AAS 15-448

SPACECRAFT UNCERTAINTY PROPAGATION USING GAUSSIANMIXTURE MODELS AND POLYNOMIAL CHAOS EXPANSIONS

Vivek Vittaldev∗, Richard Linares†, Ryan P. Russell‡

Polynomial Chaos Expansion (PCE) and Gaussian Mixture Models (GMM) are combined ina hybrid fashion to quantify state uncertainty for spacecraft. The PCE approach models theuncertainty by performing an expansion using orthogonal polynomials (OPs). The accuracyof PCE for a given problem can be improved by increasing the order of the OP expansion.Due to the ”curse of dimensionality” the number of terms in the OP expansion increasesexponentially with dimensionality of the problem, thereby reducing the effectiveness of thePCE approach for problems of moderately high dimensionality. Including a GMM with thePCE (GMM-PC) is shown to reduce the overall order required to achieve a desired accu-racy. The initial distribution is converted into a GMM, and PCE is used to propagate eachof the elements. Splitting the initial distribution into a GMM reduces the size of the covari-ance associated with each element and therefore, lower order polynomials can be used. TheGMM-PC effectively reduces the function evaluations required for accurately describing anon-Gaussian distribution that results from the propagation of a state with an initial Gaussiandistribution through a nonlinear function. Several examples of state uncertainty are propa-gated using GMM-PC. The resulting distributions are shown to efficiently capture shape andskewness.

INTRODUCTION

The increased population in space of both controllable and uncontrollable Space Objects (SOs) creates dif-ficult challenges for Space situational awareness (SSA). Some of the components of SSA such as conjunctionassessment, orbit determination (OD), and track correlation and prediction require an accurate knowledge ofthe uncertainty in the states of the SOs as a function of time. Due to the nonlinear dynamics of SOs and thelatency in observations due to the limitation of the surveillance network, non-Gaussian qualification is a keychallenge for SSA.1

A Gaussian initial distribution for the state uncertainty of SOs at an epoch is a frequent assumption becauseconventional OD techniques only output a state estimate and covariance of the estimated error. Commonfiltering techniques used for OD such as the Extended Kalman Filter (EKF) or the Batch Least Squares(BLS) work on the assumption that the state uncertainty has a Gaussian distribution. However, as the stateof the SO is propagated in time, the assumption of a Gaussian distribution is no longer valid due to the non-linearity of the dynamics of orbital motion. Changing the coordinate set in which the SO state is expressedcan have an effect on the length of the flight time the Gaussian assumption remains valid. Orbit element setssuch as Kepler elements, Equinoctial elements (EE), and Modified Equinoctial elements (MEE) absorb someof the non-linearity of orbital motion and therefore, the uncertainty distribution expressed in these states canbe assumed to be Gaussian for longer times of flight.2, 3, 4 However, the equations of motion for all elementsets are nonlinear and therefore, the Gaussian assumption is always only an approximation. The Gaussian

∗PhD. Candidate, Department of Aerospace Engineering and Engineering Mechanics , The University of Texas at Austin, Austin, TX78712.†Postdoctoral Research Associate, Space Science and Applications, ISR-1, Los Alamos National Laboratory, MS D466, Los Alamos,NM 87545.‡Assistant Professor, Department of Aerospace Engineering and Engineering Mechanics , The University of Texas at Austin, Austin, TX78712.

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assumption in all coordinate sets degrades with increasing initial state uncertainty, fidelity of perturbations,degree of mismodeling, and time of flight.3, 5

A common, computationally intensive method of propagating orbit uncertainty is to use Monte Carlo (MC)simulations.6, 7 Randomly generated samples from the initial uncertainty distribution propagated through theinitial distribution requires on the order of millions of propagations to generate statistically valid solutions.Parallelizing the computations on multiprocessor CPUs or on Graphics Processing Units (GPUs) reducesthe runtime of the simulations significantly8, 9, 10 at the cost of increasing the difficulty of implementation.11

Reducing the number of sample points required for a result with satisfactory confidence bounds is possiblethrough importance sampling. Although the computational cost can be prohibitive for most applications dueto the slow convergence, the generality of MC techniques make it an ideal benchmark to compare othermethods.

A spectrum of techniques exists that propagate the state and uncertainty of an initially Gaussian distributionthrough a non-linear function, such as orbit propagation.12 Computational cost is traded for accuracy of thefinal probability density function (PDF). Using the first order Taylor series expansion of the dynamics tolinearly propagate the covariance matrix through the dynamics lies on an extremity; while the MC simulationlies on the other. Two techniques that occupy a range within this spectrum are Gaussian Mixture Models(GMMs) and Polynomial Chaos expansion (PCE).

