IDENTIFICATION OF A TETHERED SATELLITE USING AN EXTENDED
KALMAN FILTER
Except where reference is made to the work of others, the work described in this thesis is
my own or was done in collaboration with my advisory committee. This thesis does not
include proprietary or classified information.
_________________________________________
Elizabeth Jo Volovecky Hayes
Certificate of Approval:
_________________________ _________________________
John E. Cochran, Jr. David A. Cicci, Chair
Professor Professor
Aerospace Engineering Aerospace Engineering
_________________________ _________________________
Robert S. Gross Joe F. Pittman
Associate Professor Interim Dean
Aerospace Engineering Graduate School
IDENTIFICATION OF A TETHERED SATELLITE USING AN EXTENDED
KALMAN FILTER
Elizabeth Jo Volovecky Hayes
A Thesis
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Master of Science
Auburn, Alabama
December 17, 2007
iii
IDENTIFICATION OF A TETHERED SATELLITE USING AN EXTENDED
KALMAN FILTER
Elizabeth Jo Volovecky Hayes
Permission is granted to Auburn University to make copies of this thesis at its discretion,
upon the request of individuals or institutions and at their expense. The author reserves
all publication rights.
______________________________
Signature of Author
______________________________
Date of Graduation
iv
VITA
Elizabeth Jo Volovecky Hayes was born on August 30, 1980 in Fairhope, AL.
Her parents are Joe and Debbie (Probst) Volovecky of Daphne, Alabama. Elizabeth
graduated from Daphne High School in May of 1998. She enrolled at Auburn University
in the fall of 1998, where she began her aerospace studies. Upon receiving her Bachelor
of Aerospace Engineering degree in May of 2002, she began her graduate studies in the
Aerospace Engineering Department at Auburn the following fall. She married her high
school sweetheart, Andy Hayes, in November 2004 and she currently works as an Intel
Analyst for SAIC in Huntsville, Alabama.
v
THESIS ABSTRACT
IDENTIFICATION OF A TETHERED SATELLITE USING AN EXTENDED
KALMAN FILTER
Elizabeth Jo Volovecky Hayes
Master of Science, December 17, 2007
(B.A.E., Auburn University, 2002)
62 Typed Pages
Directed by David A. Cicci
Recent studies involving a tethered satellite system(s) (TSS) have increased due
to the importance of accurately identifying and analyzing the motion of a TSS. If the
motion of a tethered satellite is not accurately identified, the satellite could be mistaken
as a ballistic threat. Standard orbit determination methods used today are unable to
identify a tracked satellite as part of a TSS, due to the non-Keplerian nature of its motion.
Accurate identification of a TSS becomes more complicated with the need to
perform this process quickly using a small set of observational data. Once this ?quick-
look? identification process is performed, it is necessary to calculate the critical orbit
determination parameters used for future TSS tracking and prediction.
An extended Kalman filter (EKF) has been developed to perform both the state
estimation and quick-look identification processes for a tethered satellite not known
vi
a priori as being part of a TSS. In the application of the EKF to a TSS, both, manual
tuning and adaptive tuning methods were used. The adaptive tuning method used is
based upon ridge-type filtering techniques involving the computation of a biasing
parameter that is used as input into the process noise matrix, which is required in tuning
the EKF.
The overall performance of the EKF is presented for varying tether lengths, tether
orientation, and observation noise levels. The results obtained from the adaptively-tuned
EKF are presented in this thesis and are compared to those obtained from a batch filter
and manually-tuned EKF presented in recent studies.
vii
ACKNOWLEDGMENTS
I would like to thank my family and friends for being so patient with me
throughout my life, but especially through my college journey. You all mean the world
to me. I would not be where I am right now without the guidance from the Aerospace
Engineering faculty at Auburn University. They have all made a lasting impression in
my life. I especially want to thank Dr. David Cicci for all of his patience and
understanding through my graduate studies. He never once turned me away even when I
thought I asked a ?stupid? question and for that, I am most thankful.
viii
Style manual or journal used: The Journal of the Astronautical Sciences ________
________________________________________________________________________
Computer software used: FORTRAN PowerStation 4.0, Microsoft Excel,
Microsoft Office Word 2003, and Microsoft Equation 3.0
ix
TABLE OF CONTENTS
LIST OF FIGURES.........................................................................................................x
LIST OF TABLES .........................................................................................................xi
1. INTRODUCTION.......................................................................................................1
2. QUICK-LOOK ORBIT DETERMINATION METHOD DESCRIPTION ...................7
2.1 Preliminary Orbit Determination Method (POD, 1
st
Stage).............................7
2.2 Identification Using an Extended Kalman Filter (2
nd
Stage)..........................10
2.3 Adaptive Tuning Method Using a Biasing Parameter ...................................14
3. PROCEDURE DESCRIPTION .................................................................................17
4. TEST CASES............................................................................................................23
5. RESULTS .................................................................................................................25
6. CONCLUSIONS .......................................................................................................32
REFERENCES..............................................................................................................35
APPENDIX A: TEST CASE DATA..............................................................................37
x
LIST OF FIGURES
FIG. 1. Tethered Satellite System (TSS) Model ..............................................................1
FIG. 2. TSS Model with Force Components and Libration Angle ....................................8
FIG. 3. TSS Center-of-Mass and Tether Length Measurements .......................................9
xi
LIST OF TABLES
TABLE 1. Parameter Variations for Scenarios..............................................................23
TABLE 2. Baseline Orbit for Data Generation...............................................................23
TABLE 3. Varying Combinations of the Biasing Function ...........................................24
TABLE 4. Comparisons for No Tether cases, ?
cm
= 0 m................................................27
TABLE 5. Comparisons for 1 km cases, ?
cm
= 909 m, LOW noise................................28
TABLE 6. Comparisons for 1 km cases, ?
cm
= 909 m, MEDIUM noise.........................28
TABLE 7. Comparisons for 1 km cases, ?
cm
= 909 m, HIGH noise ...............................28
TABLE 8. Comparisons for 10 km cases, ?
cm
= 9091 m, LOW noise ............................29
TABLE 9. Comparisons for 10 km cases, ?
cm
= 9091 m, MEDIUM noise.....................29
TABLE 10. Comparisons for 10 km cases, ?
cm
= 9091 m, HIGH noise .........................29
TABLE 11. Comparisons for 50 km cases, ?
cm
= 45455 m, LOW noise ........................30
TABLE 12. Comparisons for 50 km cases, ?
cm
= 45455 m, MEDIUM noise.................30
TABLE 13. Comparisons for 50 km cases, ?
cm
= 45455 m, HIGH noise .......................30
TABLE 14. Comparisons for 1 km UP cases, ?
cm
= -909 m, LOW noise.......................31
TABLE 15. Comparisons for 1 km UP cases, ?
cm
= -909 m, MEDIUM noise................31
TABLE 16. Comparisons for 1 km UP cases, ?
cm
= -909 m, HIGH noise......................31
TABLE 17. Test Case Data: No Tether (?
cm
= 0) .........................................................38
TABLE 18. Test Case Data: ? = 1 km (?
cm
= 909 m), LOW noise ...............................39
xii
TABLE 19. Test Case Data: ? = 1 km (?
cm
= 909 m), MEDIUM noise ........................40
TABLE 20. Test Case Data: ? = 1 km (?
cm
= 909 m), HIGH noise ..............................41
TABLE 21. Test Case Data: ? = 10 km (?
cm
= 9091 m), LOW noise............................42
TABLE 22. Test Case Data: ? = 10 km (?
cm
= 9091 m), MEDIUM noise ....................43
TABLE 23. Test Case Data: ? = 10 km (?
cm
= 9091 m), HIGH noise...........................44
TABLE 24. Test Case Data: ? = 50 km (?
cm
= 45455 m), LOW noise..........................45
TABLE 25. Test Case Data: ? = 50 km (?
cm
= 45455 m), MEDIUM noise ..................46
TABLE 26. Test Case Data: ? = 50 km (?
cm
= 45455 m), HIGH noise......... ...............47
TABLE 27. Test Case Data: ? = 1 km up (?
cm
= -909 m), LOW noise ......... ...............48
TABLE 28. Test Case Data: ? = 1 km up (?
cm
= -909 m), MEDIUM noise .. ...............49
TABLE 29. Test Case Data: ? = 1 km up (?
cm
= -909 m), HIGH noise ........................50
1
1. INTRODUCTION
A tethered satellite system (TSS) refers to a system that includes two or more
satellites, or space bodies, which are connected by a tether, or cord. One idea behind
tethered satellites is to control the motion of one satellite by attaching it to another
satellite. In a two-bodied TSS, the primary satellite is generally the larger of the two and
is referred to as the ?parent? satellite. The second satellite, usually smaller than the
parent satellite, is referred to as the ?daughter? satellite. Either satellite may be located in
a higher or a lower orbit with respect to the system?s center-of-mass or they can be in the
same orbit. A two-bodied TSS model is shown in Fig. 1 below.
FIG. 1. Tethered Satellite System (TSS) Model
The study of tethered satellites originated in the late 1800?s and has continued on
through today with actual ?in space? applications. In general, tethered satellites can offer
many important uses, such as: providing power between satellites or other space vehicles
for transfer of energy or momentum purposes, providing support to astronauts during an
Earth Surface
Parent
Daughter
Tether
2
Extra-Vehicular Activity (EVA) maneuver, which connects them to the spacecraft, or
providing aid in the control of a space vehicle?s motion. It has only been recently that
studies have addressed the orbit determination problem of tethered satellites. In one
study, a satellite, which was a member of a TSS, was incorrectly identified to be a re-
entry object?s trajectory [1]. This is due to the tether force perturbing the motion of the
tracked satellite and causing it to behave differently than the motion of an untethered
satellite [2].
The orbit determination of any space object involves the ability to accurately
track, identify, and predict the motion of the object of interest. The need to perform a
proper orbit determination analysis of a TSS has increased, since its future use has
become important to both the private and public industries. If a TSS is inaccurately
analyzed, its motion could result in a tethered satellite being incorrectly identified as a
possible threat, resulting in many unnecessary and costly measures being taken to
counteract the threat.
