for
geopotential
coefficients may be in error;
2.
Additional
sixteenth
order
geopotential
coeffi-
cients beyond the sixteenth through twenty first
degree coefficients may be required by the sharpness of
the resonance
and
the
low
altitude
of
SV10299;
3.
The Harris-Priester Density Model may be inaporooriate for use with the GEM 9 Gravity Model in this
share resonance situation; and
4.
The
of drag
coefficient
with poor
in the
tests
erformance may have been biased, either
by not using
a time
varying
density
Harris-Priester
which
estimated
may be too far from
or by using
model,
table
for
F1 0
the real values
the
7=1001
of about
35.
It is interesting
to consider
[19] two
facts
about
the
GEM 9 gravity modelling process [36]:
1.
The
GEM
9 coefficient
ing data
tion
did
from any
use
solution
16 rev/day
several
did
not
use
satellites;
satellites
with
track-
the solu-
orbital
frequencies ranging from 12 revs/day to 15 revs/
day;
173
2.
The
coefficient
error
estimates
GEM
9 solution
showed a
oroduced
sharo
increase
16th and higher order coefficients,
with
the
error
estimates
by the
for
for the
in comparison
the
12th
order
through 15th order coefficients.
While
these
sixteenth
of errors
results make the possibility
order
GEM
coefficients
9 geopotential
in the
at least
plausible, clearly more work is required before any credible
conclusions can be drawn.
In oarticular, the whole question
of
must
atmospheric
modelling
be
carefully
investigated,
both in terms of general model errors and in terms of the
effects of the small geomagnetic
storm that occured during
the predict san.
5.2
Real Data Test Case Filter Results
This
examining
data
test
section
the
presents
erformance
case
the
of
formulated
results
the
EKF and
above.
of
several
ESKF
These
for
tests
the
results
real
are
of
interest for several reasons:
1.
Real observational
data of a low altitude
earth
satellite was processed, so that real-world errors
in
the
observations,
the
gravitational
force models, and the event
and
drag
times and coordinate
transforms are present;
2.
rocessed so that the filters
data was
achieved a steady state, in spite of the large
Enough
initial errors; and
174
3.
The
rocess noise strength was computed using the
model developed in Appendix A, so that the success
of
that
model
can
be
judged
from
the
filter
histories.
The results from five filter tests are reported in this
Four of the tests are ESKF tests; there is only
section.
one
EKF
reflecting
test,
the fact
that
the a
riori
covari-
rocess noise have already been selected.
ance and the
The
four ESKF tests investigate the effects of the integration
length
grid
and
the
force
model
on
ESKF for this long arc test case.
tigated
in Chanter
4
for
results presented there
the
the
erformance
of
the
These issues were inves-
short
arc
test case.
indicate that ESKF accuracy
The
has a
slight dependence on certain force model truncations and a
much greater dependence on the integration grid length used;
these
issues were discussed
question
of whether
after a station
designed
or
not to uodate
ass.
to extend
in Chanter
those
The
4 in terms of the
the
nominal
trajectory
SKF tests in this section were
results
to a long
arc case.
The results from all of the filter tests are shown in
Table
5-3.
This table presents the force model truncation
and integration grid length options selected for each ESKF
run, the final coefficient
and the RS
between
truth
of drag estimate
trajectory error statistics
a one
day filter estimate
ephemeris.
These
results
statements:
175
for each run,
for the difference
prediction
support
and the CDC
the
following
Table
5-3
Real Data Filter Performance Results
Inout
ESKYF Test
The ET(F
Runs
Parameters
1
Force
Model
Test Run
A
3
C
T
1
1
1
2
2 Integration
Grid Length
43200.
29000.
9600.
21000.
Radial RMS*
O.040
0.053
0.056
0.046
0.047
Cross RMS*
0.072
0.079
0.81
0.057
0.048
Along RS*
0.405
1.073
1.646
0.262
0.507
1.89
1.99
2.06
1.88
1.93
10.
Performance
Estimated Drag
Coefficient
* units are kilometers
Notes:
1. The
3 1 matrix
was
comguted
in all
2. All
filter
tests
included
estimation.
3. The E'F used the transformed
and
rocess
noise
of the
SKF' runs.
drag
coefficient
initial conditions
SKF runs.
4. The ESKF force model otions are:
option 1: same as Semianalvtical
Truth,
Table
5-5
option 2:
imoroves on otion
1 by taking all
first order short oeriodics to fourth
order in e, and including J 2-raq
drag-drag
coupling in the AOG.
and
4
176
1.
The accurate estimation of the coefficient of drag
is essential for ensuring the prediction accuracy
of
a
low
shown
altitude
that
a
coefficient
satellite
oercent
five
can
orbit.
change
along
produce
It can be
in
track
the
drag
errors
of
several kilometers after one day;
2.
Neglecting
the
contaminated
then
all
errors
as
by drag coefficient-induced
of
performance
track
along
the
filter
to within
tests
show
the tolerance
being
errors,
equivalent
of the SDC and
CDC truth ephemeris agreement (see Table 5-6);
3.
Each
of
the
filters
reproduces
essentially
the
estimation results of the CDC truth ephemeris.
Six aspects of the filter test runs of Table
5-8 are now
considered in detail.
5.2.1
The Effects of Drag Coefficient Errors
The effects of drag coefficient errors can be assessed
either directly or indirectly.
A direct assessment
can be
derived by considering how the effect of a semimajor axis
rate propagates
perturbation
through the mean motion,
in the mean
anomaly;
into a resulting
a perturbation
in the mean
anomaly maps directly into an along track error.
An indirect assessment of the effects of a drag coefficient error is presented here, by comparing the ESKF run D
with
the EKF
run.
Figures
5-1 and
177
5-2
show
the
along
track
. t
4
.4
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e
2
.'a,
S
2
S
2
I"!.4
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179
l
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?
differences between the CDC truth ehemeris
ESKF filter
track
predictions
differences
are
for
August
signed,
and
truth minus the given filter
31,
are
and the EKF and
1977.
commuted
rediction.
zero
after one day, growing
difference.
Some
of
this
as
these
difference
mates
initial
orbital
CDC
from an essentially
error
element
the
KF and ESKF of
is due
to
orbital element differences between the EKF and ESKF
tions;
along
These two figures
imply a total along track error between the
1.5 kilometers
The
differences
initial
redicand
the
between their resoesctive drag coefficient esti-
are
given
in Table
5-9.
All
of
the
differences
are
very small.
The critical differences
are
the
error.
semilnajor axis
for along
error
and
track error growth
the
drag
A simole two body dynamical analysis shows that the
semimajor
axis error
accounts
for about
300 meters
final 1.5 kilometer trajectory difference;
accounted
for by semimajor
of
coefficients
differ by only three
acceptable
error
rag
axis error
coefficient
difference.
in a
rag
results justify essentially
results
coefficient
in
Table
5-3
when
of the
the remainder
is
rate induced by the
Notice
that
the
drag
ercent, which is a quite
coefficient
neglecting
comparing
estimate.
These
the along track RMS
the
oerformance
of
various filters.
5.2.2
Preliminary Timing Results
The results of Table 5-3 indicate that the
KF and ESKF
recover from large initial errors to essentially reproduce
the CDC truth ephemeris estimates.
This conclusion is
interesting for two reasons:
180
Table
5-9
EKF and ESKF Element Differences
Element
Difference /EKF-ESKF D/
a
2.1 meters
e
4 x 10-
i
0.7 microradians
1.6 microradians
X
CD
0.5 microradians
0.05
= 3%
Table 5-10
Real Data Test Case Timing Estimates
Filter Run
CPU Execution Time*
ESKF 3
0:17.10
ESKF D
0:25.53
1:57.24
EKF
* units are minutes:seconds
1'81
1.
CDC
and
SDC
tests
starting
with
the
same
EARLYORB based initial elements and constrained to
process the whole day of observations in one batch
convergence can be obtained
of observational
data
are
could not converge;
-when shorter
arcs
processed
first
to reduce
the
size
of
the
initial
errors.
