APPLICATION OF THE
EXTENDED SEMIANALYTICAL KALMAN FILTER
TO SYNCHRONOUS ORBITS
by
Elaine Ann /agner
B.S., Texas A&M University
(1980)
Submitted
to the Department
of
Aeronautics and Astronautics
in Partial
Fulfillment
of the
Requirements of the Degree of
MASTER OF SCIENCE IN AERONAUTICS AND ASTRONAUTICS
at the
MASSACHUSETTS INSITUTE OF TECHNOLOGY
June 1983
©
The Charles Stark Draper Laboratory, Inc.
-
Signature
·-
-
of Author:
Department of Aeronautics andstronautics
/I7
A ,23 February 1983
A
-
Certified
by:
Dr.
Accepted
Paul J
efol$, Thesis Supervisor
Lecturer
by:
A/trofessor Harold J. Wachman
Chairman, Aeronautics and Astronautics Graduate Committee
Archives
MAC#srS
WSTITUR
APR 2 7 1983
LIRRARES
APPLICATION OF THE
EXTENDED SEMIANALYTICAL KALMAN FILTER
TO SYNCHRONOUS ORBITS
by
Elaine Ann Wagner
Submitted to the Department of Aeronautics and Astronautics
on 23 February 1983 in partial fulfillment
of the requirements for the Degree
of Master of Science in Aeronautics and Astronautics.
ABSTRACT
The MILSTAR and ANIK Al studies served to evaluate the performance
for synchronous satellites of the semianalytical orbit determination
system.
After a summary of the development- and current operation of
Semianalytical Satellite Theory (SST) and the associated semianalytical
batch and sequential estimators, the study test procedures and results are
presented.
The MILSTAR study was used to explore the use of the semianalytical
orbit determination system to support near-autonomous operation of equatorial and inclined synchronous satellites. It was demonstrated that SST
can propagate from given initial conditions an orbit that is quite similar
to that produced by special perturbations techniques. The semianalytical
force model can, moreover, be readily explicitly truncated, so that its
adjunct filter can efficiently produce state estimates of the requisite
accuracy.
The semianalytical orbit determination system can be used onboard
an Airborne Command Post aircraft to estimate synchronous satellite state
sufficiently accurately that ground-based users can acquire the satellite. Even with limited-accuracy satellite observations only one or two
days a week, ephemeride transmission to users only monthly, and with force
model and station location errors, the Extended Semianalytical Kalman Filter was able to give user acquisition errors substantially less than 10 km
in position and 1 m/sec in velocity.
The ANIK Al study, while demonstrating the flexibility of the Goddard Trajectory System as a research tool, provided an opportunity for
demanding evaluation of all four GTDS-based estimators in a real data test
case. Telesat Canada provided high-accuracy observations of its geostationary communications satellite for estimator processing.
The Cowell and Semianalytical Differential Corrections batch estimators produced epoch solve-for states that resulted in quite similar
observation residual statistics when used in their respective propagators. These residuals were, more significantly, very much like those of
the Telesat filter estimator, which is extensively specialized for synchronous satellites.
The estimation accuracies of the GTDS-based
2
Extended Kalman and Extended Semianalytical Kalman filters were also
remarkably similar. As currently implemented, filter estimation accuracy
compared quite favorably with the Telesat extended Kalman filter.
The semianalytical orbit determination system, after slight
augmentation, could be used in a satellite tracking network. It promises
to be very much more efficient than the basically comparable specialperturbations-based extended Kalman filter.
The role of semianalytical partial derivatives matrices in batch
and sequential estimators was explored in both cases.
The studies
provoked and helped clarify areas of future work.
Thesis Supervisor:
Paul J. Cefola, Ph.D
Section Chief
Computer Science Division
The Charles Stark Draper Laboratory
Lecturer
Department of Aeronautics and Astronautics
3
ACKNOWLEDGEMENTS
I wish to thank all of those at Draper
whose
support
both
tangible
and intangible has especially helped make this work possible: Dr. Mark
Slutsky, for his collaboration in verifying the third body short periodic
functions, Mr. Leo Early, for helping me make my way through the maze that
is GTDS, and, not unimportantly,
suggesting
ways
to
fix the several
bugs
that I encountered, Mr. Wayne McClain, for the many, many helpful comments
and for the commiseration with MACSYMA work, and Dr. Paul Cefola, for
whose
support
as thesis
advisor
in this work
and at MIT I am particularly
grateful. Finally, to Mr. Stephen Taylor I extend the sincerest special
gratitude.
No words here could express my humble thankfulness for his
help and many sacrifices.
Being
fruitful,
A
associated with
and has culminated
note
of
thanks
this
group
in a strong
is
extended
has
been
enjoyable,
greatly
sense of fulfillment.
to
Mr.
Franz
Kes
and
Miss
Claire
Belanger of Telesat Canada for providing the tape of ANIK Al observations
and the annotated filter history. Mr. Kes has been especially gracious in
answering my many queries.
It has been my special pleasure to work with Elba Santos, with her
excellent support throughout the MILSTAR project. I thank her especially
now for the beautiful
and timely typing of this thesis.
The work in this thesis was supported
F04701-82-C-0088.
support.
Permission
in part by Air Force Contract
The National Science Foundation was providing personal
is
hereby
granted
by
Laboratory,
Inc.,
to the
Massachusetts
reproduce any or all of this thesis.
4
The
Charles
Institute
of
Stark
Draper
Technology
to
TABLE OF CONTENTS
Chapter
1.
2.
INTRODUCTION
Orbit Determination...........................................12
Overview
of Thesis
..........................................
12
SEMIANALYTICAL SATELLITE THEORY AND THE TWO SEMIANALYTICAL
ESTIMATORS.
........................................................
14
Orbit Propagation with Semianalytical Satellite Theory ........
14
The Semianalytical Differential Corrections Estimator .........
18
The Extended Semianalytical Kalman Filter .....................19
THE MILSTAR ORBIT DETERMINATION EVALUATION STUDY
3.2
3.3
3.4
3.5
3.6
3.7
...................
22
MILSTAR Background.................................
....... 22
Study Overview................................................ 23
Verification of Semianalytical Satellite Theory........
Semianalytical Force Model Tailoring for MILSTAR....... ....... 26
Extended Semianalytical Kalman Filter Evaluation
.......34
for MILSTAR
.................................
Assessment of MILSTAR User Acquisition Errors................. 44
Effects of Third Body A- and B 1 -Matrices on Extended
Semianalytical Kalman Filter Estimation for MILSTAR.... ....... 58
THE ANIK Al ESTIMATOR EVALUATION STUDY............................ 64
4.1
4.2
4.3
4.4
4.5
4.6
4.7
5.
12
1.2
3.1
4.
.......................................................
1.1
2.1
2.2
2.3
3.
Page
Study Motivation.-..................................... .......64
....... 64
ANIK Al Background..................................
....... 67
Study Preliminaries.................................
Cowell Differential Corrections Estimator Evaluation
for ANIK Al ............................................
......
71
Semianalytical Differential Corrections Estimator
Evaluation for ANIK Al ........................................80
Extended Kalman Filter Evaluation for ANIK Al.......... ....... 89
Extended Semianalytical Kalman Filter Evaluation
for ANIK Al................................................... 98
CONCLUSIONS AND FUTURE WORK
....................
Conclusions...........
5.2
Future Work...........
....................................
.................................... .... 110
5.1
LIST OF REFERENCES
1...........
08
....108
...............................................
156
5
LIST OF APPENDICES
Appendix
Page
A.
Third Body Short Periodic Function Evaluation Package..............112
B.
Third Body A- and B 1 -Matrices
.....................................
140
6
LIST OF FIGURES
Figure
Page
3.4.1
Geopotential Zonal Short Periodic Coefficients for
Truth Semianalytical Force Model ..............................
31
3.4.2
Geopotential Zonal Short Periodic Coefficients for
Tailored Semianalytical Force Model ..........................
32
3.5.1
Averaged Element Rates for Truth Semianalytical
Force Model ..................................................
37
3.5.2
Averaged Element Rates for Tailored Semianalytical
Force Model...................................................38
3.6.1
SYNCX Range Residuals at Beale (at Tracker) ...................
46
3.6.2
SYNCX Range-rate Residuals at Beale (at Tracker).
3.6.3
SYNCX Range Residuals at Omaha .
3.6.4
SYNCX Range-rate Residuals at Omaha
3.6.5
SYNCX Range Residuals
3.6.6
SYNCX Range-rate Residuals at Alaska .........................51
3.6.7
SYNCI Range Residuals at Beale (at Tracker).
3.6.8
SYNCI Range-rate Residuals at Beale (at Tracker) ..............
53
3.6.9
SYNCI Range Residuals at Omaha .
...............................
48
..........................
49
at Alaska ...............................
SYNCI Range Residuals
at Alaska
50
..................
52
...............................
54
3.6.10 SYNCI Range-rate Residuals at Omaha .
3.6.11
.............
47
..........................
55
............................... 56
3.6.12 SYNCI Ranqe-rate Residuals at Alaska ..........................57
4.3.1
Segment of Telesat Extended Kalman Filter History.............68
4.4.1
Cowell Differential Corrections Range Residuals
for
4.4.2
ANIK
Al ...................................................
Cowell Differential Corrections Azimuth Residuals
for ANIK Al ............................
4.4.3
; .....................
74
Cowell Differential Corrections Elevation Residuals
for ANIK Al1
................................
4.4.4
73
75
Cowell Differential Corrections Distance Residuals
for ANIK Al...................................................
7
76
LIST OF FIGURES (CONT'D)
Figure
Page
4.5.1
Semianalytical Differential Corrections Range
Residuals for ANIK Al .........................................82
4.5.2
Semianalytical Differential Corrections Azimuth
Residuals for ANIK Al .........................................83
4.5.3
Semianalytical Differential Corrections Elevation
Residuals for ANIK Al ... ..... ...... ..........
4.5.4
Semianalytical Differential Corrections Distance
Residuals for ANIK Al .........................
......84
.........
..85
4.6.1
Extended Kalman Filter Range Residuals for ANIK A1
4.6.2
Extended Kalman Filter Azimuth Residuals for ANIK A1 .........
92
4.6.3
Extended Kalman Filter Elevation Residuals for ANIK A1........93
4.6.4
Extended Kalman Filter Distance Residuals for ANIK A1 .........94
4.6.5
Long-arc Extended Kalman Filter Range Residuals
for ANIK Al .... .. o......
4.6.6
.....................
.........
96
Long-arc Extended Kalman Filter Azimuth Residuals
for ANIK Al1 ...
4.7.1
.. . .... ..
........91
...
.............................................
97
Extended Semianalytical Kalman Filter Range
Residuals
for ANIK A1
.......
...
...........................................
99
4.7.2
Extended Semianalytical Kalman Filter Azimuth
Residuals for ANIK A1 .................................... ...1
00
4.7.3
Extended Semianalytical Kalman Filter Elevation
Residuals
4.7.4
4.7.5
Extended Semianalytical Kalman Filter Distance
Residuals for ANIK A1 ..........
...
......
........101
...........................
102
Long-arc Extended Semianalytical Kalman Filter
Range
4.7.6
for ANIK A1 ......................
Residuals
for ANIK Al .....
.....
104
.......................
Long-arc Extended Semianalytical Kalman Filter
Azimuth Residuals for ANIK Al ... ,.oo
...........
8
...................
105
LIST OF TABLES
Table
3.3.1
3.3.2
3.3.3
Page
Geostationary Satellite used in Semianalytical Satellite
Theory Verification .................
..............
.......
.....24
Force Model used in Semianalytical Satellite Theory
Verification
................. .........
....* ...................
25
Summary of Semianalytical Satellite Theory Verification
Test
..........................................................
25
3.3.4
Typical Short Periodic Function Magnitudes for
3.4.1
Geosynchronous Satellites used in MILSTAR Orbit
Determination System Evaluation ............................ 27
3.4.2
Summary of Precise Conversion of Elements Testing
Synchronous
Satellites..............
of MILSTAR
SYNCX
.........
.................
26
..............................................
30
3.4.3
Tailored Force Model for MILSTAR Geosynchronous Satellites....33
3.5.1
Tracking Data for MILSTAR Filter Runs .........................35
3.5.2
Baseline Perturbations for Filter Runs
3.5.3
Process Noise Matrix used in Filter Runs ......................36
3.5.4
Summary of Extended Semianalytical Kalman Filter
Results for MILSTAR SYNCX .........................
3.5.5
... ........
0......35
...........
40
Comparison of Semianalytical Differential Corrections
and Extended Semianalytical Kalman Filter Estimators
for
SYNCX
.....................................................
41
.................
42
3.5.6
Range Bias Solve Capability for MILSTAR SYNCX
3.5.7
Summary of Extended Semianalytical Kalman Filter Results
for MILSTAR SYNCI ................
...
.........
..................
43
3.6.1
Summary of Observation Space Evaluation of Tracker
3.7.1
Effects of A-Matrix on Extended Semianalytical Kalman
Filter Estimation for MILSTAR ...
......................................
60
Effects of B1 -Matrix on Extended Semianalytical Kalman
Filter Estimation for MILSTAR
.......
......................
62
3.7.2
4.2.1
Estimate.
o. . . .. .................................................... 45
Allan Park TT&C Antenna
.................................
9
65
LIST OF TABLES (CONT'D)
Table
Page
4.3.1
Telesat Filter History Residual Statistics....................69
4.4.1
Telesat Filter State used in Cowell Differential
Corrections Initialization ..........................................
71
4.4.2
Baseline Cowell Differential Corrections Solved-for
Epoch State
..................
......................
..........
72
4.4.3
Baseline Cowell Differential Corrections Residual
Statistics........ *...... .......... .g...........
4.4.4
Solved-for Epoch State for Cowell Differential
Corrections with New Frame Transformation Data
.
.....
.........
72
...........
78
4.4.5
Residual Statistics for Cowell Differential Corrections
with New Frame Transformation Data ............
...............
78
4.4.6
Residual Statistics for Baseline Cowell Differential
Corrections without Tropospheric Corrections
..................
79
4.4.7
Baseline GEM9 Cowell Differential Corrections
Residual Statistics .. .............. ..............
80
4.5.1
Force Model used with Semianalytical Estimators
4.5.2
Baseline Semianalytical Differential Corrections
Residual Statistics............................................
86
4.5.3
Residual Statistics for Semianalytical Differential
Corrections with Time-independent Third Body Model
.........
o..86
4.5.4
Comparison of Baseline Cowell and Semianalytical
Differential Corrections Epoch Solved-for States ..............
87
4.5.5
Effects of A-Matrix on Semianalytical Differential
Corrections Estimation for ANIK A1
..........
.
.............
81
.
................
88
4.5.6
Effects of A-Matrix on Semianalytical Differential
Corrections Computation Time ..................................89
4.6.1
Baseline Extended Kalman Filter Residual Statistics ...........
90
4.6.2
Comparison of GTDS and Telesat Extended Kalman Filter
Steady-state
Variances
........ ................................
95
4.6.3
Long-arc Extended Kalman Filter Residual Statistics o........98
..
4.7.1
Baseline Extended Semianalytical Kalman Filter
Residual
Statistics
....
10
...............
........................
103
LIST OF TABLES (CONT'D)
Table
4.7.2
Page
Long-arc Extended Semianalytical Kalman Filter
Residual Statistics..............................
........103
4.7.3
Effects of Third Body B 1 -Matrix on Extended Semianalytical
Kalman Filter Estimation for ANIK A ..........................106
4.7.4
Effects of Third Body B1 -Matrix on Extended Semianalytical
Kalman Filter Computation Time
........................... 106
4.7.5
Comparison of Extended Kalman Filter and Extended
Semianalytical Kalman Filter Timing Estimates...........................
107
11
Chapter
1
INTRODUCTION
1.1
Orbit Determination
Orbit determination processors incorporate the capability both of
propagating a satellite orbit given initial conditions and of estimating
and updating the ephemeris using satellite observations. The orbit propagator being developed at The Charles Stark Draper Laboratory is the accurate,
computationally efficient
[1].
SST serves
batch
estimator--the Semianalytical Differential Corrections
processor--and
to
a
provide
more
Semianalytical Satellite
the necessary
recently
dynamics
developed
tors were
designed
Parameters
to complement
equations
of
motion.
SST's
special
Although
modeling
sequential
Extended Semianalytical Kalman Filter (ESKF) [3].
the
Theory
for
(SST)
both
a
(SDC) 12]
estimator--the
Both of these estima-
form
of
older
the Variation
SDC
has
been
of
more
extensively characterized, both estimators have established capabilities
of providing
and
accurate
simulated
data
ephemerides
test
cases.
for a variety
Estimation
of satellites
accuracy
and
in both real
computational
efficiency of both SST-based estimators have compared well with those of
special-perturbations-based estimators.
1.2
Overview
of Thesis
Two recent studies have provided the opportunity to further characterize
both
SDC and ESKF for synchronous
satellites,
for the purposes
of
future enhancement and promotion of the semianalytical orbit determination
system.
Both MILSTAR and Telesat ANIK A
information
on the strengths
studies yielded interesting new
and limitations
of the two estimators.
Both
studies provided opportunities to test modest estimator augmentations.
The single-node military satellite control facility is one of the
most vulnerable segments of a complete satellite system.
navigation,
acquired
then,
largely
in
which
the
independently
of
spacecraft
such a
ingly desirable.
12
operates
facility,
Near-autonomous
effectively
is
becoming
and
is
increas-
One example of near-autonomous navigation is the operating scenario
developed by Draper for the new MILSTAR command communications system.
Military satellite operators could well appreciate the consequent lessening of demand for routine tracking information and perhaps even a lessening of some responsibility for satellite configuration management.
Sat-
ellite users gain when spacecraft acquisition becomes more local, requiring little more than an antenna and a small processor.
The semianalytical
orbit determination system, with its efficiency and highly modular dynamics, supports near-autonomous navigation.
Military and civilian satellite system operators alike benefit in
a
number of ways from ever-greater automation of
analyst
time is
reduced and
orbit determination;
computational resources
freed.
Canada, a civilian operator of such communications satellites as
is responsible for an enlarging group of spacecraft.
Telesat
NIK Al,
Telesat has already
benefited from such greater automation, through local real-time orbit
determination.
Minimizing needs for both time and hardware resources also
appeals strongly to those involved in thorough space surveillance.
The
accurate and efficient semianalytical system enhances orbit determination
automation.
The results presented here deal with the evaluation of the performance
MILSTAR
of
the
semianalytical
and ANIK Al
lites in general.
system--evaluation
satellites
in
particular
of
its
and for
estimators
synchronous
for
satel-
Members of many communities will find the results of
interest.
As well as giving a brief introduction to Semianalytical Satellite
Theory
and its
two estimators
in Chapter
2, this
thesis
chronicle of the estimator characterization test
summary
of
results
for
the
MILSTAR
ANIK Al study, in Chapter 4.
study,
in
will
serve
procedures and
Chapter
3,
and
as a
as
for
a
the
Chapter 5 summarizes the most important
conclusions and proposes future work.
13
Chapter
2
SEMIANALYTICAL SATELLITE THEORY.AND THE TWO SEMIANALYTICAL ESTIMATORS
2.1
Orbit Propagation with Semianalytical Satellite Theory
Orbit propagation theories may be divided into several categories.
Special Perturbations Theories are those that feature application of a
high-precision numerical integrator to one of several relatively complete
and
accurate
formulations
of accelerations
acting
on
the
satellite
[4].
Such theories are highly accurate but computationally quite inefficient.
In order to capture reasonably fine details of geosynchronous satellite
motion, for example, a theory of this type with Cowell equations of motion
requires
integration
stepsizes
predictor-corrector integrator.
proposed.
of
less
than
600 seconds
in a high-order
Attempts to increase efficiency have been
The perturbation model can be tailored for specific applica-
tions, "stabilizing," "regularizing," or "smoothing" transformations can
be applied, and additional recursive formulations can be incorporated.
These
changes
have
so far resulted
in only moderate
improvements
ciency, with some bringing increased expression complexity.
in effi-
On the whole,
special perturbations theories are inflexible with respect to truncation,
and capturing
explicitly
and to a given
accuracy
level
only the essential
dynamics content is not possible.
General Perturbations Theories are alternatives.
theories
features explicit manual
term-by-term formulation in
elements of the effects of disturbing functions [5].
transformations are
then applied
Development of such
to
eliminate
orbital
Canonical analytic
short-
and
long-period
element variations, leaving only expressions for the secular motion that
can be solved analytically.
The elements
at any time can be found
immed-
iately, avoiding costly step-by-step numerical integration.
General perturbations theories are inflexible and simplified, and
consequently less accurate than special perturbations theories.
These
limitations have resulted from the need to express all models analytically, even those for atmospheric density and
third body ephemerides.
Perhaps more importantly in this regard, however, general perturbations
14
models are costly to develop and difficult to verify.
Increasing demands
for high-accuracy orbit determination have, however, already made several
additional models almost obligatory; the resultant proliferation of model
expressions
can only
tax computational
resources
of time
and memory.
In
spite of the inaccuracies, general perturbations theories are currently
the propagators
of choice in installations
in which
the number
of objects
tracked prohibits use of special perturbations techniques [6].
Semianalytical Satellite Theory is an alternative to these orbit
generators.
Here,
perturbations
that
can
be
expressed
in
terms
of
a
disturbing potential have that potential cast in terms of singularity-free
equinoctial
elements
[7] in such a way as to maximize
sive evaluations in the formulation.
the number
of recur-
These perturbations are then put in
Lagrangian Variation of Parameters (VOP) form, so that only small perturbations from two-body motion need be considered.
The equinoctial element
rates.corresponding to non-potential perturbations--specifically drag and
solar radiation pressure--are then expressed in Gaussian VOP equations,
using directly the disturbing accelerations modeled in accurate special
perturbations theories [5].
The formulation of both potential and non-
potential perturbations in Semianalytical Satellite Theory is designed for
force model flexibility and accuracy, and for speedy orbit propagation.
SST then employs the asymptotic Generalized Method of Averaging
[8,9]
to
separate
in
the
VOP
equations
-long-period
and
secular--or
so-
called mean--components of satellite motion from those with relatively
short period.
It suffices in most cases to effect this separation only to
first order in small parameters
of the modeled
perturbations.
For potential-type perturbations, this separation is accomplished
primarily by averaging the effects of all perturbations over the satellite
fast angular variable throughout a full satellite orbital period, although
additional averaging may take place over other relatively fast angular
variables
of the total
dynamic
system.
For
the other
perturbations,
the
Generalized Method of Averaging may still be applied; here the averaging
is over the time of a complete
an angular variable.
orbital
period,
rather than over a cycle of
This averaging, accomplished either analytically or
numerically, results in isolation of the smooth mean satellite motion.
15
Further transformations are then bypassed, reducing theory complexity and
eliminating the possibility of creating artificial singularities.
Integration of the averaged motion may proceed with stepsizes in a
low-order integrator on the order of a day for geosynchronous satellites.
Unlike propagation in general perturbations theories, that in SST does
rely on large-stepsize numerical integration, and on interpolators to give
accurate orbital element or position and velocity information as needed.
More importantly, however, SST retains the capability of high-accuracy
orbit determination.
Reversing the averaging transformation is accomplished by using the
mean elements at the time of interest to evaluate the short periodic portion of the motion
that is simply
satellite osculating state.
the full dynamics
added
to the mean
motion
to obtain
the
Because knowledge of the mean motion allows
to be recovered,
it suffices
without
loss of accuracy
to
compute the short periodic content only when osculating output is desired
or when observations are being processed in the adjunct estimators.
The
averaged motion may be considered an expression of essential satellite
dynamics.
The SST orbit propagation theory may be considered to be composed
of Averaged Orbit Generator
(AOG), Short Periodic Generator
attendant integrators and interpolators.
(SPG), and
The AOG is that portion of the
theory that yields the mean--or averaged--satellite motion via averaged
element
rates corresponding
gation accuracy.
to the forces necessary
to give desired
propa-
The short periodic functions are in the form of Fourier
series expansions in fast satellite angular variables of the short period
content of needed force models.
The SPG portion of SST serves to supply
the required series coefficients corresponding to current satellite mean
state and to evaluate the expansion.
SST
is
currently deeply embedded in Draper's
implementation of
the Research and Development version of the Goddard Trajectory Determination System (GTDS), a multi-purpose computer system formulated originally to support space missions and various research and development project requirements at NASA Goddard Space Flight Center.
Although a system
with automatic proper semianalytical force model selection is envisioned,
the user
of the current GTDS-based SST has both
16
the freedom and
the
responsibility of choosing explicitly the force model needed in a given
application, particularly that for the short periodic motion.
tion can be quite complicated.
This selec-
The perturbations corresponding to con-
servative forces are expressed in terms of several embedded sums; each
might have the capability of being truncated in its expansion variable.
To compute those perturbations corresponding to non-conservative (necessarily numerically averaged) forces, quadrature order must be given, and,
for weak-time-dependent formulations of the short periodic functions,* the
number of
time derivatives must be specified.
In addition, the short
periodic functions can be truncated according to order of Fourier series.
The force models in both AOG and SPG have been continually developed [11]. The AOG as currently implemented incorporates recursive analytic formulations of the geopotential zonal and tesseral resonance** perturbations, and of both single- and double-averaged*** third body perturbations.
