Robust Orbit Determination: A Machine Learning Approachmstl.atl.calpoly.edu/~bklofas/Presentations/DevelopersWorkshop2015/... · Robust Orbit Determination: A Machine Learning Approach

Robust Orbit Determination:A Machine Learning Approach

Srinagesh Sharma, James Cutler

April 24, 2015

Table of Contents

Introduction

Problem Statement

Machine Learning Approach

Results

Robust Orbit Determination 2 / 24

introduction

Introduction

The Following are true regarding state of CubeSat Development

I The number of cubesats per deployment has been steadily increasingin recent years, including cubesats to deep space.

I Autonomous Ground Station Networks with software-defined radios(SDRs) have become popular for decoding.

I Orbit Determination is difficult for large number of satellitedeployments, and generally accomplished using linearized estimatorsand filters.

Robust Orbit Determination Introduction 4 / 24

Introduction

However,

I Orbit Determination using ground stations requireI Requires an initial estimate in the linear region of the orbit

determination systemI Hard to generalize for arbitrary gravitational potential mapsI Stochastic parameters need to have nice probabilistic assumptions -

such as gaussian probability distributions on process noise

I Autonomous orbit determination for multiple CubeSats over groundstation networks is relatively unexplored ground


Motivation

With these considerations, would it be possible to

I Identify spacecraft whose communication characteristics are knowna-priori

I Perform Orbit Determination whose orbit characteristics do not havenice initial estimates and whose stochastic parameters havedistributions which may be non-parametric.

I Generalize such a system to operate over nG ground stations, over NPpasses.

Essentially, can we build an autonomous system that learns to doorbit determination?


problem statement

Problem Statement

I Consider Two Spacecraft, labeled 1, 1.I Orbit parameters j = {aj , ej ,j , Ij , j ,Mj}, j = 1, 1.I Say we know that these orbit parameters are drawn from a probability

distribution Pj and that j are bounded.

Robust Orbit Determination Problem Statement 8 / 24

Problem Statement

I Spacecraft RF transmissions have certain characteristics such asmodulation, center frequency instability, randomness of data, etc,which are known prior to launch.

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I Observe transmissions from nG ground stations over time period Tp,and extract feature vectors from the received signal.


Problem Statement

Assumptions

I The received RF signals are distinguishable almost everywhere.

I There exists a model of the system such that given an example orbit,example transmissions of these spacecraft can be generated (forexample, a perturbation model and orbit propagator).

Given such a system, for any nT test feature vectors, can we classifytransmissions and estimate the orbits 1, 1?


machine learning approach

Learning Approach

I Generate n training datasets {{xij , yij}nij=1, i ,1, i ,1}ni=1I i,1: example orbits of the first spacecraftI i,1: example orbits of the second spacecraftI xij : RF Feature Vectors of a spacecraft in the ith orbitI yij : label of xij

I Training datasets can take into consideration very subtle variations inRF transmission characteristics, drifts in center frequency, variation oftransmission rates etc.

I P1 ,P1 induces a probability distribution on the space of probabilitydistributions of the RF Feature Vectors.

Robust Orbit Determination Machine Learning Approach 12 / 24

Kernel Methods

Kernel Methods have become popular in learning theory due to theirability to estimate non-linear functions.

I S: Compact Space

I k : S S R, a symmetric, positive definite function

I Reproducing Kernel Hilbert Space(RKHS): k generates a functionspace1 Hk such that f Hk , s S, f (s) = f , k(, s)

I When universal kernels are used, the search space is a dense subset ofC (S), the space of all continuous bounded functions on S.

I They can also be used to embed probability distributions on S ontoHk , (PS) =

S k(, s)dPS

1Nachman Aronszajn. Theory of reproducing kernels. In: Transactions of theAmerican mathematical society (1950), pp. 337404.


Classification

I Say every feature vector represents only one RF source.

I One of the characteristics of RF transmissions is that the centerfrequency varies due to Doppler Shift, and therefore their signals canoverlap.

I When Transmitting Sources are identical in all aspects exceptP1 ,P1 , even to separate single RF transmissions, we consider allthe RF transmissions received over the entire time period (i.e, theprobabilistic embedding)

I This leads to Transfer Learning2

I The sources are separated by a hyperplane in this infinite dimensionalspace.

2Gilles Blanchard, Gyemin Lee, and Clayton Scott. Generalizing from several relatedclassification tasks to a new unlabeled sample. In: Advances in neural informationprocessing systems. 2011, pp. 21782186.