It is possible for this type of scenario to occur since tethered satellites behave
differently than untethered ones. This is due to the force present in the tether, which acts
on all satellites connected to the tether. Each satellite in the TSS is perturbed by the
tether force, causing it to vary from the Keplerian-type motion, typically found in
untethered satellites [1,2]. When a tethered satellite is stationed in a higher orbit than the
center-of-mass of the TSS, its velocity will be larger in magnitude than the results
predicted by classical Keplerian motion due to the presence of the tether force. Likewise,
if a tethered satellite is in a lower orbit than the center-of-mass of the TSS, its velocity
will be smaller than that indicated from classical Keplerian motion due to the tether force.
3
Classical orbit determination methods are not capable of distinguishing between a
tracked tethered satellite and an untethered satellite; therefore the satellite may be
incorrectly identified as an untethered one if the classical methods are used. Incorrect
identification will also result in an inaccurate prediction of the satellite?s future motion.
In order to prevent a TSS from being incorrectly identified, the need to quickly
perform the identification process is very important. This process of identification,
referred to as ?quick-look? identification, is performed by processing measurements of
the satellite?s motion made from tracking stations. These measurements are acquired by
the use of radar, infrared, radio and/or optical techniques and include parameters of the
observed satellite, such as: range, range-rate, azimuth, elevation, azimuth-rate and/or
elevation-rate. The measurements are then used in the orbit determination process to
estimate the satellite?s orbit. The first few measurements obtained are used in the
preliminary orbit determination (POD) procedure in order to establish a set of initial
conditions, which will then be used in the filtering process. In order to more accurately
determine the set of initial conditions, a larger set of observational measurements,
accumulated from one or more tracking stations, must then be processed.
The need to accurately estimate the state of a tethered satellite, which can include
parameters, such as: position, velocity, dynamical constants, etc., involves the differential
correction process. Once the POD provides the set of initial conditions, the differential
correction process improves the accuracy of the solution acquired by the POD procedure.
These improvements are made by processing all of the observational data available in the
filtering process. There are several filtering techniques available to use in the differential
correction process. The three most commonly used are batch-type filters, Kalman (or
4
sequential) filters, and extended Kalman (or extended sequential) filters. Batch-type
filters process an entire set of observational data at once in order to estimate the state for
a specified epoch. Kalman and extended Kalman filters process observations as they are
received. A Kalman filter provides the estimate of the satellite?s state at each observation
time where an extended Kalman filter updates the reference trajectory at each observation
to reflect the best estimate of the true trajectory. Each filter has its distinct advantages
and disadvantages; however this study will utilize the extended Kalman Filter (EKF) and
compare its results to previous studies using batch-type and sequential estimators.
A POD method was recently developed [3,4] for the use of both tethered and
untethered satellites. Following this method, several different batch-type filters for the
estimation of the state of a tethered satellite were presented and their performances were
compared [5]. In this study a two-dimensional dynamical model of a TSS was
considered. The model maintained a vertical orientation and as a result did not possess
the capabilities to include out-of-plane motion of the system, or any apparent oscillatory
motion of the TSS. There have been additional studies where out-of-plane libration of a
TSS was modeled, in an enhanced batch filter [6].
The accuracy of quick-look TSS identification was improved by the use of ridge-
type estimation methods [7]. Once a satellite is identified as being a tethered satellite,
more sophisticated models of a TSS [8-12] can then be used to predict its long-term
motion. When long arcs of observational data are available, these enhanced dynamical
models and filtering techniques are more useful.
A more recent presentation involved a method that combined all of the desired
characteristics needed in both the quick identification and the prediction of long-term
5
motion of a TSS. The three-stage TSS identification and orbit determination
methodology [13] included: a 1
st
Stage POD procedure, a 2
nd
Stage ridge-type filter, and
a 3
rd
Stage long-term prediction filter. The performance of this methodology was
demonstrated using a series of simulated cases with varying TSS geometry and
observation noise levels, as well as on real TSS data obtained from the Tether Physics
and Survivability Experiment (TiPS) [14].
An extended Kalman filter (EKF) was recently developed in order to address the
TSS identification problem using extended sequential processing of the observational
data [15]. The primary emphasis is on the quick-look aspects of tethered satellite
identification rather than on the long-term orbit prediction aspects referred to in [13].
The manually-tuned EKF utilizes the POD results from [13] as initial conditions, and
short arcs of observations are processed in order to determine the best estimate of the
satellite?s state. The performance of the EKF is evaluated through the analysis of
simulated data using differing tether lengths, tether orientations, observational error
levels, and observation arcs.
An adaptive or automated tuning methodology, used in angles-only tracking and
intercept problems [16], was applied to the EKF to improve overall filter performance
and to make the entire filtering process less tedious. The tuning method involves the use
of a biasing parameter that is computed within the EKF. The biasing parameter is an
integral part of ridge-type estimation techniques, which have shown improved accuracy
in batch and sequential solutions of ill-conditioned orbit determination problems. The
biasing parameter provides a measure of the overall solution error and is input into the
process noise matrix for tuning the EKF. The performance of the filter depends on
6
correct propagation of the covariance matrix added with the a priori covariance, the
observational covariance, and the process noise [16]. A ?tuned? extended Kalman filter
means that the best possible filter performance is achieved.
The elements of the process noise covariance and measurement noise covariance
matrices need to be properly determined through the tuning process in order to improve
the filter?s ability to provide accurate state estimates. The results from the adaptively-
tuned EKF are presented in this study and are compared to the results obtained using the
batch filter from the 2
nd
Stage of the TSS methodology presented in [13] along with
those results from the manually-tuned EKF presented in [15]. Conclusions for their use
are also provided.
7
2. QUICK-LOOK ORBIT DETERMINATION METHOD DESCRIPTION
The suggested orbit determination method for TSS involves two different
dynamic models. The first stage of the methodology is a POD strategy that uses a simple
dynamic model, while an enhanced dynamic model is used in the EKF of the second
stage quick-look identification process. The related TSS models are specifically designed
for those stages in order to yield the most accurate results for a given span of
observations. The 1
st
Stage uses only a few observations in its process where the results
are used as input into the 2
nd
Stage. The 2
nd
Stage performs an adaptive, sequential
analysis using up to 15 minutes of the observational data. Each stage, along with their
models, is described in more detail in the following paragraphs.
2.1 Preliminary Orbit Determination (POD, 1
st
Stage)
The TSS model used in the POD stage consists of a ?daughter? satellite, m, and a
?parent? satellite, m
p
, where both are considered to be point masses. These satellites are
connected by a massless tether, as illustrated in Fig. 2. Also shown in Fig. 2, is the
effective tether force that acts upon the daughter satellite. The radial and tangential force
components, F
r
and F
t
, respectively, which make up the tether force, are shown acting on
the daughter satellite, since its motion is the one being observed. Depending on where
the satellites are located in their orbit and in relation to each other, their velocities will be
affected due to the change in acceleration imposed by the force components. For
8
example, the radial force component will create a radial acceleration, a
r
, which acts on
the daughter satellite, and will cause its velocity to decrease if it lies in a lower orbit than
the parent satellite, but will cause its velocity to increase if it lies in a higher orbit than
the parent satellite. Likewise, the tangential force component will create a tangential
acceleration, a
t
, on the daughter, causing its velocity to decrease if it precedes the parent,
but causing its velocity to increase if it trails the parent. In the case where the parent
satellite is being observed, the opposite of these dynamical characteristics will be true.
FIG. 2. TSS Model with Force Components and Libration Angle
One satellite preceding or ?leading? another satellite depends on the in-plane
libration angle, ?, as shown in Fig. 2. The value of ? will also determine the directions
of the tether force components, where the positive radial direction is defined from the
center of Earth toward the daughter satellite and the positive tangential direction is that of
the direction of orbital motion.
The radial acceleration of a satellite?s motion can be obtained using a POD
method along with other information, including the gravitational parameter, ?.
Earth
?
F
r
F
t
orbit
direction
m
m
p
9
Specifically, a modified gravitational parameter, ?*, can be calculated during the POD
process by a relationship between the parameters ? and a
r
, which is presented in [3,4] as:
2
* ra
r
?= ??
(2.1)
Where, r, is the distance from the center of the Earth to the daughter satellite. Upon
obtaining ?*, it can be used to find an approximate value of the distance from the
daughter satellite to the center-of-mass of the TSS, which is designated by, ?
cm
, and
illustrated in Fig. 3..
FIG. 3. TSS Center-of-Mass and Tether Length Measurements
This distance is measured along the tether length and can illustrate whether the daughter
satellite is above or below the parent satellite. When libration is present in the TSS, ?
cm
will represent the projection of the tether length (to the center-of-mass) in the radial
direction and will be denoted by the parameter, ?
cm
*, and is presented in [3,4] as:
()
r
cm ?
?
?
?
?
?
+
?
=
*2
*
*
??
??
? (2.2)
Earth
CM
m
p
m
?
cm
?
cm
*
?
10
If ?
cm
* is positive, then the daughter satellite is below the parent and ?* < ?. If it is
negative, then the daughter is above the parent satellite where ?* > ?, which indicates
that the satellite being observed is above the parent satellite. In addition, the value of ?
cm
will approach the actual tether length as the ratio of the point masses, m/m
p
, approaches
zero.
Several classical POD methods were modified to include the capabilities to
determine ?* for a TSS and were presented in [3,4]. Due to superior convergence
characteristics, the 9
th
order f and g series method proved to perform best and those
results were used in this study. The calculated value of ?* can be used quickly to
determine whether the observed satellite is part of a TSS due to the few observations that
are used in the POD process. The output from the 1
st
Stage POD method, in the form of
position, velocity, and a
r
, is used as input into the 2
nd
Stage Extended Kalman Filter. The
POD results provide no information regarding a
t
, therefore a
t
is initially assumed to be
zero.
2.2 Identification Using an Extended Kalman Filter (2
nd
Stage)
The enhanced dynamical TSS model used in the ?quick-look? identification
process, presented in [6], is similar to the model presented in the 1
st
Stage, but includes
additional dynamical effects. The model for the 2
nd
Stage considers the tether to be
inextensible and allows for oblate Earth effects, as well as, in-plane libration in the TSS
dynamics.