To obtain the mean element
rates corresponding to
the non-
conservative force models, the averaging over the time of the satellite
period takes place via numerical quadrature; the contributions to averaged
motion of first- and second-order drag and of the coupling between the J2
gravity harmonic and drag (and drag/J2 )****--as well as solar radiation
pressure--are all determined through this numerical averaging.
Finally,
*When the disturbing function has more than one rapidly varying quantity,
a time-dependent formulation of its associated short periodic functions
will take into account the interaction of the different fast variables
[2].
If
the
time
rate
of
change
of
the
additional
variable
is
small
relative to the satellite mean motion, the so-called weak-time-dependent
formulation for the Fourier series coefficients is indicated.
**Tesseral resonance is the phenomenon whereby what are effectively repeating satellite ground tracks give rise to long-period satellite motion.
***Double-averaged
takes place, this
models
[10] are those in which a second averaging
time over the rotating earth's Greenwich Hour Angle or
the satellite-dependent angular position of a third body.
****The averaged element rates in SST are, in general, expansions to first
order in "small" perturbation parameters.
The averaged element rate
corresponding to multiple perturbations is then the sum of the effects on
the averaged equations of motion of each perturbation taken separately.
When the parameters are not truly small, then a second-order expansion of
the
averaged
element
rates
is more
appropriate,
with
some
terms
second-
order in a single parameter and some with products of different perturbation parameters.
These
last are the "coupling"
17
terms.
explicit analytic formulas exist in the AOG for the second-order effects
of J2 and for the drag/J
2
coupling.
The
integrator
used
in
propagating
the satellite mean elements is the fourth-order Runge-Kutta scheme.
The SPG can evaluate with recursive analytic formulas the shortperiod effects of geopotential zonals, tesseral m-dailies* and other nonresonant
tesserals as well
models.
The short-period manifestations of drag, solar radiation pres-
sure,
and
the
as
single-averaged
single- and double-averaged third body
third
quadrature-type formulation.
body
model
also
are
expressed
The contribution of second-order
in a
J2
and
J2 -m-daily coupling is couched in an explicit analytic formula.
The strength of a Fourier series expression for the short periodic
functions
is that an interpolation
scheme
for the relatively
slowly
vary-
ing series coefficients may be developed, minimizing the number of costly
function evaluations.
A Lagrangian interpolator currently serves in this
role.
2.2
The Semianalytical Differential Corrections Estimator
An essential part of any orbit determination system is the effective use of observation
data
to improve
knowledge
perhaps of certain force model parameters.
provides a weighted-least-squares fit to
span,
giving
an estimate
of
time, usually at data epoch.
portion
of
the
orbit
estimation process.
the quantities
of
satellite
state
and
The typical batch estimator
tracking data over a
of
interest
at some
certain
specified
The dynamics modeling and satellite theory
determination
system
is
a fundamental
part
of
the
The coupling of familiar differential corrections
estimation with Semianalytical Satellite Theory has yielded the SDC.
*The
tesseral
m-daily
terms
are
those
geopotential
terms
that
produce
effects in the satellite motion with a frequency of m cycles per day.
This is a result
solely of earth
rotation
satellite motion).
18
rate
(and is thus independent
of
Green formulated and tested the SDC and presented its fundamental
algorithmic
structure
[2].*
The
SDC has
a
solve
vector
with
both
the
non-state dynamic parameters of interest and the satellite mean equinoctial elements
of
this
iance.
time
at epoch.
solve
vector,
Input to this estimator
and,
optionally,
some
is some initial
corresponding
estimate
initial
covar-
The current best estimate of initial state is propagated to the
of
expected
the observation
observation.
and is
used
in an
observation
The value of the observation
model
residual,
to
give
an
that is, the
difference between actual and expected observation values, is then computed.
The matrix of the partials
of the observations
with
respect
to the
epoch state (the F-matrix discussed later) is evaluated at best epoch
state estimate.
An estimate of the epoch state correction that is optimal
in the least-squares
sense uses this matrix in addition
to the observation
residuals, the input covariance, the difference in input epoch state and
its current best estimate, and a weighting matrix that gives estimated
uncertainties
of all observations
in
the batch.
Implied
in
this
proce-
dure, because of linearization and other formulation errors, is an iteration on the epoch
state estimate
until
the correction
applied
falls
below
a specified threshold.
2.3
The Extended Semianalytical Kalman Filter
The
sequential
filter
estimator
continually processes
incoming
tracking data to yield an improving current-time state estimate.
determination
facilities
filters
[11,12],
and
are
use
in
of
use
in
filters
a
can
number
only
of
satellite
become
more
Orbit
control
widespread
because of their timely, accurate, and efficient estimation capability.
In order to complete the semianalytical orbit determination system,
a sequential estimator was designed.
Orbit propagation is eminently non-
linear, but a linear minimum-mean-square-error filter, although suboptimal
for orbit estimation, recommends itself because of
familiarity.
Taylor
[3],
its simplicity and
The Extended Semianalytical Kalman Filter, as formulated by
represents
the
application
of
the
ideas
of
the
successful
*The following discussion is intended to present only the basic estimator
concepts;
Green's
work
gives
complete
analytic
development
and
expressions.
19
Extended Kalman Filter
(EKF) to Semianalytical Satellite Theory, along
with certain very convenient simplifications that arise naturally in the
semianalytical theory.
What follows is only a brief summary of the ESKF
operation.*
The ESKF equations
are very
reminiscent
of those
of
the EKF.
The
nearly linear dynamics of the estimated filter state vector based on best
current vector information are propagated to the time of the incoming
observation; this yields the best estimate of the osculating satellite
state, which is used to predict a computed observation and to linearize
the observation model, this last resulting in the so-called observation
partials.
The filter gain is calculated using these partials, as well as
the measurement variance and an estimated filter covariance (which uses in
turn the semianalytical
state
transition
matrix
and a process
noise
term).
The gain weights the observation residual to give an update to the state
correction.
The ESKF solve vector is a natural one:
along with any non-
state dynamic parameters of interest, it includes the mean equinoctial
elements for the satellite that are the SST integrator variables.
There is a twist introduced in the manner in which satellite state
is
propagated
in
the ESKF,
a central
simplification
of EKF
ideas.
mean equinoctial elements propagate nearly linearly in time.
The
This fact
of semianalytical dynamics has proven perennially useful and is one of
the theory's chief strengths.
As a direct consequence of this near lin-
earity, there is no need to continually relinearize the semianalytical
dynamics using the best possible state estimate obtained from the processing of the most recent observation.
Instead, the very nearly optimal
ESKF state prediction uses dynamics that are linearized around a more
global nominal mean satellite state.
The best osculating state used in
computing the observation residuals and partials
propagation
to
correction,
the nominal
motion
observation
corresponding
time
state,
of
the
sum
of
the
and the calculated
to the sum of the first
comes
last
short
two terms.
then
from
estimated
periodics
Rather
the
state
of
the
than incur
costly short periodic function evaluations, the functions are normalized
around the nominal trajectory.
In this way, only the contribution to the
*Taylor's work gives a very thorough analytic treatment of ESKF structure
and development.
20
function due to the last estimated mean state correction need be added
(via the B 1-matrix discussed
later).
The fact that optimal state prediction here need not result from
continual
relinearization
of
the
semianalytical
dynamics
means
that
the
integration of nominal satellite mean elements may proceed with the same
efficient
had
taken
large time stepsizes
place
in
the
as would be possible
interim.*
The
integration
when no state updating
of
the
nominal
mean
elements assumes a background role relative to the ESKF sequential estimation, with an update
tion
step using
nominal
the accuracy
not affected
*Further
the best
state occurs
become large.
of nominal
state
state
before
accuring
estimate
the small
at the end of the integra-
at that
time.**
nonlinearities
This
update
of
of the mean dynamics
The estimation thus becomes more nearly linear, enhancing
of
this
suboptimal
filter convergence
enhancing
filter;
the assumption
of
linearity
has
in any filter tests to date.
the efficiency
of the filtering
process
is the fullest
use of interpolators--especially for state transition matrices and short
periodic coefficients.
**A potentially very useful property of the ESKF is then that the
integration-associated algorithms need reside in core only infrequently.
21
Chapter
3
THE MILSTAR ORBIT DETERMINATION EVALUATION STUDY
3.1
MILSTAR Background
MILSTAR
is
satellite network.
the
next-generation
defense
command
communications
The system is being designed so that satellite acqui-
sition by users may proceed in the absence of frequent contact with the
vulnerable satellite tracking network that currently has primary control
of satellite configuration.
In one operating scenario studied by Draper,
after one-time pass-off of satellite elements from the Air Force Satellite
Control
Facility, orbit determination,
including ephemeris estimation,
would proceed onboard an Airborne Command Post (ABCP) aircraft parked at a
surveyed site while observing the spacecraft.
The ephemeris data
so
obtained would then be distributed to other satellite system users to
enable acquisition.
Most users would receive accurate updated satellite
ephemerides only every few weeks.
Such ephemerides would be in the form
of coefficients both for a certain set of orthogonal polynomials to describe the satellite's long-period motion during the ensuing month and for
a collection of trigonometric functions that contribute monthly periodic
dynamics.
These polynomials and trigonometric functions, together with an
interpolation scheme and tracking model, would be employed in small user
microprocessors to effect the acquisition.
The anticipated computational resources onboard the tracking aircraft are limited.
To decrease the computational burden associated with
tracking system orbit propagation and estimation, Semianalytical Satellite Theory and its associated Extended Semianalytical Kalman Filter have
been proposed as the tracker orbit determination system.
well-developed
system
has
much
to
recommend
it
for
this
This already
application.
The SST orbit propagation scheme has established accuracy and efficiency,
and, although not yet thoroughly characterized, the ESKF has so far demonstrated
timee,
tion.
excellent performance
while
accurate satellite state
delivering
estimates so
efficiently
needed in
the
real-
this applica-
Moreover, this orbit determination system, with its averaged equi-
noctial elements as
integrator variables, yields immediately a set of
22
orbital elements
that
describe
the secular
and
long-period satellite
motion--motion that not only reflects the essential dynamics but that is
also easily well approximated by low-order polynomials.
Earlier work had
suggested that this orbit determination system would demonstrate in the
application
MILSTAR
comparable
performance
3.2
large
efficiency
computational
to that of any other
gains
in
offering
system.
Study Overview
The
MILSTAR
study
was
one.
a preliminary
The
proposed
SST/ESKF
system resides currently in a large mainframe, as perhaps 250 of more than
1000 subroutines composing the Research and Development version of the
Goddard Trajectory Determination System; excising the required subroutines
promised to be a time-consuming process.
The evaluative studies described
here would thus not indicate absolute efficiency of the final operating
system but would demonstrate capability in performing orbit determination
The
in the basic operating scenario.
studies would show for MILSTAR
spacecraft and the potentially numerous MILSTAR-type satellites operating
in a similar scenario
·
that a compact and accurate tailored semianalytical force
model could be readily constructed
·
that the ESKF could provide requisite filtering accuracy at
the tracker location
that these tracker estimation accuracies would lead to
acceptable acquisition accuracies at user locations
The results will be applicable in the broader domain of synchronous
orbit determination when computational resources and tracking resources
and accuracies are restricted.
3.3
Verification of Semianalytical Satellite Theory
As
a
fundamental
starting
place
for
work
with
Semianalytical
Satellite Theory, it was demonstrated that SST is in fact a valid and
accurate description of satellite dynamics.
23
The Precise Conversion of
Elements (PCE)* procedure with SST was used to demonstrate basic propagator soundness in a conventional fashion.
Compared with
the analogous
features of special-perturbations-type orbit propagators, the form of the
equations of motion and of the fundamental satellite state in Semianalytical Satellite Theory are significantly different.
It is desirable, how-
ever, to have SST propagation compare favorably with that of Cowell-based
special perturbations theory, an accepted accuracy standard.
The PCE verification procedure for SST involved several tasks:
·
Generating
a "truth" ephemeris
* Least-squares fitting of semianalytical theory to the
Cartesian state information in the truth ephemeris using the
Semianalytical Differential Corrections estimator
*
Comparing the resultant semianalytical ephemeris with the
truth model
The GTDS-based Cowell special perturbations propagator was used to
generate 28 days of position and velocity information for the geostationary satellite that Table 3.3.1 describes.
Table
3.3.1
Geostationary Satellite used in Semianalytical Satellite
Theory Verification
Epoch
30 January 1979
12h Om Os
Elements
a
e
.1603E-4
i
0
.1420E-3
Q
148.80
w
M
Spacecraft
Parameters
42163 km
A
m
°
8.09820
345.610
(mean 1950 coordinates)
9.09 m2
568 kg
The force model used in the Cowell propagator was one recognized to
be the best necessary for satellites of this type (4x4 GEM9 geopotential,
*Precise conversion of elements is used in the sense of minimum-leastsquares fitting of position and velocity information with an ephemeris.
24
lunar-solar point-mass gravity, solar radiation pressure in a 12th-order
Cowell-Adams predictor-corrector with 600 sec stepsize).
cal
force
model used
to fit all 28 days
of
The semianalyt-
this data was
that below,
in
Table 3.3.2.
Table 3.3.2
Force Model used in Semianalytical Satellite Theory Verification
AOG Model
SPG Model
Zonals:
Zonals:
(4x0), e2
(4x0), GEM9
Tesseral Resonance:
(2,2) through
Second-order
J2 2,
M-dailies:
(4x4), e2
(4,4)
e
Tesserals:
(4x4),
el
Lunar-solar point mass:
Parallax--moon=8,sun=3,
e2
Solar Radiation Pressure
Second-order
J2 2, e
Lunar-solar point mass:
8 frequencies,
2 time derivatives
Solar Radiation Pressure:
3 frequencies,
1 time derivative
After the SST-based PCE procedure had
converged, statistics were
computed for the comparison over the entire fit span of the resulting
semianalytical ephemeris with the truth ephemeris.
Table 3.3.3 shows the
results of this propagator verification test.
Table 3.3.3
Summary of Semianalytical Satellite Theory Verification Test
Compare Statistics
(over 28-day PCE fit span)
Component
Radial
Cross-track
Along-track
Total
Position RMS Error
(km)
.670E-3
.154E-2
.394E-2
.428E-2
25
Velocity RMS Error
(cm/sec)
.281E-3
.113E-3
.630E-4
.309E-3
The total position errors in this PCE test on the order of 4 m and a
total velocity error much less than a single centimeter per second argue
strongly
for
the
Satellite Theory.
fundamental accuracy
and
validity of
Semianalytical
The theory has displayed the capability of producing
the fine details of satellite motion.
By incurring the various short
periodic functions listed in Table 3.3.4, ever-finer detail is added.
Table 3.3.4
Typical Short Periodic Function Magnitudes for Synchronous Satellites [13]
Perturbation
3.4
Magnitude
(km)
Zonal
1.6
Lunar-solar
1
Solar Radiation Pressure
Tesseral
M-Daily
3E-2
1.6E-2
4E-4
J22
1.3E-4
Semianalytical Force Model Tailoring
The first specific studies in the MILSTAR program were designed to
show that the modular semianalytical dynamics could be effectively abbreviated to give efficient but accurate estimation.
In order to explore
semianalytical force model requirements for MILSTAR-type orbit determination, batch least-squares estimation was used to indicate explicitly the
effects of semianalytical force model truncation on propagation of satellite state.
The test procedure used here included
·
Generating
·
Simulating corrupted observations
·
Least-squares fitting of semianalytical theory to the truth
a "truth" ephemeris
ephemeris using the Semianalytical Differential Corrections
estimator
·
Comparing the resultant semianalytical ephemeris with the
truth model
·
Truncating subsequently the semianalytical force models
26
Note
that corrupted tracker-type observations, rather than uncorrupted
Cartesian state information, were produced from the Cowell ephemeris in
this tailoring task.
With semianalytical force model a given during the process, leastsquares batch fitting of corrupted observations using SST is equivalent to
imposing the averaged element representation of SST on the truth satellite
dynamics, with some uncertainty because of observation and force model
unknowns.
This uncertainty in least-squares batch estimation could be
considered roughly equal to that in sequential estimation given the same
dynamics and observations.
Thus, a force model chosen here in this test-
ing could be used in the forthcoming filter evaluation.
In batch estima-
tion, however, the modeling issues of long-term trajectory accuracy could
be more effectively addressed, since the complications of filter initial
state, process noise, and observation nonlinearities could be eliminated.
The effects
on propagation
accuracy
of
truncation
of
the
semianalytical
force model could thus be more explicitly seen with a batch estimation
procedure.
To begin
the force model
tailoring,
two satellites
of the general
type contemplated for the MILSTAR constellation were selected.
These
satellite orbit were geosynchronous and near circular, with inclinations
of approximately 0 °
and 600 --SYNCX
and SYNCI, respectively.
Table 3.4.1
describes these satellites.
Table
3.4.1
Geosynchronous Satellites used in
MILSTAR Orbit Determination System Evaluation
6
SYNCX
SYNCI
Epoch
30 January 1979
Oh Om Os
Elements
a
e
42163 km
.4368E-4
42166
i
Q
.2564E-1
°
263.57
63.000
w
214.100
349.990
286.940
M
204.220
189.360
A
m
2.2 m 2
200 kg
2.2 m2
Spacecraft
parameters
I
W
29 January 1979
Oh Om Os
27
I
I
I
I
200 kg
__ __
-
km
.3367E-4
__ _
I
For both satellites, the initial state was propagated for several
weeks with the same special perturbations propagator used above.
multi-week ephemeris became known as the truth ephemeris.
This
During the
orbit determination system evaluation study, the truth ephemerides--by
necessity
in
the
absence
of real
satellite
observations--would
serve
as
models of actual satellite motion.
The force model tailoring for both geosynchronous satellites was
based primarily on batch least-squares estimation for SYNCX.
Thus, in
order to simulate observations from the SYNCX truth ephemeris, preliminary
tracker location, and measurement types and accuracies were subsequently
chosen.
The preliminary tracker was located at a surveyed site in Beale,
°
Ca. (40
N, 2370 E), and it measured with reasonably characteristic accu-
racy [14] unbiased range (20 m accuracy), range-rate (10 cm/sec accuracy),
azimuth and elevation
(both .050 accuracy).
The
frequency with which
users would obtain updated ephemerides from the tracker was very much
uncertain at this time; nine days was the chosen interval between updates.
Total position and velocity RMS errors based on comparison over this nineday span of
the truth ephemeris with propagation of the SDC converged
epoch state were selected as suitably comprehensive preliminary performance measures for the SST-based estimators.
A preliminary error budget for user acquisition had been explored.
This budget
*
accounted
for two error
sources:
Tracker estimation errors, due to indeterminate and truncated
physical models, observation inaccuracies, and basic estimator
uncertainties
·
Ephemeris generation errors in user terminals, due to interpolation in an approximate orbit representation
Until a firmer error budget could be established, the interim allowable
RMS tracker estimation errors were considered to be a few kilometers in
position
and less than a meter per second in velocity
[15].
The Semianalytical Differential Corrections estimator was used at
this point to process several different schedules of simulated observation
data.
The different schedules were designed to illustrate the impact on
the RMS error performance criterion of both observation span and rate as
28
well as small changes in both AOG and SPG models.
results.
Here
observation
span
was
relatively
Table 3.4.2 gives test
much
more
important
observation rate in determining satellite epoch state, with
than
estimation
errors decreasing much faster than span was lengthening for early span
increases.
In addition, compared with the conservative semianalytical
force model used initially,
simple.
Considerable
possible,
based
in
the final force model
force
large
model
part
truncation
on
a
in
in this study was fairly
the
straightforward
SPG
especially
order-of-magnitude
analysis of short periodic function Fourier coefficients.
Figures
3.4.1 and
3.4.2 illustrate
the effects
of
was
truncation
For example,
in the cen-
tral body zonal short periodic representation made in the earliest model
to achieve the final tailored version, with approximation down to less
than meter level in radial and along-track position.
marizes the final tailored SYNCX force model.
29
Table 3.4.3 sum-
Table 3.4.2
Summary of Precise Conversion of Elements Testing of MILSTAR SYNCX
6 hours
every min
Pos/Vel
Total RMS Errors
SPG Model
AOG Modelt
Observation
span, rate
6.2 km,
.45 m/sec
(4x0) geopotential with Analytical:
geopotential -2
resonant tesseral
terms through
(GEM9)
2
(4x4)
1
terserols
J2
J2, e
lunar-solar gravity 4
(parallax
2
zonals (4x0), e
m-dailies (4x4), e
= 8,4), e
(4x4), e
e
Numerical:
lunar-solar gravity 7 frequencies,
1 time derivative
solar radiation
pressure 3 frequencies
0 time derivatives
30 minutes,
180 km,
as above
as above
13 m/sec
every min
6 hours,
every 10 min
as above
as above
8.2 km,
.59 m/sec
12 hours,
every 10 min
as above
Numerical:
increased number of
lunar-solar gravity
4.4 km
.32 m/sec
time derivatives
12 hours,
every 10 min
as above
to 2
Analytical:
removed m-dailies,
2
4.4 km
.32 m/sec
0
tesserals, J2 e
truncated zonals
Numerical:
truncated solar
radiation pressure
to 2 frequencies
truncated lunar-solar
12 hours,
every 20 min gravity
2
(parallax=6,3), e
truncated
(3,1),
1.6 km,*
.12 km/sec
4 frequencies
to
resonance terms
(2,2),
Numerical:
truncated lunarsolar gravity to
(3,3)
truncated solar
radiation pressure
to 1 frequency
tAll runs used the 4th-order Runge-Kutta scheme with half-day stepsize.
in light of the
*It should be noted that such good results were unexpected
results for the case immediately preceding, which are doubtless more characte-
The result is a peculiar one which would probably change if the
ristic.
observation schedule were slightly changed.
30
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C
E
Table 3.4.3
Tailored Force Model for MILSTAR Geosynchronous Satellites
AOG Model
SPG Model
Zonals:
(4x0), GEM9
Zonals:
(4x0)
Tesseral Resonance:
(2,2), (3,1),
-
M-dailies:
None
(3,3)
Second-order J2 2 , e
I
Tesserals:
None
Lunar-solar point mass:
Parallax--moon=6, sun=3, e
Lunar-solar point mass:
4 frequencies,
Solar Radiation Pressure
2 time derivatives
Solar Radiation Pressure:
1 frequency
0 time derivatives
The final semianalytical force model chosen was undoubtedly not as
abbreviated as possible.
The order-of-magnitude truncation technique used
here meant that all elements were propagated using all frequencies of the
perturbation
elements
expansion up
were
rather
through the truncation order,
insensitive
to
some
of
the
although some
expansion
terms.
The
current semianalytical force model is insufficiently explicitly modular
for element-wise truncation.
Moreover, truncation using
an
order-of-
magnitude analysis for the averaged element rates themselves, although
possible later in the MILSTAR study, was not used.
Finally, the effects
of particular perturbations will, of course, change with time, although
this knowledge could not be used in a straightforward way in the force
model tailoring.
Typical observation accuracies and tracking schedules for MILSTAR
had become
much
firmer at the end of this series
Command Post had been proposed as the tracker.
of
tasks.
The Airborne
Observations were thus
reduced to the range and range-rate observable with its communications
antenna, with scheduling of only a few several-minute periods one or two
days a week.
Observation accuracies for the ABCP tracker were estimated
to be 150 m in range and 39 cm/sec in range-rate [16].
The error bud-
get for user acquisition had also been better established.
33
Users now
were expected to receive update information from the tracker every thirty
days,
and with
the constraints
on
range
and range
rate
becoming
the most
stringent, the user's allowable acquisition errors in these two observables were
10 km and 1 m/sec, respectively
In order to begin
nario,
some reasonable
needed.
The errors
the final
tailored
actual
a priori
resulting
force
at
model's
[171.
evaluation
initial
of ESKF in this operating
state
estimate
for the filter was
the end of the nine-day
epoch
solution
were
sce-
propagation
extrapolated
for
to the
thirty-day point to give an estimate of update initial condition uncertainties for the tracker filter.
accuracies were
being
Because estimates of tracker observation
continually downgraded, however,
the
calculated
thirty-day worst-case estimator initial condition errors were doubled to
serve as initial condition errors for the ensuing ESKF estimation runs.
3.5
Extended Semianalytical Kalman Filter Evaluation for MILSTAR
The filter testing
rocedure had several parts:
* Producing MILSTAR-typical observations
* Processing of observations with the ESKF tailored-model-based
dynamics (with or without force model or other errors)
* Propagating the converged end-of-observation-span estimate
' Comparing the resulting ephemeris with the truth ephemeris
A
variety of SYNCX and SYNCI filter processing runs were subse-
quently made, the entire collection serving to evaluate filter performance
under reasonable operating circumstances.
used the tailored
Table 3.5.1.
remainder
force model above and observations
of the type listed in
Note that the nominal tracking schedule used throughout the
of the MILSTAR
study included
of observations.
It should be
for
satellite
SYNCI),
At the outset of testing, each
that
noted,
four twenty-minute
however,
observability
may
(and as will
severely
number of observations taken in a tracking interval.
34
periods
limit
be
on days
the case
the
actual
Table
3.5.1
Tracking Data for MILSTAR Filter Runs
Site Location
Latitude (geodetic)
Longitude
Altitude (above MSL)
Beale,
400
2370
130
Ca.
