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Ballistic Coefﬁcient Prediction for Resident Space Objects

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Ballistic Coefﬁcient Prediction for Resident Space ObjectsDr. Ryan Russell, Nitin Arora, Vivek Vittaldev
University of Texas at Austin
Dr. David Gaylor, Jessica Anderson
Emergent Space Technologies, Inc.
Abstract
Recent improvements in atmospheric density modeling now provide more conﬁdence in spacecraft ballistic coefﬁcient(BC) estimations, which were previously corrupted by large errors in density. Without attitude knowledge, forecastingthe BC for accurate future state and uncertainty predictions remains elusive. Our objective is to improve the predictivecapability for ballistic coefﬁcients for Resident Space Objects (RSOs), thus improving the existing drag models andassociated accuracy of the U.S. Space Object Catalog. To achieve this goal we implemented a two-pronged strategythat includes elements of time series analysis and physics based simulations.Two empirical time series prediction methods were applied and tested on simulated and measured BC time series data:a multi-tone harmonic model and an autoregressive (AR) model. Both the multi-tone harmonic model and the ARmodel were subjected to multiple levels of optimizations resulting in highly optimized ﬁnal models that were tunedspeciﬁcally with the 205 BC time series data provided by the Air Force. Two versions of the AR model were developedbased on the model prediction methodology. The second version of the AR model performed approximately as wellas the optimized multi-tone model. The proposed algorithms automatically select the ﬁtting function and durationof the ﬁt span tuned for the best prediction performance based on past known data. The results demonstrated theability to robustly and automatically ﬁt past BC data in order to predict forward for 1-10 days with improved accuracy.These improvements were mapped to position error predictions for typical satellites, demonstrating the utility of suchpredictions for conjunction analysis and other catalog applications.An archive of simulated BC data is generated using custom 6DOF high ﬁdelity simulations for RSOs using platemodels for shapes. The simulator includes force and torque perturbations due to the nonspherical Earth, third-bodyperturbations, SRP, and atmospheric drag. The simulated BC proﬁles demonstrate signiﬁcant variation over short timespans (due primarily to varying frontal areas), providing motivation to improve future BC estimation strategies. The6DOF modeling is intended to provide a physics-based BC data set to complement the BC data set provided by theAF. The improved performance of the time series prediction algorithms applied to the physics-based simulated datasuggests that the actual estimated BCs include other signatures.
Introduction
Atmospheric drag is the major source of error in the orbit determination and prediction for RSOs in low Earth orbits.In general, the drag force experienced by a satellite varies primarily due to changes in the ballistic coefﬁcient (BC)and atmospheric density. When the attitude proﬁle of an RSO is known, drag modeling is difﬁcult primarily due tothe inherent temporal and spatial uncertainties in atmospheric density. The new Air Force Space Command (AFSPC)density model coupled with current better solar predictions is addressing many of the deﬁciencies in drag modeling[1, 2, 3, 7, 8]. However, for RSOs exhibiting large frontal area variations, changes in the BC can be a major sourceof orbit prediction error. Real time ballistic coefﬁcients are currently computed from the daily differential orbit cor-rections obtained on all RSOs. AFSPC would like to predict BC changes over the period of a week from the epochtime. Corrections to thermospheric density from current models are now accurate enough to allow reasonably gooddetermination of these BCs. This ability to separate density and BC values paves the way for signiﬁcantly improvingRSO orbit prediction.In this project, we pursued a two-pronged strategy that included elements of time series analysis and physics-based sixdegree of freedom (6DOF) simulations. State-of-the-art empirical time series prediction methods were applied on BC
time series and tested using both simulated data and real data provided by the Air Force. Also, an archive of simulatedBC data was generated using custom 6DOF high ﬁdelity simulations.For the empirical time series prediction algorithms, a variety of approaches were considered and two prediction modelsshowed the most promising performance: a multi-tone harmonic model and an Auto-Regressive (AR) model. Boththe multi-tone harmonic model and the AR model were subjected to multiple levels of optimizations resulting inhighly optimized ﬁnal models that were tuned speciﬁcally with the 205 BC time series provided by the USAF. Bothalgorithms exhibited similar levels of performance. The chosen algorithms automatically select the ﬁtting function andduration of the ﬁt span to obtain the best prediction performance based on past data. The results obtained using thesealgorithms demonstrated that both algorithms have the ability to automatically ﬁt real BC data and to predict futureBCs for 1-10 days with improved accuracy over the current constant model approach. The constant model