To determine the dynamical TSS characteristics, including: acceleration
components, libration angle, etc., a batch-type filter has been used [5] to generate an
11
estimation of the daughter satellite?s state vector. This state vector includes the satellite?s
position and velocity components and the acceleration components due to the tether
force. The acceleration components, a
r
and a
t
, are both assumed to be constant over short
observation arcs. This implies that the libration angle will remain constant through the
observation arc as well. Since these parameters are tether-specific, including them in the
filter allows for the satellite to be identified as tethered or untethered. A satellite is found
to be untethered when its acceleration components are determined to be zero. This means
that there is no tether force perturbing the acceleration components. Likewise, nonzero
acceleration terms indicate that the satellite being observed is tethered. No other forces,
such as: thrust, drag, which creates orbit decay, etc. are considered in this study. Once
the acceleration terms are obtained, they can be used to calculate the libration angle of the
TSS.
In order to perform the 2
nd
Stage process in a sequential manner, an extended
Kalman filter (EKF) is used to quickly identify the observed satellite with a short arc of
observational data. Classic EKF equations are used in this study and are explained in
more detail below. To better describe the EKF, it is appropriate to summarize the process
of the Kalman (or sequential) filter first and then provide a comparison. Swerling
originally developed the sequential algorithm in 1958 [17], yet Kalman and Bucy have
been recognized more for their work with the algorithm since 1961 [18]. The most
important difference between the two filters is that the sequential algorithm processes
observational data as it is received, while the EKF does the same and in addition, updates
the reference trajectory after each observation is processed. The disadvantage of the
sequential algorithm is the significant amount of errors due to neglecting higher order
12
terms in the linearization procedure. The EKF is used to decrease the effects of those
errors, which allows for more rapid convergence. When using the sequential algorithm,
if the true trajectory and the reference trajectory are too far apart, which is often the case
at the beginning of a simulation, the estimation process may diverge due to the errors
from the linearization process, previously described. The benefit in using the EKF is that
the best estimate of the state will be reached more quickly. Below is a description of the
EKF algorithm:
1. Given the following:
- 1?
?
kX , the estimate of the state vector at tk-1, (n x 1);
- P
k-1
, covariance matrix at time t
k-1
, (n x n);
- Y
k
, p-vector of observations taken at time t
k
, (p x 1);
- R
k
, observation covariance at time t
k
, (p x p);
2. Integrate from t
k-1
to t
k
,
()ttXFX ),(=
&
, () 1
1
?
?
?
= k
k
XtX (2.3)
()
()
)(
),(
tX
ttXF
tA
?
?
= (2.4)
)()()(
11
tQtAPPtAP
T
kkk
++=
??
&
(2.5)
where )(tX is the updated state vector (n x 1), F( )(tX , t) is the system dynamics
function (n x 1), A(t) is the state sensitivity matrix (n x n), and where Q(t) is the
process noise matrix (n x n).
3. Compute,
( )
)(
),(
k
kk
k
tX
ttXG
H
?
?
= (2.6)
()
1?
+=
k
T
k
k
k
T
k
k
k
RHPHHPK (2.7)
13
()k
kkk
PHKIP ?= (2.8)
( )
kkkk
ttXGYy ),(?= (2.9)
kkk
yKx =
?
(2.10)
kkk
xXX
??
+= (2.1)
where H
k
is the measurement sensitivity matrix (p x n), ( )
kk
ttXG ),( is the
observation-state relationship vector (p x 1), K
k
is the Kalman gain (n x p), P
k
is
the covariance matrix (n x n) associated with the best estimate of the (n x 1) state
vector kX
?
, y
k
is the observation residual vector (p x 1), Y
k
is the current
observation, and
k
x
?
is the state correction vector (n x 1). All variables are
computed at time t
k
.
4. Replace k with k-1, return to step 2 and substitute. Repeat until all observations
have been read.
The algorithm above assumes that the process noise matrix, Q(t), is known. This will be
discussed in more detail in the following section.
The state vector, kX
?
,
used in the 2
nd
Stage process will include the observed
satellite?s position, velocity, and tether acceleration components. Upon obtaining the
initial estimate of the state vector at each observation time, several determinations can be
made relative to the tether acceleration components [13], such as:
1. If the values of a
r
and a
t
are found to be zero, it can be assumed that the observed
satellite is an untethered one and standard techniques can be used to analyze its
motion.
2. If the values of a
r
and a
t
are nonzero, then the following applies:
a. The libration angle, ?, can be calculated from,
?
?
?
?
?
?
?
?
=
?
r
t
a
a
1
tan? (2.12)
14
where the signs of a
r
and a
t
will determine the appropriate quadrant for ?.
b. The magnitude of the acceleration due to the tether force can be calculated
from,
()
22
tr
T
aa
m
F
+= (2.13)
c. The value of ?
cm
*
, the radial projection of the tether length to the center-
of-mass of the TSS, can be approximated through the use of a
r
and
Equations 2.1 and 2.2.
3. In the case where one of the acceleration components is equal to zero and the
other is not, then the following applies:
a. For a
t
= 0 and a
r
? 0, the satellite will be tethered and the system?s
orientation will be vertical. This will be the case when ? = 0? and a
r
> 0
or when ? = 180? and a
r
< 0.
b. For a
r
= 0 and a
t
? 0, the satellite will be tethered and the system?s
orientation will be horizontal. This will be the case when ? = 90? and a
t
>
0 or when ? = 270? and a
t
< 0.
The methodology from the 2
nd
Stage process combined with the implementation of the
tuning method described in the following section will complete the quick look
identification method.
2.3 Adaptive Tuning Method using a Biasing Parameter
Before describing the adaptive tuning algorithm it is appropriate to stress the
importance of ?tuning? an EKF in general. An EKF must be properly tuned in order to
prevent filter divergence. This is achieved by adding adequate values of the a priori
covariance and the observational covariance, if known, to the propagation of the
covariance to properly generate the filter?s performance envelope; any ?process noise? is
also added. A major disadvantage to tuning an EKF is that the results are usually
15
achieved manually, which is very tedious and allows for significant error. These
problems present the need for an aid which ?automatically? tunes an EKF. In this
presentation, an adaptive tuning method presented by Cicci [16], typically used in angles-
only tracking and intercept problems, was applied to the EKF described in the previous
section; yielding an adaptively-tuned EKF.
The adaptive tuning method used in this study is based upon ridge-type estimation
methods and requires the calculation of a biasing parameter, which represents the overall
error in the solution and is used as input into the process noise matrix. The process noise
matrix, Q(t), defined above in the 2
nd
Stage description, is necessary to successfully tune
the EKF and can be expressed as a function of the biasing parameter and written as Q(k),
as described by Cicci [16]. The biasing parameter provides the following advantages to
the EKF:
1. The diagonal terms of the process noise matrix are updated after each
observation time as opposed to remaining constant for all of the
observations.
2. Adaptive tuning allows the user to define how the biasing parameter is
implemented at the beginning of the simulation, eliminating manual
changes to the process noise.
The form of the biasing parameter, k, is determined through converting the batch solution
into a sequential solution, described in [16]. The sequential form of the biasing
parameter, k, is presented below:
() ()()[]
()[]xHIRDDRDDDHx
xHIRDDRDDDHxRDDtr
k
mmmmmm
TT
mmmmmm
TT
mm
??
?2?
22
222
+
++
= (2.14)
where D
m
is a normalizing diagonal matrix (p x p) and its ith diagonal term is defined as:
16
T
m
HPH
D
1
= (2.15)
As stated in [16], the inspection of equation 2.14 above shows that k will always be
greater than one. As the covariance matrix, P , decreases with each processed
observation, the biasing parameter, k, also decreases in value and approaches one. As
described before, k represents the overall error in the solution and Q(t) represents the
process noise, therefore k is used to compute the Q(k) matrix, which includes the effects
of k-biasing. With each observation k is computed and then used to update the
appropriate Q(k) terms. Only those Q(k) terms that will effect the acceleration terms of
the propagated covariance matrix [16] will be updated with the biasing parameter, since
that is where most of the error exists.
In summary, the description of the biasing parameter and its use within the EKF
concludes the quick-look orbit determination process. Implementation of the process,
test cases and results are described in the following sections.
17
3. PROCEDURE DESCRIPTION
At this point it is necessary to expand on the methods previously explained by
summarizing the overall procedure that was implemented in this thesis study. Beginning
with the POD (1
st
Stage) process, only the output data obtained from the POD
presentations referenced in [3,4] was used as input into the adaptive EKF (2
nd
Stage); no
additional POD processes were considered. The output data from the POD process
provides a reference trajectory for the tracked satellite with results in the form of the
position vector, R , velocity vector, V , and ?*, which includes the standard gravitational
parameter, ?, and the radial acceleration tether force component acting on the satellite.
Using equation 2.1, this radial acceleration component, a
r
, can be calculated since the
remaining variables are known. With the zero tangential acceleration component, a
t
,
these parameters form the starting state vector for the daughter satellite which is used as
input into the EKF (2
nd
Stage).
To implement the adaptive EKF in this study, the FORTRAN program developed
for [15] was modified to include the adaptive parameters. The entire estimation process
used in this study will be described in the paragraphs below, however the form that the
state vector assumes for this study, must be explained first. The state vector includes the
parameters from the POD process, satellite position, velocity, and acting tether
18
acceleration force. The state variables related to the dynamics of the satellite?s motion
are defined in equation 2.16, below:
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
==
8
7
6
5
4
3
2
1
)(
X
X
X
X
X
X
X
X
a
a
z
y
x
z
y
x
tXX
r
t
&
&
&
(2.16)
The equation of motion, including perturbation effects acting on the satellite, used in this
study and those being compared to this study, is provided in equation 2.17. Definitions
of the perturbation forces, p , which represents the effects of oblateness, and,
t
F , which
represents the tether acceleration effects on the satellite, are provided in equations 2.18
and 2.19, respectively.
t
Fp
R
R
R ++
?
=
3
?
&&
(2.17)
?
?
?
?
?
?
?
?
?=
2
2
3
2
2
3
1
2 R
z
R
RJ
p
e
?