N
E
m
Observation Schedule
1 or 2 days of data
3 range measurements taken 10 minutes apart
16 range-rate measurements 80 seconds apart
both starting
at
and
02:00
08:00
14:00
20:00
hr
hr
hr
hr
I
I
Observation Statistics
All observations unbiased
I
Orange = 150 m*
aranqe-rate =
39 cm/sec
*The effect
of the truncated
short periodics
was
partially accounted for in the estimation process by
RSS'ing typical magnitudes [13] for the neglected
perturbation
with the range observation
noise.
I
Input
an
a
priori
model errors.
errors
to the filter
covariance,
Table
calculated
runs consisted
and
process
3.5.2 gives
above
in
of perturbed
noise
to help
the so-called
the mean
account
baseline
equinoctial
Table 3.5.2
Perturbations for Filter Runs
Element Perturbation
Aa
Ah
Ak
26 m
1 E-5
1 E-5
A p
1 E-5
A q
1 E-5
AX
.020
Equivalent Position Perturbations
radial
.4 km
cross-track
.8 km
along-track
13 km
35
elements,
for
dynamic
initial-condition
perturbations
actual input.
Baseline
initial
that
were
In all cases, the initial covariance was assumed strictly diagonal, with
non-zero entries equal to the squares of the above perturbations.
state covariance was thus relatively well known.
plied
was
calculated
after
Taylor
[3] as
The process noise sup-
the square
rate errors multiplied by data-outage period.
The
of
averaged
element
In calculating the filter
process noise, the averaged element rates were obtained, as
shown in
Figures 3.5.1 and 3.5.2, for both the conservative truth semianalytical
model used in the first batch least-squares tests and the final tailored
model.
The small difference in averaged element rates for the two models
was calculated for each of the elements at some representative point in
the observation processing span, and was subsequently squared and multiplied by an expected MItSTAR data outage period (during observation processing) of six hours.
The resulting process noise matrix was similar for
both SYNCX and SYNCI.
The matrix used for both satellites was diagonal,
with the entries for both h and k and both p and q taken to be the calculated
maximum
value
of
the
respective
pair.
Table
3.5.3
gives
the
process noise matrix.
Table 3.5.3
Process Noise Matrix used in Filter Runs
Use of process
noise
good success in the past.
mean
calculated
in this way has
met with
apparent
Because ESKF process noise reflects errors in
(as opposed to osculating) element dynamics, it can reasonably be
expected to be near constant.
For both SYNCX and SYNCI, the ESKF was evaluated on the basis of
its performance when operating with uncertainties of the type and magnitude expected in the MILSTAR application.
ing
The battery of filter process-
runs for SYNCX featured so-called baseline estimation as well
as
filtering in the presence of simulated solar radiation force errors and
station location offsets.
Simulated data with
twice the
150 m
range
observation uncertainty above were also processed.
In addition, to deter-
mine
might
whether
an
extension
of
the
36
solve
vector
improve
filter
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33
performance,
simulated
being
several
solar
force
somewhat
developed).
processed
SYNCX
model
primitive
observation
similar
processing
errors
at
runs attempted
(the appropriate
this
time
but
to solve
force
being
model
for
in GTDS
currently
further
Satellite observability issues were explored by doubling the
span from one to two days.
The evaluative
was
filter
to
that
performance
used
runs of the tailoring
criterion
in the
task.
for each of these
Semianalytical
filter
Differential
The filter gave converged
runs
Corrections
elements
at the end
of the processed observation span, and these elements were then propagated
throughout
epoch.
son
the remainder
of a period
of
thirty
days
beginning
at filter
Position and velocity RMS error statistics resulted from compari-
over
the
ephemeris.
entire
thirty-day
period
of
this
ephemeris
with
the truth
Table 3.5.4 shows estimator performance in all SYNCX filter
runs.
Several
remarks concerning these results
can be
offered.
The
along-track position errors were paramount in almost all trials in which
one
day of observations
were
processed.
When
the span was
doubled,
the
cross-track errors became dominant, not having been much reduced with the
additional input although along-track position errors were substantially
diminished.
tively
The radial
unimportant.
general much
meanwhile
position
As for velocity
reduced with
subsiding
errors
little;
were
in
errors,
almost
all
the radial
cases
component
was in
additional observations, cross-track
errors
this
behavior
is
consistent
with
the
correlation of along-track position and radial velocity errors.
track velocity errors were invariably quite small.
three
runs,
it
should
be
noted
that
Cr
solve
were supplied for it was a conspicuous failure.
ties
could
at
given,
estimator
do only as well
all.
values
the
Moreover,
for the parameter
magnitude.
The effects
when
as when
when
supplied
with
the mismodeled
attempting
in both
of
rela-
to solve
cases
the solar
when
no
input
statistics
With accurate uncertaintwo
days
of
observations
was not estimated
for Cr, the filter
that were wrong
force,
among those modeled, were simply not observable here.
39
Alonq-
Considering the last
parameter
radiation
large
by
obtained
two orders
the smallest
of
force
Table
3.5.4
Summary of Extended Semianalytical Kalman Filter Results for MILSTAR SYNCX
Compare Statistics
(30-day total estimation
and prediction
span)
Type of Run
Position
RMS Errors
(km)
Baseline perturbations
1-day observation
Velocity RMS Errors
(cm/sec)
Radial
Cross-track
.544E-1
.572
Along-track
.102E+1
.417E-4
.399E-5
Total
.117E+1
.840E-4
Baseline perturbations
2-day observation
R
C-T
A-T
Total
.458E-1
.565
.256
.622
.171E-4
.412E-4
.336E-5
. 447E-4
Baseline perturbations
R
C-T
.749E-1
.583
.872E-4
observation uncertainty
1-day observation
A-T
.122E+1
Total
.135E+1
Baseline perturbations
R
C-T
.334E-1
.571
observation uncertainty
2-day observation
A-T
Total
.592
.823
Baseline perturbations
station misplaced
5 m vertically and
30 m horizontally
1-day observation
R
C-T
A-T
.544E-1
.572
.877
Total
.105E+1
Baseline perturbations
R
C-T
A-T
.415E-1
.564
.269
.411 E-4
2-day observation
Total
.627
.451 E-4
Baseline perturbations
R
C-T
A-T
Total
.279E+1
.679
.149E+2
.152E+2
.103E-2
R
C-T
.413E-1
.564
.183E-4
A-T
.271
Total
.627
.303E-5
.451E-4
twice
twice
the range
the range
10% error in Cr
(solar radiation
coeff.)
10% error Cr solve-for
no Cr input statistics
2-day observation
Baseline perturbations
10% error Cr solve-for
Cr input statistics
2-day observation
40
.728E-4
.425E-4
.548E-5
.971E-4
.419E-4
.416E-4
.247E-5
.591E-4
.624E-4
.417E-4
.399E-5
.752E-4
1 82E-4
.305E-5
. 495E-4
. 204E-3
.105E-2
·411 E-4
Later,
relative
as a check
to batch processing,
baseline filter
models,
tests
better
(the usual
In
procedure),
that
designed
of
ESKF
general,
to correspond
the iterative
the satellite
sequential
SDC is
to the
observation
expected
orbital elements than
to
the
However, when no initial covariance was supplied to the
span simply expanded
So it seems
SDC runs were
estimate of
sequential ESKF.
effectiveness
(same state perturbations, force and
and observations).
provide a
SDC
of the general
it
failed
to two days,
the tracker
to
converge.
With
the
the SDC gave only a very
must both be supplied
observation
poor
solution.
and must maintain
some
state covariance in order that orbit determination proceed effectively.
When
the filter's
initial
covariance
was
also
supplied
to the SDC in the
baseline two-day processing case, the comparison statistics for the two
estimators were quite comparable.
Table 3.5.5 summarizes this comparison.
Table 3.5.5
Comparison of Semianalytical Differential Corrections and
Extended Semianalytical Kalman Filter Estimators for SYNCX
Compare Statistics
(30-day total estimation and prediction span)
Type of Run
Position
RMS Errors
(km)
The following
Velocity
RMS Errors
(cm/sec)
SDC runs hal no input covariance:
SDC
Baseline perturbations
No convergence
1-day observation
SDC
Baseline perturbations
2-day observation
The following
Radial
Cross-track
Alonq-track
Total
runs used the same input
.132E+3
.126E+4
.262E+3
.129E+4
.958E-2
.920E-1
.966E-2
.930E-1
covariance:
SDC
R
.458E-1
.166E-4
Baseline perturbations
2-day observation
C-T
A-T
Total
.565
.249
.619
.412E-4
.335E-5
.445E-4
ESKF
Baseline perturbations
2-day observation
R
C-T
A-T
Total
.458E-1
.565
.256
.622
.171E-4
.412E-4
.336E-5
.447E-4
41
Questions
arose
concerning
the desirability
ter's solve vector to include measurement biases.
of
extending
the
fil-
The ESKF has no capa-
bility at present to solve for biases, but the demonstrated comparability
of
SDC and ESKF estimation accuracies when
identical covariances were
supplied suggested that the SDC's bias-solve capability would adequately
approximate corresponding ESKF performance.
made.
Two SDC runs were subsequently
In one, range observations supplied were biased 600 m.
The next
bias-solve run attempted to zero out an input range bias of
the
same
magnitude.
*
The input
element perturbations for both runs were baseline for
SYNCX, and range observation uncertainty was the larger 300 m
value.
To
increase observability of the range bias, the observation span was four
days of MILSTAR scheduling.
Table 3.5.6 shows the results of these runs,
and compares them with those of the above baseline SDC run with input
covariance processing two days of observation data.
Table 3.5.6
Range Bias Solve Capability for MILSTAR SYNCX
Compare Statistics
(30-day total estimation and prediction span)
Type of Run
Position RMS Errors
(km)
These
Velocity RMS Errors
(cm/sec)
runs all used the S C estimator:
Baseline perturbations
2-day observation
R
C-T
A-T
Total
.458E-1
.565
.249
.619
.166E-4
.412E-4
.335E-5
.445E-4
Baseline perturbations
600 m range bias input
4-day observation
R
C-T
A-T
.647E-1
.575
.976E+1
.710E-3
.420E-4
.464E-5
Total
.977E+1
.712E-3
Baseline perturbations
R
.648E-1
.803E-3
600 m range bias solve
C-T
.575
.420E-4
4-day observation
A-T
Total
.110E+2
.111E+2
.464E-5
.805E-3
*Solving for range bias using observations with unsuspected 600 m range
errors is essentially equivalent to solving for bias using uncorrupted
observations but supplying the filter an initial guess of 600 m--and
expecting a zero final bias estimate.
42
Range
bias
gave, instead
ing
to
axis--and
not
simply
of the zero value
for
solve
estimator.
is
The
range
lack of
thus radial
and model parameters
expected,
bias
simply
removed
helpful
by both
tracking
the
bias-solve
of 488 m.
into
to solve
the relatively
If either
span.
SDC
Attempt-
constraints
directly
Trying
and along-track--errors.
is made difficult
and
value
a bias
observability mapped
and the very short
observations
here,
observable
in
the
semimajor
for biases
poor-quality
aspect of track-
ing were significantly upgraded, further evaluation of such parameter solve
would
be justified.
considerable experience with SYNCX had been obtained, ESKF
After
line
force
The first SYNCI runs used
for SYNCI was evaluated.
performance
perturbations,
model.
the truncated
The
input
comparison
resonance
and process
covariance,
terms
were
statistics
noise,
and the tailored
By restoring
unacceptable.
in the 4x4 geopotential
SYNCX base-
field
and the short
periodics corresponding to full-field geopotential and second-order J2 and
J 2-m-daily coupling perturbations, estimation results were
improved.
Table
3.5.7 shows the results
of the SYNCI
dramatically
filter processing.
Table 3.5.7
of
Extended
Semianalytical
Kalman Filter Results for MILSTAR SYNCI
Summary
Compare Statistics
(30-day total estimation
and prediction
span)
Type of Run
Position
RMS Errors
(km)
The following
run used the tailored
Baseline perturbations
1-day observation
Velocity
RMS Errors
(cm/sec)
force model:
Radial
Cross-track
Along-track
Total
.896
.587
.117E+2
.117E+2
.844E-3
.234E-4
.598E-4
.846E-3
The following run used the tailored force model
with restored tesserals and short periodics:
Baseline perturbations
2-day observation
R
C-T
A-T
.870
.402
.791E+1
.566E-3
.294E-4
.636E-4
Total
.797E+1
.571E-3
43
Considering only these limited results, one might expect that SYNCI
estimation by the ESKF will result in performance considerably poorer than
for
SYNCX,
essential
the
SYNCI
geometry
accuracies
observability
the Beale
of
This
increase
just
above
is a problem
tracker-SYNCI
only one twenty-minute
seen during
SYNCX.
the
although
in outage
reflected in process noise.
seem
is made
system:
period
more uncertain
much
the
of four
of
by
could
satellite
per day instead
time by a factor
Whether
acceptable.
be
the four
should have
for
been
Movinq the tracker would in any event surely
improve orbit determination accuracies.
For
this
tracker
estimator,
the characters
somewhat different from those of SYNCX.
vations,
the
velocity
error
negligibly
alonq-track
position
is dominant,
small.
Only
by
is
quite
in absolute
determining
errors
are
Even with two days of input obser-
error
although
of the SYNCI
large,
terms
the impact
and
this
of these
the
radial
last error
errors
on
is
the
actual user, however, can their acceptability be effectively determined.
3.6
Assessment of MILSTAR User Acquisition Errors
Having demonstrated an easily tailored semianalytical force model
for the MILSTAR application and an effective tracker orbit determination
system, it remained to explore the effects of tracker orbit determination
accuracy--using SST/ESKF--on best-possible user acquisition.
The so-called
observation space evaluation of the tracker estimate incorporated several
tasks:
' Generating a long-arc error-free observation file
corresponding to each of several user locations
* Using the observations in a one-iteration differential
corrections estimator with solve-for vector resulting from
filter processing as initial guess
* Computing the statistical properties of the observation
residuals
at each distinct user site
The observation space evaluation of the tracker estimate would give the
most realistic demonstration of best acquisition accuracy for the groundbased user that could be expected from use of the SST/ESKF orbit determination system onboard the tracker aircraft.
44
Propaqation for thirty days of
the
filter
solve-for
vector--the
best
information
of
satellite
state
available to the user--gives the satellite's expected position at any time
of acquisition between occasions of information updating.
By generating
from the truth ephemeris a file of uncorrupted observations of the satellite
as
seen
from
a given
user
location,
a
record
of
actual
motion as seen by a perfectly informed user is established.
satellite
Observation
residuals during the first iteration of batch estimation thus directly
indicate acquisition errors when the user is realistically well informed.
°
The user sites chosen were Beale, Omaha (410 N, 2630
E), and a typi-
ical location in Alaska (660 N, 2090 E).
SYNCX is located near the equator
over the Pacific at 1940 E longitude.
Figures 3.6.1 through 3.6.6 give
expected
acquisition
range
and range-rate
sites for SYNCX, as do Figures
errors at each of the three user
3.6.7 through
3.6.12
for SYNCI.
For SYNCX,
each range residual plot shows the expected secular error increase as the
tracker update ages.
The range-rate residuals are oscillatory but with a
non-expanding envelope.
For SYNCI, both range and range-rate degraded
as time from update increased.
Table 3.6.1 summarizes results of this
observation-space evaluation of the baseline filter estimates for SYNCX
and SYNCI.
Table
3.6.1
Summary of Observation Space Evaluation of Tracker Estimate
RMS Residual
Errors
at User Locations
during the 30-day period between user updates
Test Case
Tracker
Location
Beale
Omaha
P
P
P
Alaska
P
P
P
SYNCX
Beale
250
.80
300
.84
140
1.0
SYNCI
Beale
1150
15.1
2300
17.0
1900
21.0
units:
P in meters
P is range
P in cm/sec
In order to explore the underlying observation qeometry for SYNCX,
the
station-to-satellite
vector
for each
45
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stations
was broken
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into the radial, cross-track, and along-track components of the satellitebased coordinate system.
The results supported the notion that, in gen-
eral, the observing station farthest away in longitude from the geostationary satellite will find along-track errors highly observable, and that
the station
errors.
nearest the
sub-satellite point will
see primarily radial
The tracker estimate has already demonstrated that its position
error is dominated by the along-track component and that its velocity error
is primarily radial, so it is not surprising then that Omaha and Alaska
would see the largest position and velocity errors, respectively.
Given the widely varying satellite observability for users in different locations, questions of optimum tracker location naturally arise.
Whether radial, along-track, or cross-track errors dominate in the estimationrprocess should influence tracker location for observation of geostationary satellites.
SYNCI observability from the tracker should definitely
be considered as well.
The most
important result
ob:tained in this
last task is
that,
even with a fairly abbreviated force model, and with relatively few input
observations, the tracker estimator does give reasonable user acquisition
errors.
For SYNCX, these errors are less then 1/3 km in range, and approx-
imately
1 cm/sec
in range
rate.
For SYNCI,
these
increase
to about
2 km
and 21 cm/sec, well under the expected acquisition error budget for tracker
estimation errors.
With further refinement of both this part of the budget
and that dealing with the inaccuracies of the user's approximate ephemeris
representation, further tracker force model truncation would be possible.
3.7
Effects of Third Body A- and B-Matrices
on Extended Semianalytical
Kalman Filter Estimation for MILSTAR
A
proposed augmentation of both
the Semianalytical Differential
Corrections and Extended Semianalytical Kalman Filter estimators was the
addition of A- and Bl-matrices due to third body perturbations.
A brief
examination
process
of
the roles
of
these
two matrices
in
the estimation
follows.
In the SDC, Green's F-matrix [2], the matrix of observation partials
with respect to the solve-for parameters at epoch, is calculated by chain
58
rule.
As
part
of
the F-matrix
calculation,
the matrix
of partial
deriv-
atives of osculating equinoctial elements with respect to the solve-for
state and parameters--the G-matrix essential to SDC estimation--is calculated via the Averaged Partial Generator (APG) and the Short Periodic
Partial Generator (SPPG). The G-matrix calculation explicitly uses several
matrices--among
others,
the following:
B 1 the partials of the short periodic functions with respect
to the mean equinoctial
* B2
the partials of the mean equinoctial elements with respect
to those at epoch
* B3
elements
(a state transition
matrix)
the partials of the mean equinoctial elements with respect
to the non-state solve-for parameters
In the APG, the A-matrix, the matrix of partials of mean equinoctial element rates with respect to the mean equinoctial elements, is used to propagate the B 2- and B3-matrices.
Like
the
SDC,
the
The B 1-matrix is computed
FSKF
employs
the A-
and
in the SPPG.
B1-matrices.
Perhaps
most importantly, state propagation used to calculate a predicted solve
vector and covariance uses the A-matrix.
to
propagate
the
B 2-
necessary
This matrix is again employed
and B 3-matrices
that
have
application
in
computing the matrix of partials of the averaged equinoctial state with
respect
to the solve-for
state
and parameters.
The B 1-matrix is used
to
calculate a predicted short periodic contribution to the osculating state
due to mean state
correction
[3].
Finally,
the ESKF
employs
a
matrix
of
observation partials, this time with respect to the averaged equinoctial
elements.
updating
This H-matrix used
the state and covariance
In
principle, the
rately--and,
for the batch
in
calculating the
employs
estimation
estimator,
filter
qain
and
in
the B 1-matrix directly.
process should
convergence
proceed
should
be most
most accurapid--if
these partials matrices accurately reflect all known forces acting on the
satellite.
segments
In practice, however, evaluating the matrices for numerous
of the force model is in general
computationally
not yield measurable benefits in the estimation.
expensive
and may
At least the two-body
contribution to the A-matrix is always used.
Because of the generally
small
however,
magnitude
of the short
periodic
functions,
regularly omitted entirely.
59
the B 1-matrix is
Both
SST-based estimators have
at
least the use
of A-
and B 1-
matrices evaluated by brute-force finite-differencing for any force model.
In order to increase computational efficiency in calculating the partials
due to the most important geopotential force other than strict two-body
attraction, however, analytic expressions for the J2
matrices have already been obtained.
bations
being
significant
contribution to both
The low-order third body pertur-
for high-altitude
(including geosynchronous)
satellites, analytic A- and Bl-matrices were computed using MACSYMA [18]
from closed-form expressions for averaged element rates and short periodic
functions
due
to the P2
perturbation.
To
the B 1
make
computation
trac-
table, its expressions were truncated to zeroth order in the eccentricity.
In order to evaluate the effect on filter performance of both third
body A-matrix and of the A-matrix in general, several baseline SYNCX filter
runs were made with various A-matrix options.
Table 3.7.1 shows the test
results.
Table 3.7.1
Effects of A-Matrix on Extended Semianalytical Kalman Filter
Estimation for MILSTAR
Type of A-matrix*
Position RMS Errors
(km)
Velocity RMS Errors
(cm/sec)
Baseline filter case: Radial
Analytic J2 ,
Cross-track
finite-differenced
Along-track
third body based on
Total
AOG model
.54405E-1
.57183
.10195E+1
.11702E+1
.72802E-4
.41664E-4
.39896E-5
.83976E-4
Finite-differenced
central body and
third body, both
based on AOG model
R
C-T
A-T
Total
.54397E-1
.57183
.10194E+1
.11701E+1
.72798E-4
.41664E-4
.39890E-5
.83971E-4
Finite-differenced
third body based
on AOG model
R
C-T
A-T
Total
.54387E-1
.57180
.10194E+1
.11701E+1
.72797E-4
.41662E-4
.39884E-5
.83971E-4
None
R
C-T
A-T
Total
.54387E-1
.57180
.10194E+1
.11701E+1
.72797E-4
.41662E-4
.39884E-5
.83971E+4
*plus ever-present two-body contribution
60
For this SYNCX case, inclusion of the P2
contribution to the A-
matrix, like inclusion of the J2 contribution, yields differences in total
RMS estimation errors on the order of centimeters.
gated
the
role
of
the
special
perturbations
May [19] had investi-
matrix
analogous
to
the
A-
matrix here, that is, the matrix used to propagate the special perturbations state transition matrix.
She found when working with a low-altitude
satellite and the GTDS-based Extended Kalman Filter that the J2 contribution was not needed for filter convergence, provided that a
reasonably
accurate initial state covariance matrix was supplied to the filter.
These
MILSTAR results are supportive of this conclusion.
Further testing of the impact of the A-matrix on estimation might
suggest that the matrix need reflect only the strictly two-body dynamics if
an accurate covariance is known to the filter.
In this specific MILSTAR
case, maintaining a good covariance in the tracker estimator, advisable for
tracker filter convergence, could mean significant reduction of the estimator computational burden.
Concerninq the role of the matrix for other estimators, May indicated
that the Cowell-based differential corrections estimator was indeed prone
to divergence when the contribution of the J2 perturbation was not included
in that matrix
analogous
to the A-matrix,
especially
longer than several satellite orbital periods.
for observation
spans
It remains to be determined
whether the SDC is similarly susceptible to the lack of J2, and, especially
for high-altitude
satellites,
of P2 contributions
to the A-matrix.
In order to evaluate similarly the effect of the B1 -matrix on filter
estimation,
a smattering
of SYNCX
and SYNCI
are shown in Table 3.7.2.
61
runs were
made.
The
results
Effects of B-Matrix
Type of Bl -matrix
Table 3.7.2
on Extended Semianalytical Kalman Filter
Estimation for MILSTAR
Position RMS Errors
(km)
Velocity RMS Errors
(cm/sec)
For SYNCX, baseline perturbations, 1-day observation
29-day prediction
300 m range observation uncertainty
Baseline filter case:
None
Radial
Cross-track
Along-track
Total
.74949E-1
.58317
.12207E+1
.13549E+1
.72802E-4
.42484E-4
.58448E-5
.97147E-4
Analytic J2 ,
finite differenced
third body based on
AOG model
R
C-T
A-T
Total
.74998E-1
.58317
.12206E-1
.13548E+1
.87181E-4
.42484E-4
.54883E-5
.97137E-4
For SYNCI, bas line perturbations, 2-day observation
28-day prediction
Baseline:
None
R
C-T
A-T
Total
.86998
.40220
.79093E+1
.79672E+1
.56646E-3
.29425E-4
.63555E-5
.57079E-3
Analytic J2
R
C-T
A-T
Total
.86992
.40220
.79097E+1
.79675E+1
.56649E-3
.29425E-4
.63551E-4
.57080E-3
Analytic J2 ,
finite differenced
third body based
on AOG model
R
C-T
A-T
Total
.87012
.40211
.79068E+1
.79647E+1
.56628E-3
.29419E-4
.63565E-4
.57059E-3
For
SYNCX
here,
the contribution
of
the B-matrix
estimation accuracy only sub-centimeter-level.
was
to increase
For SYNCI, the contribution
was slightly more significant, particularly when both J2 and third body
perturbations contributed to B1 -matrix calculation.
that the number
of SYNCI
observations
is quite
It should be recalled
limited;
the effects
of B1
on the estimation are expected to be greater when the observation outage
time
is large.
But
the centimeter-level
improvement
for SYNCI
would
not
justify the inclusion of the matrix in the MILSTAR tracker orbit determination system.
62
An
end-to-end check of
third body A- and B-matrices
the accuracy and burden
of the analytic
is reserved for later SDC and ESKF tests in
which their impact will be enhanced.