approachassumes that the BC remains constant at the most recently estimated value over the entire prediction interval. For 7-day predictions, our best model provided relative root mean squared (RMS) errors of approximately 9%, compared toapproximately 15% for the constant model. These improved predictions could be tremendously valuable in improvingconjunction analysis and other catalog applications. While the performance of the algorithms is promising, there isstill room for improvement from this initial proof-of-concept demonstration.The 6DOF simulations included force and torque perturbations due to the non-spherical Earth, third-body pertur-bations, SRP, gravity gradient, and atmospheric drag. The simulated BC proﬁles demonstrated signiﬁcant variationover short time spans, providing motivation to improve BC prediction strategies. The 6DOF modeling provided aphysics-based BC data set to complement the BC data set provided by the Air Force. Our forecasting algorithms weretested using the physics-based simulated BC data and highly accurate prediction capabilities were demonstrated. Theimproved performance of the time series prediction algorithms applied to the physics-based simulated data suggeststhat actual estimated BCs include effects not accounted for in the 6DOF simulations. These results provide furthermotivation for improvements in the estimation of BCs. Accordingly, in future work, we plan to leverage our 6DOFsimulations to investigate potential improvements to the time series prediction models and the direct estimation of theBC series.
BC Time Series Analysis
The BC of an object in space determines its orbital decay behavior, especially in low Earth orbit (LEO), and henceis of prime importance during drag force computation. If atmospheric forces (and all other dominant forces) can besuitably modeled, the BC, as shown in Eq. 1, may be estimated.
BC
=
mC
D
·
A
(1)Whilethemass,
m
, drag coefﬁcient,
C
D
, and area,
A
, remain inseparable, themain variation of BC isdueto changesinfrontal area. For a non-spherical tumbling object, this variation can be rapid with large amplitudes. The current spacecatalog tracking process assumes ﬁxed BC values over long durations and, therefore, does not effectively consideror model these potentially rapid BC variations. This has led to the growing need to model and predict the ballisticcoefﬁcient more effectively.One of the proposed solutions is to estimate future values of the BC using time series prediction techniques. Thesetechniques do not use complicated dynamics models, and, instead, model and predict the BC data by ﬁtting previousdata to high-order basis functions (such as Fourier series or sum-of-sine series). The time series data are smoothedusing a trend analysis in order to remove random noise. The smoothed data are then used to estimate the coefﬁcientsof the ﬁtting model. In the current work, we considered three types of ﬁtting models: non-linear Matlab based models,the multi-tone harmonic model and autoregressive (AR) model. The performance of the non-linear Matlab basedmodel was found to be inferior to the other two models and hence was not considered in later part of this work.The terminology adopted for the the BC time series analysis is deﬁned in the next section. Next, a brief overviewof the multi-tone harmonics approach is presented, followed by a brief overview of the AR approach. Finally, theevaluation of both methods using the 205 BC time histories provided by the Air Force is presented. The time seriesdata srcinated from actual orbit estimations for 205 different satellites over a time spans on the order of hundreds of days.
BC Time Series Terminology
Considering that we are analyzing multiple BC data ﬁles with varying time spans and observation frequencies, wedeﬁned the time series terminology given in Fig. 1.Fig. 1: Time Series TerminologyIn Fig. 1,
t
is the total span of the input BC time series. Epoch span,
e
, was deﬁned as 85% of the total span of theinput BC time series. In the current study, we assumed
e
was our known data span and tried to predict BC values inthe region beyond
e
. Eighty-ﬁve percent was chosen arbitrarily for this study. The calibration region,
c
, is the regionover which various model parameter optimizations are performed. The ﬁt span,
f
, deﬁnes the span of the ﬁtted model.The ﬁt span satisﬁes the following condition:
c
≤
f <
(
e
−
c
)
(2)For a consistent measure of the prediction performance of the ﬁnal model, the amplitude relative error is deﬁned in inEq. 3.amplitude relative error
=
(true value - predicted value)amplitude of the BC series over the epoch span (3)The true value of the BC is known in simulation (when the epoch span ends prior to the end of the time series history),but would be unknown when applying the algorithm in practice.The amplitude relative error served as our metric throughout the model generation/optimization process. Using theamplitude relative error instead of the standard relative error during model generation gives more weight to long termand well-deﬁned trends, which, in turn, helps reduce over-ﬁtting of the BC time series data. Standard relative error wasused for quantifying prediction models’ capabilities in the ﬁnal performance evaluation section of this report. Prior toﬁtting, the BC time series were prepared as described in the next section.