GMMs approximate any PDF using a weighted sum of Gaussian distributions with the approximation im-proving in an L1-norm sense with increasing number of elements.13 When the initial distribution is Gaussian,the GMM has spatially distributed means and each element has smaller uncertainty (i.e. differential entropy).Each Gaussian element is propagated through the nonlinear function using State Transition Tensors (STTs),14

sigma-point based methods,15, 16 quadrature, or cubature.17 Each element has a smaller uncertainty than theinitial Gaussian distribution at epoch and therefore, the Gaussian assumption for each element holds for flighttimes that are at least as long, or longer. The weighted sum of the Gaussian elements after propagation ap-proximates the resulting non-Gaussian PDF. GMMs have been successfully used in orbital mechanics foruncertainty propagation,18, 19 orbit determination,20, 21 and conjunction assessment.22, 23

PCE24 uses orthogonal polynomial (OP) expansions as a surrogate model for quantifying uncertainty. Themost suited polynomial is chosen using the Wiener-Askey scheme and depends on the initial uncertaintydistribution.25 It is also possible to compute optimal orthogonal polynomials for arbitrary PDFs that arenot part of the Wiener-Askey scheme using arbitrary PCE (aPCE).26 For Gaussian distributions, Hermitepolynomials are optimal.24, 25 The coefficients of the multivariate polynomials have to be computed, thatmap a multivariate independent and identically distributed (i.i.d.) random variable to the final PDF. Once thepolynomial coefficients are computed, sampling from the PCE generally has a lower computational cost thana full-blown MC run. PCE has been used in many fields for uncertainty quantification of computationallyintensive models.27, 28, 29, 30 In orbital mechanics, PCE has been previously used for uncertainty propagation31

and conjunction assessment.32, 33

The motivation for combining PCE and GMMs (GMM-PC) for uncertainty propagation arises from acomparison with the finite element method (FEM). The initial uncertainty is considered to be an object ofinterest and the non-linear function is an applied loading. In FEM, a mesh grid is generated over the object todiscretize it into smaller and simpler geometries. The exact solution is obtained at the nodes, but between thenodes inside each element we assume some a polynomial interpolant to approximately obtain the functionalform of displacements. Increasing the number of elements by reducing the size of each element is knownas h-refinement and increasing the order of the interpolant is p-refinement. Selectively modifying both thesize and number of elements, and the order of the polynomial interpolant is hp-refinement.34, 35 Splittingthe initial Gaussian distribution into Gaussian distributions with a smaller differential entropy is analogousto h-refinement. Using a higher order accurate method of propagating the uncertainty of a single Gaussianelement is p-refinement. The lower end of p-refinement is to use the function derivative to linearly propagatethe mean and covariance. Methods of increasing accuracy are higher order STTs,14 the sigma-point methodsof varying accuracy levels, and PCE with increasing polynomial orders. A proof of concept is presented hereto demonstrate that a hp-refinement method of using PCE for the propagation of a GMM is applicable for

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orbit uncertainty propagation by increasing accuracy while having a lower computational cost than a MCsimulation.

The theoretical background for PCE, GMMs, and the hybrid combination of both is first presented. Theperformance of the GMM-PC is then compared to using only a PCE in three highly nonlinear orbit uncertaintypropagation test cases.

POLYNOMIAL CHAOS EXPANSION

For a PCE, the uncertainty in variables through a transformation is represented by a series of orthogonalpolynomials.

u(ξ, t) =

∞∑i=0

ci(t)Ψi(ξ) (1)

In Eq. (1) ξ is a random variable. The orthogonal polynomials Ψi are defined by the following inner productin a Hilbert space: ∫ ∞

−∞Ψm(ξ)Ψn(ξ)w(ξ) = 0 (2)

Based on the distribution of the random variable, the orthogonal polynomial type and weighing function,w(ξ) from Eq. (2), are chosen from the Weiner-Askey25 scheme.

