The Pennsylvania State UniversityThe Graduate SchoolCollege of Engineering

A PROBABILISTIC APPROACH FOR UNCERTAIN LAMBERT

PROBLEM AND REACHABILITY SET CALCULATIONS

A Thesis inAerospace Engineering

byZachary Hall

© 2018 Zachary Hall

Submitted in Partial Fulfillmentof the Requirementsfor the Degree of

Master of Science

December 2018

The thesis of Zachary Hall was reviewed and approved∗ by the following:

Puneet SinglaAssociate Professor of Aerospace EngineeringThesis Advisor

David B. SpencerProfessor of Aerospace Engineering

Amy R. PritchettProfessor of Aerospace EngineeringHead of the Department of Aerospace Engineering

∗Signatures are on file in the Graduate School.

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AbstractThe uncertain Lambert problem and computation of reachability sets are in manyways complimentary problems with important applications in Space SituationalAwareness (SSA). Formulating the solution to these problems in a stochastic frame-work is an important intermediate step to enabling the next generation of collisionassessment, satellite tracking and characterization capabilities. Traditionally, ac-curately computing solutions to these problems in a probabilistic manner waseither computationally infeasible due to the large number of simulations requiredto capture the statistical properties of the solution, or was only accurate nearthe nominal solution due to the inability of linear variational methods to capturehigher order statistical moments of the solution. Using a Taylor series expansionof the deterministic solution, a polynomial approximation method is proposed tocalculate the higher order sensitivity matrices. An orthogonal polynomial basisfunction expansion is used to represent the solution of the uncertain Lambert andreachability problems. The coefficients of these orthogonal polynomials representthe higher order sensitivity matrices. The Conjugate Unscented Transformation(CUT) method is utilized for the purposes of computing the multi-dimensionalexpectation integrals required to determine the least squares coefficients, and offersextensive computational savings compared to other quadrature methods. Addition-ally, the CUT method allows the computation of these sensitivity matrices withoutthe cumbersome need to explicitly take higher order partial derivatives, as wellas having the advantage of achieving higher accuracy over a larger input domainthan was previously available using other numerical integration methods. Themathematical framework for the polynomial approximation method is laid out andtest case simulations for both the uncertain Lambert problem and the reachabilityset problem are presented. Discussion of results, as well as the relative advantagesand limitations of the method for each simulation is included.

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Table of Contents

List of Figures vi

List of Tables viii

Acknowledgments ix

Chapter 1Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Chapter 2Mathematical Background 52.1 The Taylor Series Expansion . . . . . . . . . . . . . . . . . . . . . . 52.2 The Least Squares Method . . . . . . . . . . . . . . . . . . . . . . . 92.3 Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Numerical Integration Methods . . . . . . . . . . . . . . . . . . . . 15

2.4.1 The Monte Carlo Method . . . . . . . . . . . . . . . . . . . 162.4.2 Gaussian Quadrature . . . . . . . . . . . . . . . . . . . . . . 18

2.5 Example Higher Order Sensitivity Problem . . . . . . . . . . . . . . 222.5.1 Alternatives to Gaussian Quadrature . . . . . . . . . . . . . 24

Chapter 3The Uncertain Lambert Problem 273.1 Problem Background . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Simulation Description . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . 313.2.2 Test Case 1: LEO . . . . . . . . . . . . . . . . . . . . . . . . 353.2.3 Test Case 2: GTO . . . . . . . . . . . . . . . . . . . . . . . 39

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Chapter 4The Reachability Set Problem 424.1 Problem Background . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2 Simulation Description . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2.1 Test Case 1: Two Burn Maneuver . . . . . . . . . . . . . . . 454.2.2 Test Case 2: Lunar Orbit Insertion . . . . . . . . . . . . . . 48

4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.3.1 Test Case 1: Two Burn Maneuver . . . . . . . . . . . . . . . 524.3.2 Test Case 2: Lunar Orbit Insertion Maneuver . . . . . . . . 53

Chapter 5Conclusions 615.1 Summary of Current Work . . . . . . . . . . . . . . . . . . . . . . . 615.2 Furture Research Direction . . . . . . . . . . . . . . . . . . . . . . . 62

Bibliography 64

v

List of Figures

2.1 Taylor Series Approximations for sin(x) . . . . . . . . . . . . . . . . 72.2 Inner Product of 2D Vectors . . . . . . . . . . . . . . . . . . . . . . 132.3 Error in Monte Carlo Approximation vs Number of MC Points . . . 172.4 Polynomial Approximation for N = 50 and Varying d . . . . . . . . 222.5 Polynomial Approximation for d = 7 and Varying N . . . . . . . . . 232.6 A Schematic of CUT Axes . . . . . . . . . . . . . . . . . . . . . . . 252.7 Comparison of Quadrature Schemes for Same Order of Accuracy . . 25