Preparing BC Data
Parsing
First, the 205 data ﬁles containing BC time series were parsed and imported into Matlab. The relevant data was thenautomatically extracted, splined, smoothed, and then passed to the ﬁtting algorithm. The data ﬁles were also stored ina Fortran readable binary, in order to be usable with a Fortran-based prediction algorithm in the future.
Splining
The BC data loaded into Matlab was irregularly spaced with varying time intervals. This irregularly-spaced datawas interpolated using splines and sampled at equal intervals, leading to a homogeneous BC time series with 1568elements for the multi-tone model and 5000 elements for the autoregressive (AR) model. Regularly spaced data helpsto maintain homogeneity and facilitates convergence of the frequency detection algorithm, as described later.
Fig. 2: Smoothed BC Time Series
Smoothing
The next step was to perform a time series decomposition analysis on the homogeneous BC data and break it intocyclic, seasonal, and random noise components. The cyclic and seasonal terms represented the long term trends in theBC time series and could be accurately modeled by the ﬁtting function once random noise was removed. All the 205BC time series were smoothed before being ﬁtted. Fig. 2 shows a representative raw (in blue) and smoothed BC timeseries (in red).Thesmoothedtimeseriesﬁtissuperiortotheﬁtoftheun-smoothedtimeseriesintermsofmeanRMSE.ThesmoothedBC series were used in the selected model ﬁtting algorithms.
Multi-Tone Harmonic Model
Matlab’s built-in non-linear ﬁtting models were srcinally considered for ﬁtting BC time series. Even though pre-liminary results were encouraging, it was quickly realized that the available Matlab models were lacking the ﬁdelityrequired to capture the relevant frequencies in the BC time series. Further, custom non-linear Matlab models showedlimited success and suffered from sensitivity to the initial guess of model coefﬁcients. To overcome these problems, amulti-tone harmonic series based model was adopted which is deﬁned in Eq. 4.
y
(
x
) =
m
+
O
i
=1
a
i
cos
(
w
i
x
) +
b
i
sin
(
w
i
x
)
(4)The BC value at time index
x
is
y
,
m
is the bias,
i
is the harmonic number,
a
and
b
are the scaling coefﬁcients for themulti-tome harmonic,
O
represents the order of the harmonic series and
w
represents the frequency corresponding tothe harmonic number.The main idea of the Multi-tone Harmonic Model approach is to calculate
w
i
using an iterative frequency estimationalgorithm and then obtain the model coefﬁcients
a
,
b
and
m
via a least squares ﬁt. This approach completely avoidedthe problem of guessing initial values for the model coefﬁcients while still capturing higher order frequencies in the BCdata. For the current study, the Quinn-Fernandes [5] algorithm was used for estimating various harmonic frequenciespresent in the BC time series data. The Quinn-Fernandes algorithm is an iterative, statistically optimal algorithmwhich may be interpreted as ﬁnding the maximum of a smoothed periodogram
∗
.After computing the model parameters (
w,m,a,
and
b
) for each
i
= 1
...i
max
, the BC predictions were computedby using Eq. 4 for a future value of time index
x
. An upper limit on
O
was initially set to 200 and then optimized foreach time series.
Model Generation
The model ﬁtting algorithm can be divided into two stages: the calibration region optimization stage and the ﬁnalmodel ﬁtting stage. The smoothing of the BC time series was performed using Matlab’s “smoothts” function. Thisprocessing of the input data helped suppress the random noise present in the data. It also improved the accuracy of theﬁtting process and increased robustness of the frequency estimation algorithm.