In any event, uncertainties as to the
role of A- and Bl-matrix partials in general ESKF estimation can only be
resolved by an investigation that would consider other types of satellites
and various observation schedules.
High-accuracy estimation provides a
scenario in which use of the matrices is expected to be especially useful.
Whether their use is truly justified in such estimation depends, of course,
on the completeness of the force model and the accuracy of the observations.
63
Chapter
4
THE ANIK Al ESTIMATOR EVALUATION STUDY
4.1
Study Motivation
Processing real observation data with the Extended Semianalytical
Kalman Filter and
the Semianalytical Differential Corrections estimator
provides the best possible demonstration and assurance that Semianalytical
Satellite Theory and its two estimators form indeed a complete, accurate,
and efficient orbit determination system.
The semianalytical system is
intended for a real operating environment.
This study, only the second
real data test case for the ESKF, thus provides an opportunity for most
prominent system display.
In addition, processing of real high-precision
data provides an opportunity to discover whether any as yet unmodeled
dynamics coupling is conspicuous.
It provides an opportunity to test an
asymptotic theory in an way most likely to reveal any asymptotic errors.
It was hoped that the results of the ANIK Al study would demonstrate:
·
that all GTDS-based estimators, including the ESKF, are very
similar in their ability to produce accurate orbit estimates
·
that their accuracies approach those of Telesat's extended Kalman
filter, which is specially tailored for synchronous satellites
4.2
ANIK Al Background
Telesat Canada currently operates five geostationary satellites, the
network providing a variety of satellite communication services primarily
to Canadian customers.
ANIK Al, launched in November 1972, was the initial
satellite of the world's first domestic comsat network.
The satellite is
cylindrical, with a diameter of 1.9 m, an overall height of 3.5 m, and a
current mass of approximately 245 kq.
All ANIK satellites are monitored
from the Satellite Command Center (SCC) in Ottawa, which is linked directly
to Telesat's main tracking station in Allan Park, Ontario.
communications
antenna
is
not
auto-tracking,
64
ANIK
Al
Because its
stationkeeping
requirements are tight:
+.100 in both longitude and inclination
[20].
Given these deadbands, actual orbit determination accuracies had been
derived by accounting for expected excursion due to known forces and dynamics, operational errors and hardware constraints, and
modeling
errors.
These required orbit determination accuracies have been refined through
continuing experience.
The Allan
Park facility
used to track
ANIK Al is the 11 m monopulse
Tracking, Telemetry, and Command Antenna [20], which for a given satellite
measures approximately six daily bursts of range, azimuth, and elevation
trios at one-second intervals for fifteen seconds.
averaged
and sent
directly
to
the
SCC
for
Each burst of data is
processing.
Tracking
station
location and measurement uncertainties for the burst-averaged measurements
were
supplied by Telesat
and are given in Table
Table
4.2.1.
4.2.1
Allan Park TT&C Antenna
Site Location
(relative to Telesat-supplied
geoid)
Latitude (geodetic)
Longitude
44.1728990 N
80.9366220 E
Altitude
298.1 m
(above MSL)
Observation Statistics
Range
anoise
Azimuth
Elevation
=
6 m
bias = 32 m
anoise = 28.8 sec
anis
= 28.8 sec
bias = -140 sec
bias = 350 sec
Until 1976, Telesat's orbit determination system was implemented in
a large shared utility mainframe.
For stationkeeping purposes, the system
included an iterative weighted-least-squares batch estimator
a priori
state
information
and a week's
observations
to estimate
that used
any of a
maximum of five parameters beyond the mandatory six flight parameters [20]:
* an impulsive thrust vector (three components)
· one to five observation
biases
* a linear correction
to the built-in
* a linear correction
to the tangential
tesseral geopotential harmonics
65
solar radiation
acceleration
force model
due to
Concerning
the
discovered early
that
theoretical
the
solar
parameter,
it
radiation
model
a
As
residuals.
observation
large
undesirably
solve-for
penultimate
produced
ameliorating
an
result,
been
had
small force correction was estimated as a function of solar declination.
As regards the last solve-for parameter above, because the limited operating range of the ANIK satellites has permitted polynomial approximations
due to the earth's
to accelerations
fine-tuned for
the
satellites,
Telesat
determination accuracy.
gravitation,
model
the force
significantly
could
increasing
be
orbit
With this batch processing system, the station-
keeping requirements for ANIK Al mentioned above--and those even tighter-were routinely achieved.
To
avoid increasing processing costs and excessive analyst time
spent in routine ephemeride estimation, a
determination system was designed.
new
minicomputer-based orbit
The new estimator is a real-time exten-
ded Kalman filter that, in estimating a particular vector, computes and
accounts for maneuvers, has proven eminently stable
(no restarts ever),
and, most importantly, has allowed stationkeeping accuracy requirements to
be
consistently
met
In addition,
[11].
only in unusual circumstances.
ally
logged and maneuvers
Along
of
is necessary
intervention
Typically, the satellite states are manu-
assessed
an array
with
analyst
and planned
attitude
once a week.
parameters,
and attitude-relevant
the current Telesat solve vector includes the inertial position and velocity augmented with azimuth and elevation biases and a solar radiation force
maps
into
than solve
Rather
correction.
lonq-term
longitude
for range
errors,
bias, a slowly drifting
an updated
estimate
simply used by Telesat as an observation correction.
bias
that
of this bias is
According to Telesat,
the solar radiation model developed with the batch processor has been even
further refined through sequential estimation, since the long data arcs
required for the old estimator precluded determination of fine model detail
[11]. Since beginning operation in 1977, the filter-based orbit determination system has allowed all stationkeeping constraints for ANIK A satellites to be met.
In addition,
this minicomputer-based
orbit
determination
system with extended Kalman filter estimator has provided economic advantages,
fast
data
turnaround
and
access
to
present
greater flexibility of operation and modification.
66
satellite
state,
and
4.3
Study Preliminaries
ANIK Al data supplied to Draper includes the following:
*
almost 24 days of complete, unsmoothed TT&C observation data
*
geopotential approximating coefficients for SAO 6x6 field
*
solar radiation model (resultant spacecraft acceleration as a
function
*
of sun declination)
satellite maneuver data for observation period
(no maneuvers)
*
filter history for 3 1/4 days during the given observation span,
including daily predicted augmented Cartesian state and its
standard deviations and correlations
*
process noise computation information--daily 1-a eror growth
rates for all augmented state vector components
Figure 4.3.1 shows a page of the annotated filter history.
The ANIK Al task, although it would lead especially to evaluation of
the ESKF, was used to characterize all GTDS-based estimators:
*
CDC
a Cowell special-perturbations-based differential
corrections estimator
*
SDC
the Semianalytical Differential Corrections estimator
*
EKF
a Cowell special-perturbations-based extended Kalman filter
·
ESKF the Extended Semianalytical Kalman Filter
Some criteria for evaluation of estimator performance were needed.
At
the outset, the goal was
to have all estimators match the Telesat
extended Kalman filter history in all ways possible, including duplication
of both Keplerian and Cartesian augmented state estimates and their statistics.
Other available information helped guide the choice of evaluative
criteria.
less
Range observation residuals were known to be typically 6 m or
[11] for the Telesat
week period.
of smoothed
used as a smoother
for the latest
two-
More specifically, Telesat supplied residuals for each trio
observations
the history.
estimator
used during
the 3 1/4 days of processing
shown
in
As Table 4.3.1 shows, residual statistics were computed for
the 19 trios there, and an approximate
distance
error was computed
from the
standard deviations obtained and some average ANIK Al elevation and range.
67
(
. iI
;I
IA
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w3
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68
Table
4.3.1
Telesat Filter History Residual Statistics
Residual
Standard Deviation
Range
12.3 m
Azimuth
53.2 sec
Elevation
42.7 sec
Distance error based on these statistics
11.6 km
Duplicating these residuals with a differential corrections estimator--and especially with a filter--would be a functionally meaningful display of the estimators' prowess.
The duplication of filter history and
residuals would be most pleasing and doubtless most successful if the wellestablished Telesat filtering process were emulated as much as possible.
In order to begin data processing with Draper's GTDS-based estimators in a way most consistent with Telesat procedures, several preparatory
tasks
were
completed.*
First,
the
data
were
format and were smoothed by burst averaging.
ion was adopted outright.
converted
to
an
acceptable
Telesat's 4-a editing criter-
In addition, because the details of the observa-
tion process were unknown, an observation model standard for trackers computing this trio of observation
model
[4].
types
was assumed,
namely,
the GTDS C-band
The measurements are corrected for tropospheric refraction
before Telesat processinq, so the corresponding GTDS capability was used.**
The
tracking
corrections
station
position
to a certain
Legendre polynomials.
was
flattened
given
relative
to
sphere were defined
a geoid
whose
shape
in terms
of a sum of
GTDS defines station Position relative only to a
flattened geosphere whose shape is described completely by a
coefficient and a mean radius.
flattening
Having adopted these two values from Tele-
sat information, the further effects on observation measurements of applying
the
geoid
shape
measurement noise.
applied
correction
containing
station
location
could
be
lost
in the
location in GTDS.
these changes
is in EAW2148.TELESAT.LOAD.
**The GTDS-based tropospheric model
corrects
measurements.
It should be noted that only
properties
were
Consequently, the Telesat qeoid corrections were not
to the station
*Software
to the
used
in
the
corrections
range
and
elevation
averaged tropospheric
model,
since
time-dependent properties had not been updated for 1982.
69
the
file
of
Telesat employs a modified true-of-date coordinate system referred
to the mean equinox, a system in which the coordinate frame is updated only
at midnight of each day, prior to daily output of the augmented Cartesian
state estimate.
GTDS can duplicate this only approximately with its true-
of-reference frame, one referred to
coordinate frame of midnight of
the mean equinox and in which
the
state epoch is retained throughout the
estimation.
Telesat propagates process noise to observation time and adds its
contribution to the prediction-phase covariance.
Although GTDS-based fil-
ters had formerly added process noise only to the final update covariance,
this procedure was altered to reflect Telesat practices.
In addition, the
GTDS process noise model was changed to a time-squared from a time-linear
formulation.
GTDS has no geopotential model tailored for synchronous satellites,
so the.6x6 SAO field that serves as the basis for Telesat's model was used
in both semianalytical and special perturbations force models.
With no
third body model particulars known, a general lunar-solar point-mass model
was used.
In addition, GTDS currently employs a very simple solar radia-
tion pressure model.*
In order to employ this model, some average projec-
ted satellite area was needed.
of
3 m2
plausible
[21].**
The known satellite geometry made a value
Solving
for Cr would
also
emulate
as
far as
possible Telesat's solar force parameter solve-for.
*The solar radiation force acting on the body is modeled as a function of
an average surface reflectivity and spacecraft area normal to the incoming
radiation, the mean solar flux at one A.U., and the sun-vehicle distance
[4].
**In order to estimate more directly some average projected satellite area,
the GTDS special perturbations propagator was used to generate and display
typical solar radiation forces for a satellite with unit projected area,
unit mass, and unit surface reflectivity constant, Cr
during the period
of supplied observations.
The nearly constant sun declination was also
displayed. Even with units checked scrupulously, these forces when multi2
plied
by
the
3 m
satellite
area,
a typical
Cr
of about
unity,
and
the
given satellite mass were higher by approximately a factor of satellite
mass than those forces supplied by Telesat. Telesat maintained that their
values were indeed not accelerations, but expressed certainty that an error
in GTDS
force
models
on
the order
of
such a
large
factor
would
be
quite
noticeable during the estimation process. Meanwhile, the 3 m was retained
for projected satellite area; solving for a Cr that was expected to be
quite observable would help render having a truly "average" satellite area
unessential.
70
4.4 Cowell Differential
Corrections Estimator Evaluation for ANIKAl
A baseline for estimation acccuracy was established by processing 11
days of smoothed
data with
the CDC.
As implemented
in GTDS,
this estimator
is known to meet well-established standards of accuracy for batch processors.
The force model used
in the Cowell
equations
of motion
was
reason-
ably conservative for satellites of this type (6x6 SAO qeopotential, lunarsolar
point-mass gravity,
solar
radiation
pressure
in
the
12th-order
Cowell-Adams predictor-corrector with 600 sec stepsize). The initial input
elements
to CDC, considered the baseline truth epoch conditions, were those
given by the Telesat filter history at the beqinninq of the first
observations processed in this task; see Table4.4.1.
Table
day of
4.4.1
Telesat Filter State used in
Cowell Differential Corrections Initialization
Epoch
17 April 1982
Oh Om Os
a
e
42165.7791
2.6752E-4
i
1.408310
a
92.5243330
X
220.711840
M
Included
km
147.584100
in the CDC solve
vector were
osculating
Cartesian
Position
and velocity, azimuth and elevation biases, and reflectivity constant.
biases
were
initialized
to
values
sec and 350 sec for azimuth
close
and elevation,
to
those
in
the history--to
respectively,
and Telesat's
range bias estimate was used as a constant observation correction.
processing
of
yielded,
after
14 iterations,
epoch that matched well that of the Telesat filter
shows
-150
32 m
This
11 days of data is the baseline CDC run.
The estimator
4.4.2
The
this.
The
remaining
a Kelerian state at
in a, e, i, and
two Keplerian elements (note their espe-
cially high uncertainties) differed substantially, although the sum
compares well.
Table
+ M
The very low eccentricity of the orbit will produce a near-
singularity in perigee position.
11 Cartesian elements were considerably
different.
71
Table
4.4.2
Baseline Cowell Differential Corrections Solved-for Epoch State
CDC State
Element
Corresponding Telesat Filter State
Standard
Element
Standard
Deviation
a
42165.783
e
2.5618E-4
km
Deviation
8.8E-4 km
a
42165.779
2.2E-6
e
2.6752E-4
i
1.40100
Q 92.5320
1.1E-3 0
4.5E-20
i
W 218.650
M
149.620
4.7E-1 0
5.0E-1 0
w
M
220.710
147.580
x
y
z
v
-7927.556
41423.273
149.853
-3.0182
km
km
km
km/sec
.041
.012
.072
8.7E-7
-.57745
km/sec
2.6E-6 km/sec
7.4755E-2 km/sec
6.OE-6 km/sec
-7912.208
41426.008
148.562
-3.0184
km
km
km
km/sec
v
-.57638
km/sec 9.1E-6 km/sec v
v
7.4372E-2 km/sec 6.OE-5 km/sec v
Y
az bias
el bias
C
5.5 sec
5.0 sec
1.3937
over
4.4.3
this
4.5E-2 0
4.0E-3 0
az bias -155 sec
el bias
351 sec
residuals
for range,
11-day span are displayed
gives
1.1E-40
4.7E-20
km
km
km
km/sec
5.1 sec
5.0 sec
6.1E-2
r
The CDC's
km
km
km
km/sec
Y
-138 sec
321 sec
2.2E-3 km
2.2E-7
1.40830
2 92.5240
x
y
z
v
.213
.099
.807
7.7E-6
km
statistics
for these
azimuth,
and elevation,
in Figures
residuals.
4.4.1
Here,
through
and distance
4.4.4.
Table
as in all differential
corrections and filter runs in the study, the residuals were essentially
unbiased.
Table 4.4.3
Baseline Cowell Differential Corrections Residual Statistics
Residual
Standard Deviation
Range
6.289 m
Azimuth
Elevation
60.44 sec
42.57 sec
Distance
RMS error over fit span
12.317 km
72
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The approximately 6 m range residuals are reminiscent of results for
the Telesat smoother, and much of the 12.3 km position error can easily be
accounted
for by the approximately
7 km of error
that is unavoidable
for a
satellite at such large range, with apparent elevation of some 300, and
with azimuth and elevation measurements of this basic accuracy.
be remarked
that the estimated
29 sec noise
of the original
It should
angle
measure-
ments was not returned, although the 6 m range noise was well isolated.
Much more significant is the fact that the CDC residuals are similar
to
those
obtained
from
the
Telesat
filter
history--6.3
12
m
in
range,
60 vs. 53 sec in azimuth,
both.
Range and azimuth determination for geosynchronous satellites are
not independent.
able:
and approximately
versus
42 sec in elevation
for
The final distance RMS error comparison is quite favor-
12 km for the CDC versus
11 km for the Telesat
filter.
The number of iterations required for estimator convergence seemed
unusually
large.
Doubtless
the
tracking
geometry
was
to
a large
extent
responsible, with a geostationary satellite observed from only one station.
Several possible error sources might be suspected in the CDC test:
Approximations in
*
coordinate frame
·
tropospheric correction model
·
station location
·
basic tracking model
Inadequate modeling of
·
geopotential
*
solar radiation force
Direct comparison of the epoch states for both Telesat and GTDS
estimators should be possible since the coordinate frames should coincide
at epoch.
There
may yet be
some
slight
discrepancy
in
frame
definition.
It was posited that the discrepancy might be eliminated with use of more
up-to-date inertial to earth-fixed frame transformation data.
In one-time
use of newer transformation data in the CDC, the states were indeed brought
much more into coincidence, as Table 4.4.4 shows.
discrepancy
was reduced
by a factor of five.
77
The maximum position
Table 4.4.4
Solved-for Epoch State for Cowell Differential Corrections
with New Frame Transformation Data
CDC State
Element
Corresponding Telesat Filter State
Standard
Deviation
Element
Standard
Deviation
a
e
42165.783 km
2.5548E-4
8.8E-4 km
2.2E-6
a
e
42165.779 km
2.6752E-4
2.2E-3 km
2.2E-7
i
1.40090
1.1E-20
i
1.40830
1.1E-4 0
Q
92.5650
4.5E-2
Q 92.524
4.7E-20
w
M
218.740
149.510
4.7E-1°
5.OE-1 0
W 220.710
M
147.580
4.5E-2 0
4.OE-3 0
x
y
z
v
-7924.404
41423.642
148.256
-3.0183
v
Y
v
r
°
.211
.098
.802
7.6E-6
km
km
km
km/sec
x
y
z
v
-7927.556
41423.273
149.853
-3.0182
km
km
km
km/sec
-.57727
km/sec 9.1E-6 km/sec v -.57745
km/sec
Y
7.4372E-2 km/sec 6.OE-5 km/sec v 7.4755E-2 km/sec
az bias
el bias
C
km
km
km
km/sec
0
-131 sec
321 sec
1.3931
5.4 sec
4.9 sec
az bias -155 sec
el bias
351 sec
.041
.012
.072
8.7E-7
km
km
km
km/sec
2.6E-6 km/sec
6.OE-6 km/sec
5.1 sec
5.0 sec
6.1E-2
~~~~~~~~~~~~~~~~~~~~~~~......
The residual statistics in this run were also different somewhat
better
than those
of the baseline
run, as seen in Table
4.4.5.
Table 4.4.5
Residual Statistics for Cowell Differential Corrections
with New Frame Transformation Data
Residual
Standard Deviation
Range
60.85 sec
Elevation
42.22 sec
Distance
How the states
6.050 m
Azimuth
RMS error over fit span
12. 274 km
can be brought
into better
agreement
is problematic.
The possibility of directly comparing estimator states seems remote at this
78
time; in any event, when comparing states that result after several days of
filter processing, the frames will be sufficiently different to preclude a
thorough comparison.
with
those
of
Henceforth the direct comparison of estimated states
Telesat
is
sacrificed, and
only
estimator
observation
residuals are examined.
Removing the tropospheric model's corrections showed the correction
to be beneficial.
The non-state solve-for parameters, particularly the
elevation bias, were significantly changed:
Cr
=
1.3946, azimuth
elevation biases -138 sec and 422 sec, respectively.
and
Table 4.4.6 shows
that the range and elevation residuals were lower than for the baseline
run, but the distance RMS error over the 11-day fit span was 62 m higher.
Why the residuals
were
lower for precisely
those observation
types
that are
explicitly corrected for tropospheric refraction is a question that should
be answered.
Table 4.4.6
Residual Statistics for Baseline Cowell Differential Corrections
without Tropospheric Corrections
Residual
Standard Deviation
Range
5.986 m
Azimuth
60.71
Elevation
42.14 sec
sec
Distance RMS error over fit span
12.379 km
Lastly, in order to address the accessible question of whether a
more accurate geopotential model would improve the estimation results, the
same
set-up
was
run with
a GEM9
15x15
model.
The
solved-for
non-state
parameters were slightly different from those of the baseline case:
1.3938, azimuth bias = -140 sec, and elevation bias = 322 sec.
Cr =
Table 4.4.7
shows that most of the observation residuals, when compared with those of
the baseline run, were actually somewhat degraded.
79
Table 4.4.7
Baseline G4E9 Cowell Differential Corrections Residual Statistics
Residual
Standard Deviation
Range
Azimuth
Elevation
Distance
4.5
6.146 m
60.52 sec
42.49 sec
RMS error over fit span
12.311 km
Semianalytical Differential Corrections Estimator Evaluation for
ANIK Al
After obtaining satisfaction that the GTDS-based CDC is an accurate
estimator,
evaluation
of the SDC followed.
The SDC too was used
to eval-
uate Cr and azimuth and elevation biases, while being supplied a conservative semianalytical force model with SAO geopotential.
the extremely
conservative
model used.
The integrator
Runge-Kutta scheme with half-day stepsize.
80
Table 4.5.1 shows
was the fourth-order
Table
~~
4.5.1
Force Model used with Semianalytical Estimators
|~~
-~~~
_-
SPG Model
AOG Model
-
w
_·
Zonals:
3
(6x0), e
Zonals:
(6x0), SAO
M-dailies:
Tesseral Resonance:
(2,2) through
(6,6)
2
Second-order J2 , e
4
(6x6),
1
e
Tesserals:
(6x6), e
Lunar-solar point mass:
4
Parallax--moon=8,
sun=4, e
3
2
J2 , e
Second-order
Solar Radiation Pressure
0
Second-order J 2-m-dailY 4
coupling, (6x6), e
Lunar-solar point mass:
8 frequencies,
2 time derivatives
Solar Radiation Pressure:
6 frequencies,
1 time derivative
_
_1_11
_
.
After giving a qood mean state
-
estimate
-
-~~_
as input
to an SDC run,
the
estimator converged within 12 iterations to give observation and position
residuals
very similar
to those of the baseline
through 4.5.4 illustrate.
for
this
so-called
baseline CDC run.
values
similar
CDC above, as Figures
4.5.1
Table 4.5.2 shows that the residual satistics
baseline
SDC
run
are
also
quite
familiar
from
the
In addition, the non-state solve-for parameters took on
to those
= 138 sec, and elevation
for the baseline
bias = 321 sec.
81
CDC:
Cr
= 1.3940,
azimuth
bias
a
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Table 4.5.2
Baseline Semianalytical Differential Corrections Estimator
Residual Statistics
Residual
Standard Deviation
Range
Azimuth
Elevation
6.399 m
60.40 sec
42.57 sec
Distance RMS error over fit span
12.313 km
The force model was changed
model.*
to include
a time-independent
third body
The SDC so supplied converged in 12 iterations to give observation
and distance residuals much higher than those of the baseline CDC run.
range residual suffered relatively most.
The
Table 4.5.3 shows this.
Table 4.5.3
Residuals for Semianalytical Differential Corrections Estimator
with Time-independent Third Body Model
Residual
Standard Deviation
Range
Azimuth
Elevation
61.16 m
65.05 sec
62.15 sec
Distance RMS error over fit span
15.300 km
The residuals plots for both baseline CDC and SDC runs summarize
most effectively that the two GTDS-based differential corrections estima-
tors produce excellent results for ANIKAl1. These plots could be overlaid
with the comparison yielding almost no perceptible differences.
Only for a
few range residuals do the plots show differences.
At all of these points
of discrepancy, the SDC yielded smaller residuals.
The potential effi-
ciency of the estimators, however, differs significantly.
*The weak-time-dependent third body model accounts for the slowly-varying
satellite-apparent movement of the third body (particularly the moon). It
had been established that using the time-independent model results in significant errors for synchronous satellites [22].
86
The
whereas
baseline
CDC
solve
that for the baseline
state
was osculating
SDC was mean state.
position
and velocity,
To determine
whether
the
two epoch solved-for states did in fact correspond, each was propagated
over
the 11-day
tive estimator.
statistics
fit span with
the integrator
and final Cr
of each respec-
Table 4.5.4 gives the RMS position and velocity comparison
for this span.
Table 4.5.4
Comparison of Baseline Cowell and Semianalytical Differential
Corrections Epoch Solved-for States
i
Compare Statistics
(over 11-day fit span)
Component
Position RMS Errors Velocity RMS Errors
(km)
(cm/sec)
Radial
Cross-Track
Along-Track
.196E-2
.159E-1
.459E-2
.230E-3
.116E-2
.152E-3
Total
.167E-1
.119E-2
The position
errors
here
are
reasonably
small,
two solved-for states were indeed comparable.
cross-track
and were due to slight disagreement
In an attempt
to
reduce
the number
The
I
I
i
i
indicating
iterations
of periqee.
required
convergence, a good initial covariance was supplied to it.
reflected
run,
differences
in input
and it led to convergence
cantly different
state known
in nine iterations
from that of the baseline
In order to investigate
formance,
and final
several
runs were
two-body contribution.
for
SDC
The covariance
from the baseline
to a state
SDC
not signifi-
run.
at last the effects
made.
the
larqest errors were
on argument
of
that
The first
of A-matrix
of these
on SDC per-
used only the strict
The second added to this matrix the contribution of
the new analytic P2 third body matrix.