In this work initial the initial distribution is assumed to be Gaussian and therefore, probabilists’ Hermitepolynomials are chosen according to the Wiener-Askey25 scheme. The probabilists’ Hermite polynomials arenormalized and the weight function is:

w(x) =1√2πe

−x22 (3)

The new weight function is the standard normal distribution, which effectively normalizes and improvesthe numerical properties. The normalized Hermite polynomials are found by using the following recursiverelation:36

ψn+1(ξ) = ξψn(ξ)− nψn−1(ξ) (4)where

ψ0 = 1 ψ1 = ξ (5)

The polynomials are further normalized after computing all the polynomials up to a given order:

ψn(ξ) = ψn(ξ)/√n! (6)

The infinite series from Eq. (1) is truncated at some order for implementation. The orthogonal normalizedunivariate Hermite polynomials up to order 5 are seen in Figure 1. The orbit uncertainty propagation problemis a multivariate problem and therefore, requires orthogonal multivariate polynomials. Multivariate polyno-mials are generated from a set of i.i.d. random variables and can be created using the multi-index notation.The multivariate polynomial is:

u(ξ, t) =

L∑i=0

ci(t)Ψαi(ξ) (7)

In Eq. (7), L is

L =(n+ l)!

n!l!(8)

where n is the dimension of ξ and l is the maximum order of the truncated univariate polynomial. A givenorder L̄ of the multivariate polynomial is the sum of the elements of the multi-index vector. The L multi-index vectors αi ∈ Rn contain the orders of the n i.i.d. univariate polynomials. Each element αi(j) ≥ 0 andall the L combinations have to be found for d ∈ {0, 1, . . . , l} such that:

n∑k=1

αi(k) = d (9)

3

The multivariate polynomial is the product:

Ψαi(ξ) =

n∏k=1

ψαi(k)(ξk) (10)

If the output is a vector function of dimension m, m × L coefficients ci(t) have to be computed. Figure 2shows two examples of bivariate Hermite polynomials.

−2 0 2

−5

0

5

ξ

ψ(ξ)

Figure 1: Normalized probabilists Hermite polynomials up to the 5th order

−20

2

−2

0

2−5

0

5

ξ1ξ2

Ψ

−20

2

−2

0

2−2

−1

0

1

2

ξ1ξ2

Ψ

Ψ = ψ1(ξ1)ψ1(ξ2) Ψ = ψ2(ξ1)ψ3(ξ2)

Figure 2: Bivariate normalized probabilists Hermite polynomials

The coefficients ci(t), determine the response surface of the surrogate model consisting of the Hermitepolynomials. The two major methods of finding these coefficients are the intrusive and the non-intrusivemethods. The intrusive method requires knowledge of the nonlinear function that determines the evolution ofthe random vector of inputs. The truncated polynomial expansions are introduced into the model equationsand solved using a Galerkin projection of the equations on the polynomial space. A system of equations issolved for ci(t). The intrusive method cannot be used with black-box dynamics because existing codes haveto be rewritten and therefore, is not considered in this work.

The non-intrusive method does not require any knowledge of the propagation function. Given that thesystem can be solved for a sample initial condition, the projection property (Galerkin Projection) for approx-imating the coefficients in Eq. (7) can be used:

ci(t) =

∫u(ξ, t)Ψαi(ξ)p(ξ)dξ (11)

4

where p(ξ) is the PDF of ξ. The coefficients can be computed by a quadrature numerical approximation ofEq. (11), MC sampling, or Least Squares (LS) regression. The results in this paper use the LS regressionfor computing the coefficients. All non-intrusive methods generate sample points from ξ ∼ N (0n, I3×3). Asquare-root factor of the covariance matrix linearly converts ξ from an i.i.d. variable to the initial multivariateGaussian distribution.

Random Sampling

Both LS and MC methods for numerically approximating of Eq. (11) are based on random sampling. TheMC approach uses N random samples that are generated form ξ ∼ p(ξ) and each coefficient ci(t) is solvedas:

ci(t) ≈1

N

N∑j=1

u(ξj , t)Ψαi(ξj) (12)

For LS regression, N random samples are generated for ξ. The coefficients minimize the squared differencebetween the propagated sample points and the PC expansion.

ci(t) ≈ arg minĉi(t)

1

N

N∑j=1

(u(ξj , t)−

L∑i=1

ĉi(t)Ψαi(ξj)

)2(13)

Equation 13 is solved by rewriting into the traditional LS form using the following matrix containing the Lmultivariate polynomials evaluated at the N nodes:

Ψ =

Ψα1(ξ1) . . . ΨαL(ξ1)... . . . ...Ψα1(ξN ) . . . ΨαL(ξN )

(14)Arranging the coefficients and function evaluations into vectors results in the following linear system:

(ΨTΨ

) c1(t)...cL(t)

= ΨT u(ξ1, t)...u(ξN , t)

(15)Using MC to solve for the coefficients results in a slow convergence rate. The LS method suffers from thecurse of dimensionality where the combination of increasing problem dimension and order of the polynomialscale the number of required evaluations in a supralinear manner. For an n-dimensional input state withpolynomials of maximum order l, the number of terms L in the multivariate polynomial is computed usingEquation 8. The number of terms in a 6 dimensional multivariate polynomial as a function of l is shown inFigure 3.