3.1 Uncertain Lambert Problem Diagram . . . . . . . . . . . . . . . . . 303.2 Polynomial Approximation Method Summary . . . . . . . . . . . . 313.3 Error in Expectation Values of x vs. N . . . . . . . . . . . . . . . . 333.4 Plot of Coefficient Order of Magnitude for Varying Approximation

Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.5 Test Case 1: V1 Spatial Representation . . . . . . . . . . . . . . . . 373.6 Test Case 1: V1 Mahalanobis Distance Representation . . . . . . . . 383.7 Plot of Coefficient Order of Magnitude for Varying Approximation

Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.8 Test Case 2: V1 Spatial Representation . . . . . . . . . . . . . . . . 403.9 Test Case 2: V1 Mahalanobis Distance Representation . . . . . . . . 41

4.1 Satellite Reachability Problem Diagram . . . . . . . . . . . . . . . . 454.2 Diagram of Maneuver Geometry VNC Frame . . . . . . . . . . . . . 464.3 Test Case 1: Nominal Orbit . . . . . . . . . . . . . . . . . . . . . . 474.4 Test Case 2: Diagram of Arrival Geometry . . . . . . . . . . . . . . 484.5 Test Case 2: Nominal LOI Maneuver . . . . . . . . . . . . . . . . . 504.6 Test Case 1: 4th Order Reachability Set Over Time . . . . . . . . . 534.7 Test Case 1: Spatial Position Error Plot . . . . . . . . . . . . . . . 544.8 Test Case 1 Mahalanobis Distance Position Error . . . . . . . . . . 554.9 Test Case 2 LNL: 4th Order Reachability Set Over Time . . . . . . 564.10 Test Case 2 LNL: Spatial Position Error Plot . . . . . . . . . . . . . 57

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4.11 Test Case 2 LNL: Mahalanobis Distance Position Error Plot . . . . 584.12 Test Case 2 HNL: 4th Order Reachability Set Over Time . . . . . . 594.13 Test Case 1 HNL: Spatial Position Error Plot . . . . . . . . . . . . 594.14 Test Case 1 HNL: Mahalanobis Distance Position Error Plot . . . . 60

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List of Tables

2.1 Table of Common Orthogonal Polynomials . . . . . . . . . . . . . . 15

3.1 Test Case 1 LEO State Vector . . . . . . . . . . . . . . . . . . . . . 323.2 Test Case 2: GTO State Vector . . . . . . . . . . . . . . . . . . . . 323.3 Statistical Moments Calculated Using CUT8 . . . . . . . . . . . . . 343.4 Test Case 1: Covariance Σ Using CUT8 Method . . . . . . . . . . . 353.5 Test Case 1: 1st Order Approximation Coefficients . . . . . . . . . . 353.6 Test Case 1: Initial Velocity Error . . . . . . . . . . . . . . . . . . . 383.7 Test Case 2: Initial Velocity Error . . . . . . . . . . . . . . . . . . . 40

4.1 Test Case 1: Nominal Transfer Orbit Parameters . . . . . . . . . . 464.2 Test Case 1: Nominal Burn Parameters . . . . . . . . . . . . . . . . 474.3 Nominal Maneuver Uncertainties . . . . . . . . . . . . . . . . . . . 474.4 Test Case 2: Nominal Orbit Elements . . . . . . . . . . . . . . . . . 494.5 Test Case 2: Input Uncertainty . . . . . . . . . . . . . . . . . . . . 504.6 Simulation Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . 524.7 Test Case 1: Error Summary . . . . . . . . . . . . . . . . . . . . . . 534.8 Test Case 2 LNL: Error Summary . . . . . . . . . . . . . . . . . . . 564.9 Test Case 2 HNL: Error Summary . . . . . . . . . . . . . . . . . . . 58

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Acknowledgments

I would like to recognize and thank those who have helped me throughout myacademic career, without whom this thesis would not be possible. First, I wouldlike to express my thanks and deepest gratitude to Dr. Puneet Singla for providingconstant support and guidance throughout my first few years as a graduate student.Working as advisee to Dr. Singla has been an incredible learning experience and atrue pleasure due to his depth of understanding, enthusiasm for engineering, andwillingness to help his students day or night. I would also like to extend thanks toDr. David Spencer and Dr. Amy Pritchett for reviewing and providing feedbackon my thesis. My sincere thanks must also be extended to the Air Force ResearchLaboratory (AFRL), and the Science Mathematics and Research for Transformation(SMART) scholarship program for funding my graduate studies.