(2.18)
V
Va
R
Ra
F
tr
t
+= (2.19)
In equation 2.18,
2
J is the oblateness constant coefficient,
e
R is the radius of the earth,
and z is the z-component of the satellite?s position. In equation 2.19,
r
a and
t
a
represents the satellite?s radial and tangential tether force acceleration terms, respectively.
R and V are the magnitudes of the satellite?s position and velocity, respectively.
19
Take the time derivative of the state vector dynamics to find X
&
and substitute
equations 2.17-2.19; yielding equation 2.20, where X
&
is a function of the state variables.
()
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
++
?
?
?
?
?
?
?
?
+???
?
?
?
?
?
++
?
?
?
?
?
?
?
?
+???
?
?
?
?
?
++
?
?
?
?
?
?
?
?
+???
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
==
0
0
2
15
2
9
1
2
15
2
3
1
2
15
2
3
1
),(
6738
4
2
3
2
2
2
2
2
3
3
5728
4
2
3
2
2
2
2
2
3
2
4718
4
2
3
2
2
2
2
2
3
1
6
5
4
8
7
6
5
4
3
2
1
V
XX
R
XX
R
XRJ
R
RJ
R
X
V
XX
R
XX
R
XRJ
R
RJ
R
X
V
XX
R
XX
R
XRJ
R
RJ
R
X
X
X
X
X
X
X
X
X
X
X
X
ttXFX
EE
EE
EE
?
?
?
&
&
&
&
&
&
&
&
&
(2.20)
The observation vector, y, as defined in equation 2.9 requires the use of an observation-
state relationship vector, ()()ttXG , , defined here in equation 2.21.
()( )()
(
() ()
()
() )()
?
?
?
?
?
?
?
?
?+?+?+?+
?+?+?+?+
=
?
?
?
?
?
?
?
?
???+?+?+
?+??+?
=
?+?+?+?+=
2
3
2
21
2
21
21321
3
32121
2121
2
2
3
2
21
2
211
cossinsincos
sincoscossinsincoscossincos
sin
cossinsincossincossinsincos
sinsincoscoscossin
tan
cossinsincos
SSS
SSS
SSS
SS
SSS
ZXYXXXXX
YXXZXXXX
aG
ZXYXXXXX
XXXYXX
aG
ZXYXXXXXG
????
?????????
?????????
??????
????
(2.21)
In equation 2.9, Y, represents the current observation?s range, azimuth, and
elevation values of the satellite as measured relative to the tracking station. In equation
2.21, ? is equal to the ?
E
at the current observation time. Xs, Ys, and Zs are the known
position parameters (in Earth-Centered Earth Fixed, ECEF) for the tracking station
location. The latitude and longitude of the tracking station, also known, are ? and ?,
respectively. The observation vector equations include the transformations between the
tracking station?s local coordinate frame and the ECEF coordinate frame.
20
The equations provided for X
&
and the observation vector, y, were also used in
presentations [3,4,15]. They are used in this study to provide consistency for comparing
results from test cases.
After explaining the forms of the state and observation vectors, the simulation
description proceeds. Initially, the diagonal values for the a priori covariance and
process noise matrices are given to begin the quick-look EKF analysis. The values for
the a priori covariance matrix used in this thesis will be explained in more detail in the
Test Cases section.
1. First, the state estimate found from the POD results are read into the
program. Processing one observation at a time, in 5 second intervals and
up to 15 minutes of data, the range, azimuth, and elevation defining the
satellite?s location relative to the tracking station is recorded. Also
processed is the observation covariance.
2. Compute EKF equations 2.5 through 2.10 and calculate the predicted state
vector and covariance matrix (with process noise and the k-biasing
function).
3. Update the state estimate and a priori covariance matrix.
4. Calculate the Root Mean Square (RMS).
5. Calculate ?
cm
* using equation 2.2.
6. Begin processing the next observation with the updated state estimate and
process noise matrix and continue until all observations are processed.
Once the k-biasing parameter is calculated using equation 2.14 it is used in a
biasing function to update the appropriate terms of the process noise matrix for use in the
next observation. The Root Mean Square (RMS) is also calculated at each observation to
measure any process error, which aids in identifying the convergence of the best solution.
The projection of the tether length (to the center-of-mass) in the radial direction, ?
cm
*, is
21
then calculated. The next observation is processed using the updated process noise
matrix (with biasing included) and the best estimate of the state vector. This procedure
continues until all observations are processed.
In this study, those terms in the process noise matrix that effect the tether
acceleration components will be the only terms considered. These terms are functions of
the k-biasing parameter used to automatically tune, or adjust, the propagation of the state
covariance matrix.
As mentioned before, the process noise matrix, Q(k), is updated with functions of
the k-biasing parameter. Only the acceleration terms of the matrix will be updated as
shown in the following form.
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
)(0000000
0)(000000
00000000
00000000
00000000
00000000
00000000
00000000
)(
2
1
kf
kf
kQ (2.2)
The functions used for f
1
(k) and f
2
(k) are dependent upon the types of scenarios that are
being analyzed. There are no limitations to the types of function that can be used, but
some effort may need to be put forth at the beginning of the analysis to make sure the
appropriate functions are chosen to use. For instance, in angles-only tracking intercept
problems, the functions should be chosen so that larger values of the process noise are
provided to the components that contain high levels of error [16]. Other cases may not
include levels of error that are significantly high; requiring a biasing function to yield
smaller values.
22
Using the biasing function in the process noise matrix, which is updated at each
observation, should provide faster tuning results to the EKF. This allows the EKF to
perform more powerfully and with high confidence of the best converging solution.
23
4. TEST CASES
Many different scenarios were considered in this thesis study to illustrate the
adaptive EKF?s performance when identifying whether the observed satellite is tethered
or untethered. In the end, over 2,000 scenarios were generated to test the EKF. This
involved scenarios, limited to the data provided from simulated data acquired from a
previous study [3], with varying tether lengths, orientations, observation noise levels, and
observation arcs; specific values and levels are summarized in Table 1 below.
TABLE 1. Parameter Variations for Scenarios
Tether Lengths (?) 0 km, 1 km, 10 km, 50 km, and 1 km UP
Orientation: In-Plane Libration Angle (?) 0?, 5?, 10? (N/A for 0 km cases)
Observation Noise Levels
LOW (5 m/0.002?), MEDIUM (25
m/0.01?), and HIGH (50 m/0.02?)
Observation Arcs 5, 10, and 15 minutes
The simulated data was generated using a baseline circular orbit [3]. The orbital
elements for this circular orbit are provided in Table 2 below.
TABLE 2. Baseline Orbit for Data Generation
Orbital Elements
a 6621 km
e 0.00
i 5.73 deg
? 5.73 deg
24
Each test case scenario used as input the a priori covariance matrix found with the
best manually-tuned results from the EKF presented in [15]. The idea behind this is to
provide commonality among both filters for comparison purposes and to ultimately
determine which filter yielded faster and more accurate results.
As mentioned when describing the procedure for implementing all presented
methodology in this study, many combinations for the k-biasing function, used in the
process noise matrix, Q(k), were also considered. Some of the functions used were taken
from [16], in order to provide a common foundation for comparing results to the adaptive
EKF. A description of the combinations used in the k-biasing functions is presented in
Table 3 below.
TABLE 3. Varying Combinations of the Biasing Function
Functions used for both, f
1
(k) & f
2
(k)
k and k
2
, from [16]
n
k
1
, where n = 1, 2, 3, 4, ?21
(k-1) and (k-1)
2
, from [16]
n
k )1(
1
?
, where n = 1, 2, 3, 4, ?9
(k+1) and (k+1)
2
n
k )1(
1
+
, where n = 1, 2, 3, 4, ?21
The results from the two best biasing functions from [16] and two from the
remaining combinations from Table 3 are presented for all test case scenarios in
Appendix A. The same function is not ideal for all test cases due to observation, process,
and user errors. Out of the four cases presented for each test case scenario, the case
yielding the best solution represents optimum results found from the adaptive EKF for
that test case.
25
5. RESULTS
From the results presented for each test case in Appendix A, a variety of different
combinations of biasing functions were used to acquire the end results for this study. The
best solutions from these test cases were taken and compared to the best solutions from
the batch filter presented in [13] and the manually-tuned EKF presented in [16]. Tables 4
through 16 organize this information from all three studies so that appropriate
conclusions may be drawn.
Overall the adaptive EKF produced the most accurate results within 15 minutes of
observed data for all tether lengths and k-biasing functions presented. The tables of data
provided observe the adaptive EKF?s performance at three different time spans (5, 10,
and 15 minutes); this provides the opportunity to monitor whether or not a test case
converges sooner than the anticipated 15 minute time span. Where this occurred, which
was in very few cases, it was determined that the early convergence was due to the proper
selection of the biasing function. Additionally, the rapid divergence that was found in the
following time span, for these cases, demonstrated that the best solution had already
occurred for the test case.
The adaptive EKF was able to produce the best results for the no tether cases
within 10 minutes of data and to match those of the manually-tuned EKF. The cause for
26
this result illustrates the ability to choose the biasing function to achieve the desired
results. In these cases, the process noise matrix values were zero for the manually-tuned
EKF, therefore the biasing function used in the adaptive EKF needed to provide small
values for the process noise so that the solutions would be similar when compared. This
means that the value of k needed to be larger when using a function similar in form to
()
n
k
?
+1 , where n is larger; the output from this biasing function is driven to the expected
limit, or floor, more quickly.
The adaptively-tuned EKF was more sensitive to observation error and libration
levels than those results found from the manually-tuned EKF. Results from the adaptive-
tuned EKF for test cases with high noise levels performed better than the low and
medium noise level cases for some tether lengths. In the cases where low levels of noise
were present the results varied depending on the amount of libration present. Higher
amounts of libration provided better results (i.e. RMS closest to 1 and ?
cm
*
closest to
desired value) for these cases.
Results from the adaptively-tuned EKF became more accurate with higher tether
lengths, which did not seem to have an affect in either the batch filter or manually-tuned
EKF processes. In all three filter cases, there was commonality among the no tether
cases diverging as observation noise increased.
The amount of error the input conditions and process noise are expected to
contain, aids in selecting the ?best? biasing function to use. Varying the k-biasing
function provided a unique look into the overall convergence to the best solution for each
case. Because many biasing functions were considered for each scenario in this study,
selecting the biasing function that provided the best solution to each test case was simple.