The LS regression method requires approximately 2L sample points to solve for the coefficients. However,many coefficients have a small value and therefore, compressive sensing techniques can generate a sparserepresentation of c(t) = [c1(t), . . . , cL(t)]T so that N < L function evaluations suffice. To generate a truesparse representation of c(t), the L0 norm of c(t) has to be minimized subject to the L2 norm conditionsbetween the function evaluations and the PC solution. However, minimizing the L1 norm instead of theL0 norm of c(t) converts the problem into a convex optimization problem that can be solved with commonsolvers such as CVX.37, 38

c(t) ≈ arg minĉ(t)‖ĉ(t)‖1 subject to ‖u(ξ, t)−Ψĉ(t)‖2 ≤ ε (16)

In Eq.(16), u(ξ, t) = [ξ1, . . . , ξN ]. Adding weights to the optimization problem takes into account some apriori information about PC characteristics in order to improve performance.

c(t) ≈ arg minĉ(t)‖Wĉ(t)‖1 subject to

∥∥∥Λ̃ (u(ξ, t)−Ψĉ(t))∥∥∥2≤ ε (17)

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In Eq.(17), W = diag (w) ∈ RL×L and Λ̃ = diag(λ̃)∈ RN×N are diagonal weight matrices. The

elements of W satisfy the following criteria:

0 < w(i) ≤ 1 (18a)

max (w(i)) = 1 (18b)

w(i) ∝n∑k=1

αi(k) (18c)

In Eq.(18), n is the dimension of the state and αi is the multi-index vector. The effect of W is to increase theabsolute value of the coefficients corresponding with higher order multivariate polynomials. Due to the con-vergence property of PC, the absolute value of a coefficient should decrease as the order of the correspondingpolynomial increases. An arbitrarily large polynomial order l can be chosen without the danger of over-fittingbecause the lower orders are given a higher priority due to the weighting. The weight matrix Λ̃ contains thePDF values of the sample points from the initial distribution and effectively reduces the influence of outliers.In this work, ξ ∼ N (0n, In×n) and therefore λ̃, which is the diagonal of Λ̃ from Eq. (17), is:

λ̃(j) = (2π)−n2 e−

12ξ

Tj ξj (19)

Quadrature Method

N points are chosen deterministically when the quadrature method is used. N , the locations of the nodesξj , and the corresponding weights qj depend on the dimension of the problem, and the accuracy and type ofquadrature rule. The coefficients ci(t) are solved as:

ci(t) ≈N∑j=1

qju(ξj , t)Ψαi(ξj) (20)

A full tensor product for the multivariate quadrature requires N = ln function evaluations and therefore, alsosuffers from the curse of dimensionality. Sparse grids compute quadrature nodes using sparse tensor productand therefore, reduce the number of function evaluations to N < ln for high dimensions. The number offunction evaluations required for using a full tensor product quadrature rule and two different sparse grids,Smolyak39 and Genz-Keister,40 is shown in Figure 3. The nodes for the bivariate full tensor product gridand the two sparse grids are shown in Figure 4. Sparse grids require more nodes than the full grid for the2-dimensional case but become increasingly efficient for higher dimensions.

GAUSSIAN MIXTURE MODELS

A GMM approximates any PDF in an L1-distance sense by using a weighted sum of Gaussian probabilitydistribution functions.13

p (x) =

N∑i=1

αipg (x;µi,Pi) (21)

N is the number of Gaussian probability distribution functions, and αi is a positive non-zero weight, whichsatisfies ∀ αi > 0 the following constraint:

N∑i=1

αi = 1 (22)

For uncertainty propagation, the initial Gaussian distribution is split into a GMM and each element ispropagated through the nonlinear function. Standard Gaussian propagation techniques such as STTs14 or

6

2 4 6 8 10 12 1410

0

105

1010

lN

um

ber

of T

erm

s

Full GridSmolyakKonrad−PattersonL

Figure 3: Terms required for multivariate polynomials and the number of quadrature points for a 6 dimen-sional input state

ξ1

ξ 2

ξ1

ξ 2

ξ1ξ 2

(a) Full Grid (b) Smolyak (c) Genz-Keister

Figure 4: Gauss-Hermite quadrature nodes of order 10 for normalized bivariate probabilists’ Hermite poly-nomials

sigma-point methods15, 16 are commonly used to approximate the Gaussian elements post propagation. Al-though each element remains Gaussian, the weighted sum forms a non-Gaussian approximation of the truedistribution. Modifications of the procedure exist required, when the nonlinear function is the solution of anODE: the weights can be updated post-propagation41 or the elements can be further split into more elementsor merged mid-propagation.18 However, these modifications are not considered for this work.