Additionally, I would like to thank the other members of the Control andAnalysis of Stochastic Systems (CASS) Laboratory. It has been an invaluable helpto have colleauges and friends who are always willing to provide advice and a freshperspective on our mutual research interests. Furthermore, I would like to thank allof the professors and students affiliated with the Astrodynamics Research Group ofPenn State (ARGoPS) for providing an academically stimulating environment andsense of community within our department.

Lastly, I would like to extend thanks to my friends and family. To Emily; Thankyou for your constant love and support and for providing me happiness throughoutmy graduate career. To my friends; Thank you all for the good times we’ve had,and for all the good times to come. To my brother Nate; Thank you for being myconfidant, my friend, and most importantly my role model. Finally, to my parentsLaura and Gerry; Thank you both for your continuous love and encouragement,and for making me the person I am today.

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Chapter 1 |Introduction

1.1 BackgroundSince the beginning of the space age in 1957 with the launch of Sputnik 1 by theSoviet Union, humanity has slowly pushed out into space and established a constantpresence in near-Earth orbit. Whether or not we realize it on a day to day basis,space has become an essential part of everyday life for most modern societies. Someexamples of space technologies that we use benefit from every day include the GlobalPositioning Satellite (GPS) constellation, telecommunication networks, scientificand environmental satellites for weather and disaster preparedness, and militarysupport satellites. In addition to direct space technologies, there are a myriadof spinoff technolgies that stemmed from the National Aeronautics and SpaceAdministration’s (NASA) space exploration program which inculde everything frommedicine to industrial advances to direct consumer products [1]. As humans expandtheir presence in space, certain commonly used orbits become crowded and theproblem of monitoring and managing the objects in space becomes exponentiallymore difficult.

Space Situational Awareness (SSA) is a term that has gained recent popularity inregards to managing this problem, and was underscored as an essential componentto protect national security and other interests in a report by the National ResearchCounsil [2]. According to the U.S. Air Force Doctrine Document 3-14, SSA isdefined as "the requisite foundational, current, and predictive knowledge andcharacterization of space objects and the operational environment upon whichspace operations depend- including physical, virtual, information, and human

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dimensions- as well as all factors, activities, and events of all entities conducting orpreparing to conduct space operations" [3]. In a broad sense, the goal of SSA canbe summarized as the maximization of knowledge about past, present, and futureorbital states, physical characteristics, and capabilites of all natural and artificalsatellites in near-Earth orbit [4]. One major facet of SSA is debris tracking for thepurposes of collision risk assessment. As of July 2018 the NASA Orbital DebrisProgram Office Reports that there are 19,137 man-made objects (active/defunctsatellites and other debris) being tracked by the U.S Space Surveillence Network(SSN) [5]. Coupled with limited sensor availability, these man-made objects as wellas the many thousands of additional natural satellites pose a real problem for anyentity with space assets it wishes to protect.

SSA becomes particularly important in the military domain, where knowledgeof the capabilities needed to protect friendly assests, and to potentially disable ordestroy enemy satellites, become matters of national security [6]. In January 2007,the Chinese intentionally destroyed one of their own defunct weather satellites withan Anti-Satellite (ASAT) missile test, creating a cloud of at least 2,087 pieces ofdebris trackable by the U.S. SSN, as well as an estimated 35,000 pieces as small as1cm in size [7]. This missle test was the single largest space debris creating eventin history, and the resulting debris continues to threaten the International SpaceStation (ISS) and other Low Earth Orbit (LEO) satellites to this day. Althoughthere has never been a publically documented military strike in which one countrydestroys the satellite of another country, the 2007 Chinese ASAT test and similartests from both the United States and Russia, demonstrates that such capabilitescurrently exist.

Unfortunately, the problem of tracking and maintaining a catalog of satellites isexacerbated by the presence of uncertainty in almost all aspects of space operations.Uncertainty in the dynamic model, measurements, orbital state, and maneuvercapabilities can all lead to errors in the ability to predict future states of thesatellite. These uncertainties can dramatically limit or completely eliminate theability to perform SSA tasks such as collision assessment, satellite tracking andcharacterization. In light of this, there is an evident need to formulate some of themost fundemental space problems in a stochasitc framework which can quantifyuncertainty and provide probabalistic solutions to inform decision making.