2.
The second reason has to do with efficiency.
filter runs
data
CDC
required
and started
truth
run
one
only
from large
required
through
the
initial errors;
the
four
pass
The
asses
and
started
from much smaller initial errors.
It
appears
accuracy
with
that
the
filters
as the batch
increases
can
achieve
differential
in efficiency
about
corrections
the
same
estimators
and without sacrifice of any
convergence properties.
Timing
estimates
Table 5-3 are
for
the
KF
and
two
resented in Table 5-10.
tests
tests
allow
for
processing
comparison
from
by the respective
the observations.
of the timing
force model options used.
tests
The times given are
based on the GO sten CPU times required
filter
ESKF
requirements
The
ESKF
of the
two
The CPU times shown indicate that
the ESKF tests are between four and seven times as efficient
as
the
corresponding
EKF
test.
This
estimate
of
efficiency advantage of the ESKF should be conservative,
indicated
in Chapter
4.
182
the
as
5.2.3
ESKF Integration Grid Length Selection
This
section discusses
the effect of the integration
grid length on ESKF performance. The results from the short
arc
test case of Chapter
4 show that shorter
integration
grid lengths do not necessarily yield improved performance.
An upper bound on the integration grid length is imposed by
the bounds
on
the
region
of
validity
for
the
linearization
assumption required by the ESKF.
Figures
5-3 through 5-7 show the filter histories of
the osculating semimajor axis error for each of the filter
tests presented in Table 5-8.
The variable DA is the actual
error, while the variable PA is a three standard deviation
bound.
The histories start after the initial condition
transient has decayed, to allow the later detail to be
seen.
The semimajor axis error was computed relative to the
CDC truth ephemeris.
The error history from the EKF is
included (as Figure 5-7) to provide a baseline against which
the impact of varying the ESKF integration grid length can
be seen.
The
integration
grid
lengths used by each
were presented in Table 5-8.
a detailed
study
ESKF
test
Using this data together with
of the EKF history
and
the histories
of the
ESKF tests yields three important observations:
1.
Each ESKF history shows transients at the end of
each
integration
grid;
the transient
at the end of
the first integration grid was always the largest,
due to the large error in the initial orbital
elements (58 kilometers).
183
REFiL DPTI TEST CFSE: SV10299
ELEMENT ERROR I! -
CR
-RIES
PR
La
to
r
EC
a)
aI)
0
H_1
.r
T
seconds
Figure 5-3.
ESKF Semimajor Axis Error History, Run A
(grid length = 43200 seconds)
184
RERL DTR
TEST CE:
S1 0299
ELEMENT ERROR HISTORIES
DR
X PR
Lf
EC
to
a)
-.0I
0000
T
seconds
Figure 5-4.
ESKF Semimajor Axis Error History, Run B
(grid length = 29000 seconds)
185
RErL
TEST
DRTR
CASE: SV10299
ELEMENT ERROR HISTORIES
DOR
PR
co
ca,
0
rH
-rq
seconds
Figure 5-5.
ESKF Semimajor Axis Error History, Run C
(grid length = 9600 seconds)
186
RERL
DRTR TEST CRSE: SV10299
ELEMENT ERRORHISTORIES
o
R
PR
U
go
an
4J
0
,
T
seconds
Figure 5-6.
ESKF Semimajor Axis Error History, Run D
(grid length = 21000 seconds)
187
REPL
TEST
DRTR
CSE:
SV 1 0299
ELEMENT ERROR HISTORIES
o DR
PR
0
u
U)
w
-P
Q)
0l
0r
,--
T
seconds
Figure 5-7.
EKF Semimajor Axis Error History
(integration stepsize = 10 seconds)
188
2.
The
ESKF
tests
grid
integration
and
or much
other
larger
much
show
and
lengths
The
transients.
smaller
B
D
used
and
show
ESKF
intermediate
the
used
tests
integration
much
lengths
grid
condition
initial
larger
smallest
transients.
Notice that the integration grid for
run A ended
in the middle
of a long data outage
5-3); this may
(see Table
artially
account
for
the large size of the transient for that test.
3.
All
from
of the filter histories
60,000
tested
seconds on.
agree quite
This
implies
closely
that
the
integration grid lengths do not cause any
appreciable accuracy differences once steady state
filter operation has been achieved:
the lineari-
zation errors for a steady state nominal trajectory are small.
The error histories
for the other orbital
elements
show a
very similar behavior, verifying these statements.
5.2.4
EKF and ESKF Steady State Performance
This section describes the steady state performance of
the EKF and ESKF.
Performance is measured by comparing the
EKF and ESKF position estimates with those of the CDC truth
ephemeris.
Recall that the filter tests use the same force
model as the CDC truth ephemeris.
is being
measured
is the ability
Thus the performance that
of the
filters
to reproduce
the batch DC estimate, in the oresence of real-world observation and force model errors.
189
Figure
histories
variable
5-8
for
DR
and
the
is
Figure
EKF
the
5-9
show
the
and
the
ESKF
test
actual
error,
while
position
test
the
error
D.
The
variable
PR
represents a three standard deviation bound.
Note that the
histories
transient has
decayed.
1.
start
after the initial condition
There are four important observations to make:
The
two
filter
identical,
histories
confirming
all
of
are
essentially
the
ESKF
design
assumptions for this test.
2.
Both of the filter tests achieve a final position
error of less than 50 meters with respect to the
CDC truth eohemeris;
the prediction error results
presented in Table 5-3 verify this accuracy of the
final filter estimates.
3.
The
in
transients
error bound
oresented
error
result
decreases
its
error
5-3.
bound
Each
reflects
reflect
the
in
and
from the observation
in Table
or
osition
the
the
history,
increase
in
a
outage;
data
orocessing
the
of
new
than
the
observations.
4.
The
actual
position
error
is greater
three standard deviation bound,
indicating either
slow filter convergence or apparent divergence.
The
osition
error
tests, test D, was
show
larger
initial
history
resented
for only one of the ESKF
here.
The other
condition-related
ESKF
tests
integration
grid
transients that are not oertinent to this discussion; note,
190
TEST CFRSE: SV10299
REFRL DTR
ELEMENT ERROR HISTORIES
Y
o0
CR
Z PR
11111111111(II·I_·
·
0
o
o
0
f .
o0r
(0
0
U)
1-4
a)
a)
0
0
0
0
0)
0,
0
(U
Cu
0
0
0
Co
0
cE
,
II
:
I
r
ll
l
oo000oo20000
l
i
b0
0ooo
oo0
T
Figure 5-8.
l
so 0oo
]
-
l
60000
70000
80000
seconds
EKF Position Error History
191
·
900oo0
I
--100000
|-
REAL
ORTR TEST
CRSE:
SV10299
ELEMENT ERROR HISTORIES
Y
Z PR
OR
0
a
0
0.
0
0
0·
a
0
a
ars
a
a
'9.
0
0
En
04
a)
0
,a) 0
r=1
0
,--I 0a
-,-
=r
0
0
en ·
a
0a
.J
a
0
C
C
c
3
1000C
2CCO'
30CCO
iCccO
SCooO
60000
70'8
800oo
seconds
Figure 5-9.
ESKF Position Error History, Run D
192
S3C3
_
__
100000
however, that the steady state performance of all the filter
in the actual
identical, both
tests were essentially
lot
trace and in the final estimate accuracy.
of the filter
The next section discusses the question
process noise modelling, which is related to the question of
apparent filter divergence, mentioned in (4) above.
5.2.5
Process Noise Model Verification
This section discusses
the
erformance of the process
noise model used in this real data test case.
The process
noise model is of interest for two reasons:
1.
The correctness of the process noise model determines
the
of
accuracy
the
filter
estimation
results; and
2.
The
value
the
of
process
computed using the method derived
The
results
from
this
strength
noise
test
in Appendix
case
reflect
was
A.
the
validity of that method.
