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An analysis of very short-arc orbit determination for low-Earth
objects using sparse optical and laser tracking data
J.C. Bennett
a,b,⇑
, J. Sang
c
, C. Smith
b
, K. Zhang
a
a
The Satellite Positioning for Atmosphere, Climate and Environment (SPACE) Research Centre, School of Mathematical and Geospatial Sciences,
RMIT University, GPO Box 2476, Melbourne, Victoria 3001, Australia
b
EOS Space Systems Pty. Ltd., Mount Stromlo Observatory, Cotter Road, Weston Creek, Australian Capital Territory 2611, Australia
c
School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079, China
Received 16 July 2014; received in revised form 15 October 2014; accepted 16 October 2014
Available online 23 October 2014
Abstract
The requirement to regularly track an increasing number of objects will result in straining existing tracking networks. This paper
investigates the orbit prediction capability of an orbit determination process using very short-arc optical and laser debris tracking data
for objects in low-Earth orbits. An analysis is carried out to determine the reduction in orbit prediction accuracy when tracking data over
5 s from each pass is only available for an orbit determination.
The results show that the reduction in accuracy is not extensive and good orbit predictions are still possible when using only 5 s of
data from the beginning of each pass. The results are achievable due to an accurate ballistic coefficient estimation and accurate tracking
data. The dependence of the results on the perigee altitude of the objects is obvious, indicating modelling error of the atmospheric mass
density in lower orbits remains the dominant source of error.
Ó2014 COSPAR. Published by Elsevier Ltd. All rights reserved.
Keywords: Short-arc orbit determination; Sparse; Debris
1. Introduction
The near-Earth orbit environment has become progres-
sively cluttered from over 50 years of space operations.
Due to the large number of objects, particularly in the
low-Earth orbit (LEO) environment, providing reliable
orbital information for these objects is a challenging task.
The most comprehensive publicly accessible source of
orbital data is in the form of two-line element (TLE) sets
available through Space-Track.org (https://www.space-
track.org). Orbital information is imperative for space
situational awareness, particularly for conjunction
assessments.
Studies into the population growth of objects in the
near-Earth orbit environment have been numerous since
Kessler and Cour-Palais (1978) predicted the onset of a col-
lisional cascade (the Kessler Syndrome), where collisions
become the dominant source of new debris. The instability
in some LEO orbits as seen in the modelling studies (Liou
and Johnson, 2006, 2008; Rossi et al., 2009; Bennett and
Sang, 2011), has sparked in-depth investigations into
mitigation and remediation scenarios to stabilise the envi-
ronment (Braun et al., 2013; Inter-Agency Space Debris
Co-ordination Committee, 2013; Liou, 2013; Mason
http://dx.doi.org/10.1016/j.asr.2014.10.020
0273-1177/Ó2014 COSPAR. Published by Elsevier Ltd. All rights reserved.
⇑
Corresponding author at: The Satellite Positioning for Atmosphere,
Climate and Environment (SPACE) Research Centre, School of Mathe-
matical and Geospatial Sciences, RMIT University, GPO Box 2476,
Melbourne, Victoria, 3001, Australia. Tel.: +61 2 6222 7905; fax: +61 2
6288 2853.
E-mail addresses: james.cameron.bennett@rmit.edu.au (J.C. Bennett),
jzhsang@sgg.whu.edu.cn (J. Sang), csmith@eos-aus.com (C. Smith), kefei.
zhang@rmit.edu.au (K. Zhang).
www.elsevier.com/locate/asr
Available online at www.sciencedirect.com
ScienceDirect
Advances in Space Research 55 (2015) 617–629
et al., 2011; Phipps, 2014; Stupl et al., 2013; White and
Lewis, 2014).
The uncertainty in debris orbit prediction (OP) yields
unreliable conjunction assessments which could result in
unexpected collisions with operational spacecraft. For
example, considering two-body dynamics, the semi-major
axis is related to the orbital period through Kepler’s third
law. An error in the semi-major axis determined from
observations results in orbital period error which means
that as the prediction period increases, the error in the cal-
culated orbit increases. Regular tracking is required to
reduce this error growth. Departing from two-body
dynamics, the two main sources of orbit estimation error
are incomplete modelling of the perturbing forces acting
on the object and sensor measurement error (Vallado,
2007, Ch. 10). Obtaining the true orbit is unlikely when fit-
ting data corrupted with measurement error. The orbit
error reduces as the tracking data precision increases.