Instead of forming a GMM approximation of the initial multivariate Gaussian distribution, a univariateGMM library of the standard normal distribution is formed.18, 20, 23, 42 The univariate library is applied alonga column of the square-root factor of the covariance matrix in order to form a GMM approximation of amultivariate Gaussian. The univariate splitting library has to be computed only once and is stored in the formof a lookup table. Finding the univariate library is converted to an optimization problem where the distancebetween the GMM and the standard normal distribution is minimized. The L2 distance is used instead of L1because a closed-form solution exists for the L2 distance between a GMM and a Gaussian distribution. Alibrary where all the standard deviations in the split are the same (homoscedastic), σ =

√1/N , and odd N

up to 39 elements is used in this work.43, 44 With increasing N , σ decreases and therefore, the differentialentropy of each element decreases as seen in Figure 5.

To apply the univariate splitting library to a multivariate Gaussian distribution pG ∼ N (µ,P), the uni-

7

0 10 20 30 40−0.5

0

0.5

1

1.5

# Elements in GMME

lem

en

t d

iffe

ren

tia

l e

ntr

op

y

Figure 5: Differential entropy per element

variate splitting library is applied along a column of the square-root S of the covariance matrix:

P = SST (23)

For an n-dimensional state, the covariance matrix of each element is:

Pi = [s1 . . . σsk . . . sn] [s1 . . . σsk . . . sn]T (24)

where sk is the desired column of S. The means of the multivariate GMM are:

µi = µ+ µisk (25)

If Cholesky or spectral decomposition is used to generate S, the possible splitting options are limited to2n directions. However, it is possible to apply the univariate splitting direction along any desired directionby generating a square-root matrix with one column parallel to the input direction.45 For extremely non-linear problems, splitting along a single direction may not account for the entire non-linearity of the problem.Therefore, splitting the initial multivariate distribution in multiple directions is required in order to betterapproximate the non-Gaussian behavior post-propagation.44, 46 In such cases the splitting library can beapplied recursively as a tensor product to split along multiple directions.

GAUSSIAN MIXTURE MODEL POLYNOMIAL CHAOS

Increasing the order of a PCE invokes the curse of dimensionality. As a PCE converges to a good ap-proximation of the model, the effect of the higher order coefficients should be smaller than that of the lowerorder coefficients. In addition to a reduced effect, the number of coefficients grows factorially as the orderis increased. Therefore, a strong incentive exists to use the lowest order PCE as possible, subject to an ac-ceptable accuracy level. The non-Gaussian behavior of the PDF post-propagation depends primarily on thenonlinearity of the function and the size of the initial uncertainty. Splitting the initial Gaussian distributioninto smaller distributions, decreases the size of the initial uncertainty of each element. Therefore, lower orderpolynomials are sufficient for an accurate answer. Increasing the number of splits in the initial distributionincreases the required number of function evaluations linearly.

The initial PDF has been previously decomposed into smaller subdomians in the Multi-Element general-ized Polynomial Chaos (ME-gPC) method.47 For a Gaussian initial distribution, the elements in the ME-gPCmethod do not have a Gaussian distribution as seen in Figure 6 for the standard normal distribution. Therefore,orthogonal polynomials are constructed for the arbitrary distributions using the aPCE technique.26 Becauseeach element in the GMM-PC technique has a Gaussian initial distribution, standard Hermite polynomials

8

can be used. Analytical recursive relations exist for Hermite polynomials and therefore, complicated multi-variate integrals do not have to be solved to compute the coefficients for orthogonal polynomials for arbitrarydistributions. In the ME-gPC method, the initial PDF is exactly represented by the elements, but the numer-ically computed polynomials are approximations. However, the coefficients of the Hermite polynomials areaccurately know but the initial GMM PDF is an approximation in the GMM-PC method.