The results are shown in Table
4.5.5 below,
those
where
they are compared
with
with its central and third body state partials.
87
of
the baseline
SDC
run,
Table 4.5.5
Effects of A-matrix on Semianalytical Differential Corrections
Estimation
for ANIK Al
Type of A-matrix*
Residuals
Baseline:
Analytic J 2,
finite-differenced
third body based on
AOG model
None
Analytic P2 third
body
Range
Azimuth
Elevation
Distance
6.399 m
60.40 sec
42.57 sec
12.313 km
R
59.63 m
Az
El
101.3 sec
74.74 sec
Dis
20.868 km
R
6.248 m
Az
61.06 sec
El
42.19 sec
Dis
12. 294 km
*plus ever-present two-body contribution
Using the strictly two-body A-matrix in the SDC produced an unacceptable solution.
however,
led to quite
that
the
versus
would
of
baseline
The addition of the new analytic P2
effective
case.
12 for the baseline
appear
solve-for
from the number
state that having
production
This
case
and
last
of an epoch
run
of iterations
the effects
state
converged
11 for the case
contribution,
in
with
comparable
15
no
and the statistics
to
iterations,
matrix.
It
of the final
of either J 2 or P 2 perturbations
in
propagating the semianalytical state transition matrix is sufficient for
an accurate batch least-squares solution.
advantage in final accuracy*
Having both
ives no apparent
although estimator convergence is slightly
enhanced.
The timing estimates below when considered from a
perspective
show
that
the last case is relatively
more
per-iteration
efficient
than
the
baseline.
*It should be remarked that the baseline CDC run used only J2 (in addition
to the two-body effects) in propagating the state transition matrix.
88
Table 4.5.6
Effects of A-Matrix on Semianalytical Differential
Corrections Computation Time
CPU Execution Time
Type of Run
3:09.39 min:sec
Baseline A-matrix
(12 iterations)
2:28.93
Two-body A-matrix
(11 iterations)
3:17.08
(15 iterations)
Two-body and
analytic P2 third
body A-matrix
4.6
blend
Extended Kalman Filter Evaluation for ANIKA1
to
urge
In
order
of
information
forward
from
next
Telesat
best
the
and
the
EKF
the best
performance,
CDC
baseline
run
above
were
The converged baseline CDC Cartesian state and corresponding
supplied it.
variances were supplied to the filter, as was the CDC's force model with
100 sec timestep in the fourth-order Runge-Kutta-Fehlberg intergrator that
is used with the EKF.
minimize
filter
This choice of initial state and covariance would
transients
at the beginning
of processing
and thus
induce
the steady-state performance that could be compared with that represented
in the Telesat filter history.
The EKF does not presently have the capa-
bility of solving for observation biases, so the CDC's final azimuth and
elevation biases were used, along with the Telesat-supplied range bias, as
constant observation corrections.
In addition, the CDC's bias standard
deviations were RSS'ed into the observation noise in order to help keep
filter gains high.
Finally, specifying process noise in a way intended to
further duplication of the operation of the mature Telesat filter, that
process noise given for the Cartesian state in the filter history was used
directly.
A number several orders of magnitude higher (1.E-24) was used
for the process noise contribution from Cr uncertainty.
The EKF performed well under these conditions.
The solved-for Cr
in this filter run was 1.3947 versus 1.3937 for the baseline CDC case.
observation
the
residuals,
processing
span,
discounting
were
very
those
similar
89
large
values
to those
of
at the beginning
the baseline
CDC
The
of
and
SDC, as Figures 4.6.1 through 4.6.4 show, and were thus comparable to those
obtained by Telesat.
Table 4.6.1 gives the residual statistics.
One espe-
cially large (40 m) range residual resulted early in the filter history,
probably from filter over-correction due to
matrix
off-diagonal
terms.
If
the omission of
this residual were
ignored,
covariance
the
range
residual standard deviation for this estimation would be a very comfortable
6.294 m.
Table 4.6.1
Baseline Extended Kalman Filter Residual Statistics
Residual
Standard Deviation
Range
Azimuth
Elevation
7.953 m
61.46 sec
42.84 sec
Distance RMS error over fit span
12.382 km
I
The integrator used with the Telesat filter was unknown.
Integrator
noise can interfere with the estimation process, so the fourth-order RungeKutta-Fehlberg integrator that is used with the EKF was used to propagate
for five days the baseline CDC's solved-for state using its special perturbations model with integration stepsizes of both 100 and 600 sec.
Result-
ant observation residuals were computed using these ephemerides and the
supplied observations.
Residuals for the 100 sec run were always higher.
The differences in residuals between the two runs were strictly increasing,
reaching
values at the end of the five days
of 6 m in range
.1 sec in azimuth and elevation, respectively.
sec is appropriate
the
100 sec
for this integrator
value--a
value
considered
satellite--might be questioned.
.4 sec and
Whether a timestep of 600
is not known;
typical
and
in any event,
for use with
this
use of
type
of
Certainly putting this EKF into operation
would oblige more complete evaluation of the effects of integrator noise.
The Cartesian state variances predicted by the EKF were typically
several
orders
of magnitude
lower
than
those
obtained
by Telesat.
Table
4.6.2 gives typical state uncertainties for both filters in steady-state
operation.
90
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Table 4.6.2
Comparison of GTDS and Telesat Extended Kalman Filter
Steady-state Variances
Variance
Element
GTDS EKF
Because
this
the
Telesat
EKF
a
e
5.3E-4 km
1.OE-6
1.9E-3 km
2.2E-7
i
]
8.9E-6 0
3.5E-4 0
1.1E-4 0
4.OE-2 0
W
M
3.4E-30
3.6E-30
4.4E-10
4.7E-20
EKF had
11-day observation
reached
steady-state
span, these
variances
operation
would
at
the
be a reflection
end
of
of the
basic satellite observability (including observation accuracies) and the
process
noise.
observability
for
shown in the baseline
for these
Telesat.
plied
As
It
process
estimators
is possible
noise,
that
CDC and SDC above,
has been
then,
approximately
barring
the process
the satellite
as good
misrepresentation
noise
contribution
of
as that
of the supazimuth
and
elevation bias in the Telesat filter were larger than was otherwise taken
into account
in the RSS'ing
in the Telesat
Process
above, and thus helped
maintain
large
variances
filter.
noise is notoriously
means
that the state
might
be suspected
will not be accurately
in this EKF run,
small
that
filter
approximations
filter
hard to choose well.
convergence
is
known.
means
Too large a value
Too
small
that the state
threatened,
and force model errors
since
the
a value,
variances
effects
as
are so
of
the
are not taken into account.
In order to determine whether degradation of filter performance over
longer
periods
was a
possibility,
the EKF
full almost 24 days of supplied data.
tics,
as
seen
in
Table
4.6.3,
show
above
residuals
to
process
the
Slight increases in residual statisthat
the
gently degraded toward the end of the span.
trate this in the filter
was used
plots
degradation is especially apparent.
95
filter
estimate
was
indeed
Figures 4.6.5 and 4.6.6 illus-
for range
and
azimuth,
where
the
cr
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Table 4.6.3
Long-arc Extended Kalman Filter Residual Statistics
Residual
Standard Deviation
Range
Azimuth
Elevation
Distance
8.360 m
64.45 sec
43.59 sec
RMS error
over fit span
12.727 km
To
operating
make
the EKF
environment
be added,
operational
as Telesat's
the effects
with
this
filter,
of integrator
satellite
the bias solve
noise
process
variances
noise
would
take
for the basically
into
Q,
capability
the
and M.
(
the
same
should
investigated,
Some rational selection
consideration
in-plane
in
on the estimation
and the process noise issue definitely addressed.
for
and
especially
Considering
small
the lack of
EKF experience and its unrefined synchronous satellite force model and
incomplete
solve
capability
at
this
time,
its performance
here
is
quite
admirable.
4.7
Extended Semianalytical Kalman Filter Evaluation for ANIK Al
The final
tion
and most important
of the ESKF.
made using
experience
the ESKF quite
The baseline
baseline
The
SDC.
In
part
of the ANIK
gained
study
in the previous
was evalua-
tasks,
however,
easy.
run used the converged
order
Al
to maximize
state and the force
comparability
of
EKF
model of the
and
ESKF,
the
a priori covariance was the diagonal Cartesian EKF initial covariance converted
to the equinoctial
state
solve.
1.3951
versus
The
coordinates
baseline
11-day
that
ESKF
1.3940 for the baseline
are
the elements
filtering
gave
a
used
Cr
SDC, and gave the residuals
in
ESKF
value
of
shown in
Figures 4.7.1 through 4.7.4.
Table 4.7.1 displays residual statistics very
much
and
like
those
of the EKF,
thus
filter.
98
comparable
to
those
of
the Telesat
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Table
4.7.1
Baseline Extended Semianalytical Kalman Filter
Residual Statistiscs
Residual
Standard Deviation
Range
Azimuth
7.942 m
61.48 sec
Elevation
42.81 sec
Distance RMS error over fit span
12.382 km
This baseline ESKF run was also given the additional 13 days of
observations to process.
It too showed slight end-of-span degradation,
as Figures 4.7.5 and 4.7.6 and Table 4.7.2 indicate.
Again, a more careful
selection of process noise should eliminate this tendency.
Table 4.7.2
Long-arc Extended Semianalytical Kalman Filter Residual Statistics
Residual
Standard Deviation
Range
Azimuth
Elevation
8.368 m
64.47 sec
43.57 sec
Distance RMS error over fit span
12.728 km
As a last test of the ESKF,
was
added
illustrates
case,
to
the
that
as seen
in
partials
the
model
beneficial
the residuals
level in total distance
the analytic
of
the
influence
over
P2 e
baseline
of
the
the 11-day
error.
103
third body B-matrix
filter.
matrix
processed
in
Table
the
span,
4.7.3
ANIK
Al
is meter-
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Table 4.7.3
Effects
of Third Body B 1-Matrix on Extended Semianalytical
Estimation
Type of B 1-matrix
Baseline filter case:
None
Analytic P2 third
body
The impact
of B 1 usage
Kalman Filter
for ANIK Al
Residuals
Range
Azimuth
Elevation
Distance
7.942 m
61.48 sec
42.87 sec
12.382 km
R
7.947 m
Az
El
61.47 sec
42.81 sec
Dis
12.380 km
on computational
time is illustrated
in Table
4. 7.4.
Table 4.7.4
Effects of Third Body B 1-Matrix on Extended Semianalytical Kalman Filter
Computation Time
Short-arc Filter Run
CPU Execution Time
No B1-Matrix
0:36.72 inin:sec
Analytic P 2 third
body
0:37.50
The best summary of the performance
contained in their residual plots.
of the two GTDS-based
filters
is
Figures 4.6.1 through 4.6.4 and 4.7.1
through 4.7.4 illustrate how effectively both filters can estimate qeosynchronous
satellite
state.
tually indistinguishable.
Moreover,
both
yield
results
that
are
vir-
As indicated previously, the ESKF does have the
advantage of providing mean satellite dynamics immediately, simplifying
maneuver planning for the tracking facility.
Table
4.7.5
shows
the strong
contrast
filters.
106
in efficiencies
of these
two
Table 4.7.5
Comparison of Extended Kalman Filter and Extended Semianalytical Kalman
Filter Timing Estimates
CPU Execution Time
Filter Run
Whether
Long-arc EKF
3:35.82 min:sec
Long-arc ESKF
0:59.38
such an advantage
importance depends on
orbit
determination.
in efficiency
for the ESKF is of paramount
the installation's resources and its demands for
In
any
event,
efficiency is certainly attractive.
107
the
potential
for
such
relative
Chapter
5
CONCLUSIONS AND FUTURE WORK
5.1
Conclusions
The conclusions that result from the MILSTAR and ANIK Al studies
must be made with awareness of the restrictions inherent in all limited
studies.
It is, of course, tempting to generalize too extensively, but
being mindful of the specifics of both studies will help keep these remarks
in the proper context.
The MILSTAR study was used to explore the use of a semianalytical
orbit determination system to support near-autonomous satellite operation.
In early work in the study, it was demonstrated that Semianalytical Satellite Theory can propagate from given initial conditions an orbit that is
quite similar to that produced by special perturbations techniques.
The
semianalytical force model can, moreover, be readily explicitly truncated,
allowing semianalytical propagator performance to be gradually, controllably downgraded so that propagator computational burden can be lessened.
The semianalytical orbit determination system can be used onboard an
Airborne Command Post aircraft to estimate synchronous satellite state sufficiently accurately that ground-based users can acquire the satellite.
For this tracking of synchronous satellites, the total observation span was
much more significant than observation rate in determining final estimation
accuracy.
Even with limited-accuracy satellite observation only one or two
days a week, ephemeride-transmission to users only monthly, and with force
model and station location errors, the Extended Semianalytical Kalman Filter, together with a truncated "tailored" force model, was able to give
user
acquisition errors
1 m/sec
substantially less than 10 km in position and
in velocity.
Estimator accuracy lagged somewhat for the inclined synchronous satellite studied.
such satellites
Although further study is needed, orbit determination for
appears
to require
a full complement
of resonance
terms
the averaged dynamics and full-field geopotential short periodics.
in
To
ensure further a sufficient number of observations of inclined satellites,
108
tracker placement to enhance observability is very important.
could be optimized
for all satellites
isolated.
Position of
acquisition
accuracies;
once
the dominant
Location
tracker
estimator
the ground-based user does significantly affect
for the geostationary
satellite
of this
study,
for
example, the dominant along-track position errors were most observable to
the user farthest away in longitude from the subsatellite point.
Some very specific conclusions of the MILSTAR study also deserve
mention.
For the tracking schedule and accuracies of the study, rapid con-
vergence of the batch Semianalytical Differential Corrections estimator-and accuracy of the final converged state--require the supplying of an
Regarding the partial deriv-
accurate initial covariance to the processor.
atives matrices used in semianalytical estimation, more than the strict
two-body
to
contribution
the A-matrix
does
not seem
to be
in
needed
In addi-
processing, at least when a good filter covariance is available.
tion, use of the B 1-matrix is basically
inconsequential
ESKF
operation,
to ESKF
although its effects are measurably beneficial for tracking with large data
outage periods.
The most important summary conclusion to the MILSTAR study is that
the
semianalytical
orbit
determination
system
is
mature
and
can
support
near-autonomous satellite acquisition.
The ANIK
Al
study
demonstrated
the flexibility
of
the Goddard
Tra-
jectory Determination System as a research tool, illustrating the ease with
which a wealth of resources can be brought in selectively to support evaluation
real
of the primary
data
test case
GTDS
functions.
provided
More
an opportunity
specifically,
for demanding
of course,
the
testing
all
of
four GTDS-based estimators.
The Cowell and Semianalytical Differential Corrections estimators
produced epoch solve-for states that resulted in quite similar observation
residual statistics when
used
in
These
their respective propagators.
residuals were also, more significantly, very like those of the Telesat
filter.
quite
The
solved-for states of the batch estimators were
similar, but they
Attempts
to
bring
the
differed significantly from
states
more
into
coincidence
themselves
those of
illustrated
Telesat.
the
very
significant impact of inertial to earth-fixed frame transformation errors
on state estimates.
109
Further probing of the SDC showed again its improved convergence
when supplied with a good initial covariance.
was
this time not significantly
iance.
dependent
State estimation accuracy
on the use of an initial
covar-
For estimation accuracy, this batch estimator was very heavily
dependent
on
use
of more
than
strict
two-body
dynamics
in the
A-matrix.
Either J2 or P2 contribution is sufficient for reasonable estimator convergence
to an accurate
epoch
state for the synchronous
satellite.
Use of
both does not significantly enhance either accuracy or convergence speed.
The estimation accuracies of the GTDS-based Extended Kalman
Extended Semianalytical Kalman filters were remarkably similar.
rently implemented, the
accuracy
and
As cur-
of both filters, in particular that of
the ESKF, compared quite favorably with that of the Telesat filter.
And
especially after processing the first few observations, the filter residuals were much like those of the two batch estimators.
One more specific conclusion should be mentioned.
of use of the P2 third body
B1-matrix
in ESKF
estimation
The evaluation
showed
that
the
matrix yielded only meter-level improvement in distance residuals.
The
semianalytical orbit determination system, after slight aug-
mentation, could be used in a satellite tracking network.
be
very
much
more
efficient
than
the
basically
It promises to
comparable
special-
perturbations-based extended Kalman filter.
5.2
Future Work
Future work is always indicated. The MILSTAR study raised many questions concerning best semianalytical force model truncation for synchronous
orbit propagation.
A more effective and direct manner of truncating both
AOG and SPG models is needed.
Element-wise truncation would be especially
valuable.
In addition, synchronous satellite observability studies are
indicated.
These would lead to optimized tracker location, optimization
particularly needed in the case of near-autonomous navigation, where there
are few trackers.
The
consummate
test of SST/ESKF-based
orbit
determina-
tion in a MILSTAR-type scenario would be actual demonstration of the semianalytical
system
in a microprocessor;
already in progress.
110
work
on
such
an implementation
is
The
ANIK
Al
study
potential development.
led to clear
isolation
of
a number
of
areas
of
Following Telesat's lead, observation bias-solve
capability should be implemented in the GTDS-based sequential estimators.
In addition, the long-awaited rational procedure for isolating truly representative estimator process noise is very much needed.
would, of course, be helpful.
Any inroads here
ESKF processing of these observations with
process noise derived using Taylor's method might be a start.
A thorough
estimator characterization also calls for convergence testing, something
not explicitly
pursued
that the poor existent
in these
studies.
solar radiation
Moreover,
force model
since
it seems
is a significant
likely
barrier
to bringing forth better performance of the GTDS-based estimators, continued development
of this model is very much indicated.
Other more minor areas of
interest include investigation of the
effects of observation tropospheric correction and integrator noise on the
estimation proces.
A more complete evaluation of the effects of third body
A- and B1-matrices awaits different satellites and different tracking scenarios.
Capping off the Telesat work would mean resolving the apparent coordinate system discrepancies between Telesat's and Draper's orbit determination
systems
so
that
true
filter
state
comparison
is
possible.
If past
experience is any guide, this coordinate system alignment would lead to
enhancement of both systems.
Afterward, the new capability of accounting
for maneuvers in Semianalytical Satellite Theory [23,24] should be tested
with the semianalytical estimators in processing the extensive Telesatsupplied ANIK B observations in the most stringent evaluation yet of filter
capability for high-accuracy orbit determination.
111
APPENDIX A
THIRD BODY SHORT PERIODIC
FUNCTION EVALUATION
PACKAGE*
An analytic formulation of the third body short periodic functions
had been substantially formulated by Kaniecki [25].
The expressions were
residing in several fundamental MACSYMA blocks comprising a user-friendly
and powerful evaluation package.
Kaniecki's original analysis had, how-
ever, several critical errors as well as certain hobbling restrictions.
Most importantly, because of several changes of expansion variable incorporating indefinite integrals, the short periodic functions failed to satisfy
a constraint fundamental in Semianalytical Satellite Theory, namely, that
the contribution of the short periodic component of the motion average to
zero over a complete satellite orbit.
Renormalizing the generating func-
tion, and thus the short periodic functions derived from it, followed in a
straightforward fashion.
The extant evaluation package also contained an
error in the auxiliary function W(t,n,s).
Several function blocks (those
for W(t,n,s) and for the generating function and the short periodics) had
singularities for zero-eccentricity orbits.
periodic
functions was not
The formulation of the short
valid for retrograde orbits.
All of these
restrictions have now been removed.
Because most users of this evaluation package will be interested in
extracting the short periodic functions Fourier coefficients directly, a
new block, the last in the third body short periodic function evaluation
package presented here, was written.
Some way of removing the third body
parallax factor which arises in the evaluation of the generating function
partials was necessary, because the factor has no finite expansion in the
eccentric longitude that is the satellite angular variable of convenience
for formulation of the third body short periodic perturbation.
of this factor took the form of a change
in an embedded
Elimination
summation
index
for
the h, k, and X partials via a basic identity.
The corrected and amplified short periodic package, that found in
MACSYMA batch-type module WAGNER; THIRD LOAD, follows.
The results of this
package have been thoroughly checked with the analytic third body short
periodic functions [26] newly implemented in GTDS.
*For suggesting a number of remedies for Kaniecki's package I am especially
grateful to Dr. Mark Slutsky.
112
/*
THE THIRD BODY SHORT PERIODIC PACKAGE CONSISTS OF THE THIRTEEN
MACSYMA BLOCKS THAT SUPPORT CALCULATION OF THE CLOSED FORM THIRD BODY
SHORT PERIODICS FORMULATED SUBSTANTIALLY BY JEAN-PATRICK KANIECKI IN
HIS MIT MASTER'S THESIS. KANIECKI'S ORIGINAL PACKAGE WAS CORRECTED,
EXPANDED FOR USE WITH RETROGRADE ORBITS, AND FREED OF SINGULARITIES.
THE REQUIRED CHANGES WERE MADE IN 1982 BY ELAINE WAGNER. IN ADDITION,
THE LAST BLOCK WAS ADDED SO THAT EXPLICIT FOURIER COEFFICIENTS FOR THE
SHORT PERIODIC FUNCTIONS COULD BE EXTRACTED.
THE FORMULATION FOLLOWING DOES HAVE ITS RESTRICTIONS. THE GENERATING
FUNCTION, FOR EXAMPLE, MAY BE COMPUTED ONLY THROUGH NINTH DEGREE BECAUSE
OF CORE SPACE LIMITATIONS IN THE CURRENT MACSYMA CONSORTIUM IMPLEMENTATION. SIMILARLY, THE SHORT PERIODIC FUNCTIONS ONLY THROUGH SIXTH DEGREE
ARE COMPUTABLE. UNDERSTAND ALSO THAT THERE IS A DIFFERENCE BETWEEN CALCULATING THE DESIRED QUANTITIES AND DISPLAYING THEM; THEREFORE, BE PREPARED
TO BE CRAFTY OR TO FOREGO DISPLAY, ESPECIALLY WITH EXPRESSIONS OF HIGHER
DEGREE. CONCERNING FURTHER MANIPULATION OF THE SHORT PERIODIC EXPRESSIONS,
PREPARE YOURSELF FOR LONG STRUGGLES WITH CORE MEMORY RESTRICTION. THE
BLOCKS MAY EASILY BE PARED DOWN TO ONLY THOSE ESSENTIAL TO A GIVEN CALCULATION, OF COURSE.
THE BLOCKS IN THE THIRD BODY SHORT PERIODIC PACKAGE ARE
THE FOLLOWING:
C(X,Y)
SI(N,X,Y)
Q(N,M,X)
V(N,M)
HANSENI[T,N,M,MAXE](H,K)
HANSEN2[N,M,MAXE](H,K)
JACOB1[N,A,B](Y)
NEWCOMB1[R,S,N,M]
POISSON(I,J)
WF(T,N,S,X)
STBF(N,THIRD)
DELTASTB(N,THIRD)
DELTASTBC(N, THIRD)
VARIABLES APPEARING IN THE THIRD BODY SHORT PERIODIC FUNCTIONS INCLUDE,
IN ADDITION TO THE SET OF EQUINOCTIAL ELEMENTS A, H, K, P. Q, AND LAM, THOSE
BELOW:
F
THE ECCENTRIC LONGITUDE
MUM, MUS
THE GRAVITATIONAL PARAMETERS OF THE MOON AND SUN,
RESPECTIVELY
II
THE RETROGRADE FACTOR
XN
THE SATELLITE MEAN MOTION
X
1
2
2
SQR1T(1 - H - K )
C
2
P+Q
2
R3M, R3S
THE DISTANCE BETWEEN THE EARTH AND THE MOON, AD
AND THE SUN, RESPECTIVELY
AL, BE, GA
THE DIRECTION COSINES OF THE THIRD BODY WITHI RESPECT TO
THE EQUINOCTIAL FRAME.
113
THE EARTH
BETD
2
2
1 + SQRT(1 - H - K )
=
1 +
X
AR
THE PARTIAL DERIVATIVE OF THE ECCENTRIC LONGITUDE WITH
RESPECT TO THE MEAN LONGITUDE
=
A/R
1
1 - H SIN(F) - K COS(F)
DBTDDH
THE PARTIAL DERIVATIVE OF BETD WITH RESPECT TO H
- H
2
2
SQRT(1 - H - K )
DBTDDK
THE PARTIAL DERIVATIVE OF BETD WITH RESPECT TO K
- K
2
2
SQRT(1 - H - K )
DALDP
THE PARTIAL DERIVATIVE OF AL WITH RESPECT TO P
-2 (Q II BE + GA)
1 +P
DALDQ
2
2
+ Q
THE PARTIAL DERIVATIVE OF AL WITH RESPECT TO Q
2 P II BE
2
2
1 + P +Q
DBEDP
THE PARTIAL DERIVATIVE OF BE WITH RESPECT TO P
2 Q II AL
2
2
1 + P +Q
DBEDQ
THE PARTIAL DERIVATIVE OF BE WITH RESPECT TO Q
-2 II (P AL - GA)
2
1 +P+Q
DGADP
2
THE PARTIAL DERIVATIVE OF GA WITH RESPECT TO P
2 AL
2
2
1 + P +Q
114
DGADQ
THE PARTIAL DERIVATIVE OF GA WITH RESPECT TO Q
-2 II BE
1+
2
P+Q
2-
DFDH
THE PARTIAL DERIVATIVE OF THE ECCENTRIC LONGITUDE
WITH RESPECT TO H
= -AR COS(F)
DFDK
THE PARTIAL DERIVATIVE OF THE ECCENTRIC LONGITUDE
WITH RESPECT TO K
= AR SIN(F)
*/ALLOC4
ALLOC(4)$
115
C(N,X,Y):=
/*
X,Y
THIS FUNCTION COMPUTES THE POLYNOMIAL C
N
= Re{(X+jY) }.