Therefore, highly accurate tracking data is one of the
necessities for debris OP accuracy and is necessary to pro-
tect space assets.
Currently, the debris laser tracking system located at
EOS Space Systems on top of Mount Stromlo, Canberra,
requires an accurate track from the optical tracking system
before the laser is fired. This limits the system operation to
two terminator sessions per day due to the need for the
debris target to be sun-illuminated and visible from the
ground station. Efforts are underway to extend the system
capability to operate outside of terminator – unaided laser
ranging. A benchmark OP accuracy requirement for
unaided laser ranging is set to 20 arc-seconds pointing
error, although this is yet to be experimentally verified.
Until non-terminator tracking is realised, the system is
operational for approximately 4 h a day. This operation
time is reduced if weather conditions are not clear. The azi-
muth and elevation data collected from the optical tracking
system has been shown to have approximately 1.5 arc-sec-
ond root-mean-square (RMS) error. The debris laser rang-
ing system range accuracy is better than 1.5 m RMS error
(Sang and Smith, 2011; Sang et al., 2012).
Optimally tasking the laser tracking is a well-
constrained problem involving pass duration, geometry, the
number of objects to track, telescope slew time, etc. There
are multiple ways to optimise an individual tracking ses-
sion based on specific campaign needs, for example, higher
priority may be assigned to objects based on a predefined
mission-related hierarchy, limited tracking opportunities,
or objects with elements likely to quickly degrade. If the
requirement for the pass duration can be minimised, the
number of objects that can be tracked during a session
increases. Tasking a network of stations is more compli-
cated and is considered in detail by Arregui et al. (2012).
In this paper, the focus is to minimise the tracking data
requirement rather than optimising the session operation.
Any reduction in tracking load of a single station should
readily extrapolate to a reduced load in a station network.
The goal is to reduce the tracking data requirements
without too much loss of OP accuracy in a data sparse
situation (i.e. 2 passes) for low LEO debris objects – where
atmospheric drag effects are the dominant source of orbit
perturbations.
If the data required to achieve a certain OP accuracy can
be reduced, this has important benefits in the maintenance
of a space debris catalogue. Although not considered in
this paper, the buildup of a catalogue of objects is not a
simple process and involves many problems including: data
association, track correlation, and initial orbit determina-
tion (IOD), often from sparse and short-arc data with no
a priori orbital information (Milani et al., 2011; DeMars
et al., 2012). Orbit and detection constraints can be used
to define admissible regions (Tommei et al., 2007) and can-
didate solutions found by sampling. Milani et al. (2012)
provide a large-scale simulation study on the creation
and maintenance of a space debris catalogue for low-Earth
debris objects (above 1100 km perigee altitude) using a net-
work of optical tracking stations. It is found that it would
take 2 months to build up a catalogue containing 98% of
the objects they considered.
Good OP accuracy has been achieved using sparse opti-
cal and laser tracking data of debris objects from a single
station (Bennett et al., 2013; Sang and Bennett, 2014;
Sang et al., 2014). Together with the accurate observational
data, the key to the OP accuracy achievements is the accu-
racy of the ballistic coefficient determined using a newly-
developed method which uses long-term TLE data (Sang
et al., 2013). Sufficient OP accuracy for laser debris orbit
manoeuvre is achievable, as shown in Bennett et al., 2013.
In what follows, an orbit determination (OD) study is
performed to determine if there is any extra benefit in short
term OP accuracy in using full passes of optical and laser
tracking data versus a scenario where only very short-arc
data is available. The situation considered throughout is
one of sparseness – the likely scenario when dealing with
debris tracking data.