The Lambert problem and the solution to a given orbit dynamical model

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represent two fundemental and complimentary problems in astrodynamics, whichhave been rigorously studied for decades. The Lambert problem is a two-pointboundary value problem, where the initial and final position of a satellite and thetime of flight are given, and the solution is the initial and final velocity whichsatisfies those conditions. This problem is complimentary to the solution of adynamic model, which solves for the final position and velocity of a satellite giventhe initial position and velocity. In the simplest two body model, there are analyticexpressions for the relationship between initial and final satellite state; however,for more complex dynamic models, numerical integration is frequently used tointegrate the equations of motion forward in time to the final state. One similaritybetween both of these problems is that the given variables are assumed to bedeterministic and known to infinite precision. When we consider the input variablesto have some uncertainty associated with them, as would be the case in realisticsatellite operation scenarios, the solution to these problems becomes stochastic.The stochastic formulation of the Lambert problem is called the uncertain Lambertproblem, and the stochastic formulation of a given dynamical model is known asthe reachability set problem.

Traditionally, any non-deterministic solutions to these two problems use firstorder sensitivity approaches to determine how the solution varies linearly withrespect to the deviation around a nominal solution. These approaches provide ameans to determine the solution in a small region around a nominal trajectory,however they quickly break down due to high non-linearities present in the problemdynamics. The goal of this thesis is to develop and implement the framework neededto develop a probabalistic solution to the uncertain Lambert Problem and thereachability set problem using higher order sensitivity matrices which is valid notonly around a nominal trajectory but over the domain of a probability distributionfunction (pdf).

1.2 Thesis OutlineThis section provides a brief summary of the chapters and overall structure of thisthesis.

Chapter 2 provides the mathematical background necessary for the developmentof the higher order sensitivity method proposed in this thesis. The background

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begins with a discussion of the Taylor series expansion and how polynomials canbe used to approximate non-linear functions. The least squares method and howto formulate minimum error polynomial coefficients with respect to a given pdfby leveraging orthogonal polynomials is presented. Next, orthogonal polynomials,their properties, and how to compute them using Gram-Schmidt Orthogonalizationis discussed. Finally, a summary is given on numerical integration techniques andmethods to compute least squares polynomial approximation coefficients.

Chapter 3 presents the uncertain Lambert problem. Background literatureand gaps in research are identified, and the improvements this thesis provides arediscussed. Two simulation test cases are described in detail with all parametersprovided so that they may be reproduced in the future, and the reults of thesesimulations for varying sensitivity matrix approximation orders are presented anddiscussed.

Chapter 4 presents the reachability set problem, using a similar analysis formatas in Chapter 3. Two test cases taken from literature are described in detail andthe results are presented and discussed for varying sensitivity matrix approximationorders.

Chapter 5 summarizes the thesis and provides concluding remarks on the relativeadvantages and limitations of the higher order sensitivity matrix method proposedin this thesis. This chapter also makes suggestions for future research to expandupon the work in this thesis.

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Chapter 2 |Mathematical Background

This chapter will present the preliminary mathematics underlying the proposedpolynomial approximation method. The concepts presented in this chapter willbe used in numerical analyses in later chapters. This chapter will cover topicsincluding the Taylor Series Expansion, The Least Squares Method, OrthogonalPolynomials, and Numerical Integration Methods.

2.1 The Taylor Series ExpansionThe Taylor series provides an infinite polynomial series expansion of a smoothfunction, f(x) as a function of perturbation δx around a point x∗. In the contextof this thesis δx is characterized as the deviation due to uncertainty in the randomvariable x. For a function to have a valid Taylor series expansion over an openinterval, the function must be infinitely differentiable and analytic over that interval.Assume for the time being, that f(x) is a scalar function. The taylor series for f(x)can be written as:

f(x) = f(x∗) + f ′(x∗)1! (x− x∗) + f ′′(x∗)

2! (x− x∗)2 + f (3)(x∗)3! (x− x∗)3 + ... (2.1)

This series can also be written in compact form given by Eq.(2.2) (see Ref. [8]).

f(x) =∞∑i=0

f (i)(x∗)i! (x− x∗)i (2.2)

Note that if x∗ = 0, the series becomes a special case known as the Maclaurin series.The derivative terms f (i)(x∗) are constant because they are evaluated at the point

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at which the expansion is about, so the Taylor series can be written as an infinitepolynomial series known as a Taylor polynomial.

f(x) = c0 + c1x+ c2x2 + c3x

3 + . . . (2.3)

Since Taylor polynomials are infinite, for practical purposes the polynomial istruncated to a finite number of terms and is referred to as the dth order Taylorpolynomial approximation.

f(x) ≈ p(d)(x) = c0 + c1x+ c2x2 + . . .+ cdx

d (2.4)

The validity of a dth order Taylor series approximation is dictated by size ofthe perturbation δx, and the smoothness of the function f(x). Taylor seriesapproximations can be used to approximate many transendental functions. Onesuch example is the sin function, for which the 7th order approximation is shownbelow:

sin(7)(x) ≈ x− x3

3! + x5

5! −x7

7! (2.5)

Figure 2.1 shows the Taylor series approximation of sin(x) about x∗ = 0 for varyingapproximation orders d on the interval [−π, π]. Notice that as the order increases,the approximation becomes more accurate over a larger portion of the interval.This is the fundemental motivation behind the polynomial approximation method.