The EKF and ESKF position error histories presented in
the
previous
the position
deviation
section
show
an apparent
errors consistently
bound.
Note
that
filter
divergence:
exceed the three standard
the
position
errors
remain
bounded, implying that the process noise model is adequate.
A more detailed
investigation can be made using the element
histories.
193
Figures 5-19 and 5-11
resent the osculating Keolerian
and mean equinoctial element histories for the EKF test and
the ESKF test D, respectively.
Variables prefixed by a 'D'
refer
to the actual error,
refer
to the corresponding
Two of
the
Keplerian
while those
refixed
by a
'P'
three standard deviation bound.
variable
names
require
The name 'CO' means capital omega, which is
explanation.
, the longitude
of the ascending node; the name 'LO' means lower case omega,
which
is
w,
the
argument
of
erigee.
11
of
the
other
variable names follow directly from the usual notation.
Figure 5-10 includes the osculating Keplerian histories
for
the
inclination,
the
longitude
of
the
node,
and
argument of perigee; each of these element histories
an
apparent
observation
only
mean
divergence
span.
element
apparent divergence
for
a
significant
portion
The mean equinoctial
element
showing
divergence.
an
anparent
Q
of Q is fundamental, since the
the
shows
of
the
is the
The
rocess
noise calculations
for both the EKF and the ESKF are based
on the process noise strength commuted in mean equinoctial
coordinates using the method given in Aendix
A.
Three possible explanations for the mismodelling of the
process noise for Q are roposed:
1.
The observations offer poor observability of Q, so
that either a longer data arc or multiple asses
through the data are required;
2.
The
filters
are
tracking
the
real
satellite
dynamics, as indicated by the observations, rather
than the truncated and aporoximate force model
employed by the CDC truth ephemeris; and
194
A
,_^L
DTA TEST Cr'2 ,
ER9RR HISTORIES
ELE'E-NT
o
\' i399
'Z
D'
PR
I "E
C-
DI
+ PI
0
,l
En
r=;
r--I
.H
T
_
·
__
·
·
·
·
h
0
08-
O
0
a,
I
-O
.
.
0
C)
U,.
'0
10000
2
000 0 3
S0000
00004000
60000
T
70000
80000'
900ooo
In
C1
rd
10
-
-
u-a
0
seCUnUIus
Figure 5-10. EKF Osculating Keplerian Element Histories
195
I
I L0000
RERL ORTR
TEST
CASE:
S10299
ELEMENT ERROR HISTORIES
A PCO
O DCO
·
1
_·
1
0
·
·
& PLO
DLO
·
·
·
·
·
DM
·
g
o
c~
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.re
Cd
(a
rd
=f
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AS
\Y
m
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Y
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_
I
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c~0
(D
'
I
I
10000
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i
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20000
30000
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40000
50000
s000
I
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70000
I
90000
80000
A iI
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rd
,(d
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A~~~~~~~~d
1
i
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=tr --tlkm--qg
v"~~~~--n
u,
10
10000
i
30000
20000
i
i
i
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i
50000
i
i
60000
i
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70000
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90000
80000
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i
i
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rd
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°
o
0
C.
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10000
20000
30000
0000
Figure 5-10.
T
5b00 '
6O0o
seconds
continued
196
700
000
'
bo
000o
o000
I'____
100000
REAL
DTR
CqSE: SV10299
TEST
ELEMENT ERROR HISTORIES
o GR
O OK
DH
I
+ PK
w
C4
0n
5-
0)
0r::
H
.-I
.!
Ii
.
I
I
!
I
I
I
I
I
i
I
I
I
I
I
I
I
MI
r
u
r
r
0,
I
.
6.
0o
I
I
0o
'0
10000
l
I
.
I
20000
.
.
30000
40000
T
.
o0000
.
.
60000
.
.
70000
.
.
80000
.
o
C
C
C
C:
seconds
Figure 5-11. ESKF Mean Equinoctial Element
Error Histories, Test D
197
A*-.
.
90000
1 10000
RE-L
CSE:
DRTR TEST
SV 0.299
ELEMENT ERROR HISTORIES
O DL
DP
I
T
·
·
·
1
__
·
·
·
·
·
__
0o
0
-0
:ooXo
!
;0
_
10000
2b0000
b00
oo
oo
00
T
sbo000000 000
U2
.r-
ra
(dr
$4
T
seconds
.Figure
5-11. -- continued198
70000
80000
I moow
3.
There
is an error
statistics
used
in either
the
in
the method
process
noise
or
the
strength
calculations of Appendix A.
The
second
specific.
The
ephemerides
proposed
can
explanation
discussion
of
the
CDC
be
and
effects
tends
of this resonance
noise
process
to
more
SDC
truth
indicated that the satellite was in very sharp
resonance with the sixteenth order geopotential
The
made
were not considered
in the
A.
filter
of Appendix
calculations
harmonics.
follow the observtions
Now
more closely
a
than does a
differential corrections algorithm, so it is possible that a
bias was introduced by the neglect of resonance in the force
model.
Certainly the error history of Q indicates that only
It is interesting to note that
a small bias is required.
resonance generally causes motions of the orbital plane, and
hence has a significant impact on the equinoctial elements P
and
.
In conclusion,
process
noise model
successful.
completely
of
5-10 and
develooed
Additional work
5-11
show
that
the
in Appendix A was basically
must
be
done
in
order
to
explain the apparent divergence of the estimate
the equinoctial
5.2.6
Figures
element
Q.
Semianalytical Modelling Errors
This section oresents evidence
indicting a very small
semianalytical force model error when compared with the real
world dynamics.
199
from
Figures
5-12 and
the
truth
final
CDC
estimates
respectively.
smooth,
5-13
ephemeris
of
the
EKF
resent
the
inclination
for one
day
redictions
test
and
the
SKF
errors
of
the
test
D,
Notice that the EKF prediction error is quite
while
the
ESKF
prediction
error
shows
an
error
residual with a twelve hour period.
Proulx,
periodic
et
al.
[18]
investigated
errors, and successfully modelled
of such errors as resulting
average
similar
element
rates
short periodics.
due
a large
from the coupling
to oblateness
12
and
hour
ortion
between the
the m-daily
Proulx's model was employed in all of the
ESKF and SDC tests conducted for this test case; the 12 hour
periodic error shown in Figure 5-13 is the residual error.
While
additional
work
is
required
to
account
for
this
residual error, its small magnitude does not make such work
an urgent requirement.
5.3
Real Data Test Case Summary
The results of this chaoter extend the conclusions of
Chaoter 4 in several important ways:
1.
The
EKF
and
observational
consistent
ESKF
were
used
data,
with
the
resulting
accuracy
that of
the
batch
CDC
and
the
integra-
with
to
orocess
real
SDC
estimators;
2.
The
tion
primary
grid
impact
length
linearization
of
was
the
choice
found
to
of
be
due
to
the
errors induced by the large initial
condition errors used in this test case;
200
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.1
*r4
3.
The EKF and ESKF were found to have very similar
and accuracy
filter histories
The
was
ESKF
in
especially
the
erformance;
region of steady state
4.
used,
were
parameters
input
when corresoonding
a
have
to
found
considerable
efficiency advantage over the EKF, even though the
computer software emoloyed for the ESKF tests has
not been optimized;
5.
The
process
noise
was basically
must
be
verified,
done
to
develooed
model
although
the
improve
in
A.opendix
work
additional
modelling
A
for one
element.
In
addition
to
these
results,
an
interesting
force
model anomaly concerning the GEM 9 sixteenth order gravitational
coefficients
additional
work
must
was
be
discovered;
done
mechanism for the anomaly.
203
to
a
identify
good
deal
of
and
rove
the
Chapter
6
CONCLUSIONS AND FUTURE WORK
The
primary
goal
of
this
thesis
has been
the design
of
improved sequential orbit determination algorithms by anolication of the computational methods of semianalytical satellite
theory
to
the
estimation
algorithms
resulting
from
sequential filtering theory.
Two new orbit determination algorithms are oresented in
this
thesis.