27
As verification that the best solution had been achieved, the very next variation of the
biasing function used in the scenario, and any thereafter, provided an immediate and
significant divergence. This pattern provided confidence throughout the entire
adaptively-tuned EKF analysis that the best solution for each test case had been
determined.
The RMS values calculated at each observation and recorded for all three time
spans provided good results in most of the cases. Where the RMS values were
significantly higher than the ideal value of one, the conclusion is that the adaptively-
tuned EKF procedure required more observations, than either the batch or manually-
tuned EKF processes needed, to find the best converged solution.
TABLE 4. Comparisons for No Tether cases, ?
cm
= 0 m
Batch Filter [13]
Manually-tuned
EKF [16]
Adaptively-tuned EKF
with k-biasing
? ?t RMS
?
cm
*
(m)
RMS
?
cm
*
(m)
RMS
?
cm
*
(m)
low 5 min: 1.021 -116 0.243 -2416 0.2431 -2416
10 min: 1.025 310 0.443 -39 0.4433 -39
15 min: 1.006 152 0.686 92 0.6862 92
med 5 min: 1.042 4246 0.233 -6559 0.2329 -6559
10 min: 1.021 624 0.363 320 0.3631 320
15 min: 1.011 708 0.709 685 0.7086 685
high 5 min: 1.033 4613 0.188 -9622 0.1879 -9622
10 min: 1.021 1400 0.362 551 0.3622 551
15 min: 1.009 1493 0.706 1297 0.7059 1297
28
TABLE 5. Comparisons for 1 km cases, ?
cm
= 909 m, LOW noise
Batch Filter [13]
Manually-tuned
EKF [16]
Adaptively-tuned EKF
with k-biasing
? ?t RMS
?
cm
*
(m)
RMS
?
cm
*
(m)
RMS
?
cm
*
(m)
0? 5 min: 1.022 277 13.211 -233381 16.03 30885
10 min: 1.022 991 3.218 9275 7.760 30051
15 min: 1.007 1077 0.570 1227 4.507 1082
5? 5 min: 1.022 230 21.458 -343299 19.65 29076
10 min: 1.022 943 4.320 10599 7.835 16121
15 min: 1.008 1028 0.599 810 4.837 -492
10? 5 min: 1.021 260 0.620 9537 1.884 -13500
10 min: 1.021 956 0.487 2668 0.842 697
15 min: 1.010 979 0.669 1006 0.785 1024
TABLE 6. Comparisons for 1 km cases, ?
cm
= 909 m, MEDIUM noise
Batch Filter [13]
Manually-tuned
EKF [16]
Adaptively-tuned EKF
with k-biasing
? ?t RMS
?
cm
*
(m)
RMS
?
cm
*
(m)
RMS
?
cm
*
(m)
0? 5 min: 1.020 -1318 6.831 693696 8.238 613209
10 min: 1.021 1001 4.017 -39847 11.36 104116
15 min: 1.008 1521 0.952 1423 3.638 566
5? 5 min: 1.020 -1364 13.007 910863 6.816 203867
10 min: 1.021 950 3.101 58191 3.533 56500
15 min: 1.008 1472 0.800 970 2.489 6855
10? 5 min: 1.021 3381 13.075 918297 6.842 204010
10 min: 1.021 1236 3.115 58121 3.541 56510
15 min: 1.010 1416 0.799 840 2.491 6734
TABLE 7. Comparisons for 1 km cases, ?
cm
= 909 m, HIGH noise
Batch Filter [13]
Manually-tuned
EKF [16]
Adaptively-tuned EKF
with k-biasing
? ?t RMS
?
cm
*
(m)
RMS
?
cm
*
(m)
RMS
?
cm
*
(m)
0? 5 min: 1.015 -892 0.473 20009 2.342 -8226
10 min: 1.023 840 0.677 7081 1.273 2355
15 min: 1.006 1003 0.594 1502 0.9346 963
5? 5 min: 1.020 -1432 8.636 1743440 4.763 -157130
10 min: 1.021 1096 2.381 90619 4.373 119816
15 min: 1.009 1950 0.680 1533 3.238 930
10? 5 min: 1.021 -8401 8.756 1784948 4.787 -161501
10 min: 1.022 1509 2.386 91278 4.380 119851
15 min: 1.011 2196 0.679 1458 3.238 744
29
TABLE 8. Comparisons for 10 km cases, ?
cm
= 9091 m, LOW noise
Batch Filter [13]
Manually-tuned
EKF [16]
Adaptively-tuned EKF
with k-biasing
? ?t RMS
?
cm
*
(m)
RMS
?
cm
*
(m)
RMS
?
cm
*
(m)
0? 5 min: 1.022 8423 17.910 -591062 6.060 -217109
10 min: 1.025 9192 1.382 22633 0.4890 8391
15 min: 1.010 9339 0.637 10154 0.6134 9632
5? 5 min: 1.021 7948 11.251 26732 11.77 53852
10 min: 1.025 8714 5.229 14630 5.429 13736
15 min: 1.065 8845 1.755 9770 2.720 12807
10? 5 min: 1.021 7444 35.889 -1443476 12.54 551718
10 min: 1.195 8406 0.370 27736 1.721 13200
15 min: 1.028 8267 0.604 9959 1.747 8343
TABLE 9. Comparisons for 10 km cases, ?
cm
= 9091 m, MEDIUM noise
Batch Filter [13]
Manually-tuned
EKF [16]
Adaptively-tuned EKF
with k-biasing
? ?t RMS
?
cm
*
(m)
RMS
?
cm
*
(m)
RMS
?
cm
*
(m)
0? 5 min: 1.020 6801 0.544 -2396 0.6496 9091
10 min: 1.022 9179 0.472 19235 0.5880 15966
15 min: 1.008 9787 0.657 12817 1.050 22614
5? 5 min: 1.020 6341 4.986 -360337 14.66 1506834
10 min: 1.021 8708 2.010 21618 7.127 68631
15 min: 1.010 9294 0.780 10351 2.398 6134
10? 5 min: 1.020 5842 4.986 -360842 14.24 1383031
10 min: 1.022 8274 2.010 20967 9.143 115028
15 min: 1.016 8855 0.779 9671 2.926 4684
TABLE 10. Comparisons for 10 km cases, ?
cm
= 9091 m, HIGH noise
Batch Filter [13]
Manually-tuned
EKF [16]
Adaptively-tuned EKF
with k-biasing
? ?t RMS
?
cm
*
(m)
RMS
?
cm
*
(m)
RMS
?
cm
*
(m)
0? 5 min: 1.015 7211 0.380 25179 0.2834 10359
10 min: 1.023 8992 0.478 14181 0.3606 10416
15 min: 1.006 9225 0.529 10105 0.4563 9186
5? 5 min: 1.020 6301 18.581 1603925 6.635 -1426637
10 min: 1.021 8888 0.997 148312 0.8132 71679
15 min: 1.010 9791 0.758 10630 0.6790 10518
10? 5 min: 1.020 5811 18.580 1603099 6.634 -1427007
10 min: 1.021 8450 0.996 147601 0.8133 71021
15 min: 1.011 9355 0.758 10014 0.6795 9873
30
TABLE 11. Comparisons for 50 km cases, ?
cm
= 45455 m, LOW noise
Batch Filter [13]
Manually-tuned
EKF [16]
Adaptively-tuned EKF
with k-biasing
? ?t RMS
?
cm
*
(m)
RMS
?
cm
*
(m)
RMS
?
cm
*
(m)
0? 5 min: 1.022 44523 7.125 -255455 3.986 -64684
10 min: 1.027 45385 1.066 51432 0.4422 45428
15 min: 1.023 45659 0.890 46422 0.6646 44962
5? 5 min: 1.017 42148 76.441 1804829 2.504 59790
10 min: 1.035 42994 12.660 -20485 1.207 43754
15 min: 1.844 43146 4.399 42729 1.220 45149
10? 5 min: 1.016 39625 270.15 -5901658 3.873 46189
10 min: 1.105 40795 14.464 33030 1.976 42401
15 min: 3.202 40910 0.503 46430 1.725 40072
TABLE 12. Comparisons for 50 km cases, ?
cm
= 45455 m, MEDIUM noise
Batch Filter [13]
Manually-tuned
EKF [16]
Adaptively-tuned EKF
with k-biasing
? ?t RMS
?
cm
*
(m)
RMS
?
cm
*
(m)
RMS
?
cm
*
(m)
0? 5 min: 1.021 42789 5.219 -302465 13.13 648210
10 min: 1.022 45482 0.619 43857 9.663 194554
15 min: 1.008 46154 0.653 49841 4.583 44944
5? 5 min: 1.019 40517 0.534 -138322 12.87 1202841
10 min: 1.021 43138 0.696 62693 8.772 144106
15 min: 1.054 43703 0.540 48204 3.384 43042
10? 5 min: 1.019 38056 0.345 -217184 1.432 -691193
10 min: 1.022 40931 0.693 58296 0.5688 45995
15 min: 1.178 41358 0.540 44817 0.7569 44486
TABLE 13. Comparisons for 50 km cases, ?
cm
= 45455 m, HIGH noise
Batch Filter [13]
Manually-tuned
EKF [16]
Adaptively-tuned EKF
with k-biasing
? ?t RMS
?
cm
*
(m)
RMS
?
cm
*
(m)
RMS
?
cm
*
(m)
0? 5 min: 1.020 42834 2.775 1758469 6.549 -1435114
10 min: 1.022 45584 0.732 142976 0.8103 107429
15 min: 1.009 46644 0.518 50734 0.6709 48181
5? 5 min: 1.019 40629 3.369 2836331 19.21 3636757
10 min: 1.021 43221 0.771 169977 0.9604 77643
15 min: 1.022 44137 0.519 48170 0.8773 45485
10? 5 min: 1.019 38232 5.070 5227532 153.7 7650025
10 min: 1.022 41013 0.938 203335 39.36 -1267781
15 min: 1.058 41908 0.528 46766 31.57 30505
31
TABLE 14. Comparisons for 1 km UP cases, ?