Two OD variants are considered: (1) Initially a least
squares OD procedure is used to fit the full pass data; (2)
The process is repeated with only 5 s of data from the
beginning of each pass used in the OD. The tracking data
observations can be taken as 1D – range only; 2D – angles
only; and 3D angles and range together. A comparison is
made between the OPs from the two OD variants using
3D observations to determine the potential loss of accuracy
when using only a small fraction of each pass. The 5 s OD
variant is then analysed comparing fitting 3D observations
with fitting 1D and 2D data to determine the importance of
3D positioning in short-arc OD. The use of TLE-generated
positions as supplemental observations is also analysed to
enhance the 5 s 1D and 2D fitting procedure OP accuracy.
In the absence of “true”orbits, the accurate optical and
Debris Laser Ranging (DLR) tracking data that falls after
the OD period (i.e. not used in the OD fitting) is used to
determine the accuracy of the OP for the two OD variants.
The tracking data distribution used in this work will be pre-
sented first, followed by the OD/OP study. The results of
618 J.C. Bennett et al. / Advances in Space Research 55 (2015) 617–629
the study are then presented and finally some conclusions
are drawn with suggestions for future research directions.
2. Tracking data distribution
In April/May 2013, EOS Space Systems carried out a
debris laser tracking campaign targeting low-Earth orbit
debris objects where one of the objectives was to determine
short term OP accuracy from sparse tracking data. This
campaign was successful and an analysis of the short-term
OP accuracy from this and other campaigns may be found
in Sang et al. (2014). The data distribution of the tracked
objects selected for the short-arc analysis is shown in
Table 1. For each of the laser passes shown in Table 1 there
is also an associated optical pass that was delivered to the
laser system. In all the OD analyses that follow, when a
“full pass”is referred to it means the full optical and laser
pass together. Likewise, a “5 s pass”means both the optical
and laser passes were at most 5 s each in length. Note: a
“full pass”referred to here is not a horizon-to-horizon
track rather a typical track collected during a tracking ses-
sion, usually in the order of a couple of minutes.
The observations are weighted in the OD processing.
Assume wq;wb;wel and wTLE are the weight factors for the
range, azimuth, elevation measurements and TLE
pseudo-observations, respectively. It is assumed here that
wb¼wel ¼1 and wq¼1e10 so essentially the weight of
the range (in metres) is the same order as that of the angu-
lar measurements (in radians) when fitting angles and range
observations together (3D). The optimal weighting factor
for fusing TLE pseudo-observations with the angles or
range observations in the OD process is determined later.
For more information on weighted least squares OD the
reader is referred to Vallado (2007).
The OP cases considered in the analysis are presented in
the next section.
3. Orbit determination and prediction
In what follows, 2-day OD windows containing 2 nights
of tracking data are set where there is subsequent data
available to determine the OP accuracy after 1 or 2 days.
For example, Object 1430 was tracked on the 24th, 25th
and 26th of April and a 2-day OD is set to start on the
24th April and span 2 days, and a 1-day OP case follows
on the 26th. So the corresponding OP case is number 1
in Table 2. This is shown graphically in red in Table 1.
To include more cases, 3-day OD processes using 2
nights of tracking data (i.e. there is data on day 1 and 3
of the OD) are also considered. For example for Object
2125 a 1-day OP case is set to start on the 5th May and
span 3 days. The corresponding case number is case
number 3, where the asterisk denotes it is not a standard
2-day, 2-pass OD process. The OP case numbers for a 1-
day OP are listed in Tables 2 and 3 lists the 2-day OP case
numbers.
In the OD and OP computations the Earth gravitational
effect is modelled using a 100 100 EIGEN-GL05C grav-
ity model (Fo
¨rste et al., 2008). The ocean tidal effects are
included through the CSR 3.0 ocean tidal model (Eanes
and Bettadpur, 1995), and the solid Earth tidal force is
computed using the specification given in McCarthy and
Petit (2004, Ch. 6). Third body gravitational forces are
computed using the DE200 planetary ephemeris. The
area-to-mass ratio of the object, determined from the bal-
listic coefficient:
BC¼CDA
m;ð1Þ
is given by A=m¼BC=CD¼BC=2:2, where BCis estimated
using the method in Sang et al. (2013),Ais the unknown
cross sectional area in the direction of motion in m2;mis
the mass in kg, also unknown, and CDis the drag
Table 1
Data distribution for the April/May 2013 tracking campaign. A “
*
”denotes a laser pass was collected on the associated date and a “
**
”denotes two passes
were collected. The perigee and apogee are in kilometres. Example of 1-day OP case number 1 (26th April) and corresponding OD (24th–25th April) are
marked in red.