N
VERSION OF 1982
MACSYMA BLOCK
*/
/*
CALLING SEQUENCE:
C(N,X,Y)
*/
/*
BLOCKS CALLED:
NONE
*/
/*
PROGRAMMER:
J-P.KANIECKI,MIT-FEBRUARY 1979
REFERENCE:
KANIECKI, JEAN-PATRICK R.,
SHORT PERIODIC VARIATIONS IN THE FIRST-ORDER SEMIANALYTICAL
SATELLITE THEORY
*/
/*
RESTRICTION:
N MUST BE AN INTEGER.
*/
REALPART((X+%I*Y)^N)$
116
SI(N,X,Y):=
/*
THIS FUNCTION COMPUTES THE POLYNOMIAL S
N
X,Y
=Im{(X+jY) }.
N
*/
/*
VERSION OF 1982
MACSYMA BLOCK
*/
/*
CALLING SEQUENCE:
SI(N,X,Y)
*/
BLOCKS CALLED:
NONE
*/
PROGRAMMER:
J-P.KANIECKI,MIT-FEBRUARY 1979
*/
/*
REFERENCE:
KANIECKI, JEAN-PATRICK R.,
SHORT PERIODIC VARIATIONS IN THE FIRST-ORDER SEMIANALYTICAL
SATELLITE THEORY
*/
RESTRICTION:
N MUST BE AN INTEGER.
IMAGPART((X+%I*Y)"N)$
117
Q(N,M,X):=BLOCK(],
/*THIS BLOCK COMPUTES
THIS BLOCK COMPUTESTHE FUNCTION Q
(X).
N,M
*/
/*
VERSION OF 1982
MACSYMA BLOCK
*/
/*
CALLING SEQUENCE:
Q(N,M,X)
*/
/*
BLOCKS CALLED:
NONE
*/
/*
ANALYSIS:
J-P.KANIECKI,MIT-FEBRUARY 1979
*/
/*
PROGRAMMER:
J-P.KANIECKI,MIT-FEBRUARY 1979
*/
/*
REFERENCE:
KANIECKI, JEAN-PATRICK R.,
SHORT PERIODIC VARIAfIONS IN THE FIRST-ORDER SEMIANALYTICAL
SATELLITE THEORY
*/
/I
RESTRICTION:
N MUST BE AN INTEGER GREATER THAN OR EQUAL TO ZERO.
*/
IF N < 0 THEN RETURN(ERROR),
IF M > N THEN RETURN(O),
IF M = 0 AND N = 0 THEN RETURN(1),
IF M = N THEN RETURN ((2*N-1)11),
IF M = N-1 THEN RETURN(X*((2*N-1)11)),
RETURN(RATSIMP(1/(N-M)*(X*(2*N-1)*Q(N-1
118
V(N,M):=BLOCK([],
/*
THIS BLOCK COMPUTES THE COEFFICIENT V
N,M
*/
VERSION OF 1982
MACSYMA BLOCK
*/
CALLING SEQUENCE:
V(N,M)
*/
BLOCKS CALLED:
NONE
a/
/e
ANALYSIS:
J-P.KANIECKI,MIT-FEBRUARY 1979
a/
/'
PROGRAMMER:
J-P.KANIECKI,MIT-FEBRUARY 1979
'/
REFERENCE:
KANIECKI, JEAN-PATRICK R.,
SHORT PERIODIC VARIATIONS IN THE FIRST-ORDER SEMIANALYTICAL
SATELLITE THEORY
a/
RESTRICTION:
N MUST BE AN INTEGER GREATER THAN OR EQUAL TO ZERO.
a/
IF N < 0 THEN RETURN(ERROR),
IF M > N THEN RETURN(O),
IF M - 0 AND N
0 THEN RETURN(1),
IF INTEGERP((M+N)/2) = FALSE THEN RETURN(O),
IF M
N THEN RETURN(((2*N-1)tl)/((2*N)I)),
RETURN((M-N+1)*V(N-2,M)/(M+N)))$
119
HANSENI[T,N,M.MAXE](H,K):=HAJSl(T,r,M,MAXE,H,K)$
HANSI(T,N,M,MAXE,H,K): BLOCK([HANS,I,SIGNMT,UPLIM],
/*
N,M
THIS BLOCK COMPUTES THE MODIFIED HANSEN COEFFICIENT Y
(H,K)
T
TRUNCATED TO ORDER MAXE IN THE ECCENTRICITY.
*/
/*
VERSION OF 1982
MACSYMA BLOCK
*/
/5
METHOD:
FINITE SUMMATION OF TERMS GUIDED BY THE DESIRED TRUNCATION
ORDER, IN WHICH EACH SUMMAND IS EXPRESSED EXPLICITLY USING THE
NEWCOMB1 BLOCK
*/
/5
CALLING SEQUENCE:
HANSEN1[T,N,M,MAXE](H,K)
/*
BLOCK CALLED:
NEWCOMB1[R,S,N,M]
,/
/,
LOCAL VARIABLES:
HANS
I
SIGNMT
UPLIM
INTERMEDIATE VALUE FOR THE CALCULATION OF
HANSEN1[T,N,M,MAXE](H,K)
SUMMATION INDEX
SIGN OF (M MINUS T)
UPPER LIMIT OF THE SUMMATION
/*
ANALYSIS:
P. CEFOLA, CSDL
E. ZEIS, MIT
*/
/
PROGRAMMER
E. ZEIS, MIT
*/
/
REFERENCE:
CEFOLA, PAUL J.,
A RECURSIVE FORMULATION FOR THE TESSERAL DISTURBING FUNCTION
IN EQUINOCTIAL VARIABLES
*/
/*
RESTRICTIONS:
T, N, M AND MAXE MUST BE INTEGERS.
MAXE MUST BE NON-NEGATIVE.
-/
/
THE FIRST TERM IN THE EXPANSION OF HANSEN1[T,N,M,MAXE](H,K) IS
AT LEAST OF ORDER ABS(M-T) IN ECCENTRICITY.
IF MAXE<ABS(M-T) THEN RETURN(O),
/*UPPER
LIMIT
OF
THE
SUMMATION
UPPER LIMIT OF THE SUMMATION
*/
120
IF INTEGERP((MAXE-ABS(M-T))/2)=TRUE THEN UPLIM:(MAXE-ABS(M-T))/2
ELSE UPLIM:(MAXE-ABS(M-T)-)/2,
INITIALIZATION
INITIALIZATION
*/
HANS:O,
/*
SIGN OF M-T
*/
IF (M-T)>O THEN SIGNMT:1
ELSE SIGNMT:-1,
/*
USING SIGNMT, TWO DIFFERENT EXPANSIONS FOR THE HANSEN COEFFICIENTS
MAY BE COMBINED.
,/
FOR I:UPLIM STEP -1 THRU 0 DO(HANS:(H^2+K^2)*(HANS
+NEWCOMB1[I+ABS(M-T),I,N,-SIGNMT*M])),
RETURN(FACTOR(HANS*(K+SIGNMT
'%
^
^
H)
I*1
^ABS(M-T)/(H 2+K 2))))$
121
HANSEN2[N,M,MAXE](H,K): HAIS2(N,M,MAXE,H,K)$
HANS2(N,M,MAXE,H,K):=BLOCK([NAXEC],
N,M
/*
THIS BLOCK COMPUTES THE MODIFIED HANSEN COEFFICIENT Y
(H,K).
0
IF MAXE IS GREATER THAN OR EQUAL TO ZERO, THEN THE COEFFICIENT IS
TRUNCATED TO ORDER MAXE IN THE ECCENTRICITY. THE EXACT EXPRESSION IS
CALCULATED FOR NEGATIVE MAXE.
./*
VERSION OF 1982
MACSYMA BLOCK
*/
/*
METHOD:
USE OF THE HANSEN1 BLOCK IF N IS NON-NEGATIVE OR IF N IS EQUAL
TO -1 AND MAXE IS NON-NEGATIVE, AND RECURSIVE FORMULATION OF A
CLOSED FORM EXPRESSION IN THE OTHER CASES
*/
/*
CALLING SEQUENCE:
HANSEN2[N,M,MAXE](H,K)
*/
/,
BLOCKS CALLED:
HANSEN1[T,N,M,MAXE](H,K)
HANSEN2[N,M,MAXE](H,K)
-/
/,
LOCAL VARIABLE:
MAXEC
MAXIMUM POWER OF ECCENTRICITY IN THE COMPLETE
EXPANSION
,/
/,
ANALYSIS:
P. CEFOLA, CSDL
E. ZEIS, MIT
,/
/*
PROGRAMMER:
E. ZEIS, MIT
,/
/,
REFERENCE:
COLLINS, SEAN K.,
COMPUTATIONALLY EFFICIENT MODELLING FOR LONG TERM PREDICTION
OF GLOBAL POSITIONING ORBITS
~~~~~~~~~~~/*
RESTRICTIONS:
N, M AND MAXE MUST BE INTEGERS.
IF MAXE IS NEGATIVE,
AND IF N IS GREATER THAN OR EQUAL TO -1, THEN THE
ABSOLUTE VALUE OF N MUST BE GREATER THAN OR
EQUAL TO THE ABSOLUTE VALUE OF M.
OTHERWISE--IF N IS SMALLER THAN -1--THEN THE
ABSOLUTE VALUE OF N MUST BE GREATER THAN THE
ABSOLUTE VALUE OF M.
GRADEF(X,H,H*X^3),
GRADEF(X,K,K* X -3),
122
/*
THE VALUE OF ANY HANSEN COEFFICIENT WITH AN ARGUMENT M<O IS THE
CONJUGATE OF THE HANSEN COEFFICIENT IN WHICH M IS REPLACED BY -M.
*/
IF M<O THEN RETURN(SUBST(-%I,%I,HANSEN2[N,-M,MAXE](H,K))),
/*
IF MAXE>=O, THEN A TRUNCATION IS PERFORMED.
*/
IF MAXE>=O THEN (IF MAXE<M THEN RETURN(ERROR),
/*
IF N>=-1 OR IF MAXE<2, THEN THE FORMULAS USED ARE
IDENTICAL TO THOSE USED IN THE HANSEN1 BLOCK.
*/
IF N>=-1 OR MAXE<2 THEN RETURN(IIANSEN1[O,N,M,MAXE](H,K)),
INITIALIZATION AND RECURSION FORMULA FOR N<INITIALIZATION AND RECURSION FORMULA FOR N<-1
*/
IF M=-(N+1) THEN RETURN(O),
IF M-(N+2) THEN RETURN(X^(-3-2*N)*((K+%I*H)/2)^(-N-2)),
RETURN(FACTOR((2*(M+1)*HANSEN2[N,M+1,MAXE+1](H,K)+(M-N)
*(K-%I*H)*HANSEN2[N,M+2,MAXE](H,K))/((-N-M-2)*(K+%I*H))))),
FOR MAXE<O, CHECK THAT THE REQUIRED CONDI
FOR MAXE<O, CHECK THAT THE REQUIRED CONDITIONS ARE REALIZED.
*/
IF
ABS(M)>ABS(N)
OR ABS(M)=ABS(N) AND N<-1 THEN RETURN("NO EXACT EXPRESSION"),
/*
ANY HANSEN COEFFICIENT WITH N>=O (AND SATISFYING THE ABOVE CONDITION)
MAY BE FULLY EXPANDED IN TERMS OF THE ECCENTRICITY.
*/
IF N>=O THEN (
/*
ORDER OF THE COMPLETE EXPANSION
*/
IF INTEGERP((N-M)/2)=TRUE THEN MAXEC:N
ELSE MAXEC:N+1,
CALL OF HANSEN WITH MAXECAS ARGUMENT
CALL OF HANSEN1 WITH MAXEC AS ARGUMENT
*/
RETURN(HANSENI[O,N,M,MAXEC](H,K))),
/
FORMULAS FOR N=-l
*/
IF N=-1 THEN (IF M=O THEN RETURN(1)
ELSE RETURN((1/X-1)*(K+%I*H)/(K^2+H^2))),
123
/*
INITIALIZATION AND RECURSION FORMULAS FOR N<-1
*/
IF M=-(N+1) THEN RETURN(O),
IF M-(N+2)
THEN RETURN(X^(-3-2*N)*((K+%I*H)/2)^(-N-2)),
RETURN(FACTOR((2*(M+1)*HANSE112[N,M+l,MAXE](1H,K)+(M-N)*(K-%I*H)
*HANSEN2[N,M+2,MAXE](H,K))/((-N-M-2)*(K+%I*H)))))$
124
JACOB1[N,A,B](Y):=JACOB(N,A,B,Y)$
JACOB(N,A,B,Y):=BLOCK([JAKOB,R],
/*
A,B
THIS BLOCK COMPUTES THE JACOBI POLYNOMIAL P
(Y), IN WHICH N, A, AND
N
B ARE INTEGERS.
*/
I
JERSION OF 1982
MACSYMA BLOCK
*/
/*
METHOD:
EXPLICIT FORMULA BASED ON BINOMIAL
a/
CALLING SEQUENCE:
JACOB1[N,A,B](Y)
*/
BLOCKS CALLED:
NONE
*/
/*
/*LOCAL
VARIABLES:
R
SUMMATION INDEX
JAKOB
INTERMEDIATE VALUE FOR THE CALCULATION OF
JACOB1[N,A,B](Y)
*/
/
ANALYSIS:
P. CEFOLA, CSDL
E. ZEIS, MIT
a/
/$
PROGRAMMER:
E. ZEIS, MIT
/
REFERENCE:
McCLAIN, WAYNE D.,
A RECURSIVELY FORMULATED FIRST-ORDER SEMIANALYTICAL ARTIFICIAL
SATELLITE THEORY BASED ON THE GENERALIZED METHOD OF AVERAGING,
VOL. II
/I
RESTRICTION:
N, A, AND B MUST BE INTEGERS GREATER THAN OR EQUAL TO ZERO.
/
CHECK THAT N, A, AND B HAVE CORRECT VALUES.
a/
IF
N<O
OR A<O
OR B<O THEN RETURN(ERROR),
INITIALIZATION
INITIALIZATION
JAKOB:O,
125
EXPLICIT FORMULA WITH THE FORM OF A FINITE SUMMATION
*/
FOR R:O THRU N DO(JAKOB:JAKOB+BINOMIAL(N+A,R)*BINOMIAL(N+B,N-R)*(Y+1)^R
*(Y-1)^(N-R)),
SIMPLIFICATION AND MULTIPLICATION BY A CONSTANT
RETURNRATSIMP2
RETURN(RATSIMP(2-(-N)*JAKO
B)))$
126
NEWCOMBI[R,S,N,M]:=NEWCMB1(R,S,N,M)$
NEWCMB1(R,S,N,M):=BLOCK([BINOM,MINRS,NEWKOMB,T],
/
N,M
THIS BLOCK COMPUTES THE NEWCOMB OPERATOR X
*/
R,S
0
VERSION OF 1982
/*4
MACSYMA BLOCK
'I4
*/
METHOD:
RECURSIVE FORMULAS WITH EXPLICIT INITIALIZATION
*/
/*
CALLING SEQUENCE:
NEWCOMB1[R,S,N,M]
*/
/
BLOCK CALLED:
NEWCOMB1[R,S,N,M]
/I
LOCAL VARIABLES:
BINOM
THE GENERALIZED BINOMIAL COEFFICIENT (3/2
MINRS
MINIMUM OF R AND S
NEWKOMB INTERMEDIATE VALUE FOR THE CALCULATION OF
NEWCOMBI[R,S,N,M]
T
SUMMATION INDEX
T)
'/
/*4
ANALYSIS:
P. CEFOLA, CSDL
E. ZEIS, MIT
'/
0
/*4
PROGRAMMER:
E. ZEIS, MIT
*/
/*4
REFERENCE:
CEFOLA, PAUL J.,
A RECURSIVE FORMULATION FOR THE TESSERAL DISTURBING FUNCTION
IN EQUINOCTIAL VARIABLES
*/
/I
RESTRICTION:
R, S, N, AND M MUST BE INTEGERS.
/0
IF ONE OF THE SUBSCRIPTS IS NEGATIVE, THEN THE RESULT IS 0.
*/
IF
R<O
OR S<O THEN RETURN(O),
/A
INITIALIZATION AND RECURSION FORMULAS FOR S-0
IF S0
THEN (IF R=O THEN RETURN(1)
ELSE (IF R1
THEN RETURN(M-N/2)
127
ELSE RETURN((2*(2M-Ntl)*NEWCOIIB1[R-1,0,N,M+1]
+(M-N)*NEWCOtMB1[R-2,0,N,M+2])/(4*R)))),
GENERAL RECURSION FORMULAS
NEWKOMB:-2*(2*M+N)*NEWCOMB1[R,S-1,N,M-1]-(M+N)*NEWCOMBI[R,S-2,N,M-2]
-(R-5*S+4+4*M+t)*NEWCOMB1[R-1,S-1,N,M],
MINRS:MIN(R,S),
IF MINRS<2 THEN RETURN(NEWKOMB/(4*S))
ELSE FOR T:2 THRU MINRS DO(IF T=2 THEN BINOM:3/8
ELSE BINOM:3*(-1)^T*(2*T-5)11
/(2^T*TI),
NEWKOMB:NEWKOMB+2*(R-S+M)*(-1)-T*BINOM
*NEWCOMB1[R-T,S-T,N,M]),
RETURN(NEWKOMB/(4*S)))$
128
POISSON(I,J):=BLOCK([],
/*
THIS BLOCK COMPUTES THE POISSON BRACKETS (I,J) OF EQUINOCTIAL
ELEMENTS VIA AN EXPANSION CLOSED FORM IN ECCENTRICITY.
*/
VERSION OF 1982
MACSYMA BLOCK
*/
/*
CALLING SEQUENCE:
POISSON(I,J)
/*
BLOCK CALLED:
POISSON(I,J)
*/
/*
ANALYSIS:
W.D. McCLAIN, CSDL
*/
/e
PROGRAMMERS:
E. ZEIS, MIT
MODIFICATIONS FOR GENERALIZATION TO RETROGRADE
CASE BY J-P.KANIECKI,MIT-AUGUST 1979
a/
/'
REFERENCE:
McCLAIN, WAYNE D.,
A RECURSIVELY FORMULATED FIRST-ORDER SEMIANALYTIC ARTIFICIAL
SATELLITE THEORY BASED ON THE GENERALIZED METHOD OF AVERAGING,
VOL. I
a/
/'
RESTRICTION:
I AND J MUST BELONG TO THE SET OF LETTERS USED TO
REPRESENT THE EQUINOCTIAL ELEMENTS: A, H, K, P, Q, AND LAM.
a/
DEFINITION OF THE POISSON BRACKETS OF THE FORM (AJ)
DEFINITION OF THE POISSON BRACKETS OF THE FORM (A,J)
a/
IF I-A THEN (
IF J=LAM THEN RETURN(-2/(XN*A))
ELSE RETURN(O)),
DEFINITION OF THE POISSON BRACKETS OF THE FORM (H,J)
IF I=H THEN (
ELSE
ELSE
ELSE
IF
IF
IF
IF
J=K THEN RETURN(-1/(XN*A^2*X))
J=P THEN RETURN(-K*P*X*(1+C)/(2*XNA^2))
J=Q THEN RETURN(-K*Q*X*(1+C)/(2*XN*A^2))
JLAM THEN RETURN(H/(XN*A^2*(X+1)))
ELSE RETURN(O)),
/I
IN THE FOLLOWING PART, WE WILL TAKE ADVANTAGE OF THE FACT THAT
(I,J)--(J,I).
129
FORM (KJ)
DEFINITION OF THE POISSON BRACKETS OF THE
DEFINITION OF THE POISSON BRACKETS OF THE FORM (K,J)
*/
IF I=K THEN (
IF J=P THEN RETURN(H*P*X*(1+C)/(2*XN*A^2))
ELSE IF J=Q THEN RETURN(H*Q*X*(1+C)/(2*XN*A^2))
ELSE IF J=LAM THEN RETURN(K/(XN*A^2*(X+1)))
ELSE IF J=H THEN RETURN(-POISSON(J,I))
ELSE RETURN(O)),
/DEFINITION
OF
THE
POISSON
BRACKETS
OF
THE
FORM
(PJ)
DEFINITION 01 THE POISSON BRACKETS OF THE FORM (P,J)
*/.
IF J=Q THEN RETURN(-X*II*(1+C)-2/(4*XN*A^2))
IF I=P THEN (
ELSE IF J=LAM THEN RETURN(P*X*(1+C)/(2*XN*A^2))
ELSE IF J=H
OR J=K THEN RETURN(-POISSON(J,I))
ELSE RETURN(O)),
/E
DEFINITION OF THE POISSON BRACKETS OF THE FORM (Q,J)
*/
IF IQ
THEN (
IF
ELSE IF
OR
OR
/DEFINITION
OF
J=LAM THEN RETURN(Q*X*(1+C)/(2*XN*A^2))
J=H
JuK
J=P THEN RETURN(-POISSON(J,I))
ELSE RETURN(O)),
THE
POISSON
BRACKETS
OF
THE
FORM
(LAMJ)
DEFINITION OF THE POISSON BRACKETS OF THE FORM (LAM,J)
IF ILAM THEN (
THEN RETURN(O)
IF J=LAM
ELSE RETURN(-POISSON(J,I))))$
130
WF(T,N,S,X):=BLOCK([],
N,S
/*
(X).
THIS BLOCK COMPUTES THE FUNCTION W
T
*/
/*
VERSION OF 1982
MACSYMA BLOCK
/*
CALLING SEQUENCE:
WF(T,N,S,X)
*/
/*
BLOCK CALLED:
JACOB1[N,A,B](Y)
,/
/*
DEFINITIONS:
BET =SQRT(H^2+K^2)/(l+SQRT(1-H^2-K^2))
=BETN/BETD
BETN=SQRT(H^2+K^2)
BETD=1+SQRT(1-H^2-K^2)
ETA=SIGN(T-S)
'/
/'
ANALYSIS:
J-P.KANIECKI,MIT-AUGUST 1979
THE BLOCK WAS MODIFIED IN 1982 BY ELAINE WAGNER (MIT)
TO INCLUDE THE ETA FACTOR WHICH WAS ERRONEOUSLY MISSING IN
KANIECK'S ORIGINAL DEVELOPMENT AND TO REMOVE SINGULARITIES
ARISING IN THE BLOCK'S USE WITH ZERO-ECCENTRICITY ORBITS.
'/
/*
PROGRAMMERS:
J-P.KANIECKI,MIT-AUGUST 1979
ELAINE WAGNER
MIT
'/
/*
REFERENCE:
KANIECKI, JEAN-PATRICK R.,
SHORT PERIODIC VARIATIONS IN THE FIRST-ORDER SEMIANALYTICAL
SATELLITE THEORY
*/
/*
RESTRICTIONS:
N MUST BE AN INTEGER GREATER THAN OR EQUAL TO ZERO.
IF ISI IS GREATER THAN ITI, THEN N-ISI MUST BE GREATER THAN
OR EQUAL TO ZERO.
IF ISI IS LESS THAN ITI, THEN N-ITI MUST BE GREATER THAN OR
EQUAL TO ZERO.
,/
IF N<O THEN RETURN (ERROR),
/*
THE SUBSTITUTION SQRT(H^2+K^2)/BETD (SQRT(H^2+K^2)*X/(1+X)) FOR BET
IS MADE TO AVOID SINGULARITIES IN WF FOR ZERO-ECCENTRICITY ORBITS.
IF
T>=
THEN
ETA
IF T >: S THEN ETA:1 ELSE ETA:-1,
13 1
RETURN(IF ABS(S) >= ABS(T) THEN
(IF N-ABS(S) < 0 THEN ERROR
ELSE (1/X)-N(-(SQRT(H^2+K^2)*X/(l+X))*SQRT(I^2+K^2)*(K-%I*ETA*H)
/(K^2+H^2))-ABS(T-S)*(N+S)!*(N-S)!/((N+T)I*(N-T)I)
*(1-BET^2)^-ABS(S)*JACOB[N-ABS(S),ABS(T-S),ABS(T+S)](X))
ELSE (IF N-ABS(T) < 0 THEN ERROR
ELSE (1/X)-N*(-(SQRT(H^2+K^2)*X/(l+X))*SQRT(H^2+K^2)
*(K-%I*ETA*H)/(K2+H^2))^ABS(T-S)*(1-BET^2)^-ABS(T)
*JACOB1[N-ABS(T),ABS(T-S),ABS(T+S)](X))))$
132
STBF(N,THIRD) :BLOCK([],
/*
THIS BLOCK COMPUTES THE REAL PART OF
th
HARMONIC OF THE
FUNCTION FOR THE N
IF THIRD = M, THEN THE THIRD BODY IS
IF THIRD = S, THEN THE THIRD BODY IS
THE CLOSED FORM GENERATING
THIRD BODY PERTURBATION.