Filter
They
(SKF) and
(ESKF).
are
called
the
Semianalytical
the Extended Semianalytical
Both of these
Kalman
Kalman Filter
algorithms were designed
with
the
objective of achieving the same accuracy as existing sequential
orbit
determination
algorithms
advantage in computational
while
retaining
the
efficiency enjoyed by semianaly-
tical satellite theory.
The
ters:
SKF
the
and
ESKF
Linearized
designs
Kalman
are
Filter
Kalman Filter (EKF), respectively.
are
typically used
based
on
(LKF)
subootimal
and
the
fil-
Extended
These suboptimal filters
in sequential orbit determination
algo-
rithms and have been found to perform quite adequately.
The
use
de-
of
these
signs, since
filters
important
for
the
SKF
and
ESKF
they allow a good deal of decouoling between
the computational
filter.
is
structures
Chapter 2 oresents
of the satellite
the mathematical
theory
and the
introductions
to semianalytical satellite theory and to sequential estima-
tion theory required
for the design of the SKF and the ESKF.
204
Chapter 3 discusses
the ESKF.
the
the actual designs of the SKF and
The SKF served as the baseline for the design of
ESKF.
The
computational
flow
of
each
algorithm
is
explicitly detailed to indicate the interaction between the
filters and the satellite theory.
that
the
solve
vector
consists
The SKF and ESKF assume
of
the
mean
equinoctial
elements generated by the semianalytical orbit generator and
any unknown dynamic
arameters, such as the coefficients of
drag or solar radiation pressure.
most
natural
accounts
for
satellite
one
all
for
of
the
the
This solve vector is the
given
ossible
satellite
theory,
and
interactions
between
the
theory and the filter solve
vector.
Chapter
3
also presents the results of three numerical tests conducted
to verify
ESKF.
simplifying
design
assumptions
for the SKF and
These design assumptions allow:
(1) neglect of the
state transition matrix
in process noise
the use of the LKF state
correction
the
use
ESKF,
periodic
and
(3)
the
corrections
coefficients
osculating
in
of
calculations,
prediction
the
elements.
calculation
equations
3 1 matrix
instead of recomputed
the
of
the
the
for
(2)
for
short
short
periodic
ESKF
redicted
Each of these assumptions has a large
impact on the overall efficiency of the SKF and ESKF; the
latter
two represent
the
interaction
theory formulation of semianalytical
of
the
erturbation
satellite
theory with
the filtering techniques employed.
The
test
cases
results
from
are presented
two
end-to-end
in Chapters
orbit
4 and
The test case of Chapter 4 used a
determination
5.
short
arc of simu-
lated data for a low altitude polar satellite to examine the
performance characteristics
of the SKF, ESKF, LKF, and EKF
205
during their transient response to a large initial condition
error.
No errors were introduced
in the filter dynamical
models, but errors were added to the observations.
istic
initial
condition
error
GTDS early orbit algorithms.
was
generated
by using the
The performance of each filter
was tested subject to several of the applicable
meters.
A real-
input para-
These parameters included:
1.
the
a
riori
covariance,
2.
the
3.
the filter force model,
4.
relinearization
rocess noise model,
strategy
for the nominal
trajec-
tory,
5.
coefficient of drag estimation,
6.
BI matrix short
eriodic linearization or trunca-
tion,
7.
the semianalytical interpolator structure, and
8.
mean
early
orbit
initialization
for
the
SKF
and
ESKF.
The
ways:
by
performance
the accuracy
of
each
test
of ephemeris
was measured
redictions
in three
based
on the
final filter estimate; by the oosition error history during
the observation
rocessing
soan;
and by the efficiency
esti-
mate given by the CPU time required for execution.
The test case of Chaoter 5 extended the results of the
short-arc
processed
state,
test case of Chaoter 4 in two ways: the filters
a sufficient
amount of data to achieve a steady
and real satellite
real-world
errors
force model.
occurred
tracking data was orocessed,
in
the
observations
and
in
so
the
Only the EKF and ESKF were used to process
206
this
data.
The
(transformed)
EKF
and
ESKF
tests
used
corresponding
initial conditions and process noise models.
The nrocess noise model used in these tests is develooed in
Appendix
tion
of
errors.
impact
.
This model is aproximate
the
effects
of
robable
real
world
force model
Several ESKF tests were conducted to determine
of
semianalytical
force model
integration stepsize on ESKF
ance
but allows considera-
measures
accuracy,
used
execution time.
erformance.
in Chaoter
the observation
4
ere
by
a
The same
used
here:
span error history,
and
the
erform-
orediction
and the CPU
The orediction accuracy and the observation
span error history were measured
provided
truncations
the
batch
relative
differential
to the baseline
corrections
(DC) orbit
determination algorithm.
The
real
data
test
case
interesting force model anomaly:
9 16th order
Priester
gravitational
Amtosoheric
Density
very
of
the
sharp
DC
algorithms.
resonance
with
uncovered
an
the interaction of the GEM
coefficients
Model
estimation caused a degradation
ance
formulation
and
with
the Harris-
drag
coefficient
in the prediction
The
given
satellite
the
16th
order
erformwas
in
a
geopotential
harmonics.
The results of several DC tests defining
anomaly are
resented in Section 5.1.
this
More work is required
to comoletely exolain this anomaly.
6.1
Conclusions
The
fundamental
presented
in this
efficiency
application
can
of
conclusion
thesis
be
made
is that
without
semianalytical
207
to be drawn from the work
substantial
loss
of
satellite
improvements
in
accuracy
by
the
theory
to
the
sequential orbit determination
supported
by
the
results
roblem.
of
the
This conclusion
short-arc
test
case
is
of
Chapter 4, used to examine filter transient response, and by
the results from the many-orbit test case of Chapter 5, used
to examine steady state filter
erformance
in the
resence
of real-world model errors.
Several other significant conclusions can be stated:
1.
The
computational
EKF
are
structures
comoatible
with
tures of semianalytical
possible
exception
of
the
2.
(3-10)
occurs
when
and Section
employed
the ESKF
the
struc-
and
the ES(F have been verified,
by
[see
4.3.2].
in the
and
is used
interpolator
assumptions
tests,
and
satellite theory; the one
The
numerical
LKF
interpolator
with the oosition and velocity
Equation
the
design
of
the SKF
both by direct
the
final
filter
Performance.
3.
Semianalytical
Satellite
Theory
offers
consider-
able flexibility for truncations in the analytical
development of the force model, with the
for large improvements
in the efficiency
otential
without
loss of accuracy.
4.
The
length selected
for the integration grid can
have a significant effect on SKF and ESKF accuracy
during
transients;
there were not any detectable
accuracy differences during steady state filtering
for the cases considered.
208
5.
The linearized
to the short
corrections
functions
obtained
important
for
the
by use of
accuracy
of
periodic
the
31 matrix
the
ESKF
are
predicted
osculating elements.
6.
The
process
filters
the
noise model
employed
by any of
impact on
tested can have a significant
resulting
estimation
accuracy.
in accounting
The
process
A was generally
noise model developed in Apendix
successful
the
for the dynamical model
errors in the real data test case.
7.
Drag coefficient estimation can be very important
for the accuracy of the ephemeris predictions
of
low altitude satellites.
8.
The Epoch Point Conversion Iteration (4-1) gives a
low-cost means of obtaining good mean equinoctial
elements.
9.
can be
The process noise and a priori covariance
successfully
coordinates
transformed
to osculating
from
mean
position
coordinates when corresponding
equinoctial
and
velocity
ESKF and EKF tests
are desired.
10.
The EKF and ESKF are able to successfully estimate
and predict satellite orbits in the presence of
real-world observation and force model errors.
11.
The
EKF and
ments in
the ESKF offer
significant
improve-
erformance over simple global lineariza-
tion algorithms
like the LKF or the SKF.
209
There
are
specific
simpler
applications,
however,
structures
computational
of
where
the LKF
SKF will make their use uniquely desirable
the
test
cases
totally unexpected.
and
[61.
is supported by the results
Each of these conclusions
from
the
considered.