NORAD ID April May Perigee Apogee
23 24 25 26 27 28 1 2 4 5 6 7 8 9 10
1430
*******
720 799
2125
**********
596 821
2621
*** * * ** *
587 677
2980
** *****
628 772
5557
* * ** * * * * *
767 840
6275
**** *** * ***
775 828
8956
** ******* *
639 683
11060
** * *
824 841
13923
* * * ****** **
786 810
17122
***** ***
663 676
23606
***** ** * *
599 607
25475
**** *** * * ***
787 791
26121
* * * * *** **
561 657
26702
* * **** * **
565 571
26703
* * ** ***
587 590
J.C. Bennett et al. / Advances in Space Research 55 (2015) 617–629 619
coefficient (assumed to be 2.2 here). When computing the
radiation pressure forces, the solar radiation pressure coef-
ficient is fixed at 1.1 and the area-to-mass ratio is taken as
A=mobtained above. The density model used is the
NRLMSISE-00 model (Picone et al., 2002). In each case
the initial state for the OD process was provided by the first
available TLE immediately before the OD window. The
errors in the mass density (typically 10–15%) and ballistic
coefficient are separate error sources in the computation of
atmospheric drag effects. Here, CD(and hence A=m) is fixed
in the orbit computation. The uncertainty in one parameter
can often be absorbed by estimating another parameter in
the OD, but as pointed out in an excellent assessment of
satellite drag and atmospheric density modelling (Vallado
and Finkleman, 2014), this can shift the uncertainty to
the other parameter, resulting in a physically unrealistic
quantity. Estimating the drag coefficient from sparse track-
ing data is unreliable and better results are consistently
achieved if it is fixed (Bennett et al., 2013).
4. OP results
In this section, the results from the OP analyses are pre-
sented where the 1-day and 2-day OP accuracy is compared
for an OD process using the whole pass information versus
using only 5 s of data from each pass. For each observation
that falls within the prediction period, the difference is cal-
culated with the pointing angles and range determined
from the OP. The overall OP accuracy is calculated as
the RMS of these differences for the along and cross track
telescope pointing directions and the range residuals.
The 5 s OD variant is then analysed by comparing 3D
observations OD fitting with fitting 1D and 2D data to
determine the importance of 3D positioning in short-arc
OD – in particular, the improvement in debris orbit
computation that is achieved by introducing laser range
measurements to purely optical observations typical of an
optical tracking network station. Due to dealing with very
short arcs in the 5 s OD variant, the use of TLE-generated
Table 2
1-day OP case numbers. A “
*
”denotes a 3 day OD.
NORAD ID April May
23 24 25 26 27 28 1 2 4 5 6 7 8 9 10
1430 1 2
2125 3
*
45
2621 6
2980 7
5557 8
*
9
*
6275 10 11 12
*
13
8956 14
*
15
*
16
11060
13923 17
*
18 19
17122 20
23606 21 22
25475 23 24
*
25
26121 26 27 28
*
26702 29
*
30 31 32
*
26703 33
Table 3
2-day OP case numbers. A “
*
”denotes a 3 day OD.
NORAD ID April May
23 24 25 26 27 28 1 2 4 5 6 7 8 9 10
1430 1 2 3
*
2125 4
*
5
2621
2980 6
5557
6275 7 8 9
*
10
*
11
*
8956 12 13 14
*
11060
13923 15 16
*
17
17122 18
23606 19
25475 20 21
*
22
*
26121 23
26702 24
*
25 26
26703
620 J.C. Bennett et al. / Advances in Space Research 55 (2015) 617–629
Fig. 1. Box and whisker plot comparison for the 1-day OP RMS residuals from ODs using all of each pass (left box plot in each subfigure) and ones using
only 5 s of observations from the beginning of each pass (right box plot in each subfigure). Note the Y-axes scales are logarithmic for ease of comparison.