cm
= -909 m, LOW noise
Batch Filter [13]
Manually-tuned
EKF [16]
Adaptively-tuned EKF
with k-biasing
? ?t RMS
?
cm
*
(m)
RMS
?
cm
*
(m)
RMS
?
cm
*
(m)
0? 5 min: 1.022 -1532 65.567 1254150 15.31 18527
10 min: 1.022 -833 5.161 -50243 7.075 26939
15 min: 1.006 -761 0.903 -1122 4.110 118
5? 5 min: 1.022 -1484 21.456 -344736 19.65 27320
10 min: 1.022 -785 4.320 8959 7.835 14476
15 min: 1.006 -712 0.600 -804 4.835 -2144
10? 5 min: 1.022 -1532 21.456 -344685 19.65 27378
10 min: 1.022 -740 4.320 9024 7.835 14542
15 min: 1.007 -688 0.600 -736 4.836 -2069
TABLE 15. Comparisons for 1 km UP cases, ?
cm
= -909 m, MEDIUM noise
Batch Filter [13]
Manually-tuned
EKF [16]
Adaptively-tuned EKF
with k-biasing
? ?t RMS
?
cm
*
(m)
RMS
?
cm
*
(m)
RMS
?
cm
*
(m)
0? 5 min: 1.020 -3121 3.225 654906 0.8382 108676
10 min: 1.021 -809 4.278 -40633 8.219 138718
15 min: 1.008 -318 0.943 -1744 8.659 -20563
5? 5 min: 1.020 -3074 8.911 638134 10.39 763366
10 min: 1.021 -757 3.134 33660 9.381 94157
15 min: 1.008 -269 0.833 -624 3.203 -606
10? 5 min: 1.020 -3025 8.911 638192 10.39 763419
10 min: 1.021 -709 3.134 33726 9.381 94216
15 min: 1.008 225 0.833 -624 3.203 -540
TABLE 16. Comparisons for 1 km UP cases, ?
cm
= -909 m, HIGH noise
Batch Filter [13]
Manually-tuned
EKF [16]
Adaptively-tuned EKF
with k-biasing
? ?t RMS
?
cm
*
(m)
RMS
?
cm
*
(m)
RMS
?
cm
*
(m)
0? 5 min: 1.020 -3194 11.369 277325 2.556 -2158
10 min: 1.021 -683 2.465 124955 2.705 204906
15 min: 1.009 -155 0.764 -1779 3.004 112275
5? 5 min: 1.020 -3149 7.018 2823610 4.762 -158748
10 min: 1.021 -637 0.728 67245 4.373 118088
15 min: 1.009 203 0.612 -1209 3.237 -1239
10? 5 min: 1.020 -3100 7.073 1424384 5.194 198245
10 min: 1.021 -593 2.486 71591 4.528 144550
15 min: 1.008 246 0.679 -901 3.265 -836
32
6. CONCLUSIONS
In summary, this study generated interesting and unique results for the quick-look
identification methodology presented for identifying a satellite as being a member of a
TSS using an EKF. The 2
nd
Stage TSS identification method used for both the manually-
tuned EKF [15] and the adaptive EKF provided correct and quick identification of the
tracked satellite. Both EKFs were sufficient in producing accurate results within the 15
minute time span. Where at least 15 minutes of data is processed, identification of a
tethered satellite as part of a TSS can almost certainly be made.
One major disadvantage to using either EKF is that the results are very sensitive
to the filter tuning parameters. Specifically, the choice of the a priori covariance and
process noise parameters greatly affect the accuracy of the outcome of the filter. For the
manually-tuned EKF cases, both parameters were manually-tuned which contributed to
the difficulty of achieving acceptable filter performance. The adaptive EKF?s filter
tuning was slightly more automated with the addition of the biasing function and took
advantage of using the a priori covariance found in the best manually-tuned cases. This
required that only the process noise matrix be determined for the adaptively-tuned cases.
Finding the appropriate biasing function to use for tuning adaptive EKF cases
depends on the level of error expected during process noise; and in this study, how
accurately tuned the a priori covariance was from the manually-tuned cases. Selecting
the appropriate biasing function form and variation of the form can be tedious, but
33
through experience with filter tuning the appropriate function can more easily be
determined. Once the proper biasing function is selected, it is more advantageous when
compared to the manually-tuned EKF because the process noise is updated each time an
observation is processed in the adaptive EKF, rather than remaining constant throughout
the analysis when using a manually-tuned filter.
Comparing the results of both EKFs to those of the batch filter, shows that a batch
filter can be used to identify a tethered satellite more quickly than the EKFs can.
Although the EKFs provided more accurate results than those found in the batch filter,
the EKFs required more observations to do so.
Several recommendations can be made for future use of EKFs in identifying a
satellite as a member of a TSS. In addition to using the most accurate data available to
achieve optimum results, it is highly advised that at least 15 minutes of observation data
be used in the 2
nd
Stage identification process. This observation data should be recorded
as often as possible, since observations are processed by the adaptive EKF at each time
step.
With the application of the adaptive EKF in this study, satisfying results were
achieved; however additional analysis can be performed to streamline the entire tuning
process. An iterative tuning process can be established when an EKF is required or
desired for use in any study. This can be achieved by establishing an iteration procedure
for any parameters needing to be tuned, such as the a priori covariance, the biasing
function form and variation or any other parameter requiring tuning. Critical output
values can be monitored within defined tolerances so that the desired filter performance
can be achieved.
34
An iterative tuning procedure would benefit in the use of EKFs since it would
alleviate the need for a trial and error process, which has traditionally been the means for
tuning an EKF. However, extensive analysis must be performed early in the study when
defining the iteration details. More effort may be required in the setup of an iterative
process of this nature, but the caliber of the results achieved along with the ease of the
tuning process would produce a useful and effective software solution where the
application of EKFs are needed.
35
REFERENCES
1. Asher, T. A., D. G. Boden, and R. J. Tegtmeyer, ?Tethered satellites: The orbit
determination problem and Missile Early Warning Systems,? AIAA Paper 88-4284,
AIAA/AAS Astrodynamics Conference, Minneapolis, MN, August 15-17, 1988.
2. Hoots, F. R., Roehrich, R. L., and Szebehely, V. G., ?Space Shuttle Tethered Satellite
Analysis,? Directorate of Astrodynamics, Peterson AFB, CO, August 1983.
3. Qualls, C., and Cicci, D. A., ?Preliminary Orbit Determination of a Tethered
Satellite,? Paper AAS 00-191, presented at the AAS/AIAA Astrodynamics Specialist
Conference, Clearwater, FL, January 23-26, 2000.
4. Qualls, C., ?Preliminary Orbit Determination of a Tethered Satellite,? AU MS Thesis,
2000.
5. Kessler, S. A., and Cicci, D. A., ?Filtering Methods for the Orbit Determination of a
Tethered Satellite,? The Journal of the Astronautical Sciences, Vol. 45, No. 3, July-
September 1997, pp. 263-278.
6. Cicci, D. A., Lovell, T. A., and Qualls, C., ?A Filtering Method for the Identification
of a Tethered Satellite,? The Journal of the Astronautical Sciences, Vol. 49, No. 2.,
April-June 2001, pp. 309-326.
7. Cicci, D. A., Qualls, C., and Lovell, T. A., ?A Look at Tethered Satellite
Identification Using Ridge-Type Estimation Methods,? Paper AAS 99-415, presented
at the AAS/AIAA Astrodynamics Specialist Conference, Girdwood, AK, August 16-
19, 1999.
8. Cochran, J. E., Jr., Cho, S., Cheng, Y-M, and Cicci, D. A., ?Dynamics and Orbit
Determination of Tethered Satellite Systems,? The Journal of the Astronautical
Sciences, Vol. 46, No. 2, April-June 1998, pp. 177-194.
9. Cho, S., Cochran, J. E., Jr., and Cicci, D. A., ?Modeling Tethered Satellite Systems
for Detection and Orbit Determination,? The Journal of the Astronautical Sciences,
Vol. 48, No. 1, January-March 2000, pp. 89-108
36
10. Cochran, J. E., Jr., Cho, S., Lovell, T. A., and Cicci, D. A., ?Evaluation of the
Information Contained in the Motion of One Satellite of a Two-Satellite Tethered
System,? The Journal of the Astronautical Sciences, Vol. 48, No.4, October-
December 2000, pp. 477-493.
11. Cho, S., Cochran, J. E., Jr., and Cicci, D. A., ?Approximation Solutions for Tethered
Satellite Motion,? AIAA Journal of Guidance, Control, and Dynamics, Vol. 24, No. 4,
August 2001, pp. 746-754.
12. Cho, S., Cochran, J. E., Jr., and Cicci, D. A., ?Identification and Orbit Determination
of Tethered Satellite Systems,? Applied Mathematics and Computation, Vol. 117,
2001, pp. 301-312.
13. Cicci, D. A., Cochran, J. E., Jr., Qualls, C., Lovell, T. A., ?Quick-Look Identification
and Orbit Determination of a Tethered Satellite,? The Journal of the Astronautical
Sciences, Vol. 50, No. 3, July-September 2002, pp. 339-353.
14. ?TiPS: Tether Physics and Survivability Satellite Experiment? web pages, Naval
Center for Space Technology, http://hyperspace.nrl.navy.mil/TiPS/data.html.
15. Cicci, D. A., Volovecky, E. J., Qualls, C., ?Identification of a Tethered Satellite
Using a Kalman Filter,? AIAA Paper 04-165, AIAA/AAS Astrodynamics Conference,
Maui, HI, March 2004.
16. Cicci, D. A., ?An Adaptive Extended Kalman Filter for Angles-Only
Tracking/Intercept Problems,? The Journal of the Astronautical Sciences, Vol. 41,
No. 3, July-September 1993, pp. 411-435.
17. Swerling, P. ?First Order Propagation in a Stagewise Smoothing Procedure for
Satellite Observations,? The Journal of the Astronautical Sciences, Vol. 6, 1959, pp.
46-62.
18. Kalman, R. E., and Bucy, R. S., ?New Results in Linear Filtering and Prediction
Theory,? ASME Journal of Basic Engineering, Vol. 83, 1961, pp.95-108.