None
of the
conclusions
is
It is desirable that additional supoort
This and other
be provided by further testing.
issues for
future work are addressed in the next section.
6.2
Future Work
in this thesis motivates
resented
The research
addi-
tional work in filtering theory, in semianalytical satellite
and
theory,
in
requirements
the
the
for
mentation of the orbit determination
Recommendations
this thesis.
algorithms
are also
imple-
operational
studied in
)resented for areas
requiring further testing, both to verify the conclusions of
this thesis and to establish the
erformance characteristics
for different orbit determination oroblems.
One
of
discovery
the
imoortant
the
sensitivity
of
results
of
process noise model emoloyed.
of
filter
Chapter
4
was
the
to
the
performance
This discovery motivated the
develooment of the process noise model of Appendix A, which
allowed
the
strength.
modelling
a
priori
Alternative
problem
the covariance
are
calculation
to
approaches
given
by
the
of
work
a
orocess
noise
the
orocess
noise
of Wright
[351 and
correction term of the Gaussian Second Order
Filter, which model gravity model errors and filter linearization
errors,
respectively.
The
application
of
these
alternate aoopproachesto the semianalytical filters discussed
210
herein
deserves
further
study.
In particular,
it may
be
possible to use the slowly varying nature of semianalytical
dynamics to develop very efficient implementations of these
methods.
The Dynamic Model Compensation
(DMC) method
[34]
may offer additional benefits.
Two extensions of Semianalytical Satellite Theory will
help support additional testing of the SKF and the ESKF.
The test cases in this thesis studied low altitude nearlycircular satellite orbits.
The development of explicit
third body short
eriodics will allow mean element initialization for high altitude satellites at low cost, by use of
an EPC
models
ness
rocedure.
for
the
The development
third
perturbation
body
of analytical 31 matrix
perturbation
(closed-form
in
and
the
for
the oblate-
eccentricity)
will
allow the ESKF to be tested efficiently with high altitude
and high eccentricity satellites, respectively.
Several
asoects
of the operational
the SKF and ESKF require
been done to establish
efficiency
when
imolementation
investigation.
Very
the tradeoffs between
various
truncations
of
of
little has
accuracy and
the
analytical
development of the semianalytical force model are made. The
tests presented in this thesis indicate that large increases
in
efficiency
with
only
small
losses
of
accuracy
are
possible.
be
The question of software ootimization should also
addressed.
The current implementation did not have
efficiency
zation
is
as a
rimary
goal,
9
ossible.
would be heloful here.
cations
for the
SKF
and
and
so a good
deal
of ootimi-
timing budget of the program flow
Finally, one of the imoortant appliESKF may
mous satellite navigation.
lie
in the
area of autono-
The standard Kalman
211
rediction
and update algorithms currently implemented should be rather
easily replaced
The
by a more stable
requirements
square root
for implementing
formulation.
semianalytical
satellite
theory in a small word length computer should be studied.
The last area for future work consists of recommendations
for additional
examined.
studied
There
orbit
are
determination
many
in this thesis.
extensions
test cases to be
to
For example,
the
test
erformance
cases
evalua-
tions for high altitude and high eccentricity satellites are
of
interest.
The question
of steady
state
accuracy when
very high accuracy observations are available is imoortant.
Equally important is the estimation
accuracy when there is
only a very soarse schedule of observations.
are required
of
the
to establish the exact oerformance
ESKF
when
the
osition
used (see Section 4.3.2).
future
study
stability
satellite
changes.
means
of
examines
use
of making
of
the
the tracking
force
ranid
or
of real
these
velocity
data
for
accuracy
and
errors,
such
anomaly
from the M1GSAT
a
as
density
data offers
In this regard,
model
is
roosed
atmospoheric
observational
tests.
force
filter
model
rooerties
interpolator
The last test case
deterministic
The
and
the effects on
maneuvers,
investigation
using
Further tests
one
further
can be made
satellite.
by
This
satellite should also have been in a very shar? 16th order
geopotential resonance during several days of its decay.
4
212
Appendix A
PROCESS NOISE MODEL SELECTION
The results from Chapter 4 show that filter estimation
accuracy can be very sensitive
used.
A
trial
and
error
to the process noise model
search
for
the
process
noise
strength giving the best estimation accuracy could be conducted
for the simulated data test of Chapter
4, since a
truth model existed for measuring that accuracy.
The real
data test case of Chapter 5 and efficiency requirements
general motivate
the development
for process noise modelling.
of more rigorous
test
The
case
and
indicates
transformation
of
extensions
the
methods
This appendix discusses
selection of the process noise strenqth
process
in
the
for the real data
for other
noise
situations.
covariance
from
equinoctial coordinates to cartesian coordinates for making
analogous EKF and ESKF runs is also presented.
A.1
Process
Noise
Analysis
for
Semianalytical
Satellite
Theory
Process noise models are used to account for the growth
of
the
true
errors.
estimation
error
due
to
dynamical
modelling
The basic requirement for stable estimation with a
filter is that the true errors correspond to the covariances
computer
by the filter; the squared errors should roughly
equal the computed variances.
small,
the
corrections
Kalman
gain
are not
is
made.
When the covariances are too
also
too
When
small,
the
so
the
covariances
required
are
too
large, the Kalman gain is correspondingly too large and over
corrections are made that can result in unstable oscillation
213
of the estimate
errors.
These
insights form the basis of
the three approaches to process noise modelling found in the
literature:
1.
Use
of
the
white
noise
model
for
the
process
noise, with the strength set either by trial and
2
error
or
[29],
[33];
Modelling
Markov
physical
the
with
selected
noise
the
as
a
[34]; and
Analytical
development
based
[20],
first
order
initial conditions
parametrically
performance
correlations
[3],
considerations
process
process,
dynamics
3.
by
of
on
for
force
geodetic
optimum
model
error
and
error
analysis
[35].
The
first
approach
near-linearity
and
is used here, taking advantage of the
slowly-varying
character
of
the
semi-
analytical dynamics.
The development of the process noise model assumes the
formal existence of the true dynamical model as well as the
known nominal model.
a
=
These are represented as
A2(a)
+
and
214
..
; a(to )
= a
and
(A-l)
aN
AN(
=
respectively.
Note
N)
=
true
model
+ u
that
the
N
(A-2)
contains
the
complete expansion of the averaged equations of motion, so
that no analytical
nominal model
approximations
are
made
at all.
The
is truncated at first order, consistent with
the Semianalytical Satellite Theory developed in Chapter 2.
Also,
the
first
order
term
AN
the errors in known force models.
noise;
it
is
to
be
selected
is not
exact,
reflecting
The term u is the process
to
minimize
the
difference
between aN and a.
Let
(A-1)
a
and
- a
-
aN
(A-2).
be
the
Write
trajectory
A 1 ()
error
) AN(
=
to explicitly
account for force model errors.
tion equation
for
a results
between
A1
A perturba-
from subtracting
(A-2) from
(A-1)
=
-a+
--
-N
+ 6A(a)+
with
initial
equation
A-
23aaN-a
condition
a
N
N
Aa
+
--
(A-3)
A(a) +
Ao
=
o
-
aNO'
This
accounts for all three sources of dynamical model
error
in the perturbation
equation
tical
filters; the second, third, and fourth terms on the
right hand side of (A-3) are due to:
215
employed
by semianaly-
1.
neglect of higher order terms in the linearization
of the equations of motion
required
by filtering
theory;
2.
first order force model errors; and
3.
truncation
of the asymptotic series expansion of
the averaged equations of motion
(see McClain [9]
for the general development of this series).
These error sources are modelled using process noise as
Aa
The new process
matching
of
-N
noise
Aa and
AN
+ v ;
aN
(A-4)
=
v is to be selected
AaN.
This
matching
for
is done
the
best
statis-
tically,
by equating covariances, in recognition of lack of
knowledge of v.