Fig. 2. Box and whisker plot comparison for the 2-day OP RMS residuals from ODs using all of each pass (left box plot in each subfigure) and ones using
only 5 s of observations from the beginning of each pass (right box plot in each subfigure). Note the Y-axes scales are logarithmic for ease of comparison.
J.C. Bennett et al. / Advances in Space Research 55 (2015) 617–629 621
positions as supplemental observations is also analysed to
aid convergence and increase accuracy in the 1D and 2D
fitting.
4.1. 1-day OP results for an OD fitting 3D (angles and
range) observations
Fig. 1 shows box plots for the RMS residuals after a 1-
day OP for the along and cross track pointing errors, and
range errors. For each box plot – and for all box plots in
the remainder of the paper – the horizontal lines in the
box section correspond to the first three quartiles, the bot-
tom and top “whiskers”correspond to the minimum and
maximum values, respectively, and the diamond locates
the mean value. These plots summarise all of the 1-day
OP results for the cases presented in Table 2. In these fig-
ures is a comparison between the OP results where the full
passes were used in the OD and when only 5 s of data was
used in each pass. The average pass length for the full pass
ODs was approximately 2.9 min. Fitting 5 s of data from
each pass yields only a minor increase in the 1-day OP
error. The medians of the pointing errors are both below
20 arc-seconds which means at least 50% of cases could
potentially be acquired with a diverged laser beam for
unaided laser ranging.
Overall there was a reduction in 1-day OP accuracy
when reducing the pass length to 5 s; however, the accuracy
achieved is still sufficient for routine tracking for catalogue
maintenance.
4.2. 2-day OP results for an OD fitting 3D (angles and
range) observations
Fig. 2 shows box plots for the RMS residuals after a 2-
day OP for the along and cross track pointing errors, and
range errors. These plots summarise all of the 2-day OP
results for the cases presented in Table 3. Similar character-
istics are seen in the 2-day OP results as that shown in the
1-day ones. Again, using only short-arc tracking data
resulted in a slightly reduced OP performance. Interest-
ingly, the medians of the pointing errors are still both
below 20 arc-seconds indicating that in many cases the
OP error isn’t increasing dramatically over longer predic-
tion periods. Also worth noting is that the median cross
Fig. 3. 1-day OP RMS residuals vs perigee altitude for the full pass OD case. The along track (AT) and cross track (CT) pointing errors have units of arc-
second, while the range is in metres.
622 J.C. Bennett et al. / Advances in Space Research 55 (2015) 617–629
track error is smaller when using the short-arc data. This is
attributed to tracking station bias being more apparent in
the longer passes.
In the next section the RMS residuals versus perigee alti-
tude are plotted to determine if the lower objects experience
larger errors since they are more sensitive to the modelling
errors in the atmospheric mass density model and the sim-
plifying assumptions on the object’s characteristic such as a
constant ballistic coefficient.
4.3. OP error vs perigee
Figs. 3 and 4 show the OP RMS errors plotted against
the perigee for 1 and 2 day predictions, respectively. Due
to the similarity between the results of the full pass OD
and the 5 s pass OD, only the full pass results are plotted.
It is clear that the lower perigee objects are the ones that
experience the larger OP errors. Fig. 3 shows 11 out of
33 cases have a combined pointing RMS error of more
the 50 arc-seconds after a 1-day OP. Unaided laser raging
would be difficult in these circumstances. The OP range
error is not important for visual acquisition purposes but
it is important when ranging to the target. Smaller residuals
increases the accuracy of the expected range band.
In the following sections it is investigated whether the
same claims of OD/OP success can be made in the 5 s
OD case when 3-D data is not available for fitting.
4.4. 1-day OP error from an OD fitting 2D (angles)
observational data
In this Section the 5 s pass OD process was repeated
with only the azimuth and elevation observations fitted,
i.e. the DLR observations were ignored. Fig. 5 shows the
results comparing the 2D case with the 3D results from
Section 4.1 for the 1-day OP errors. This shows that when
the DLR observations were ignored, the OP accuracy
reduced significantly with the median errors increasing by
a factor of more than 4. Due to the reduction in accuracy,
unaided laser ranging would most likely fail in the majority
of cases if 5 s of angles-only data from 2 passes was used in
the OD fitting.