THE MOON.
THE SUN.
*/
/*
VERSION OF 1982
MACSYMA BLOCK
*/
/*
CAI.LINGSEQUENCE:
STBF(N, THIRD)
,/
/,
BLOCKS CALLED:
C(N,X,Y)
SI(N,X,Y)
WF(T,N,S,X)
V(N,M)
Q(N,M,X)
HANSEN1[N,M,-1](H,K)
/*
DEFINITIONS:
STB1, STB2, AND STBSTAR ARE INTERMEDIATE VALUES IN THE
CALCULATION.
I/
ANALYSIS:
J-P.KANIECKI,MIT-AUGUST 1979
THE BLOCK WAS MODIFIED IN 1982 BY ELAINE WAGNER (MIT) TO
REMOVE SINGULARITIES ARISING IN USE WITH ZERO-ECCENTRICITY
ORBITS.
*/
/*
PROGRAMMERS:
J-P.KANIECKI,MIT-AUGUST 1979
ELAINE WAGNER
MIT
5/
RESTRICTION:
N MUST BE AN INTEGER GREATER THAN OR EQUAL TO TWO.
AS OF 1982, THE MACSYMA CONSORTIUM IMPLEMENTATION WOULD
COMPUTE THE GENERATING FUNCTION (IN THIS FORMULATION)
ONLY THROUGH ELEVENTH DEGREE.
5/
IF N<2 THEN RETURN (ERROR),
STB1(M):=(IF M = 0 THEN 1 ELSE 2)*(C(M,AL,BE)+%I*SI(M,AL,BE)),
STB2(T,N,M):=IF T = 0 THEN 0 ELSE WF(T,N+l,-M,X)*%E^(%I*T*F)/(%I*T),
STBSTAR(N):=SUM(V(N,M)*Q(N,M,GA)*STBI(M)*((F-LAM)*HANSEN2[N,-M,-l](H,K)
+WF(1,N+1,-M,X)*((-K*%I/2)+(H/2))
+WF(-1,N+1,-MX)*((K*%I/2)+(H/2))
+SUM(STB2(T,N,M),T,-N-1,N+1)),M,O,N),
RETURN((IF THIRD=M THEN MUM ELSE MUS)/(XN*(IF THIRD=M THEN R3M ELSE R3S)
^(N+1))*A^N*REALPART(STBSTAR(N))))$
133
DELTASTB(N,THIRD):=BLOCK([ST,DSDA,DSDH,DSDK,DSDP,DSDQ,DSDLAtI],
/*
THIS BLOCK COMPUTES THE VARIATIONS OF THE SIX EQUINOCTIAL
ELEMENTS ASSUMING THAT THE DISTURBANCE IS DUE ONLY
th
TO THE N
HARMONIC OF THE THIRD BODY PERTURBATION.
IF THIRD = M, THEN THE THIRD BODY IS THE MOON.
IF THIRD = S, THEN THE THIRD BODY IS THE SUN.
*/
/,
VERSION OF 1982
MACSYMA BLOCK
*/
/*
CALLING SEQUENCE:
DELTASTB(N,THIRD)
*/
/*
BLOCKS CALLED:
STBF(N,THIRD)
POISSON(I,J)
*/
/*
LOCAL VARIABLES:
th
ST=GENERATING FUNCTION FOR THE N
HARMONIC OF THE
THIRD BODY
DSDA=PARTIAL DERIVATIVE OF ST WITH
RESPCT TO A
DSDH=PARTIAL DERIVATIVE OF ST WITH
RESPECT TO H
DSDK=PARTIAL DERIVATIVE OF ST WITH
RESPECT TO K
DSDP=PARTIAL DERIVATIVE OF ST WITH
RESPECT TO P
DSDQ=PARTIAL DERIVATIVE OF ST WITH
RESPECT TO Q
DSDLAM=PARTIAL DERIVATIVE OF ST WITH
RESPECT TO LAM
t/
/*
DEFINITIONS:
DBTDDH=PARTIAL DERIVATIVE OF BETD WITH
RESPECT TO H
=-H/SQRT(1-H^2-K^2)
DBTDDK=PARTIAL DERIVATIVE OF BETD WITH
RESPECT TO K
=-K/SQRT(1-H^2-K^2)
DALDP=PARTIAL DERIVATIVE OF AL WITH
RESPECT TO P
=-2*(Q*II*BE+GA)/(1+P^2+Q^2)
DALDQ=PARTIAL DERIVATIVE OF AL WITH
RESPECT TO Q
=2*II*P*BE/(1+P^2+Q^2)
DBEDP=PARTIAL DERIVATIVE OF BE WITH
RESPECT TO P
=2*II*Q*AL/(1+P^2+Q^2)
DBEDQ=PARTIAL DERIVATIVE OF BE WITH
RESPECT TO Q
=-2*II*(P*AL-GA)/(I+P^2+Q^2)
DGADP=PARTIAL DERIVATIVE OF GA WITH
RESPECT TO P
=2*AL/(1+P^2+Q^2)
134
DGADQ=PARTIAL DERIVATIVE OF GA WITH
RESPECT TO Q
=-2*II*BE/(1+P^2+Q^2)
AR=PARTIAL DERIVATIVE OF F WITH
RESPECT TO LAM
=1/(1-H*SIN(F)-K*COS(F))
DFDH=PARTIAL DERIVATIVE OF F WITH
RESPECT TO H
=-AR*COS(F)
DFDK=PARTIAL DERIVATIVE OF F WITH
RESPECT TO K
=AR*SIN(F)
*/
/*
ANALYSIS:
J-P.KANIECKI,MIT-AUGUST 1979
THE BLOCK WAS MODIFIED IN 1982 BY ELAINE WAGNER (MIT) TO
REMOVE SINGULARITIES ARISING IN USE WITH ZERO-ECCENTRICITY
ORBITS. IN ADDITION, CHANGES WERE MADE TO ENSURE THE BLOCK'S
APPLICABILITY FOR RETROGRADE ORBITS.
*/
/*
PROGRAMMERS:
J-P.KANIECKI,MIT-AUGUST 1979
ELAINE WAGNER
MIT
*/
/0
REFERENCE:
KANIECKI, JEAN-PATRICK R.,
SHORT PERIODIC VARIATIONS IN THE FIRST-ORDER SEMIANALYTICAL
SATELLITE THEORY
,/
/
RESTRICTIONS:
N MUST BE AN INTEGER GREATER THAN OR EQUAL TO TWO.
AS OF 1982, THE MACSYMA CONSORTIUM IMPLEMENTATION WOULD
COMPUTE THE THIRD BODY SHORT PERIODICS IN THIS FORMULATION
ONLY THROUGH SIXTH DEGREE. IN THESE TESTS, THE GENERATING
FUNCTION WAS NOT PRE-EVALUATED.
IF N<2 THEN RETURN (ERROR),
ST:STBF(N,THIRD),
ST:SUBST([BET-SQRT(H^2+K^2)*X/(I+X)],ST),
/,
THE DERIVATIVES OF BET WITH RESPECT TO H AND K ARE SINGULAR FOR
ZERO-ECCENTRICITY ORBITS.
S/
GRADEF(X,H,H*X^3),
GRADEF(X,K,K*X^3),
GRADEF(BETD,H,DBTDDH),
GRADEF(BETD,K,DBTDDK),
GRADEF(AL,P,DALDP),
GRADEF(AL,Q,DALDQ),
GRADEF(BE,P,DBEDP),
GRADEF(BE,Q,DBEDQ),
GRADEF(GA,P,DGADP),
GRADEF(GA,Q,DGADQ),
135
GRADEF(F,LAM,AR),
GRADEF(F,H,DFDH),
GRADEF(F,K,DFDK),
DSDA:DIFF(ST,A),
/*
NOTICE THAT IMPLICIT DIFFERENTIATION OF XN WITH RESPECT TO A IS
INAPPROPRIATE BECAUSE THE GENERATING FUNCTION ALREADY INCLUDES
THE (1/XN) TERM THAT IS SIMPLY A MULTIPLIER OF THE ENTIRE
(POISSON BRACKET * PARTIAL OF THE GENERATING FUNCTION) SUM
IN McCLAIN'S FORMULATION OF THE SHORT PERIODIC FUNCTIONS.
*/DSDHDIFFSTH
DSDH:DIFF(STH)
DSDK:DIFF(ST,K),
DSDP:DIFF(ST,P),
DSDQ:DIFF(STQ),
DSDLAM:DIFF(ST,LAM),
/
DEFINITIONS OF THE VARIATIONS OF THE EQUINOCTIAL ELEMENTS
*/
DELTA%A:-POISSON(A,LAM)*DSDLAM,
DELTA%H:-POISSON(H,K)*DSDK-POISSON(H,P)*DSDP
-POISSON(H,Q)*DSDQ-POISSON(HLAM)*DSDLAM,
DELTA%K:-POISSON(K,H)*DSDH-POISSON(K,P)*DSDP
-POISSON(KQ)*DSDQ-POISSON(K,LAM)*DSDLAM,
DELTA%P:-POISSON(P,H)*DSDH-POISSON(P,K)*DSDK
-POISSON(P,Q)*DSDQ-POISSON(PLAM)*DSDLAM,
DELTA%Q:-POISSON(Q.H)*DSDH-POISSON(Q,K)*DSDK
-POISSON(Q,P)*DSDP-POISSON(QLAM)*DSDLAM,
DELTA%LAM:-POISSON(LAM,A)*DSDA-POISSON(LAMH)*DSOH
-POISSON(LAM,K)*DSDK-POISSON(LAM,P) DSDP
-POISSON(LAM,Q)DSDQ-3*ST/(XN*A^2),
/,
THE RESULTS
,/
RETURN([DELTAA,DELTAH,DELTA%K,DELTA%P,DELTA%QDELTALAM]))$
136
DELTASTBC(N,THIRD):=BLOCK([ST,DSDH,DSDK,DSDP,DSDQ,DSDLAM],
/
THIS BLOCK COMPUTES THE VARIATIONS OF THE SIX EQUINOCTIAL
ELEMENTS ASSUMING THAT THE DISTURBANCE IS DUE ONLY
th
TO THE N
HARMONIC OF THE THIRD BODY PERTURBATION.
IF THIRD = M, THEN THE THIRD BODY IS THE MOON.
IF THIRD
S, THEN THE rHIRD BODY IS THE SUN.
*/
VERSION OF 1982
MACSYMA BLOCK
*/
/
CALLING SEQUENCE:
DELTASTBC(N,THIRD)
/$
BLOCKS CALLED:
C(lN,X,Y)
SI(N,X,Y)
WF(T,N,S,X)
V(N,M)
Q(N,M,X)
HANSEN1[N,M,-1](H,K)
POISSON(I,J)
*/
/I
LOCAL VARIABLES:
th
ST-GENERATING FUNCTION FOR THE N
HJRMONIC OF THE
THIRD BODY
DSDA=PARTIAL DERIVATIVE OF ST WITH
RESPCT TO A
DSDH=PARTIAL DERIVATIVE OF ST WITH
RESPECT TO H
DSDK=PARTIAL DERIVATIVE OF ST WITH
RESPECT TO K
DSDP=PARTIAL DERIVATIVE OF ST WITH
RESPECT TO P
DSDQ=PARTIAL DERIVATIVE OF ST WITH
RESPECT TO Q
DSDLAM=PARTIAL DERIVATIVE OF ST WITH
RESPECT TO LAM
*/
/I
DEFINITIONS:
DBTDDII-PARTIALDERIVATIVE OF BETD WITH
RESPECT TO H
=-H/SQRT(l-I^h2-K^2)
DBTDDK=PARTIAL DERIVATIVE OF BETD WITH
RESPECT TO K
=-K/SQRT(1-H^2-K^2)
DALDP=PARTIAL DERIVATIVE OF AL.WITH
RESPECT TO P
=-2*(QII*BE+GA)/(1+P^2+Q^2)
DALDQ=PARTIAL DERIVATIVE OF AL WITH
RESPECT TO Q
=2*II*P*BE/(I+P^2+Q^2)
DBEDP=PARTIAL DERIVATIVE OF BE WITH
RESPECT TO P
=2*II*Q*AL/(l+P^2+Q^2)
137
DBEDQ=PARTIAL DERIVATIVE OF BE WITH
RESPECT TO Q
--2*II*(P*AL-GA)/(1+P^2+Q^2)
DGADP=PARTIAL DERIVATIVE OF GA WITH
RESPECT TO P
-2*AL/( 1+P^2+Q^2)
DGADQ=PARTIAL DERIVATIVE OF GA WITH
RESPECT TO Q
--2*II*BE/(I+P^2+Q^2)
STB1, STBP2, STBS2, AND STBSTAR ARE INTERMEDIATE VALUES IN
THE CALCULATION.
*/
/*
ANALYSIS:
J-P.KANIECKI,MIT-AUGUST 1979
DELTASTB WAS MODIFIED IN 1982 BY ELAINE WAGNER (MIT) TO
REMOVE SINGULARITIES ARISING IN USE WITH ZERO-ECCENTRICITY
ORBITS. IN ADDITION, CHANGES WERE MADE TO ENSURE THE BLOCK'S
APPLICABILITY FOR RETROGRADE ORBITS.
DELTASTBC INCORPORATES CHANGES MADE TO DELTASTB TO ENABLE THE
ISOLATION OF FOURIER COEFFICIENTS FOR THE SHORT PERIODIC FUNCTIONS. IN PARTICULAR, AN EXPANSION IDENTITY WAS USED TO ELIMINATE A/R, THE PARTIAL DERIVATIVE OF THE ECCENTRIC ANOMALY WITH
RESPECT TO THE MEAN ANOMALY, A TERM THAT HAS NO FINITE FOURIER
SERIES EXPANSION IN THE ECCENTRIC ANOMALY.
*/
/*
PROGRAMMERS:
J-P.KANIECKI,MIT-AUGUST 1979
ELAINE WAGNER
MIT
*/
/*
REFERENCE:
KANIECKI, JEAN-PATRICK R.,
SHORT PERIODIC VARIATIONS IN THE FIRST-ORDER SEMIANALYTICAL
SATELLITE THEORY
/
RESTRICTIONS:
N MUST BE AN INTEGER GREATER THAN OR EQUAL TO TWO.
AS OF 1982, THE MACSYMA CONSORTIUM IMPLEMENTATION WOULD
COMPUTE THE THIRD BODY SHORT PERIODICS IN THIS FORMULATION
ONLY THROUGH FOURTH DEGREE.
,/
IF N<2 THEN RETURN (ERROR),
STB1(M):-(IF M
0 THEN 1 ELSE 2)*(C(M,AL,BE)+%I*SI(M,AL,BE)),
STBS2(T,N,M):-IF T
0 THEN 0 ELSE SUBST([BET=SQRT(H^2+K^2)*X/(1+X)],
WF(T,N+,-MX))*%E^(%I*TF)/(%I*T),
STBP2(TN,M):=SUBST([BET=SQRT(H^2+K^2)*X/(t+X)],WF(T,N,-M,X))*%E^(%I*T*F),
STBNN,M) : =SUBST([BET=SQRT(H^2+K'2)*X/(I+X)],WF(1,N+1,-M,X)*(-K*XI/2+H/2)
+WF(-1,N+l,-M,X)*(K*%I/2+H/2)),
STBSTAR(N):=SUM(V(N,M)*Q(N,M,GA)*STBI(M)*((K*SIN(F)-H*COS(F))
*HANSEN2[N,-M,-1](H,K)
+SUBST([BET=SQRT(H2+K^2)*X/(1+X)] ,WF(1,N+I,-M,X)
*((-K*%I/2)+(H/2))+WF(-1,N+I,-M,X)*((K*%I/2)+(H/2)))
+SUM(STBS2(T,N.M),T,-N-1,N+1)),M,O,N),
ST:(IF THIRD=M THEN MUM ELSE MUS)/(XN*(IF THIRD-M THEN R3M ELSE R3S)
^ ( N+I) ) *A^NREALPART(
STBSTAR(N
) ),
138
/*
THE SUBSTITUTIONS ABOVE ARE NECESSARY TO AVOID SINGULARITIES FOR ZEROECCENTRICITY ORBITS OF BET'S DERIVATIVES WITH RESPECT TO H AND K.
*/
GRADEF(X,H,H*X^3),
GRADEF(X,K,K*X^3),
GRADEF(BETD,H,DBTDDH),
GRADEF(BETD,K,DBTDDK),
GRADEF(AL,P,DALDP),
GRADEF(AL,Q,DALDQ),
GRADEF(BE,P,DBEDP),
GRADEF(BE,Q,DBEDQ),
GRADEF(GA,P,DGADP),
GRADEF(GA.Q,DGADQ),
DSDA:DIFF(ST,A),
/*
NOTICE THAT IMPLICIT DIFFERENTIATION OF XN WITH RESPECT TO A IS
INAPPROPRIATE BECAUSE THE GENERATING FUNCTION ALREADY INCLUDES
THE (1/XN) TERM THAT IS SIMPLY A MULTIPLIER OF THE ENTIRE
(POISSON BRACKET * PARTIAL OF THE GENERATING FUNCTION) SUM
IN McCLAIN'S FORMULATION OF THE SHORT PERIODIC FUNCTIONS.
*/
DSDP:DIFF(ST,P),
DSDQ:DIFF(ST,Q),
DSDH:SUBST([F-LAM=K*SIN(F)-H*COS(F)],(IF THIRD-M THEN MUM ELSE MUS)
/(XN*(IF THIRD=M THEN R3M ELSE R3S)
-(N+I))*A-N*REALPART(SUM(V(NM)*Q(N,MGA)*STB1(M)
*(DIFF(SUM(STBS2(TN,M),T,-N-1,N+1+STBN(N,M),H)
+SUM(STBP2(T,N,M),T,-N,N)*-COS(F)+DIFF((F-LAM)
*HANSEN2[N,-M.-1](H,K),H)),M,O,N))),
DSDK:SUBST([F-LAM=K*SIN(F)-H*COS(F)],(IF THIRD=M THEN MUM ELSE MUS)
/(XN'(IF THIRD=M THEN R3M ELSE R3S)
-(N+1))*A-N*REALPART(SUM(V(NM)*Q(N,M,GA)*STBI(M)
*(DIFF(SUM(STBS2(T,N,M),T,-N-1,N+1)+STBN(NM),K)
+SUM(STBP2(T,N,M),T,-NN)*SIN(F)+DIFF((F-LAM)
*HANSEN2[N,-M,-1](H,K),K)),M,O,N))),
DSDLAM:(IF THIRD=M THEN MUM ELSE MUS)/(XN*(IF THIRD=M THEN R3M ELSE R3S)
^(N+I))*A-N*REALPART(SUM(V(NM)*Q(N,M,GA)*STBI(M)
*(SUM(STBP2(T,NM),T,-N,N)-HANSEN2[N,-M,-1](HK)),M,O,N)).
/*
DEFINITIONS OF THE VARIATIONS OF THE EQUINOCTIAL ELEMENTS
*/
DELTA%A:-POISSON(A,LAM)*DSDLAM,
DELTA%H:-POISSON(H,K)*DSDK-POISSON(H,P)*DSDP
-POISSON(H,Q)*DSDQ-POISSON(HLAM)*DSDLAM,
DELTA%K:-POISSON(K,H)*DSDH-POISSON(K,P)*DSDP
-POISSON(KQ)*DSDQ-POISSON(K,LAM)*DSDLAM,
DELTA%P:-POISSON(P,H)*DSDH-POISSON(P,K)*DSDK
-POISSON(P,Q)*DSDQ-POISSON(PLAM)*DSDLAM,
DELTA%Q:-POISSON(QH)*DSDH-POISSON(QK)*DSDK
-POISSON(Q,P)*DSDP-POISSON(QLAM)*DSDLAM,
DELTA%LAM:-POISSON(LAM,A)*DSDA-POISSON(LAMH)*DSDH
-POISSON(LAM,K)*DSDK-POISSON(LAM,P)*DSDP
-POISSON(LAM,Q)*DSDQ-3*ST/(XN*A^2),
/*
THE RESULTS
RETURNDELTAADELHDELTAKD
RETURN([DELTA%ADELTA%HDELTA%KDELTA%PDELTAQDELTA%LAM))$
139
APPENDIX B
THIRD BODY A- AND B1 -MATRICES
What follow are the final analytical expressions for both A- and
B1 -matrix entries due to the third body P2 perturbation.
The A-matrix is
closed form in the eccentricity, the Bl-matrix is truncated to order zero.
All entries were generated using MACSYMA, and the B-matrix
computation
employed the third body short periodic package in Appendix A.
Using an EPHEM for the low-eccentricity SYNCX, the GTDS-implemented
analytic A-matrix routine was verified by comparing matrix entries with
those
model.
produced
by
finite
differencing
a P2
e0
single-averaged
The matrices agreed to within quadrature error.
third
body
In addition, by
using the same EPHEM, the e0 Bl-matrix was checked by comparing it with
that produced by
short
periodic
significant
finite differencing P2 quasi-e0
functions.
figures.
substituting directly
matrices.
The
Output
in
the
The
of
results
both
agreed
analytical third body
to
routines was
more
also
than
three
verified
MACSYMA-implemented expressions
for
by
both
expressions for the matrix entries reside currently in
loadfile-type files WAGNER; AMAT FINAL and WAGNER; B1MAT FINAL.
140
A-MATRIX PARTIALS, THAT IS,
2
AVERAGED ELEMENT RATES WITH RESPECT TO
THE FOLLOWING ARE THE THIRD BODY P
PARTIALS OF THE THIRD BODY P
2
AVERAGED EQUINOCTIAL ELEMENTS. THEY ARE CLOSED FORM IN THE
ECCENTRICITY. AS FOR THE LABELING, DADOTDH SIGNIFIES THE PARTIAL OF
THE A RATE WITH RESPECT TO AVERAGED H.