Transfer Learning

I For (1, 1) (J 2,F2J ), (X ,Y ) (X Y,FX FY), A probabilitydistribution Pi on i , i = 1, 1, induces on the space of probabilitydistributions, BXY such that

P(i)(X ,Y |1 = i,1, 1 = i,1) = (P1 P1 ) 1

xij , yij P(i)(X ,Y |1 = 1,i , 1 = 1,i ) is a function of 1, 1 and a measure on BXY . 1 is its pre-image.

I Due to this, Transfer Learning can be applied, therefore, for a given lossfunction l , it is possible to find a function h : X BX Y such thaty = sign(h(PX ,X ))).

I Let Hk be the RKHS associated with kernel k1 : X X R. For (BX ),the set of mean embeddings associated with BX , let HkP be the RKHSassociated with the kernel kP : (BX )(BX ) R (The mean embeddingis defined as (PX ) =

X k2(, x)dPX ). We seek an estimate hH of h such

that the following criteria is satisfied

hH = arg minhHk

EPXY,(X ,Y )PXY [L(h((PX ),X ),Y )] (2)


Orbit Determination

I When the following existI An invertible mapping between the distribution of the feature vectors

and the distribution of the center frequency in TP .I A unique mapping between the orbit parameters in the support of P

and the center frequency, given a communication system.

Then, there exists a mapping between the distribution of the featurevectors and the orbit parameters.

I When the kernel used in an embedding is universal, the mappingbetween the RKHS and the Probability distribution is unique

I Therefore, when the above two criteria are satisfied, there exists amapping between the kernel embedding of the RF feature vectors andthe orbit parameters.

I Use a second universal kernel operator over the space of embeddingsto estimate this function!


Distribution Regression

I P induces a probability distribution = Pi on (1,PX |y=1)and on (1,PX |Y=1).

I We estimate a function3 fl such that, for the mean embedding(PX ) =

X k(, x), dPX , and for the kernel operator

K : (PX )(PX ) J ,

f2,l = arg minflHkP

E(fl((PX )) l2J ) + 2fl2HkP (3)

For l = 1, 1I This is an estimate of the function which maps the embedding of RF

Feature Vectors to the orbit parameters!

I Applied to the test dataset to determine orbits!3Barnabas Poczos et al. Distribution-Free Distribution Regression. In:

Proceedings of the Sixteenth International Conference on Artificial Intelligence andStatistics. 2013, pp. 507515.


results

Results - Classification

Two Spacecraft with BPSK transmissions, 10Kbps bandwidth, center frequencies

separated by 10 Khz at 437.5MHz, signals received at 10dB SNR at one ground

station over one pass. Synthetic Data, feature vectors were RF Signatures (3

dimensional) by Bkassiny et. al4

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Support Vectors for datasets in red

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Support VectorsClass 1Class 1

Classification Method % ErrorTransfer Learning 2.93

Pooled Classification 6.83Training: 40 orbitsTesting: 10 orbitsSVM with bias

4Mario Bkassiny et al. Blind cyclostationary feature detection based spectrumsensing for autonomous self-learning cognitive radios. In: Communications (ICC),2012 IEEE International Conference on. IEEE. 2012, pp. 15071511.

Robust Orbit Determination Results 19 / 24

Results - Orbit Determination

One Spacecraft with BPSK transmissions, Two Ground Stations (AnnArbor and Chicago) 10dB SNR at receiver in an AWGN channel,Receiving over two passes (2 hour interval), Synthetic Data. Probability oftransmission 0.03, uniform over the passes in TP . Mean Motion was keptconstant at 14.7732. Priors:

e 10e1103(N (4 104, 1 108) +N (4 104, 1 108)) N (16/9, (/18)2)I N (/4, (/182)) N (3/2, (/36)2)M N (/4, (/1800)2)

Training with 1080 orbit insertions, testing with 20 orbits.


Results

Results based on a small cross validation set indicates initial convergence

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Future Work

I Upper and Lower bounds on performance of successive classificationand orbit determination.

I Integration of two loss functions into one single machine learningsystem.

I Experimental Implementation into networked global ground stations.

I Extension to Geolocation systems.


Acknowledgements

I JPL Strategic University Research Partnership FY2015


Thank You

Questions?


IntroductionProblem StatementMachine Learning ApproachResults

Documents

Robust Orbit Determination: A Machine Learning Approachmstl.atl.calpoly.edu/~bklofas/Presentations/DevelopersWorkshop2015/... · Robust Orbit Determination: A Machine Learning Approach