The maximum RMS errors increased significantly, par-
ticularly in the along track direction where an increase of
a factor of 10 is seen in the maximum. Including the range
Fig. 4. 2-day OP RMS residuals vs perigee altitude for the full pass OD case. The along track (AT) and cross track (CT) pointing errors have units of arc-
second, while the range is in metres.
J.C. Bennett et al. / Advances in Space Research 55 (2015) 617–629 623
information in the OD fitting improves the 1-day OP med-
ian RMS by approximately 76%, 83%, and 70%, in the
along, cross track and radial directions, respectively.
4.4.1. Including TLE pseudo-observations in the OD
TLE pseudo-observations – generated using SGP4 prop-
agation (Vallado et al., 2006) from TLE/s falling inside the
OD window – can be used to help constrain the OD process
so that the OP accuracy improves. For each TLE, pseudo-
observations were generated every 10 min using SGP4,
starting 3 days before the TLE epoch and finishing half
an orbital period after the TLE epoch.
The question arises of how to weight them versus the
angular data in an OD process. An optimal relative weight-
ing factor c1is sought that results in the best OP accuracy,
i.e. in the OD processing the TLE weight is set to
wTLE ¼c1wb¼c1.
To determine the optimal weighting factor, all of the
OD computations for the 1-day OP cases displayed in
Table 2 were performed for a series of relative TLE
Fig. 5. Box and whisker plot comparison for the 1-day OP RMS residuals from ODs using only 5 s of observations from the beginning of each pass using
(1) azimuth, elevation, and range (3 dimensional – left box plot in each subfigure); and (2) azimuth and elevation only (2 dimensional – right box plot in
each subfigure). Note the Y-axes scales are logarithmic for ease of comparison.
Fig. 6. 1-day average OP RMS residuals vs relative TLE pseudo-observation weight c
1
. The along and cross track errors are in arc-seconds, while the
range error is in metres. The results are averaged over all objects.
624 J.C. Bennett et al. / Advances in Space Research 55 (2015) 617–629
weighting factors and the average 1-day RMS OP error
was compared. Fig. 6 shows the average 1-day RMS error
for each TLE weight which is relative to the angular obser-
vations which have unit weight. The lower the error, the
more optimal the weighting factor. Overall the optimal
weighting factor is found to be c1¼1e16, i.e. for the best
results the TLE pseudo-observations are weighted by
1e16 in the OD computations.
Fig. 7 shows a comparison between the 1-day OP results
of using both the optical and DLR data in the OD (3D
case) and that using only the optical data (2D) with the
optimally weighted TLE pseudo-observations. For the
2D OD fitting, including the optimally weighted TLE
pseudo-observations decreases the median 1-day OP error
in the along, cross track and radial directions by approxi-
mately 70%, 49%, and 55%, respectively, when compared
to the results displayed in Fig. 5. However, it fails to meet
the accuracy of fitting 3-dimensional observations (with no
TLEs) as seen in Fig. 7.
Similarly, in the next section the 1-day OP accuracy is
determined for the 5 s case when there is no angular data
available, i.e., only the DLR observations are used.
4.5. 1-day OP error from an OD fitting only 1D range
observations
The short-arc OD was performed for the 1-day OP cases
in Table 2 assuming no azimuth and elevation data was
available. This was to simulate the effects where unaided
laser ranging is assumed to be possible, i.e. tracking data
is delivered to the OD procedure from a system where
the laser fires without the need for an optical track. In
the 5 s pass case, only cases 3, 4, 19, 23 converged. In this
situation it is clear there is insufficient data for a reliable
OD procedure and that supplementary data is required.