ξ

p(ξ

)

Element 1

Element 2

Element 3

ξ

p(ξ

)

Element 1

Element 2

Element 3

GMM-PC ME-gPC

Figure 6: The difference in a three element refinement of the standard normal distribution between theGMM-PC and the ME-gPC techniques

RESULTS

Three test cases that benefit from using GMM-PC are presented in this section. The first two cases arespace objects in an eccentric Medium Earth Orbit (MEO) and a Molniya orbit with two-body dynamics. Thefinal test case is an object in a Geosynchronous Transfer Orbit (GTO) under the influence of perturbations.Cartesian coordinates in the ECI frame are used to express all the uncertainties.

The two-sample univariate Cramér-von Mises test is used to compare the performance of the PCE andGMM-PC with respect to an MC simulation:48

CvM =N1N2N1 +N2

∫ ∞−∞

[FN1(x)−GN2(x)]2 dHN1+N2(x) (26)

where FN1(x) is the empirical distribution of the surrogate model with N1 samples, GN2(x) is the empiricaldistribution of the MC simulation with N2 samples, and HN1+N2(x) is the empirical distribution functionof two samples together. The univariate test is computed along each of the individual unit reference framedirections of the Radial-Intrack-Crosstrack (RIC) reference frame centered at the mean of the MC simulation.

The univariate splitting library used to generate the initial GMMs is from Vittaldev et. al.23 The PCE coef-ficients are computed using LS where the number of function evaluations is twice the number of multivariatecoefficients, L.

Medium Earth Orbit

The first test case is a MEO with two-body dynamics propagated for 3 days. The initial state and uncertaintyat epoch are found in Table 1. The highest nonlinearity and therefore, the non-convergence is along the initialvelocity direction for the MEO case as seen in the values of the univariate coefficients in Figure 7.

The coefficient values corresponding to the order of univariate polynomials within the multivariate poly-nomials from Eq. (10) for a one element GMM-PC are plotted in Figure 7. The square-root of the initial

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a e i Ω ω ν σrR σrI σrC σvR σvI σvC

24, 475km 0.5 0◦ 0◦ 0◦ 0◦ 1 km 1 m 1 m 0.1 m/s 0.1 m/s 0.1 m/s

Table 1: Initial osculating orbit elements and uncertainty expressed in the RIC coordinate frame for a MEOobject

covariance is constructed such that ξ1 is along the velocity vector vv. Figure 7(a) shows that this case ishighly non-linear along the velocity direction because increasing the order of the PCE does not result in a fastreduction in the values of the coefficients. The expansion along the other directions does converge, as shownby the reducing coefficient size in Figure 7(b). Applying a univariate split along ξ1 reduces the size of theuncertainty and therefore, polynomials of a lower order can be used to achieve convergence.

(a). ξ1: Velocity direction (b). ξ2 − ξ6: Other directionsFigure 7: Coefficient values for the corresponding univariate polynomial orders for the directions of the i.i.d.state for a one element GMM-PC used to propagate uncertainty of an object in MEO

The CvM values for the samples in the RIC frame with respect to the MC simulation is shown in Figure8. Using only PCE, which is a one element GMM-PC, does not result in a converged solution in the vIdirection. As the number of splits along the velocity direction is increased, a lower computation cost resultsin an increased accuracy.

The initial Gaussian distribution is split into a GMM with up to 9 elements along the vv direction. Thesum of L of all the elements is used as a proxy for the computation cost. The GMM-PC is propagated for thedesired flight time of 3 days and is sampled. The CvM values for the samples in the RIC frame with respectto a MC simulation is shown in Figure 8. Using only PCE, which is a one element GMM-PC, does not resultin a converged solution with low error for all the univariate directions except for vI . As the number of splitsalong the velocity direction is increased, a lower computation cost results in a more accurate representationof the uncertainty. Sampled point clouds from the MC simulation, the PCE, a 9 element GMM split alongvv using the second order Divided Difference Transform,16 and a 9 element 6th-order GMM-PC are found inFigures 9 and 10.

Figures 9 and 10 clearly show the accurate representation of the MC simulation by the GMM-PC. Pointclouds have a tendency to exaggerate the visual effect amount of outliers because point density is not easilyrepresented. There are many outliers present in the pure PCE and GMM solutions, especially in the radialdirection. The GMM approximation is very jagged because it is similar to plotting a circle with only 9 points.Each element still must approximate a curve with a line, so the semi minor axis of the post ellipse gets inflatedto accommodate for curvature of the true sub-distribution each element is fitting.