DADOTDA
0
DADOTDH
0
DADOTDK
0
DADOTDP
0
DADOTDQ
0
DADOTDL
0
DHDOTDA
2
2
- 9 MU (GA K (BE (II (K - 4 H - 1) Q + 5 H K P)
2
+ AL ((4 K
2
2
2
2
- H + 1) P - 5 H II K Q)) X + (BE - 4 AL + 1) K
3
- 5 AL BE H)/(4 A R3 X XN)
DHDOTDH
2
2
- 3 MU (GA H K (BE (II (K - 4 H 2
+ AL ((4 K
+ (- GA
+(-
1) Q + 5 H K P)
4
2
- H + 1) P - 5 H II K Q)) X
K (AL (5 II K Q + 2 H P) + BE (8 H II Q - 5 K P))
3
2
2
2
2
BE + 4 AL - 1) H K + 5 AL BE H ) X - 5 AL BE)/(2 R3 X XN)
DHDOTDK
2
- 3 MU (GA K
2
+ AL ((4 K
2
-H
2
2
(BE (II (K - 4 H - 1) Q + 5 H K P)
4
+ 1) P - 5 H II K Q)) X
141
2
+ (GA (BE (II (3 K
2
- 4 H
- 1) Q + 10 H K P)
2
2
2
2
2
+ AL ((12 K - H + 1) P - 10 H II K Q)) - (BE - 4 AL + 1) K
2
2
2
3
+ 5 AL BE H K) X + BE - 4 AL + 1)/(2 R3 X XN)
DHDOTDP
2
2
- 3 MU ((2 AL K (GA II Q + BE) (II (K - 4 H - 1) Q + 5 H K P)
2
2
2
2
- AL ) ((4 K - H + 1) P - 5 H II K Q)
- 2 K (BE GA II Q + GA
2
+ (C + 1) GA K (AL (4 K
2
- H
2
+ 1) + 5 BE H K)) X
2
2
+ 10 II (2 AL BE K + (BE - AL ) H) Q + 2 GA (8 AL K + 5 BE H))
/(2
3
(C + 1) R3 X XN)
DHDOTDQ
- 3 II
MU ((2 K (-
2
2
2
2
AL GA P + GA - BE ) (II (K - 4 H - 1) Q
2
2
+ 5 H K P) + 2 BE K (GA P - AL) ((4 KK - H
+ 1) P - 5 H II K Q)
2
2
2
+ (C + 1) GA K 'BE (K - 4 H - 1) - 5 AL H K)) X
2
2
- 10 (2 AL BE K + (BE - AL ) H) P + 2 GA (2 BE K - 5 AL H))
/(2
3
(C + 1) R3 X XN)
DHDOTDL
0
DKDOTDA
2
2
9 MU (GA H (BE (II (K - 4 H - 1) Q + 5 H K P)
2
+.AL ((4 K
2
2
- (4 BE
2
- AL
2
- 5 AL BE K
+ 1) P - 5 H II K Q)) X
- H
3
-1) H)/(4 A R3 X XN)
DKDOTDH
2
3 MU (GA H
2
+ AL ((4 K
2
2
(BE (II (K - 4 H - 1) Q + 5 H K P)
2
- H
4
+ 1) P - 5 H II K Q)) X
142
2
2
+ (GA (BE (II (K - 12 H - 1) Q + 10 H K P)
2
2
+ AL ((4 K - 3 H + 1) P - 10 H II K Q)) + 5 AL BE H K
2
+ (4 BE
2
- AL
2
2
2
2
3
- 1) H ) X - 4 BE + AL + 1)/(2 R3 X XN)
DKDOTDK
2
2
3 MU (GA H K (BE (II (K - 4 H -.1) Q + 5 H K P)
2
2
4
+ AL ((4 K - H + 1) P - 5 H II K Q)) X
+ (GA H (BE (2 II K Q + 5 H P) + AL (8 K P - 5 H II Q))
2
+ (4 BE
2
- AL
2
- 1) H K) X
2
+ 5 AL BE K
3
- 5 AL BE)/(2 R3 X XN)
DKDOTDP
2
2
3 MU ((2 AL H (GA II Q + BE) (II (K - 4 H - 1) Q + 5 H K P)
2
2
2
2
- 2 H (BE GA II Q + GA - AL ) ((4 K - H + 1) P - 5 H II K Q)
2
2
2
+ (C + 1) GA H (AL (4 K - H + 1) + 5 BE H K)) X
2
2
+ 10 II ((BE - AL ) K - 2 AL BE H) Q + 2 GA (5 BE K - 2 AL H))
/(2
3
(C + 1) R3 X XN)
DKDOTDIQ
2
3 II
MU ((2 H (- AL GA P + GA
2
- BE ) (II
2
2
(K - 4 H - 1) Q
2
2
+ 5 H K P) + 2 BE H (GA P - AL.) ((4 K - H + 1) P - 5 H II K Q)
2
2
+ (C + 1) GA H (BE (K - 4 1
2
- 1) -
5 AL H K)) X
2
2
- 10 ((BE - AL ) K - 2 AL I3E H) P - 2 GA (5 AL K + 8 BE H))
/(2
3
(C + 1) R3 X XN)
DKDOTDL
O
143
DPDOTDA
2
2
9 (C + 1) GA (BE (K - 4 H - 1) - 5 AL H K) MU X
3
8 A R3 XN
DPDOTDH
2
2
- 3 (C + 1) GA MU X (H (BE (K - 4 H - 1)
2
- 5 AL H K) X
3
- 5 AL K - 8 BE H)/(4 R3
XN)
DPDOTDK
2
2
- 3 (C + 1) GA MU X (K (BE (K - 4 H - 1)
2
- 5 AL H K) X
3
+ 2 BE K - 5 AL H)/(4 R3
XN)
DPDOTDP
2
- 3 MU (GA (AL II
2
+ AL BE (K
(K
2
- 4 H - 1) Q + 5 BE H II K Q)
2
2
2
3
- 4 H - 1) + 5 (GA - AL ) H K) X/(2 R3 XN)
DPDOTDQ
2
2
- 3 II MU (- GA (AL (K - 4 H - 1) + 5 BE H K) P
3
2
2
2
2
+ (GA - BE ) (K - 4 H - 1) + 5 AL BE H K) X/(2 R3 XN)
DPDOTDL
0
DQDOTDA
2
2
9 (C + 1) GA II (AL (4 K - H + 1) + 5 BE H K) MU X
3
8 A R3 XN
DQDOTDH
2
2
2
3 (C + 1) GA II MU X (H (AL (4 K - H + 1) + 5 BE H K) X
+ 5 BE K - 2 AL H)/(4
3
R3 XN)
144
CQDOTDK
2
2
2
3 (C + 1) GA II MiU X (K (AL (4 K - H + 1) + 5 BE H K) X
3
+ 8 AL K + 5 BE H)/(4 R3
XN)
DQDOTDP
2
2
- 3 II MU (GA (BE II (4 K - H + 1) Q - 5 AL H II K Q)
2
2
2
2
3
+ (GA - AL ) (4 K - H + 1) - 5 AL BE H K) X/(2 R3 XN)
DQDOTDQ
2
2
3 MU (GA (BE (4 K - H + 1) P - 5 AL H K P)
2
2
2
2
3
- AL BE (4 K - H + 1) + 5 (GA - BE ) H K) X/(2 R3 XN)
DQDOTDL
0
DLDOTDA
2
2
- 3 MU (3 GA (BE (II (K - 4 H - 1) Q + 5
2
+ AL ((4 K
2
- H
+ 1) P - 5 H II K Q)) X (X + 1)
2
2
- 1) (3 K
- (3 GA
K P)
2
+ 3 H + 2) (X + 1)
2
2
2
2
- 15 ((BE - AL ) (K - H ) - 4 AL BE H K) X
2
2
2
2
2
2
2
2
- 3 (BE (4 K - H ) - AL (K - 4 H ) - K - 10 AL BE H K - H ))
3
/(4 A R3
(X + 1) XN)
DLDOTDH
2
2
3 MU (- GA H (BE (II (K - 4 H - 1) Q + 5 H K P)
2
2
3
2
+ AL ((4 K - H + 1) P - 5 H II K Q)) X (X + 1)
2
+ GA (AL (5 II K Q + 2 H P) + BE (8 H II Q - 5 K P)) X (X + 1)
2
+ H (- AL
2
2
2
2
2
2
2
3
(4 K - H ) + BE (K - 4 H ) + K - 10 AL BE H K + H ) X
2
- 2 (10 AL BE K - (3 GA
2
2
2
- 5 BE + 5 AL - 1) H) X
145
2
2
2
- 6 (5 AL BE K - (2 GA - 2 BE + 3 AL - 1) H) X - 10 AL BE K
2
2
- BE
+ 2 (3 GA
2
3
2
+ 4 AL - 2) H)/(2 R3 (X + 1) XN)
DLDOTDK
2
2
3 MU (- GA K (BE (II (K - 4 H - 1) Q + 5 H K P)
2
2
+ AL ((4 K
- H
2
3
+ 1) P - 5 H II K Q)) X (X + 1)
2
- GA (BE (2 II K Q + 5 H P) - AL (5 H II Q - 8 K P)) X (X + 1)
2
2
+ K (- AL
(4 K
3
2
2
2
2
2
2
- H ) + BE (K - 4 H ) + K - 10 AL BE H K + H ) X
2
2
2
- 5 AL
2
2
2
- 1) K - 10 AL BE H) X
2
- 2 AL
+ 3 BE
+ 6 ((2 GA
+ 2 (3 GA
2
+ 5 BE
+ 2 ((3 GA
- 1) K - 5 AL BE H) X
2
3
2
- AL - 2) K - 10 AL BE H)/(2 R3 (X + 1) XN)
2
+ 4 BE
DLDO
2
2
- 3 MU ((2 AL (GA II Q + BE) (II (K - 4 H - 1) Q + 5 H K P)
2
- 2 (BE GA II Q + GA
2
2
2
- AL ) ((4 KK - H + 1) P - 5 H II K Q)
2
2
+ (C + 1) GA (AL (4 K - H + 1) + 5 BE H K)) X (X + 1)
- 4 (2 (5
II
(AL K + BE H) (BE K - -AL H) Q
2
+ GA (AL (4 K
2
- H
+ 1) + 5 BE H K)) X + 5 II (AL K + BE H) (BE K
2
- AL H) Q + GA (AL (4 K
2
- H
+ 2) + 5 BE H K)))
3
/(2 (C + 1) R3 (X + 1) XN)
DLDOTDQ
2
2
2
2
- 3 II MU ((2 (- AL GA P + GA - BE ) (II (K - 4 H - 1) Q
2
2
+ 5 H K P) + 2 BE (GA P - AL) ((4 K - H + 1) P - 5 H II K Q)
2
2
+ (C + 1) GA (BE (K - 4 H - 1) - 5 AL H K)) X (X +
2
2
- 4 ((2 GA (BE (K - 4 H - 1) - 5 AL I K)
146
)
-
10 (AL K + BE H) (BE K - AL H) P) X - 5 (AL K + BE H) (BE K
2
2
- AL H ) P - GA (BE (K - 4 H -
2) - 5 AL H K)))
3
/(2 (C + 1) R3
(X + 1) XN)
DLDOTDL
0
147
THE FOLLOWING ARE THE THIRD BODY P
PARTIALS OF THE THIRD BODY P
B1-MATRIX PARTIALS, THAT IS,
2
SHORT PERIODIC FUNCTIONS WITH RESPECT TO
2
AVERAGED EQUINOCTIAL ELEMENTS. THEY HAVE BEEN TRUNCATED TO ORDER ZERO
IN THE ECCENTRICITY. THE LABELING OF THE EXPRESSIONS IS STRAIGHTFORWARD:
DELTAH MEANS THE PARTIAL OF THE SHORT PERIODIC VARIATION OF A WITH RESPECT
TO AVERAGED H.
DELTAA
2
6 (2 AL BE SIN(2 EL) - BE
2
COS(2 EL) + AL
COS(2 EL)) MUM
3
2
R3M XN
DELTAH
2
(A SIN(EL) (6 GA
2
2
2
2
- 9 (BE - AL ) - 2) - 3 A (BE - AL ) SIN(3 EL)
3
2
- 6 A AL BE COS(3 EL) - 18 A AL BE COS(EL)) MUM/(2 R3M XN )
DELTAK
2
(A COS(EL) (6 GA
2
2
+ 9 (BE - A ) - 2) + 6 A AL BE SIN(3 EL)
2
2
3
2
- 3 A (BE - AL ) COS(3 EL) - 18 A AL BE SIN(EL)) MUM/(2 R3M XN )
DELTAP
- 3 A ((- AL DBEDP - BE DALDP) SIN(2 EL)
3
2
+ (BE'DBEDP - AL DALDP) COS(2 EL)) MUM/(R3M XN )
DELTAQ
- 3 A ((- AL DBEDQ - BE DALDQ) SIN(2 EL)
3
2
+ (BE DBEDQ - AL DALDQ) COS(2 EL)) MUM/(R3M XN )
DELTAL
2
2
3 A ((BE - AL ) SIN(2 EL) + 2 AL BE COS(2 EL)) MUM
3
2
R3M XN
DELTHA
2
2
2
2
2
- 3 (SIN(EL) (- 6 GA - 9 (BE - AL ) + 2) + (BE - AL ) SIN(3 EL)
3
2
+ 2 AL BE COS(3 EL) - 18 AL BE COS(EL)) MUM/(4 A R3M XN )
148
DELTHH
2
2
2
(COS(2 EL) (1 - 3 (GA - BE + AL )) - 6 AL BE SIN(4 EL)
2
3
2
2
+ 3 (BE - AL ) COS(4 EL) - 6 AL BE SIN(2 EL)) MUM/(8 R3M XN )
DELTHK
- MUM (SIN(2 EL) (3 (C + 1) (BE DBEDQ Q + DBEDP P)
2
2
2
+ AL (- DALDQ Q - DALDP P)) - 3 (GA + 5 (BE - AL )) + 1)
- 3 COS(2 EL) (10 AL BE - (C + 1) (AL DBEDQ Q + DBEDP P)
2
2
+ BE DALDQ Q + DALDP P))) + 3 (BE - AL ) SIN(4 EL) + 6 AL BE COS(4 EL))
3
2
/(8 R3M XN )
DELTHP
- (- 3 SIN(EL) (2 DGADP GA + 3 (BE DBEDP - AL DALDP))
+ (BE DBEDP - AL DALDP) SIN(3 EL) + (AL DBEDP + BE DALDP) COS(3 EL)
3
2
- 9 (AL DBEDP + BE DALDP) COS(EL)) MUM/(2 R3M XN )
DELTHQ
- (- 3 SIN(EL) (2 DGADQ GA + 3 (BE DBEDQ - AL DALDQ))
+ (BE DBEDQ - AL DALDQ) SIN(3 EL) + (AL DBEDQ + BE DALDQ) COS(3 EL)
3
2
- 9 (AL DBEDQ + BE DALDQ) COS(EL)) MUM/(2 R3M XN )
DELTHL
2
2
2
- (COS(EL) (- 6 GA - 9 (BE - AL ) + 2) - 6 AL BE SIN(3 EL)
2
2
3
2
+ 3 (BE - AL ) COS(3 EL) + 18 AL BE SIN(EL)) MUM/(4 R3M XN )
DELTKA
2
2
2
- 3 (COS(EL) (- 6 GA + 9 (BE - AL ) + 2) - 2 AL BE SIN(3 EL)
2
2
3
2
+ (BE - AL ) COS(3 EL) - 18 AL BE SIN(EL)) MUM/(4 A R3M XN )
DELTKH
- MUM (SIN(2 EL) (1 - 3 ((C + 1) (BE DBEDQ Q + DBEDP P)
2
+ AL (- DALDQ Q - DALDP P)) + GA
2
2
- 5 (BE - AL )))
+ 3 COS(2 EL) (10 AL BE - (C + 1) (AL DBEDQ Q + DBEDP P)
149
2
2
+ BE DALDQ Q + DALDP P))) + 3 (BE - AL ) SIN(4 EL) + 6 AL BE COS(4 EL))
3
2
/(8 R3M XN )
DELTKK
2
2
2
- (COS(2 EL) (1 - 3 (GA + BE - AL )) - 6 AL BE SIN(4 EL)
2
3
2
2
+ 3 (BE - AL ) COS(4 EL) + 6 AL BE SIN(2 EL)) MUM/(8 R3M XN )
DELTKP
- (COS(EL) (9 (BE DBEDP - AL DALDP) - 6 DGADP GA)
+ (- AL DBEDP - BE DALDP) SIN(3 EL) + (BE DBEDP - AL DALDP) COS(3 EL)
3
2
- 9 (AL DBEDP + BE DALDP) SIN(EL)) MUM/(2 R3M XN )
DELTKQ
- (COS(EL) (9 (BE DBEDQ - AL DALDQ) - 6 DGADQ GA)
+ (- AL DBEDQ - BE DALDQ) SIN(3 EL) + (BE DBEDQ - AL DALDQ) COS(3 EL)
3
2
- 9 (AL DBEDQ + BE DALDQ) SIN(EL)) MUM/(2 R3M XN )
DELTKL
2
(SIN(EL) (- 6 GA
2
2
2
2
+ 9 (BE - AL ) + 2) + 3 (BE - AL ) SIN(3 EL)
3
2
+ 6 AL BE COS(3 EL) + 18 AL BE COS(EL)) MUM/(4 R3M XN )
DELTPA
2
2
9 (C + 1) MUM (COS(2 EL) (2 (BE - AL ) P
- (C + 1) (AL DBEDQ + BE DALDQ) II) + SIN(2 EL)
3
2
(- 4 AL BE P - (C + 1) (BE DBEDQ - AL DALDQ) II))/(16 A R3M XN )
DELTPH
(C + 1) MUM ((SIN(3 EL) + 9 SIN(EL))
2
2
(2 (BE - AL ) P - (C + 1) (AL DBEDQ + BE DALDQ) II)
+ (COS(3 EL) + 9 COS(EL)) (4 AL BE P + (C + 1) (BE DBEDQ - AL DALDQ) II)
3
2
- 6 (C + 1) DGADQ COS(EL) GA II)/(8 R3M XN )
150
DELTPK
2
2
(C + 1) MUM (COS(3 EL) (2 (BE - AL ) P
- (C + 1) (AL DBEDQ + BE DALDQ) II) - 9 COS(EL)
2
2
(2 (BE - AL ) P - (C + 1) (AL DBEDQ + BE DALDQ) II)
+ 3 SIN(EL) (12 AL BE P + (C + 1) (2 DGADQ GA
t
3 (BE DBEDQ - AL DALDQ)) II)
3
+ SIN(3 EL) (- 4 AL BE P - (C + 1) (BE DBEDQ - AL DALDQ) II))/(8 R3M
2
XN )
DELTPP
2
3 MUM (SIN(2 EL) (BE (- 4 AL (2 P
- (C + 1) (4 DBEDQ II + DALDP) P
+
+ C + 1)
(C + 1) D2BDQP II))
+ (C + 1) (AL (4 DALDQ II - DBEDP) P + (C + 1) D2ADQP II)
- (C + 1) DBEDP DBEDQ - DALDP DALDQ) II))
2
2
2
+ COS(2 EL) (2 (BE - AL ) (2 P + C + 1)
- (C + 1) (AL (4 DBEDQ II + DALDP) P + (C + 1) D2BDQP II)
+ BE (4 DALDQ II - DBEDP) P + (C + 1) D2ADQP II)
+ (C + 1) DALDP DBEDQ
t
3
2
DALDQ DBEDP) II)))/(16 R3M XN )
DELTPQ
3 MUM (SIN(2 EL) (- (C + 1) (BE (4 DBEDQ II Q + DALDQ P)
+ (C + 1) D2BDQQ II) + AL ((- C - 1) D2ADQQ II - 4 DALDQ II Q - DBEDQ P))
2
2
+ (C + 1) DBEDQ - DALDQ ) II) - 8 AL BE P Q)
2
2
+ COS(2 EL) (4 (BE - AL ) P Q - (C + 1)
(AL (4 DBEDQ II Q + DALDQ P) + (C + 1) D2BDQQ II)
+ BE (4 DALDQ II
Q - DBEDQ P) + (C + 1) D2ADQQ II)
3
2
+ 2 (C + 1) DALDQ DBEDQ II)))/(16 R3M XN )
DELTPL
2
2
- 3 (C + 1) MUM (SIN(2 EL) (2 (BE - AL ) P
- (C + 1) (AL DBEDQ + BE DALDQ) II) + COS(2 EL)
3
2
(4 AL BE P + (C + 1) (BE DBEDQ - AL DALDQ) II))/(8 R3M XN )
151
DELTQA
2
2
9 (C + 1) MUM (COS(2 EL) (2 (BE - AL ) Q
+ (C + 1) (AL DBEDP + BE DALDP) II) + SIN(2 EL)
3
2
((C + 1) (BE DBEDP - AL DALDP) II - 4 AL BE Q))/(16. A R3M XN )
DELTQH
2
2
(C + 1) MUM (SIN(3 EL) (2 (BE - AL ) Q
+ (C + 1) (AL DBEDP + BE DALDP) II) + 9 SIN(EL)
2
2
(2 (BE - AL ) Q + (C + 1) (AL DBEDP + BE DALDP) II)
+ 3 COS(EL) (12 AL BE Q + (C + 1) (2 DGADP GA - 3 (BE DBEDP - AL DALDP)) II)
3
2
+ COS(3 EL) (4 AL BE Q - (C + 1) (BE DBEDP - AL DALDP) II))/(8 R3M XN )
DELTQK
2
2
(C + 1) MUM (COS(3 EL) (2 (BE - AL ) Q
+ (C + 1).(AL DBEDP + BE DALDP) II) - 9 COS(EL)
2
2
(2 (BE - AL ) Q + (C + 1) (AL DBEDP + BE DALDP) II)
+ SIN(3 EL) ((C + 1) (BE DBEDP - AL DALDP) II - 4 AL BE Q)
- 3 SIN(EL) ((C + 1) (2 DGADP GA + 3 (BE DBEDP - AL DALDP)) II - 12 AL BE Q))
3
2
/(8 R3M XN )
DELTQP
3 MUM (COS(2 EL) ((C + 1) (BE (4 DBEDP Q + DALDP II P)
+ (C + 1) D2ADPP II) + AL ((C + 1) D2BDPP II - 4 DALDP Q - DBEDP II P))
2
2
+ 2 (C + 1) DALDP DBEDP II) + 4 (BE - AL ) P Q)
+ SIN(2 EL) (- (C + 1) (AL (4 DBEDP Q + DALDP II P) + (C + 1) D2ADPP II)
+ BE (4 DALDP Q - DBEDP II P) + (- C - 1) D2BDPP II)
2
2
3
2
- (C + 1) DBEDP - DALDP ) II) - 8 AL BE P Q))/(16 R3M XN )
DELTQQ
2
2
2
3 MUM (COS(2 EL) (2 (BE - AL ) (2 Q + C + 1)
+ (C + 1) (AL (4 DBEDP II - DALDQ) Q + (C + 1) D2BDPQ II)
+ BE (4 DALDP II + DBEDQ) Q + (C + 1) D2ADPQ II)
152
+ (C + 1) DALDP DBEDQ + DALDQ DBEDP) II))
2
+ SIN(2 EL) (- 8 AL BE Q
- (C + 1) (BE
(- 4 DBEDP II - DALDQ) Q + (- C - 1) D2BDPQ II + 4 AL)
+ AL (4 DALDP II + DBEDQ) Q + (C + 1) D2ADPQ II)
3
2
- (C + 1) DBEDP DBEDQ - DALDP DALDQ) II)))/(16 R3M XN )
DELTQL
2
2
- 3 (C + 1) MUM (SIN(2 EL) (2 (BE - AL ) Q
+ (C + 1) (AL DBEDP + BE DALDP) II) + COS(2 EL)
3
2
(4 AL BE Q - (C + 1) (BE DBEDP - AL DALDP) II))/(8 R3M XN )
DELTLA
2
2
9 MUM (SIN(2 EL) (7 (BE - AL ) - (C + 1)
(BE DBEDQ Q + DBEDP P) + AL (- DALDQ Q - DALDP P)))
+ COS(2 EL) (14 AL BE - (C + 1) (AL DBEDQ Q + DBEDP P)
3
+ BE DALDQ Q + DALDP P))))/(8 A R3M
2
XN )
DELTLH
2
- MUM (- COS(EL) (13 (6 GA
2
2
- 9 (BE - AL ) - 2)
- 6 (C + 1) (GA (2 DGADQ Q + 2 DGADP P)
- 3 (BE DBEDQ Q + DBEDP P) + AL (- DALDQ Q - DALDP P))))
2
2
+ COS(3 EL) (13 (BE - AL ) - 2 (C + 1)
(BE DBEDQ Q + DBEDP P) + AL (- DALDQ Q - DALDP P)))
- 2 SIN(3 EL) (13 AL BE - (C + 1) (AL DBEDQ
+ DBEDP P)
+ BE DALDQ Q + DALDP P))) - 18 SIN(EL)
(13 AL BE - (C + 1) (AL DBEDQ Q + DBEDP P) + BE DALDQ Q + DALDP P))))
3
2
/(8 R3M XN )
DELTLK
MUM (SIN(EL) (6 (C + 1) (GA (2 DGADQ Q + 2 DGADP P)
+ 3 BE DBEDQ Q + DBEDP P) - 3 AL DALDQ Q + DALDP P))
2
- 13 (6 GA
2
2
+ 9 (BE - AL ) - 2)) + SIN(3 EL)
153
2
2
(13 (BE - AL ) - 2 (C + 1) (BE DBEDQ Q + DBEDP P)
+ AL (- DALDQ Q - DALDP P))) + 2 COS(3 EL)
(13 AL BE - (C + 1) (AL DBEDQ Q + DBEDP P) + BE DALDQ Q + DALDP P)))
- 18 COS(EL) (13 AL BE - (C + 1) (AL DBEDQ Q + DEDP
P)
2
3
+ BE DALDQ Q + DALDP P))))/(8 R3M XN )
DELTLP
- 3 MUM (COS(2 EL) (AL (2 P DBEDQ Q + DBEDP P)
+ (C + 1) D2BDQP Q + D2BDPP P) + (C - 13) DBEDP)
+ BE (2 P DALDQ Q + DALDP P) + (C + 1) D2ADQP Q + D2ADPP P)
+ (C - 13) DALDP) + (C + 1) (dALDP DBEDQ + DALDQ DBEDP) Q
+ 2 DALDP DBEDP P)) + SIN(2 EL) (AL (- 2 P DALDQ Q + DALDP P)
- (C + 1) D2ADQP Q + D2ADPP P) + (13 - C) DALDP)
2
+ BE ((C + 1) D2BDQP Q + D2BDPP P) + 2 DBEDQ P Q + 2 DBEDP P
+ (C - 13) DBEDP) + (C + 1) (dBEDP DBEDQ - DALDP OALDQ) Q
2
2
3
2
+ DBEDP - DALDP ) P)))/(8 R3M XN )
DELTLQ
2
- 3 MUM (COS(2 EL) (BE (2 DALDQ Q + (C + 1) D2ADQQ Q + D2ADPQ P)
+ 2 DALDP P Q + (C - 13) DALDQ) + AL (2 Q DBEDQ Q + DBEDP P)
+ (C + 1) D2BDQQ Q + D2BDPQ P) + (C - 13) DBEDQ)
+ (C + 1) (2 DALDQ DBEDQ Q + DALDP DBEDQ + DALDQ DBEDP) P))
+ SIN(2 EL) (BE (2 Q DBEDQ Q + DBEDP P) + (C + 1) D2BDQQ Q + D2BDPQ P)
+ (C - 13) DBEDQ) + AL (- 2 Q DALDQ Q + DALDP P)
- (C + 1) D2ADQQ Q + D2ADPQ P) + (13 - C) DALDQ)
2
+ (C + 1) (DBEDQ
2
- DALDQ ) Q + DBEDP DBEDQ - DALDP DALDQ) P)))
3
2
/(8 R3M XN )
DELTLL
2
2
3 MUM (COS(2 EL) (7 (BE - AL ) - (C + 1)
(BE DBEDQ Q + DBEDP P) + AL (- DALDQ Q - DALDP P)))
154
+ SIN(2 EL) ((C + 1) (AL DBEDQ Q + DBEDP P) + BE DALDQ Q + DALDP P))
3
2
- 14 AL BE))/(4 R3M XN )
155
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