Equation (A-4) is a linear equation and so has a state
transition
(A-4)
matrix
solution
AaN(t)
=
*(t,t ).
The
solution
to
is
t
f
t
o
(t,T)
216
(T)
dT
(A-5)
Now
(and
semianalytical
linear,
so the
other
dynamics
VOP)
= I
(t,to)
approximation
as discussed in Section 3.4.
are
almost
is quite
good,
The trajectory error becomes
t
AaN (t)
f
=
t
dT
V(T)
(A-6)
0
Using the white noise model for the process noise and assuming it has constant strength gives the result verified
in
Section 3.4.1
A(t,to0 )
(t
( - to0))
Q
=
(A- 7)
for A can be derived
On the other hand, another expression
using
semianalytical
heuristics.
Since v is a vector
unmodelled mean element rates, it is slowly varying.
errors
models.
in deterministic
since
modelled,
also be deterministically
It can
it results
The new expression
of
from
for A
assumes that v is not random; thus the mean square value of
V
is
t
A(t,to)
t
f
=
t
f
t
0
V(T)
()
dT da
o
(A- 8)
I
vT
*
(t - t
217
o
)
Equating this result with (A-7) gives
Q
This
result
=
is
v v
at
approximate,
First, no time argument
errors,
v.
(A-9)
as
indicated
in
two
ways.
is given for the mean element rate
Rather, these rates will be computed based on
knowledge of the truncation, force model, and initial condition errors; these error rates are assumed (and verified) to
be sufficiently slowly varying.
the
prediction
the
best
interval
fit between
quadratic model
value
Second, some mean value for
must
t-t0
be
computed,
the time-linear model
(A-8).
to
give
(A-7) and
the
The mean data outage time is a good
for At, since filter divergence usually starts during
an outage.
A.l.1
Real Data Test Case Process Noise
The process
(A-1)
through
noise model
(A-9) was
initially
data test case in Chapter 5.
that
test
case
density model
was
errors
presented
above
developed
in Equations
for the
real
The process noise strength for
selected
to
account
and geopotential
for
atmospheric
coefficient
errors.
The computations giving the process noise used for the real
data test case are summarized here, step by step.
1.
At was
taken
to be
the mean
data
outage
time.
Outages on August 30, 1977 ranged from 30 seconds
to 4.5 hours.
The mean outage time was 1.125
hours or 4050.0 seconds.
218
2.
A
was
search
literature
Reference
errors.
coefficient
geoootential
to determine
conducted
[36]
gives the geoootential coefficients and estimated
for
errors
the GEM
and
coefficients
Table
A
their
Typical
in GTDS.
used
errors
are
resented
in
over
all
geopotential
-1.
value
mean
value
average
of
are1
of
was
coefficients
This
9 field
for
desired,
by means
was
comouted
the
geonotential
of
commutation
of a
v.
weighted
errors.
coefficient
The formula used was
N-
n
=
N,
rel
n=O
The numerator
in
the
errors
dard
6
arel =
3.
A
7+5
=O
due
the resulting
to the
The denominator
nrm
deviation
geonotential
2
I
represents
geoootential
(A-10)
n
to make
random
coefficient
scales
this stan-
it relative
perturbation.
variances
For
to the whole
an
8x3
field,
2.4 * 10-4 resulted.
literature
search
[381,
[371,
[39]
into
atmo-
soheric density model errors resulted in an error
standard
deviation
choice
219
of
are
= 20%.
Table A-1
Coefficients and Errors
Geootential
Taken from Gravity Model Imorovement Using Geos-3
(GEM 9 & 10)
GSFC, 1977
n,
Cn,m
1-6
6n,m
1E-6
1E-9
rel
2,0
-434.2
---
1
2E-6
3,0
0.953
---
1
1E-3
4,0
0.542
---
1
2E-3
5,0
0.068
---
2
3E-2
6,0
-0.151
---
2
1E-2
7,0
0.093
---
2
2E-2
3,0
0.051
---
2
4E-2
15,0
0.001
---
5
5E-0
17,0
0.016
---
5
3E-1
2,2
2.434
-1.393
3
1E-3
3,1
2.023
0.252
5
2E-3
3,2
0.392
-0.622
8
7E-3
4,1
-0.533
-0.465
5
7E-3
4,2
0,353
0.663
5
7E-3
4,3
0.933
-0.203
4
4E-3
..
Notes:
n,m
C
nm'
S
nm
.
= geopotential coefficient
6
= coefficient
nm
error
6
are-1 -
nn_
2_
C 2 + S2
nm
nm
220
= relative error
4.
Mean element rate histories were generated over a
12 hour
arc with
the
semianalytical
integrator.
Mean element rate contributions due to the geopotential and drag were printed separately.
gravity
field was used.
Table A-2.
calculated
An 8x8
The rates are given
in
The relative error standard deviations
in
2)
and
3)
are
used
to
compute
the
probable errors in the mean element rates of Table
A-2; these errors and the resulting total probable
error are shown in Table A-3.
5.
A diagonal
process
noise
strength
matrix
Q
was
calculated, using
Qi=
ii
v2 t
=v i At
(A-11)
The resulting value of Q is
Q = diag[2.E-8,4.E-19,2.E-16,1.E-17,2.E-17,8E-16]
The elements h and k and the elements p and q have
similar geometry.
give
these corresponding elements
strength.
Chapter
The matrix Q is normalized
to
the same noise
The value of Q used in the test case of
5 is
Q = diag[2.E-8,2.E-16,2.E-16,2.E-17,2.E-17,8E-16]
221
Ln
W
03
·*.0
l
~o
I
I
I
E O0
CO 4J- 4
I
E~
N
CO
1
_
CO
I
Lii
Bi
-0
°
fN*
I
coI
C)
I
I
C)
I
ON
d4
.
I
C)
r\
'
C)
I
CY.
C.)
I
rLnOC
J )
.
N
I
I
*II~
*
I
c I)
Cro
4
ci:'
.
C
I
'n
r4
C)
I CI
-o
E
a
i
-0gO
I'
2I
C)
H
C)
*I
i
L
I
C)OC)'o
I 4JrH
a l
l*
I
N
C
I
Li
Ol
,-I
I
I
O
(N
N
-
H
I
Li
oI3
I
Co
I
Lii
[
Li
I C)
I
C r*
IH
a)
C13
rL0H L )-4
CO
a)
I
-
.I
I !
-t
I
rH
L]
C.
2I
N
0
lJcOr
C)
-I
rH
II+
+I
O
o
o
)o
c
CJ
o
Io
-o
C
0
'Cs
-0
>
aP 3
0O
(tI C
L. cl,
C
c
-,- ,-~
O E'
-
k
tS-
(
(1)
a)
c
O
O
0
·5
,
NC
N
)
cL
)
m
O~
222
.C)
O
(t14-)r
-
r4U)Li
OrIC
X R2
,
c
U)
I
Table
-3
Mean Element Rate Probable Errors
-
V
C A *
=
--
rel
v Drag
v Grav
a
-2.E-6
0.
h
-3.E-12
2.4E-12
k
2.E-10
-2.4E-13
o
-2.E-12
4.SE-1ll
q
2.E-12
7.2E-ll1
-1.E-12
A
-4.
.
.
.
.
,
.
Total Rate Errors
-2 .E-6
-1.E-12
=
2 .E-10
5.E-11
7.E-11
-5 .E-10
223
.
!
E-10
, !
A.1.2
Process Noise Modeling Extensions
The method
used for the process noise computation
for
the real data test case of Chapter 5 can be used to model
the
process
noise
linearization)
requirement
terms
errors
due
and
to
initial
truncation
condition
errors.
(i.e.,
The
basic
is the computation of the mean element rates v
due to the error source.
The state dynamics bias correction
term from the Second Order Gaussian Filter can be used as an
estimate for v due to initial condition errors.
[9] equations
McClain's
for higher order terms in the averaged equa-
tions of motion must serve when modelling the process noise
due to mean element rate truncation.