10

L

101

102

103

104

CvM

10-2

100

102

104

rr

ri

rc

vr

vi

vc

L

101

102

103

104

105

|CvM

| 2

10-2

100

102

104

N = 1

N = 3

N = 5

N = 7

N = 9

(a). CvM metric for PCE (b). 2-norm of vector of CvM metrics for GMM-PCFigure 8: Two-sample univariate Cramer-von-Mises metric in the RIC frame for GMM-PC, split along

velocity, with respect to an MC simulation of 1, 000, 000 samples of an object in MEO. L is the total numberof coefficients required per direction for the multivariate polynomial and is analogous to the compute cost.

(a). MC (b). GMM: N = 9

(c). PCE (d). GMM-PC: l = 6, N = 9

Figure 9: 1,000,000 samples for position in the Radial-Intrack plane from the MC, a PCE, and GMM-PCand GMM simulations with an initial split applied along vv for the uncertainty propagation of a MEO object

11

(a). MC (b). GMM: N = 9

(c). PCE (d). GMM-PC: l = 6, N = 9

Figure 10: 1,000,000 samples for velocity in the Radial-Intrack plane from the MC, a PCE, and GMM-PCand GMM simulations with an initial split applied along vv for the uncertainty propagation of a MEO object

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Molniya Orbit

The second test case is an orbit from Jones et. al.31 to show the benefits of using GMM-PC. The initialstate and uncertainty for the Molniya orbit are found in Tables 2 and 3, respectively. The orbit is propagatedfor 10 days to compare the final statistics. This case is known to not converge when Cartesian coordinatesare used to represent the uncertainty.31

a e i Ω ω ν

26, 562km 0.741 63.4◦ 90◦ −90◦ 0◦

Table 2: Initial osculating orbit elements for an object in a Molniya orbit31

σx σy σz σvx σvy σvz

10 m 10 m 10 m 1 m/s 1 m/s 1 m/s

Table 3: Initial uncertainty expressed in Cartesian coordinates in the ECI frame for the object in a Molniyaorbit

Figure 11(a) shows that this case is again highly non-linear along ξ1, velocity, direction because increasingthe order of the PCE does not result in reducing values of the coefficients. The expansion along the otherdirections does converge, as shown by the reducing coefficient size in Figure 11(b).

(a). ξ1: Velocity direction (b). ξ2 − ξ6: Other directionsFigure 11: Coefficient values for the corresponding univariate polynomial orders for the directions of the

i.i.d. state for a one element GMM-PC used to propagate uncertainty of an object in a Molniya orbit

The initial Gaussian distribution is split into a GMM with up to 9 elements along the vv direction. TheGMM-PC is propagated for the desired flight time of 10 days and is sampled. The CvM values for thesamples in the RIC frame with respect to a MC simulation is shown in Figure 12. Using only PCE, which is aone element GMM-PC, does not result in a converged solution with low error for all the univariate directionsexcept for rC . As the number of splits along the velocity direction is increased, a lower computation costresults in an increased accuracy.

Because only two-body dynamics are used and the initial uncertainty in the position coordinates is only 10m, the final distribution is thinly spread along the orbit. Therefore, the resulting distributions of the MC, PCE,

13

GMM using , and GMM-PC are shown as histograms in the RIC frame of the MC mean in Figure 13. Thesamples from the 9 element 6th-order GMM-PC clearly form a good approximation of the MC histograms inall the univariate directions of the RIC frame. PCE and the pure GMM fail to approximate the tails of thedistributions.

L

101

102

103

104

CvM

10-2

100

102

104

106

rr

ri

rc

vr

vi

vc

L

101

102

103

104

105

|CvM

| 2

10-2

100

102

104

106

N = 1

N = 3

N = 5

N = 7

N = 9

(a). CvM metric for a one element GMM-PC (b). 2-norm of vector of CvM metrics for GMM-PCFigure 12: Two-sample univariate Cramer-von-Mises metric in the RIC frame for GMM-PC, split along

velocity, with respect to an MC simulation of 1, 000, 000 samples of an object in a Molniya orbit. L is thetotal number of coefficients required per direction for the multivariate polynomial and is analogous to thecompute cost.