4.5.1. Including TLE pseudo-observations in the OD
If TLE pseudo-observations are used to help constrain
the OD the convergence rate improves. The question again
arises of how to weight them versus the range data. Similar
to Section 4.4.1,wTLE ¼c2wq¼c2is set and all of the OD
computations for the 1-day OP cases displayed in Table 2
were performed for a series of relative TLE weighting fac-
tors and the average 1-day RMS OP error was compared to
determine the optimal weighting factor. Fig. 8 shows the
average 1-day RMS error with different relative TLE
weights. Fig. 8 shows the minimum occurs when the TLE
pseudo-observations are weighted as 1e6 (i.e.
c2¼1e6) relative to the DLR observations which have
unit weight. This was subsequently selected as the optimum
weighting factor. Incidentally, the independently deter-
mined optimal weighting factors in this section and Section
4.4.1 to fuse the angles only or range only data with TLE
pseudo-observations, verifies the results from the authors’
previously determined optimal weighting for fusing the
optical and laser measurements. This means to process all
Fig. 7. Box and whisker plot comparison for the 1-day OP RMS residuals from ODs using only 5 s of observations from the beginning of each pass using
(1) angles and range (3 dimensional – left box plot in each subfigure); and (2) angles + TLE (right box plot in each subfigure). Note the Y-axes scales are
logarithmic for ease of comparison.
J.C. Bennett et al. / Advances in Space Research 55 (2015) 617–629 625
observation types at once the weighting would be
wb¼wel ¼1;wq¼1e10, wTLE ¼1e16.
Fig. 9 shows a comparison between the 1-day OP results
of using both the optical and DLR data in the OD (3D
case) and that using only the DLR data with the optimally
weighted TLE pseudo-observations. The 1D + TLE case
performs quite well, nearing the accuracy of the 3D
5 s case. Comparing Figs. 7 and 9 shows the 1D + TLE
case outperforms the 2D + TLE case, highlighting the
importance of the accurate range data. This importance
is easier to see when examining the median values as will
be considered in the next section.
5. Summary of OP performance
The 1-day results from the previous sections are summa-
rised in Table 4 including the number of OD cases in
Table 2 that converged. The TLE pseudo-observations’
ability to aid convergence is most noticeable when
comparing the 1D 5 s case to the 1D + TLE 5 s case, where
Fig. 8. 1-day average OP RMS residuals vs relative TLE pseudo-observation weight c
2
. The along and cross track errors are in arc-seconds, while the
range error is in metres. The results are averaged over all objects.
Fig. 9. Box and whisker plot comparison for the 1-day OP RMS residuals from ODs using only 5 s of observations from the beginning of each pass using
(1) angles and range (3 dimensional – left box plot in each subfigure); and (2) range + TLE (right box plot in each subfigure). Note the Y-axes scales are
logarithmic for ease of comparison.
626 J.C. Bennett et al. / Advances in Space Research 55 (2015) 617–629
the number of OD convergences increased from 4 up to 28.
Due to only 4 cases converging the medians were not calcu-
lated in the 1D 5 s case. There were 5 cases where there
were no TLEs available for the OD, which is evident in
the 1D + TLE summary results.
From Table 4 the following changes in the median 1-day
OP RMS error can be seen:
3D full vs 3D short-arc: Only considering short-arc 3D
observations in the OD fitting increases the error by
approximately 145%, 1%, and 78% in the along, cross
track and range, respectively;
3D short-arc vs 2D short-arc: Only considering short-arc
optical observations in the OD fitting increases the error
by approximately 313%, 494%, and 232% in the along,
cross track and range, respectively;
–3D short-arc vs 2D short-arc +TLE: At best, the
error growth is reduced to approximately 24%,
202%, and 49% in the along, cross track and range,
respectively, by including TLEs;
3D short-arc vs 1D short-arc: The 1D OD variant with
5-s passes is unreliable;
–3D short-arc vs 1D short-arc +TLE: At best, the
error growth is approximately 34%, 97%, and 43%
in the along, cross track and range, respectively, by
including TLEs;
Using full 3D passes produces the best results. In the short-
arc cases, using 3D passes is the best, followed by
1D + TLE, then 2D + TLE, and finally 2D. The depen-
dence of OP accuracy on the TLE weighting factor is clear.
The improvements gained by including range observations
in the OD are substantial.
6. Conclusions
The results in this paper indicate that for short term OP, it
is not necessary to collect long pass data. In a tracking
station network this means more objects can be routinely
tracked with less data necessary to maintain the orbital
elements at similar short term accuracy as reported here.