37
APPENDIX A: TEST CASE DATA
38
TABLE 17. Test Case Data: No Tether (?
cm
= 0)
Noise Level f(k) ?t RMS
?
cm
*
(m)
LOW (k-1)
2
5 min. 1.948E+7 -7408104
10 min. 26720 -1.358E+9
15 min. 15512 -1.227E+8
k 5 min. 9508 -4.909E+10
10 min. 536754 -3.489E+7
15 min. 776503 -1.381E+7
1/(k+1)
20
5 min. 0.2431 -2416
10 min. 0.4431 -39
15 min. 0.6860 93
1/(k+1)
21
5 min. 0.2431 -2416
10 min. 0.4433 -39
15 min. 0.6862 92
MEDIUM (k-1) 5 min. 1.010E+8 -3.033E+9
10 min. 304318 -4.504E+9
15 min. 20177 -1.558E+9
k 5 min. 145829 -8.204E+8
10 min. 7.976E+7 -1.397E+9
15 min. 5595 -1.013E+9
1/(k+1)
20
5 min. 0.2329 -6559
10 min. 0.3631 320
15 min. 0.7086 686
1/(k+1)
21
5 min. 0.2329 -6559
10 min. 0.3631 320
15 min. 0.7086 685
HIGH (k-1) 5 min. 0.1572 258513
10 min. 0.2502 556325
15 min. 0.4262 613046
k 5 min. 0.1531 31660
10 min. 0.2779 1264131
15 min. 0.8096 473790
1/(k+1)
19
5 min. 0.1879 -9622
10 min. 0.3622 551
15 min. 0.7058 1297
1/(k+1)
20
5 min. 0.1879 -9622
10 min. 0.3622 551
15 min. 0.7059 1297
39
TABLE 18. Test Case Data: ? = 1 km (?
cm
= 909 m), LOW noise
? f(k) ?t RMS
?
cm
*
(m)
0? k
2
5 min. 79246 -2.629E+7
10 min. 54037 -6.210E+9
15 min. 15664 -1.004E+9
(k-1) 5 min. 9.770 185433
10 min. 18598 -8.038E+9
15 min. 90578 -8320845
1/(k+1)
8
5 min. 11.18 63636
10 min. 5.368 12295
15 min. 3.782 1397
1/k
8
5 min. 16.03 30885
10 min. 7.760 30051
15 min. 4.507 1082
5? k 5 min. 1.582E+7 -3.289E+10
10 min. 844865 -4796600
15 min. 34161 -4.853E+8
k
2
5 min. 208974 -1.162E+8
10 min. 1.429E+7 -4.033E+8
15 min. 5381568 -2.482E+7
1/(k+1)
6
5 min. 16.25 31850
10 min. 6.801 24515
15 min. 4.047 -766
1/(k+1)
8
5 min. 19.65 29076
10 min. 7.835 16121
15 min. 4.837 -492
10? (k-1) 5 min. 41970 -1.039E+10
10 min. 3316391 -8.432E+9
15 min. 1190777 -1.641E+8
k 5 min. 0.9509 -3578792
10 min. 9710738 -1.894E+8
15 min. 860739 -3.679E+7
1/(k+1)
19
5 min. 2.055 -14873
10 min. 2.096 -662
15 min. 0.697 1027
1/(k+1)
18
5 min. 1.884 -13500
10 min. 0.842 697
15 min. 0.785 1024
40
TABLE 19. Test Case Data: ? = 1 km (?
cm
= 909 m), MEDIUM noise
? f(k) ?t RMS
?
cm
*
(m)
0? k 5 min. 103.1 1818062
10 min. 0.4188 46684
15 min. 105657 -4.427E+8
(k-1) 5 min. 22.26 -2.336E+7
10 min. 2.181 -84327
15 min. 5230 -2.208E+8
1/k
6
5 min. 4.723 161874
10 min. 4.876 74596
15 min. 3.059 506
1/(k+1)
13
5 min. 8.238 613209
10 min. 11.36 104116
15 min. 3.638 566
5? k 5 min. 5.621 250947
10 min. 3.045 325449
15 min. 28066 -6080792
(k-1) 5 min. 5.628 358120
10 min. 2.998 449543
15 min. 1.092 486845
1/(k+1)
5
5 min. 7.538 -225738
10 min. 3.755 44240
15 min. 2.625 7176
1/(k+1)
4
5 min. 6.816 203867
10 min. 3.533 56500
15 min. 2.489 6855
10? (k-1) 5 min. 5.629 250800
10 min. 3.046 324929
15 min. 6938 -7375386
k 5 min. 5.638 358068
10 min. 3.001 448444
15 min. 1.139 486206
1/k
7
5 min. 6.662 121095
10 min. 3.468 44970
15 min. 2.491 7993
1/(k+1)
4
5 min. 6.842 204010
10 min. 3.541 56510
15 min. 2.491 6734
41
TABLE 20. Test Case Data: ? = 1 km (?
cm
= 909 m), HIGH noise
? f(k) ?t RMS
?
cm
*
(m)
0? k
2
5 min. 215727 -1.252E+8
10 min. 60705 -3.087E+8
15 min. 47348 -1.666E+8
(k-1) 5 min. 2.286E+7 -3.890E+8
10 min. 40120 -4.000E+8
15 min. 17875 -1.082E+8
1/k
16
5 min. 1.789 -3662
10 min. 1.131 2921
15 min. 0.7322 891
1/k
20
5 min. 2.342 -8226
10 min. 1.273 2355
15 min. 0.9346 963
5? k 5 min. 4.319 418352
10 min. 4.249 196265
15 min. 78.05 -6102592
(k-1) 5 min. 4.358 384777
10 min. 4.206 202737
15 min. 30.01 -724931
1/k
5
5 min. 4.748 -259218
10 min. 4.354 113148
15 min. 3.245 3201
1/(k+1)
3
5 min. 4.763 -157130
10 min. 4.373 119816
15 min. 3.238 930
10? k 5 min. 4.331 417880
10 min. 4.253 196247
15 min. 71.67 -6049962
(k-1) 5 min. 4.370 376580
10 min. 4.213 202907
15 min. 29.36 -654351
1/k
5
5 min. 4.772 -263485
10 min. 4.362 113210
15 min. 3.245 3014
1/(k+1)
3
5 min. 4.787 -161501
10 min. 4.380 119851
15 min. 3.238 744
42
TABLE 21. Test Case Data: ? = 10 km (?
cm
= 9091 m), LOW noise
? f(k) ?t RMS
?
cm
*
(m)
0? k
2
5 min. 9333 -1.945E+9
10 min. 16554 -4.025E+7
15 min. 16904 -5.611E+9
(k-1)
2
5 min. 10.79 187180
10 min. 4.014E+7 -6.773E+7
15 min. 9369 -3.998E+8
1/k
14
5 min. 34.75 -393739
10 min. 5.419 17539
15 min. 4.153 9784
1/k
9
5 min. 6.060 -217109
10 min. 0.4890 8391
15 min. 0.6134 9632
5? k
2
5 min. 1384498 -1.951E+7
10 min. 205806 -7.384E+8
15 min. 29305 -4.252E+9
(k-1)
2
5 min. 9.251 174365
10 min. 27515 -7.132E+7
15 min. 38093 -2.057E+9
1/k
6
5 min. 13.29 42465
10 min. 6.268 15103
15 min. 3.298 4459
1/k
5
5 min. 11.77 53852
10 min. 5.429 13736
15 min. 2.720 12807
10? k 5 min. 17918 -7.760E+8
10 min. 25090 -1.508E+7
15 min. 211137 -2.656E+9
(k-1) 5 min. 46356 -1.644E+8
10 min. 556165 -3.057E+8
15 min. 24391 -1.139E+7
1/(k+1)
9
5 min. 45.71 -1787109
10 min. 1.496 15401
15 min. 1.254 10057
1/(k+1)
8
5 min. 12.54 551718
10 min. 1.721 13200
15 min. 1.747 8343
43
TABLE 22. Test Case Data: ? = 10 km (?
cm
= 9091 m), MEDIUM noise
? f(k) ?t RMS
?
cm
*
(m)
0? k 5 min. 17853 -2.590E+11
10 min. 184594 -1.598E+8
15 min. 10008 -4.272E+8
k
2
5 min. 1.557E+7 -3.012E+9
10 min. 157668 -1.627E+8
15 min. 107549 -4732578
1/(k+1)
12
5 min. 1.006 65842
10 min. 0.6383 15992
15 min. 0.8980 10545
1/(k+1)
6
5 min. 0.6496 9091
10 min. 0.5880 15966
15 min. 1.050 22614
5? (k-1)
2
5 min. 3.197 256493
10 min. 17718 -8.348E+8
15 min. 5936 -3.960E+9
(k-1) 5 min. 3.191 254015
10 min. 1.979 802715
15 min. 44311 -3.185E+8
1/(k+1)
13
5 min. 14.24 1383702
10 min. 9.139 115611
15 min. 2.928 5340
1/k
21
5 min. 14.66 1506834
10 min. 7.127 68631
15 min. 2.398 6134
10? k 5 min. 3.158 362342
10 min. 2.540 836567
15 min. 449341 -2.630E+9
(k-1) 5 min. 3.191 253351
10 min. 2.109 1030993
15 min. 9.151 -1709413
1/k
20
5 min. 15.07 1567105
10 min. 4.800 6979
15 min. 2.038 14877
1/(k+1)
13
5 min. 14.24 1383031
10 min. 9.143 115028
15 min. 2.926 4684
44
TABLE 23. Test Case Data: ? = 10 km (?
cm
= 9091 m), HIGH noise
? f(k) ?t RMS
?