A.2
Process Noise Transformations
The desire to compare SKF and ESKF performance with the
LKF and EKF implemented in the RD GTDS FILTER program raised
the question of the validity of such comparisons.
That is,
the semianalytical filters have parameters and inputs given
in mean
equinoctial
coordinates, while
have cartesian inputs and parameters.
the Cowell filters
The transformation of
the initial state and covariance are straightforward and are
discussed in Chapter 4.
The equations for transforming the
process noise are developed here.
The process
noise covariances
in mean equinoctial and
cartesian elements are
T
t
A (t,t
a oaa
f
(tT)
0
224
Qa(T)
a(t,T)
dT
(A-12)
and
t
=
Ax(tto)
x
0
T
I
t
x(t, T)
Qx(T)
(t, T)
dT
(A-13)
0
respectively, where
3a(t)
a(t,T)
=
(A-14)
_
aa(T)
and
3x(t)
)
%x(tT
=
(A-15)
ax(T)
The process noise strengths Q are related in the same manner
as the initial covariances.
Thus
ax(T )
Q
(T )
=
[ -
ax(T)
)
] * Qa(T) ·
aa(T)
225
[-]
aa(T)
(A-16)
The
rule
partial
derivatives
in terms
of
px can
be expanded
of
Substitution
chain
a by
ax(T)
ax(t)
%x(t,T)
by the
=[
-
aa(t)
(A-17)
]
and
*a(t,)
(A-16)
*
-
[]
aa( T)
into
(A-17)
(A-13) gives
the
desired transformation
ax(t)
ax(t)
AX(tt)
=
There are two comments.
velocity
[
aa(t)
]
a(to t o)
[
a(t)
]
(A-18)
First, the partials of position and
are time varying,
so the process noise covariance
calculated in (A-18) is not linear in time, contrary to GTDS
assumptions.
Second, the implementation of (A-18) requires
the computation of the position and velocity partials; these
partials
can
be
expanded
by
the
chain
rule
using
the
osculating equinoctial elements as intermediate variables as
done in (2-34).
Only the two body partials are used; the
B 1 matrix of short peridic partials are neglected.
226
Appendix
B
SOFTWARE OVERVIEW
The
SKF
and
ESKF
were
implemented
in
provided by the version of RD GTDS resident
Semianalytical
the
testbed
at CSDL.
The
Satellite Theory developed at CSDL has been
implemented in this version of RD GTDS.
This version of RD
GTDS also contains an LKF and EKF capability, implemented by
the FILTER program.
used
by
the
SKF
SKF and
ESKF
is
Many of the FILTER subroutines are also
and
ESKF.
summarized
The
program
below
development
as well
as
the
set
of
the
up
for
execution.
B.1
Software Description
Four new subroutines were written and forty-four exist-
ing subroutines were modified in this thesis work.
one of these routines were written or modified
ESKF
implementation,
nine
for
test
case
Twenty-
for SKF and
support,
and
eighteen for bug correction and code clarification.
B.l.1
SKF and ESKF Subroutine Descriptions
Short
descriptions
of
the
subroutines
written
and
modified for filter implementation follow:
COREST:
Updates
the
integration
nominal
trajectory
with
the current filtered correction; modified to allow
SKF and ESKF updates and CDRAG and CSOLAR solve.
227
ESTSET:
Sets
switches
to set
modified
GVCVL:
for APG
and SPPG model
SKF and
selection;
ESKF options.
Sets output titles; modified to include mean equinoctial variable names.
KF:
The Kalman Filter executive
routine; modified
to
eliminate unnecessary computations to allow efficiency tests.
KFEND:
Does
end of
final
the
filtering
nominal
estimate
trajectory
and
by making
processing
update
to
covariance
and
the
the
propagating
report
time;
modified for SKF and ESKF operation.
KFHIST:
A
new
routine
that
generates
filter
state
and
covariance
histories at observation times in car-
tesian,
Keplerian,
and
mean
equinoctial
coordinates.
KFOBS:
Controls
acceptance
of
the
next
observation,
propagation of the nominal trajectory, and computation
of
the
predicted
Filter operation;
tics
on
edited
modified
observation
for
to accumulate
observations
and
Kalman
statis-
observation
residuals.
KFPRED:
Implements the Kalman Filter state and covariance
propagation
equations; modified
to interact with
ORBITV for semianalytical state transition matrix
computation and ESKF state prediction.
228
KFSTRT:
Initializes
the
filter
observation
statistics;
correction
modified
vectors
to
and
initialize
observation residual statistics.
KFUPDT:
Implements
the Kalman Filter state and covariance
update equations; modified to accumulate observation residual statistics and to control state and
covariance history output.
OBSPRT:
Controls the computation of observation
modified
for
SKF
and
ESKF
velocity
to account
to osculating
for the mean equinoctial
and
computations
partials;
transformation
partials
position
(partials
computation is controlled by ORBITV).
ORBITV:
The
executive
for
the
Semianalytical
Theory equations of motion
tions.
Satellite
and variational
Controls AOG and SPG computations
and APG and SPPG computations
equa-
(SPORB)
(SKFPRT); modified
to control SKF and ESKF nominal trajectory updates
and
ESKF state correction
prediction
and updated
state computation.
OUTSLV:
Prints the filter estimate and covariance
initial report time; modified
at the
to allow mean equi-
noctial variables.
RESINV:
Performs initialization
tasks
for the
Semianaly-
tical integrator by initializing the state and the
partials; modified to initialize the mean equinoctial state transition matrix.
22 9
RKINTG:
The variable order Runge Kutta integrator for the
Semianalytical equations of motion and variational
equations,
written by A. Bobick
integrate
the
inverse
of
the
[20]; modified to
state
transition
matrix.
RPTEST:
Prints
the filter estimate and covariance
at the
final report time; modified to allow mean equinoctial variables.
SKFMAT:
A
new
routine
from
partials
that
computes
the mean
the
transformation
plus dynamic
equinoctial
parameters solve vector to the corresponding osculating equinoctial variables
(i.e., implements the
SPPG).
SKFPRT:
A
new
routine
that controls
state
Semianalytical
matrix
transition
computation
transition
of the
(uses the
matrix
inverse interpolator set up by
RKINTG) and the computation of the Semianalytical
solve
to osculating position
vector
transformation partials
and velocity
(implements a local inter-
polator) required by OBSPRT.
SKFUDT:
A
new
routine
that
updates
the
Semianalytical
nominal trajectory and reinitializes
tor at the end of an integration
SNGSTP:
Generates
the
first
periodic coefficient
tical Runge
Kutta
integration
the integra-
grid.
grid
and
short
interpolators for Semianaly-
integrator; modified to set up
the state transition matrix interpolator.
230
SPORB:
Controls
of
calculation
the osculating
position
and velocity and mean equinoctial elements for the
integrator at output
Semianalytical
the
integration
current
times within
Implements
grid.
the
local position and velocity interpolator and uses
short
the
modified
interpolators;
coefficient
periodic
to make ESKF updated state computations
more efficient.
Test Case Support Subroutine Descriptions
B.1.2
Modifications
support
three
to- nine
aspects
were
subroutines
of the SKF and
ESKF
required
testing;
to
summaries
of the requirements and the subroutine descriptions follow.
The coefficient of drag was estimated in two test cases
to improve estimation performance; one subroutine was modified
to support
AERO:
this
for the LKF and
capability
atmospheric
Computes
drag
Two of
the
station-specific
stations.
estimation
test
cases
Four
LKF and
by the
required
observation
and
partial
to calculate CDRAG partials
derivatives; modified
for CDRAG
forces
EKF.
the
EKF.
to have
capability
statistics for C-Band tracking
subroutines
were
modified
for
this
capability:
DSPEXC:
This
routine
capability
the
is
the
executive
for
the
in RD GTDS; it was modified
simulation
of observations
with different error statistics.
231
DATASIM
to allow
of the same type
OUTDSl:
Prints the observation statistics summary for the
DATASIM program; modified
to account for station-
specific statistics.