In the general case, the polynomial described by Eq. (2.4) can be rearrangedinto any linear combination of cixi to express the polynomial p(d)(x) in terms ofnew coefficients and arbitrary polynomial basis functions φi(x) of maximum poweri for i = 0, 1, . . . d.

p(d)(x) = cTφ(x) = ciφi(x) (2.6)

Notice that for 1D systems the total number of basis functions is M = d + 1,however, this linear growth breaks down in higher dimensions. Consider a generalmulti-dimensional system with (n× 1) input vector x, and (m× 1) output vectory = f(x).

x =

x1

x2...xn

, y =

y1

y2...ym

(2.7)

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-3 -2 -1 0 1 2 3

x

-3

-2

-1

0

1

2

3

f(x)

1st Order Taylor Series

3rd Order Taylor Series

5th Order Taylor Series

7th Order Taylor Series

sin(x)

Figure 2.1. Taylor Series Approximations for sin(x)

Consider the Taylor polynomial approximation for this multi-dimensional system.It is neccessary that the polynomial approximation contains all permutations ofthe input vector for each power, so the total number of basis functions M followsthe factorial relationship given by Eq.(2.8) [9].

M = (n+d)Cd = (n+ d)!d!n! (2.8)

It follows naturally then, that the number of basis functions contained in just theith approximation order follows the relationship given by:

bi = (n+i)Ci − (n+i−1)C(i−1) = (n+ i)!− i(n+ i− 1)!i!n! (2.9)

The dth order polynomial approximation for y must now contain all permuta-tions of x up to a maximum order d. Thus, the general case for the polynomialapproximation is:

yj ≈ c(0)j + cj

(1)Tφ(1) + cj(2)Tφ(2) + . . . cj

(d)Tφ(d) (2.10)

Where yj is the jth output for j = 1, 2 . . .m, φ(i) is a (bi × 1) vector containing

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all ith order permutations of x, and c(i)j is the vector of corresponding coefficients.

In this thesis c(i)j are refered to interchangably as approximation coefficients and

sensitivity matrices. The c(0)j terms represent the mean of yj and the cj

(1) termsrepresent the first order sensitivity terms (similar in a sense to the state transitionmatrix). Consider the example where d = 2 and φ(i) are ith order monomials.

cj(1) =

c

(1)j,1

c(1)j,2...

c(1)j,b1

φ(1) =

x1

x2...xn

, cj(2) =

c

(2)j,1

c(2)j,2...

c(2)j,b2

φ(2) =

x21

x1x2...

x1xn

x22

x2x3...

x2xn...x2n

(2.11)

Similar to the single dimensional case, cj(i) and φ(i) can be rewritten into any

arbitrary polynomial basis functions of maximum power i and new coefficients. Apolynomial approximation for each output element in y can now be written as

yj ≈ cj,iφi fori = 1, 2, . . .Mj = 1, 2, . . .m

(2.12)

y ≈ Cφ(x) (2.13)

where C is an (m ×M) matrix with elements cj,i in the jth row and ith column,and φ(x) is an (M × 1) vector of all arbitrary polynomial basis functions. Equation(2.12) is the most general expression for a Taylor polynomial approximation of amulti-dimensional system.

One additional thing to note on the use of polynomials in general is scaling. Ingeneral a dth order polynomial will become numerically ill conditioned if d is largeand if x << −1 or 1 << x. An ill conditioned polynomial will cause numericalerrors to accumulate due to finite precision computing capabilities. To circumventthis issue, it is common to express the variable x in terms of the normalized

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parameter ζ such that:x = x∗ + Sζ (2.14)

Where S is a scaling Matrix that is dependent on the type of weighting functionbeing considered. The two most common distributions are the uniform and theGaussian distribution. For a uniform distibution ζ is sampled on the interval [−1, 1]and S is a diagonal matrix with terms (bi − ai) where xi is uniform on the interval[ai, bi]. For a Gaussian distribution, ζ is is sampled from a unit variance, zeromean normal distribution, and S =

√Σ where Σ is the covariance matrix of x. By

scaling x in this manner, we can greatly reduce numerical errors in our polynomialapproximation. To this end, the coefficients will be calculated with respect to thenormalized variable ζ such that

y ≈ Cφ(ζ) (2.15)

Direct computation of Taylor series is cumbersome because coefficients cj,i requiresthe computation of higher order derivatives. Even when f(x) is known, computingcj,i can be a tedious process. Furthermore, the Taylor series is valid only in a smallneighborhood of x∗. In the next section we present a least squares approach tocompute these coefficients without explicitly computing derivatives. In addition toeliminating the need to take higher order derivatives, the least squares process alsoguarantees that error is minimized over the desired domain.