∗
HTTP://escape.library.uq.edu.au/eserv.php?pid=UQ:10626&dsID¯comparison-t.pdf
For each BC time series, stage 1 starts by decomposing the series into three parts. The prediction region,
p
, is ﬁxedat 5 days. Values of 1, 2, 3, and 5 days were considered for the calibration region,
c
. For each
c
value, a two leveloptimization was carried out by adjusting the ﬁt span,
f
, and the harmonic order,
O
. The purpose of this optimizationwas to minimize the amplitude relative error in the calibration region. An optimized value of
f
was found to bebeneﬁcial for improving the predictive capability of the ﬁtted model over long intervals. The combination of
f
and
O
that gave the minimum amplitude relative error in the calibration region was then selected. Next the model was reﬁtusing the selected combination of
f
and
O
, including the calibration region in the optimized ﬁt span (
f
new
=
f
+
c
).We then calculated the maximum amplitude relative error in the prediction region,
p
, for a 5 day prediction interval.The
c
value corresponding to the minimum error over the 5 day prediction region was ﬁnally selected and stored.After the calibration region was selected for the 205 data ﬁles, the stage 2 algorithm decomposed each time series intothree parts, the ﬁt span,
f
, the calibration region,
c
, and the future prediction region. If this algorithm were used inreal time to predict BCs, the future prediction region would correspond to a future period of time for which BC datawas not yet available. As in stage 1, values of
f
and
O
were selected to minimize the amplitude relative error in thecalibration region. The ﬁnal model was then created by adding the calibration region to the selected ﬁt span and thenreﬁtting over the new ﬁt span (
f
new
=
f
+
c
). This process was carried out for each of the 205 BC time series dataﬁles. This ﬁnal model is our working model and is used to predict the future BC values in the future prediction region.
Autoregressive Model
The autoregressive model (AR) is a linear prediction model which attempts to predict future outputs of a system basedon the previous outputs. AR models are frequently used in analysis and forecasting of time series data in the ﬁeldof econometrics. These models assume that the underlying series is approximately stationary, meaning that the meanremains relatively constant over the time span (a condition satisﬁed by BC data). The general form of an AR modelfor an input
y
(
t
)
is given in Eq. 5.
y
(
t
) =
q
t
=1
a
i
y
(
t
−
1) +
e
(
t
)
(5)The autoregressive coefﬁcients are
a
i
and
q
is the order of the AR model. The noise term,
e
, is assumed to be Gaussianwhite noise. We used Matlab’s built-in "ar" function with the default settings, to compute the AR model coefﬁcients fora given
q
value. The order
q
depends on the input series and is generally identiﬁed using an autocorrelation analysis.Autocorrelation is the cross-correlation of a signal or a series with itself with some lag parameter. It may be describedas the similarity between various observations as a function of the lag in the independent variable (i.e. time, in a BCprediction problem). This mathematical tool helps to ﬁnd repeating patterns buried under noise, or in identifying theharmonic frequencies in a signal.
Model generation
Typically, before autoregressive model estimation, an autocorrelation analysis on the time series is performed. Therewas signiﬁcant cross correlation between various values of the BC time series, so the order,
q
, could not be determinedby directly inspecting the autocorrelation plot. To overcome this problem, we performed an optimization of
q
duringthe estimation of the AR model. The order
q
was allowed to vary between 1 and 35 during the optimization phase.As with the multi-tone harmonic model, we also performed an optimization of the ﬁt span,
f
, which resulted in a twolevel optimization strategy for generating the AR model.
Prediction
The future predictions using the AR model employed MATLAB’s built-in "predict" function. We implemented alinear AR prediction model which uses the latest predicted data along with the past data to make future predictions.Our approach is just one of various possible approaches to AR prediction and it may be valuable to look into otherpossible approaches for future work.
Time Series Fitting Models Performance Evaluation
The predictive capability of the selected time series ﬁtting models (both the multi-tone and AR) was evaluated overprediction intervals of approximately 1, 3, 5, 7, 9 and 11 days. For each of these predictions, the RMS relative errorsand max relative errors were computed and plotted. Upon investigation it was found that the multi-tone model is not

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