Applications requiring high precision such as conjunction
assessments should proceed with caution. The short-arc data
analysis considered has not yet received an error covariance
reliability assessment. This will be the part of future studies,
particularly when laser debris tracking data from multiple
stations is available. Recently, laser ranging to space debris
was also demonstrated in Graz, Austria (Kirchner and
Koidl, 2013) and Shanghai, China (Zhang et al., 2012).
For the cases considered, TLE pseudo-observations are
not required for convergence in the 5 s pass OD when 3D
data is used. The results are dependent on the availability
of an accurate ballistic coefficient, with similar accuracy
(10%) as the method described in Sang et al. (2013),
and accurate tracking data. Ansalone and Curti (2013)
highlighted the importance of accurate optical data in their
genetic algorithm IOD method from a too short arc optical
observation pass. They considered a single 60 s optical pass
and found that processing very short-arc data required the
sensor accuracy to be good. The results of the short-arc
analyses in this paper are applicable to stations with similar
tracking accuracy.
The other important factor is the separation time
between the very short-arc observation passes. If the two
5 s passes were close to each other then the same accuracy
would not be achieved.
A TLE was used for the initial state to begin the OD
process, although any comparable state could have been
Table 4
1-day OP results summary. ~
AT;~
CT;~
Rng are the median RMS values of the 1-day along, cross track and range errors, respectively.
OD variant Data type Rel. Weight # Converged ~
AT (”)~
CT (”)~
Rng (m)
Full
3D 33 4.4 9.7 38.6
5s
3D 33 10.8 9.8 68.7
2D 32 44.6 58.2 228.2
1D 4
2D + TLE
1e10 33 66.6 154.8 378.2
1e12 33 45.2 74.1 218.2
1e14 33 14.2 30.5 112.3
1e16 33 13.4 29.6 102.5
1e18 33 35.7 54.5 154.1
1e20 32 40.0 58.2 216.7
1D + TLE
1e00 28 55.9 96.0 292.4
1e02 28 43.6 78.5 171.5
1e04 28 29.0 50.4 135.4
1e06 28 14.5 19.3 98.5
1e08 28 18.5 23.2 108.8
1e10 27 61.9 136.2 287.1
J.C. Bennett et al. / Advances in Space Research 55 (2015) 617–629 627
used. This means the results of this paper are applicable
to objects that have been previously tracked and for cat-
alogue maintenance where the short-arc observation data
is used to correct a pre-existing state, rather than cata-
logue development. The 5 s of data was selected from
the beginning of the passes to be representative of what
would result in current operational practices. The results
show the accuracy achievable from 5 s of observations
from 2 passes is sufficient to maintain orbital elements
for a catalogue. Future analyses will extend this method
to the whole LEO region.
The short-arc OP success reported is reliant on the avail-
ability of the associated angular observations from the
optical tracking system. With no angular data the OD
diverges in most cases. Similarly, if the DLR data is left
out of the short-arc fitting procedure the OP error quadru-
ples. This shows the importance of 3-dimensional position-
ing data. This has important consequences in short-arc
OD/OP success if unaided laser ranging is realised for
debris objects. Including TLE pseudo-observations in the
short-arc OD process improved the convergence rate for
the 1D and 2D cases. However, it failed to reach the accu-
racy of the short-arc 3D case even with optimally weighted
TLE pseudo-observations, with increases in the cross track
error of 202% and 97% in the 2D and 1D cases, respec-
tively. Given the average pass length considered was
2.9 min, only requiring 5 s is a savings of over 97% in track-
ing load.
The main limiting factor in low LEO OP reliability and
accuracy is accurately modelling the atmospheric mass
density. Efforts are underway to develop effective methods
to improve the density model accuracy using tracking data.
Also, instead of assuming a simple spherical shape for deb-
ris objects, a taxonomy should be created that fuses all pos-
sible data from the tracking sensors to create a realistic
representation of each object. These characterisations
should ideally be dynamic and feature in the catalogue.
These aspects will be part of the focus of future studies
to improve debris OP.
Acknowledgements
The authors would like to gratefully acknowledge the
Australian government and EOS Space Systems Pty. Ltd.
research grant support for this project through the Enter-
prise Connect Researcher in Business Grant and the
research grant support from the Australian Research
Council’s Linkage Projects scheme (Project ID:
LP130100243).
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