2.2 The Least Squares MethodLeast squares is a statistical method for solving an overconstrained system whichprovides the globally minimum error between a polynomial fit and an unknownfunction f(x). The method was first published in 1805 by Adrien Marie Legendre,however the discovery is often attributed to Carl Friedrich Gauss for having devel-oped the method as early as 1795 [10]. In general a best fit polynomial can be foundfor any non linear function, and the accuracy of this polynomial approximation willincrease as the order of the polynomial increases. We begin by defining the errorto be

ej = yj − cj,iφi (2.16)

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In the general case, the cost function J can be defined as the inner product of errorsquared

J = 12 〈ej, ej〉ρ

= 12 〈yj − cj,iφi, yj − cj,iφi〉ρ

(2.17)

Where 〈·, ·〉ρ denotes the inner product with respect to the probability distributionfunction (pdf) ρ(x) associated with x. Note that this formulation can be used forthe inner product of either continuous or discrete variables. For the discrete case,the inner product of two (N × 1) vectors v(x) and u(x) is defined to be

〈v(x),u(x)〉ρ =N∑i=1

ρ(xi)v(xi)u(xi) (2.18)

Geometrically, the discrete inner product is a measure of the projection of thecomponents of vector v onto the components of vector u. A good discussion onvector projections and their relationships to orthogonality can be found in [11].In higher dimensional space, it is difficult to represent vectors graphically, so thenotion of an inner product is easier to represent algebraically. As a vector increasesin length (N → ∞), the discrete vectors become continuous functions. In thecontinuous case, the inner product becomes an integral over the domain of the pdf

〈v,u〉ρ =∫

Ωv(x)u(x)ρ(x)dx (2.19)

Returning to our cost function, we attempt to find the coefficients which minimizeJ . The 1st order necessary condition is:

mincj,k

J = ∂J

∂cj,k= 0, for k = 1, 2, . . .M (2.20)

Taking the partial derivative of Eq. (2.17) with respect to the coefficient vectorgives the following set of M equations known as Normal equations:

∂J

∂cj,k= 0 =

⟨ej,

∂ej∂cj,k

⟩ρ

0 = 〈(yj − cj,iφi),−φk〉ρ

(2.21)

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Because the inner product is a linear operator, we can regroup Eq. (2.21) as:

〈yj, φk〉ρ = cj,k 〈φi, φk〉ρ (2.22)

This can also be written in Matrix form where A is a (m×M) matrix, and B is aM ×M matrix.

A = CB

Aj,k = 〈yj, φk〉ρ , Bi,k = 〈φi, φk〉ρ(2.23)

Eq. (2.23) is known as the normal equation. It is important to note that thenormal equation expressed using inner products is a generalization, which canbe evaluated for either continuous or discrete cases. Up until this point φi werearbitrary polynomial basis functions, however, if we select them specifically to beorthogonal with respect to ρ(x), B becomes a diagonal matrix. See Section 2.3 fordetails.

Bi,k =〈φi, φi〉ρ for i = k

0 for i 6= k(2.24)

A diagonal B matrix allows for easy computation of the inverse, and causes thecoefficients to become linearly independent. The solution for the best fit coefficientscan now be written as:

C = AB−1

cj,i =〈yj, φi〉ρ〈φi, φi〉ρ

(2.25)

The computation of terms in B can be done offline for any pdf through application ofthe Gram-Schmidt Process. The major challenge in this problem lies in computingthe multi-dimensional continuous inner products of the generalized non-linearfunction yj found in the A matrix. Detailed discussions of both Gram-Schmidtorthogonalization and numerical integration methods can be found in the followingsections.