cm
*
(m)
0? (k-1)
2
5 min. 86174 -2.786E+11
10 min. 222068 -3.655E+8
15 min. 520551 -1.107E+9
k
2
5 min. 74891 -2.003E+8
10 min. 46188 -7.888E+7
15 min. 21370 -1.053E+8
1/(k+1)
21
5 min. 0.2406 11202
10 min. 0.4266 9307
15 min. 0.7085 8814
1/k
15
5 min. 0.2834 10359
10 min. 0.3606 10416
15 min. 0.4563 9186
5? k 5 min. 4.735 513295
10 min. 2.552 250591
15 min. 3.387 545368
(k-1) 5 min. 4.336 79536
10 min. 2.047 375226
15 min. 1.050 343727
1/(k+1)
6
5 min. 9.441 -800925
10 min. 0.8150 54226
15 min. 0.7375 11919
1/k
10
5 min. 6.635 -1426637
10 min. 0.8132 71679
15 min. 0.6790 10518
10? k 5 min. 4.735 512591
10 min. 2.553 249809
15 min. 3.558 564122
(k-1) 5 min. 4.336 78918
10 min. 2.048 374348
15 min. 1.074 340145
1/(k+1)
6
5 min. 9.441 -801358
10 min. 0.8149 53564
15 min. 0.7352 11274
1/k
10
5 min. 6.634 -1427007
10 min. 0.8133 71021
15 min. 0.6795 9873
45
TABLE 24. Test Case Data: ? = 50 km (?
cm
= 45455 m), LOW noise
? f(k) ?t RMS
?
cm
*
(m)
0? (k-1)
2
5 min. 5.395 231609
10 min. 801420 -4.4474E+11
15 min. 1.693E+7 -2.602E+8
k
2
5 min. 10994 -5502491
10 min. 16017 -1.009E+9
15 min. 264641 -3.167E+7
1/k
15
5 min. 2.604 57065
10 min. 0.4719 44662
15 min. 0.8280 45715
1/k
16
5 min. 3.986 -64684
10 min. 0.4422 45428
15 min. 0.6646 44962
5? k
2
5 min. 1.446E+8 -2.454E+11
10 min. 193956 -3.943E+9
15 min. 2604 -7301015
(k-1)
2
5 min. 5.396 228756
10 min. 42646 -3.877E+8
15 min. 51900 -5525326
1/k
7
5 min. 2.438 64344
10 min. 1.186 45427
15 min. 1.186 49021
1/k
8
5 min. 2.504 59790
10 min. 1.207 43754
15 min. 1.220 45149
10? k
2
5 min. 3.548 447265
10 min. 8726927 -2.807E+8
15 min. 3381018 -2.204E+8
(k-1)
2
5 min. 864520 -1.155E+7
10 min. 4.396E+7 -3.929E+8
15 min. 76714 -4.701E+7
1/k
6
5 min. 0.8651 50501
10 min. 0.7711 47243
15 min. 0.5045 47104
1/k
10
5 min. 3.873 46189
10 min. 1.976 42401
15 min. 1.725 40072
46
TABLE 25. Test Case Data: ? = 50 km (?
cm
= 45455 m), MEDIUM noise
? f(k) ?t RMS
?
cm
*
(m)
0? k 5 min. 3.175 404529
10 min. 1.878 225780
15 min. 22242 -1.102E+7
k
2
5 min. 2.675 675655
10 min. 1315 -7.271E+7
15 min. 2152 -9912064
1/(k+1)
13
5 min. 10.448 753723
10 min. 8.351 147276
15 min. 3.649 49440
1/(k+1)
14
5 min. 13.13 648210
10 min. 9.663 194554
15 min. 4.583 44944
5? (k-1) 5 min. 3.192 291176
10 min. 1.655 798882
15 min. 2.886 892367
k 5 min. 3.175 401593
10 min. 1.569 690194
15 min. 6.579 469003
1/k
20
5 min. 12.86 1324856
10 min. 5.037 49065
15 min. 2.499 52389
1/(k+1)
13
5 min. 12.87 1202841
10 min. 8.772 144106
15 min. 3.384 43042
10? (k-1)
2
5 min. 3.184 284810
10 min. 1.354E+7 -1.763E+9
15 min. 10182 -7.710E+7
k
2
5 min. 2.455 726058
10 min. 515473 -7.197E+7
15 min. 6302 -1.803E+7
1/(k+1)
11
5 min. 44.32 3140658
10 min. 2.368 -77223
15 min. 1.104 46131
1/(k+1)
7
5 min. 1.432 -691193
10 min. 0.5688 45995
15 min. 0.7569 44486
47
TABLE 26. Test Case Data: ? = 50 km (?
cm
= 45455 m), HIGH noise
? f(k) ?t RMS
?
cm
*
(m)
0? (k-1) 5 min. 4.350 117136
10 min. 2.048 439609
15 min. 51874 -3881839
k 5 min. 4.750 556965
10 min. 2.553 300542
15 min. 2.055 665919
1/k
14
5 min. 41.33 3.437E+7
10 min. 4.218 -115323
15 min. 1.140 48648
1/k
10
5 min. 6.549 -1435114
10 min. 0.8103 107429
15 min. 0.6709 48181
5? k 5 min. 3.583 400650
10 min. 665978 -2.094E+8
15 min. 3679 -9128196
(k-1) 5 min. 4.119 291700
10 min. 1.945 529531
15 min. 1.295 356237
1/k
13
5 min. 15.90 2601556
10 min. 0.7851 95354
15 min. 0.9108 46854
1/(k+1)
8
5 min. 19.21 3636757
10 min. 0.9604 77643
15 min. 0.8773 45485
10? (k-1)
2
5 min. 19.54 -2344955
10 min. 3.616 231902
15 min. 26125 -4.056E+8
(k-1) 5 min. 8.483 -1325513
10 min. 34868 -1.097E+9
15 min. 2647 -8848180
1/k
20
5 min. 152.9 1.221E+7
10 min. 40.52 -1325599
15 min. 33.98 20924
1/(k+1)
12
5 min. 153.7 7650025
10 min. 39.36 -1267781
15 min. 31.57 30505
48
TABLE 27. Test Case Data: ? = 1 km up (?
cm
= -909 m), LOW noise
? f(k) ?t RMS
?
cm
*
(m)
0? k 5 min. 19052 -2.120E+8
10 min. 34387 -3.174E+8
15 min. 21728 -9.479E+8
(k-1) 5 min. 12.19 186597
10 min. 1.384E+7 -2.584E+8
15 min. 50100 -5.576E+8
1/k
15
5 min. 61.00 -255588
10 min. 24.29 33578
15 min. 14.07 -2049
1/(k+1)
5
5 min. 15.31 18527
10 min. 7.075 26939
15 min. 4.110 118
5? (k-1) 5 min. 12.74 187621
10 min. 8.131E+7 -1.886E+9
15 min. 38714 -2.556E+8
(k-1)
2
5 min. 11.79 177763
10 min. 36909 -4.148E+7
15 min. 1849706 -5546073
1/(k+1)
6
5 min. 16.25 30146
10 min. 6.801 22862
15 min. 4.044 -2528
1/(k+1)
8
5 min. 19.65 27320
10 min. 7.835 14476
15 min. 4.835 -2144
10? k
2
5 min. 174303 -5196165
10 min. 10543 -3.563E+11
15 min. 43525 -1.850E+9
(k-1)
2
5 min. 11.79 177828
10 min. 82339 -7.497E+8
15 min. 407830 -2.402E+8
1/(k+1)
6
5 min. 16.25 30206
10 min. 6.801 22928
15 min. 4.044 -2450
1/(k+1)
8
5 min. 19.65 27378
10 min. 7.835 14542
15 min. 4.836 -2069
49
TABLE 28. Test Case Data: ? = 1 km up (?
cm
= -909 m), MEDIUM noise
? f(k) ?t RMS
?
cm
*
(m)
0? k 5 min. 4.040 1711062
10 min. 332479 -5.537E+8
15 min. 9223 -2.146E+8
(k-1)
2
5 min. 9463021 -3.204E+10
10 min. 4138 -1.782E+9
15 min. 762 -5482947
1/k
12
5 min. 4.414 426439
10 min. 15.24 106843
15 min. 53.03 -54076
1/(k-1)
3
5 min. 0.8382 108676
10 min. 8.219 138718
15 min. 8.659 -20563
5? (k-1) 5 min. 6.645 252247
10 min. 3.381 232577
15 min. 59352 -1.100E+7
k 5 min. 6.614 358337
10 min. 3.326 337699
15 min. 1.071 492506
1/(k+1)
14
5 min. 14.62 699033
10 min. 12.01 164880
15 min. 4.301 -9869
1/(k+1)
13
5 min. 10.39 763366
10 min. 9.381 94157
15 min. 3.203 -606
10? k
2
5 min. 6.614 493022
10 min. 766.6 -1.042E+8
15 min. 153437 -6506473
k 5 min. 6.614 358405
10 min. 3.326 337777
15 min. 1.065 492748
1/(k+1)
4
5 min. 6.934 119266
10 min. 3.643 56365
15 min. 2.568 4557
1/(k+1)
13
5 min. 10.39 763419
10 min. 9.381 94216
15 min. 3.203 -540
50
TABLE 29. Test Case Data: ? = 1 km up (?
cm
= -909 m), HIGH noise
? f(k) ?t RMS
?
cm
*
(m)
0? (k-1) 5 min. 3.266 1274895
10 min. 2.674 1734528
15 min. 3874 -9.433E+7
k 5 min. 2.351 633480
10 min. 5228083 -2.215E+9
15 min. 6232 -5.042E+7
1/(k-1)
4
5 min. 2.042 -8681
10 min. 2.923 191447
15 min. 3.063 101456
1/(k-1)
2
5 min. 2.556 -2158
10 min. 2.705 204906
15 min. 3.004 112275
5? k 5 min. 4.319 416463
10 min. 4.250 194422
15 min. 208.4 -6786659
(k-1) 5 min. 4.358 382826
10 min. 4.207 200965
15 min. 31.04 -846237
1/k
4
5 min. 4.599 342338
10 min. 4.329 140397
15 min. 3.207 991
1/(k+1)
3
5 min. 4.762 -158748
10 min. 4.373 118088
15 min. 3.237 -1239
10? k 5 min. 4.501 445663
10 min. 4.313 195575
15 min. 44.87 -5428452
(k-1) 5 min. 4.601 386415
10 min. 4.246 202680
15 min. 29.96 -702344
1/k
5
5 min. 5.374 -307660
10 min. 4.571 115868
15 min. 3.322 -796
1/k
4
5 min. 5.194 198245
10 min. 4.528 144550
15 min. 3.265 -836