SETDC:
One of the RD GTDS control card processors; modified
to
read
in
station-specific
observation
statistics.
WEIGHT:
Computes
the weight
assigned
residual
by differential
to each observation
corrections
and
filter
programs; modified to account for station-specific
observation statistics.
Implementation
of
the
between mean equinoctial
process
noise
and osculating
transformation
position and velo-
city frames required modifying three routines.
was
modified
to
support
the
real
data
Semianalytical
mean
One routine
process
noise
computation.
AVRAGE:
Computes
the
modified
to print
element
the rate history
tional and drag perturbations
rates;
for gravita-
for the real data
process noise calculation.
ANOISE:
Computes the process noise covariance contribution
to the Kalman
fied
to
Filter prediction
include
the
equations;
equinoctial
to
modi-
cartesian
process noise transformations.
SETAPC:
Initializes the Kalman Filter a priori and process
noise covariances; modified to account for process
noise transformation initialization.
232
SETFIL:
The control card
rocessor for the RD GTDS FILTER
program; modified
to allow control card input of
the process noise transformation option.
Software Bug Removal
B.1.3
Eighteen subroutines were modified to remove errors or
Since the corrections are specific to the
clarify the code.
particular
are given
names
ELERD,
EO,
RSETRK,
RPTIME,
here.
KFINIT,
only the subroutine
implementations,
subroutine
The modified
OBEDIT,
OBS,
SPCOTO,
SETRUN,
routines
are EDITOR,
OBSWF,
OBSWT,
OUTPAR,
SPJ2PR,
VARSP,
VRSPAN,
VRSPFD, and WFCONT.
B.1.4
Interaction Diagrams
Software
tion B.1.1
interaction diagrams for key routines of Sec-
and
.1.2 are
shown
in Figure
B-1.
See reference
[3] for additional information.
B.2
Program Execution
This
section describes
the setun of the JCL, RD GTDS
control cards, and software flags in subroutines ESTSET and
HWIRE as required for SKF and ESKF program runs.
B.2.1
JCL Setup
Up to three data sets may be required
for an SKF or
ESKF run.
The Linkage Loader control cards describing the overlay
structure are currently in
233
0-)
rl)
-H
a)
4J
rd
a)
m
.1
0)
Cd
H
a)
(1)
234
cn
0
C14
(N
a)
cd
t12
fa
Ei
rd
rs
r4
C)
0
"-4
4-)
U
a,
pQ)
.
-I
H
)
p
d
Il
ro
0
(12
LI
m
.4-
m
rz
.H
235
C)
rt1
so-I
-lj
0m
0)
rd
.I
CD
-I
rH
a)
FZ4
236
SPT1 2 4 4.SKF .GTDS .OVERLAY.OBJ
The source code,
INCLUDE statements, and load modules
for the updated routines of Section B.1 are in
FORT
SPT1244. GTDS.UPDATE. OBJ
LOAD
respectively.
The state and covariance histories may be generated on
output
data
cussed below)
sets
for plotting
by setting the flags
IWUPD, IWPRD, and IPRFIL.
(dis-
Data sets of the
required format can be generated by the ALLDS command.
B.2.2
RD GTDS Control Cards
The
RD GTDS control
cards directly
impacting the SKF
and ESKF operation as well as the new control card
mented
for
station-specific
observation
imple-
statistics
are
described here.
The selection of an extended-type
Kalman Filter versus
a linearized-type filter remains unchanged from the previous
FILTER
implementation
but
is repeated
The pertinent control card is
col 1-8
FILTER
9-11
12-14
I
J
237
here
for
emphasis.
The variable
I=l selects either
selects the EKF or ESKF.
the LKF or SKF, while
The variable
I=2
J=l indicates that
process noise covariances should be computed.
Selection
semianalytical
between
and
Cowell
filters
(e.g., LKF vs. SKF or EKF vs. ESKF) for the same FILTER card
is accomplished by the ORBTYPE card.
This card selects the
orbit generator type; it has the format
The
col 1-8
9-11
12-14
15-17
18-28
39-59
ORBTYPE
I
J
K
S
T
pertinent
the
selects
Cowell
size;
are
variables
Semianalytical
version.
are
S,
and
T.
Setting
I=10 selects
Filter while
S=10 for Cowell and
The variable
integrations.
I=5
the
S sets the integrator step-
The variable
typical values
Semianalytical
I,
S=21600 for
T=1 is required
for semianalytical runs.
The process
frame
to
the
noise
transformation
cartesian
frame
for
from the equinoctial
making
a
LKF
or
EKF
run
analogous to a SKF or ESKF run is set by the INPUT card; the
format of this card is
col 1-8
INPUT
The variable
while
9-11
I
I=l selects inertial cartesian
I=2 triggers
the transformation.
process noise,
The process noise
strength is input by SPNOISE cards as described in [31; when
I=2
the
input
strength
is
taken
coordinates.
238
to
be
in
equinoctial
The new control card implementing the station-specific
observation statistics is called the "station card zero"
card by analogy with GTDS terminology for the control cards
describing
network.
other
properties
of
The new card has the form
col 1
/
2-8
9
statname
0
the
tracking
10-11,12-14,15-17
II
12
station
18-38,39-59,60-80
13
R1
R2
R
3
where
statname
= a legal RD GTDS station name
I1,I 2, I 3
= RD GTDS observation type
R!,R 2 ,R
=
3
the
corresponding
observation
standard
deviation
The
appropriate
units are meters,
centimeters
per
second,
and arc seconds for range, range rate, and angle measurements, respectively.
B.2.3
Software Switches
The software switches required for SKF and ESKF opera-
tion are set in the subroutine ESTSET, but interaction
the
ESKF
with
the
position
and
velocity
interpolator
requires the discussion of the subroutine HWIRE.
implementation
equations
of
Before the
of SKF software, these subroutines
the semianalytical
variational
motion
(AOG
+
respectively.
235
equations
SPG)
of
selected
(APG + SPPG) and
force
model
options,
The flags added to ESTSET affect filter operation and
intermediate
output.
Table
There
B-1.
The output
are three
flags are
summarized
flaqs that determine
in
SKF and
ESKF operation and one flag that enables the LKF to emulate
SKF operation by relinearization of the nominal trajectory.
The
flag
analytical
IUPD=l
causes
relinearization
nominal trajectory
of
the
Semi-
for the SKF and ESKF at the
end of an integration grid by adding the filter correction
to the
final
grid
linearization
The
inverse
state;
when
IUPD=2
the
semianalytical
is global over the observation span.
flag INTINV=l turns on the state transition matrix
interpolator;
otherwise,
the
required
inverse
is
computed explicitly.
The flag ILKFUP=l causes LKF relinearization
vals specified
by DTLKF.
Otherwise,
at inter-
the linearization
is
global.
The
selection
of
by the flag IESKF.
the
ESKF versus
the SKF
is controlled
When IESKF=2, the SKF is selected.
The
choice of the ESKF observation prediction equations changes
according to whether or not the local position and velocity
interpolator
is
being
used.
The
different
equations
employed in the two cases are discussed in Chapter 3.
The
position and velocity interpolator is switched on and off in
the
routine
INTPOS=2
HWIRE;
INTPOS=l
turns it off.
The
turns
the
corresponding
interpolator
on,
settings of the
flag IESKF are IESKF=3 and IESKF=1.
Both ESKF implementations assume that the short periodic coefficient interpolator is on.
240
Table B-I
Print Flag Settings
(1
= ON;
2
= OFF)
ISLVPT
orint the nominal trajectory elements
before and after adding in the filter
correction
IWPRD
nrint the filter state and covariance
mean
and
keplerian,
cartesian,
in
equinoctial (for SKF/ESKF) elements as
oredicted at the current observation
time
I'UPD
the
print
covariances
update
IPRFIL
this
set
corresponding
the
after
the outout
history (e.g., =43
FT43FOO1)
241
FRN
for
implies
and
state
measurement
the
filter
output
on
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