Geosynchronous Transfer Orbit

The third and final test case is a GTO with perturbations due to atmospheric drag, non-spherical Earth ofdegree and order 8, Solar Radiation Pressure (SRP), and third-body attraction of the Sun and the Moon. Theinitial state and uncertainty for the GTO are found in Tables 2 and 3, respectively. The highest nonlinearityand therefore, the non-convergence is again along the initial velocity direction as seen in the values of theunivariate coefficients in Figure 14.

a e i Ω ω ν

24, 475km 0.731 7◦ 250◦ 8◦ 0◦

Table 4: Initial osculating orbit elements for a GTO orbit object

σx σy σz σvx σvy σvz

5 m 5 m 5 m 0.5 m/s 0.5 m/s 0.5 m/s

Table 5: Initial uncertainty expressed in Cartesian coordinates in the ECI frame for the GTO object

The CvM values for the samples in the RIC frame with respect to the MC simulation is shown in Figure15. Using a one element GMM-PC, does not result in a converged solution in the vI direction. As the numberof splits along the velocity direction is increased, a lower computation cost results yields a higher accuracy.The behavior of the point clouds and the histograms are similar to the Molniya case and are therefore notpresented.

14

Figure 13: Histograms in the RIC frame of 1,000,000 samples from the MC , a PCE, and a GMM-PC andGMM with an initial split applied along vv for the uncertainty propagation of an object in a Molniya orbit

15

(a). ξ1: Velocity direction (b). ξ2 − ξ6: Other directionsFigure 14: Coefficient values for the corresponding univariate polynomial orders for the directions of the

i.i.d. state for a one element GMM-PC of a GTO

L

101

102

103

104

CvM

10-2

100

102

104

106

rr

ri

rc

vr

vi

vc

L

101

102

103

104

105

|CvM

| 2

10-2

100

102

104

106

N = 1

N = 3

N = 5

N = 7

N = 9

(a). CvM metric for PCE (b). 2-norm of vector of CvM metrics for GMM-PCFigure 15: Two-sample univariate Cramer-von-Mises metric for PCE and GMM-PC, split along velocity,

with respect to an MC simulation of 1, 000, 000 samples. L is the total number of coefficients required perdirection for the multivariate polynomial and is analogous to the compute cost.

16

CONCLUSION

A Polynomial Chaos Expansion forms a surrogate model for uncertainty propagation through a non-linearfunction, which is computationally more efficient to sample from when compared to a full Monte Carlo sim-ulation. The performance of the PCE method depends on the nonlinearity of the function and the size ofthe initial uncertainty. For the three cases of uncertainty propagation in Cartesian coordinates shown here,increasing the order of the PCE results in slow convergence and a large computation cost. Splitting the initialGaussian distribution into a GMM with the weighted Gaussian distributions along the most influential direc-tion, which is initial velocity vector for the cases presented here, reduces the uncertainty along that direction.Therefore, a more accurate description of the propagated uncertainty is found at a lower computation cost,compared to simply increasing the order of the PCE. However, the GMM-PC technique is only recommendedfor scenarios where a PCE does not converge such as the three orbit uncertainty propagation test cases fromthis paper.

Comparing uncertainty propagation to FEM makes the GMM-PC method a hp-refinement method wherethe order of the PCE and the number of splits are independently adapted. When compared to a ME-gPCmethod, the GMM-PC technique is easier to implement because the PDF of each element remains Gaussian.Therefore, the analytical Hermite polynomials are used instead of developing a gPC framework to find or-thogonal multivariate polynomials with respect to arbitrary PDFs. In the ME-gPC method, errors may creepinto the computation of the orthogonal polynimials, while the error in the GMM-PC technique arises due tothe approximation of the initial PDF with a finite number of GMM elements. Therefore, the best possibleperformance of the GMM-PC has an upper bound which is equivalent to the approximation accuracy of theGMM splitting library. For extending a PCE implementation to the GMM-PC, only the weights, means, andstandard deviations of a univariate splitting library are required, which can be precomputed and stored in atabular manner.

Many avenues for future work open up from the combination of GMMs and PCE into a combined GMM-PC framework. The types of initial orbital conditions that fail to converge with a traditional PCE, but performsatisfactorily with a GMM-PC should be investigated and classified. Adaptively picking the order of the PCEper GMM-PC element could require fewer function calls for certain problems. Currently, the Least Squaresmethod does not reuse function evaluations when the initial splitting is changed, which could become aWeighted Least Squares problem. The GMM splitting and PCE order become the post-processing phase,depending only on the computational budget, and not an a priori decision.

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http://cvxr.com/cvxhttp://stanford.edu/~boyd/graph_dcp.htmlhttp://stanford.edu/~boyd/graph_dcp.html

IntroductionPolynomial Chaos ExpansionRandom SamplingQuadrature Method

Gaussian Mixture ModelsGaussian Mixture Model Polynomial ChaosResultsMedium Earth OrbitMolniya OrbitGeosynchronous Transfer Orbit

Conclusion