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2.3 Orthogonal PolynomialsThe property of orthogonality stems from the definition of the inner product and isdefined as:

〈v(x), u(x)〉ρ = 0 (2.26)

Where v(x), u(x) are orthogonal functions of the variable x. Gram-Schmidtorthogonalization is a method for computing polynomials that satisfy this propertyfrom a non-orthogonal basis set. A discussion of the Gram Schmidt Process canbe found in [11, 12]. For a single variable (n=1), take the set of non orthogonalpolynomials ψ of maximum power d to be:

ψ(d) =

1x

x2

...xd

(2.27)

We seek to find the orthogonal set of polynomials φ, from non orthogonal basis setψ. Since we are not restricted on what the first orthogonal polynomial φ0 can be,we select it to be φ0 = 1 for simplicity. To find the next orthogonal polynomialwe substract the first orthogonal polynomial multiplied by some scaling constant cfrom the second non-orthogonal polynomial.

φ2(x) = ψ2(x)− cφ1(x) (2.28)

This guarantees to eliminate all components of φ1 which have a non-zero projectionon φ2. Taking the inner product of both sides of this equation with φ1 and applyingthe definition of orthogonality, we can guarantee that φ1 and φ2 are orthogonal.

〈φ1(x), φ2(x)〉ρ = 〈φ1(x), (ψ2(x)− cφ1(x)〉ρ = 0 (2.29)

The constant coefficient can be solved for as shown in Eq. (2.30)

c =〈ψ2(x), φ1(x)〉ρ〈φ1(x), φ1(x)〉ρ

(2.30)

12

Figure 2.2. Inner Product of 2D Vectors

Figure 2.2 shows graphically how eliminating the projection of ψ2 on φ1 canguarantee the orthogonality with Eq. (2.28). Continuing in this manner, we canalways determine the next orthogonal polynomial φk(x) by eliminating projectionsof the previous orthogonal polynomials on the next non-orthogonal polynomial. Thisleads to the recursive relationship given Eq. (2.33) by known as the Gram-Schmidtformula [11].

φk = ψk −k−1∑i=1

〈ψk, φi〉ρ〈φi, φi〉ρ

φi for k = 1, 1 . . . d (2.31)

It is important to note that these orthogonal polynomials are non-unique, and canbe linearly scaled by any constant value sk and remain orthogonal.

〈φ1, φ2〉ρ = 〈s1φ1, s2φ2〉ρ = 0 (2.32)

Scaling the polynomial is also known as normalization and is given by Eq.(2.33).

φ′k = skφk (2.33)

Where φ′k is the normalized polynomial. Various normalizations are used for differentorthogonal polynomials, but two commonly used normalizations are orthonormal-ization and monic normalization [8]. Orthonormalization scales the polynomial suchthat the inner product of the polynomial with itself is 1, and monic normalizationscales the polynomial such that the leading coefficient is 1. Orthonormalization is

13

given by:sk = 1

‖φk‖, 〈φ′k, φ′k〉ρ = 1 (2.34)

Where ‖φk‖ is defined as the 2-norm of the polynomial:

‖φk‖2 =∫

Ωφk(x)2ρ(x)dx (2.35)

and monic normalization is given by:

φk(x) = ckxk + ck−1x

k−1 + . . . c0, sk = 1ck

(2.36)

Assume that we would like to determine the set of orthogonal polynomials withrespect to the uniform pdf ρ(x) = 1

2 . The first few orthogonal polynomials arecalculated in the following manner:

φ0 = ψ0 = 1

φ1 = ψ1 −〈ψ1, φ0〉ρ〈φ0, φ0〉ρ

φ0 = x

φ2 = ψ2 −〈ψ2, φ0〉ρ〈φ0, φ0〉ρ

φ0 −〈ψ2, φ1〉ρ〈φ1, φ1〉ρ

φ1 = x2 − 13

φ3 = ψ3 −〈ψ3, φ0〉ρ〈φ0, φ0〉ρ

φ0 −〈ψ3, φ1〉ρ〈φ1, φ1〉ρ

φ1 −〈ψ3, φ2〉ρ〈φ2, φ2〉ρ

φ2 = x3 − 35x

(2.37)

Since we can use any normalization factor for the normalization in equation (2.33)and the polynomials will remain orthogonal, If we choose to normalize these poly-nomials such that skφk(1) = 1 they become the well known Legendre Polynomials:

φ0 = 1

φ1 = x

φ2 = 12(3x2 − 1)

φ3 = 12(5x3 − 3x)

(2.38)

Another important property of orthogonal polynomials is that they can be computedrecursively without performing the Gram-Schmidt process. This type of relation isknown as a three-term reccurance relation. In general, any orthogonal basis set

14

Orthogonal Polynomial Interval ρ(x) ak bk ck

Legendre [−1, 1] 1 2k+1k+1 0 k

n+1

Hermite (−∞,∞) e−x2

2 1 0 k

Ultraspherical [−1, 1] (1− x2)λ− 12 2k+λ

k+1 0 k+2λ−1k+1

Laguerre [0,∞) e−xxα − 1k+1

2b+α+1k+1

k+αk+1

Chebyshev (First Kind) (−1, 1) (1− x2)− 12 2 0 1

Chebyshev (Second Kind) [−1, 1] (1− x2) 12 2 0 1

Table 2.1. Table of Common Orthogonal Polynomials

can be found using a three-term recursive relationship of the form:

φk+1(x) = (akx+ bk)φk(x)− ckφk−1(x) (2.39)

There are many different types of othogonal polynomials for which this relationshipis valid, but the most commonly used are Legendre Polynomials for uniformdistributions and Hermite Polynomials for Gaussian distributions. Table 2.3 isadapted from Ref. [8] and summarizes a select few orthogonal polynomials, theirweighting functions, and their reccurance relation coefficients. See Chapter 18 inRef. [8] for details on more complex orthogonal polynomials. The next sectioncombines many of the concepts discussed in this chapter, and demonstrates howorthogonal polynomials relate to numerical integration and computation of thehigher order sensitivity matrices.

2.4 Numerical Integration MethodsThis section will present various numerical integration methods and discuss thestrengths and weaknesses of each. In general, the integral of a function f(x) withrespect to the pdf ρ(x) can be expressed as:

F =∫ b

a

f(x)ρ(x)dx (2.40)

Note that for a standard unweighted integral the pdf is a uniform density functionρ(x) = 1. All numerical integration techniques are variations of the same basicmethod, which seek to approximate the integral of a function as the summation of

15

a finite number of weights wi, multiplied by the function evaluated at N points xi.

F ≈N∑i=1

wif(xi) (2.41)

Note that a constraint on the weights is that the sum of wi must equal 1.

N∑i=1

wi = 1 (2.42)

The method by which the points and weights are selected is what separates numericalintegration techniques from one another.

2.4.1 The Monte Carlo Method

The Monte Carlo Method is a very commonly used method for performing numericalintegration. The method involves randomly sampling points xi from the pdf of xand assigning equal weight to all points.

F ≈ 1N

N∑i=1

f(xi) (2.43)

To illustrate this method, an example covariance matrix for a Gaussian pdf willbe computed. Determining the covariance matrix requires the computation of thefollowing multi-dimensional integral:

Σ = E[(x− µ)(x− µ)T ] =∫ ∞−∞

(x− µ)(x− µ)Tρ(x)dx (2.44)

Where ρ(x) = e−12 xTx is a standard unit variance zero mean Gaussian distribution.

Assume that x is a (3× 1) random vector pdf ρ(x) such that the true covariancematrix of x is given by equation (2.45).

Σtrue =

σ2x 0 0

0 σ2y 0

0 0 σ2z

=

1 0 00 1 00 0 1

(2.45)

16

100 102 104 106 108

N

10-4

10-3

10-2

10-1

100

101

|Σtr

ue-Σ

MC

|

Figure 2.3. Error in Monte Carlo Approximation vs Number of MC Points

The covariance can be computed numerically using the Monte Carlo method byrandomly sampling xi from the standard Gaussian pdf ρ(x), and computing theintegral with equation (2.46).

ΣMC ≈1N

N∑i=1

xixTi (2.46)

This method was used to compute the covariance matrix for values of N rangingfrom 10 to 100,000,000. The resulting covariance matrix for N = 100, 000, 000 isshown below.

ΣMC ≈

1.0001 1.6357× 10−4 3.8732× 10−5

1.6357× 10−4 0.9998 −7.7672× 10−5

3.8732× 10−5 7.7672× 10−5 0.9999

The error e was also computed for each value of N as the matrix norm of truecovariance minus calculated covariance, and plotted in Figure 2.3.

e = |Σtrue − ΣMC | (2.47)

From Figure 2.3, it is clear that a reasonable approximation of the covariance

17

integral can be found numerically with a very large number of Monte Carlo samples.However, there is significant burden associated with evaluating 100,000,000 samplepoints. Additionally, observing Figure 2.3 it can be seen that the error is reduced byapproximately 1 order of magnitude for every 2 orders of magnitude of additionalMC points. This trend gives a exponentially diminishing return on reducing errorby adding more MC points. Clearly, we would like to avoid the computationalburden associated with Monte Carlo numerical integration if possible.

2.4.2 Gaussian Quadrature

In General, deterministic alternatives to the random sampling numerical integrationscheme are known as quadrature rules. A fundemental assumption of the quadraturemethod is that the function being integrated can be reasonably approximated bya polynomial. There are many different quadrature rules that are formulated tocarefully leverage symmetries in a probability distribution function such that usinga specific set of weights and points allows us to exactly recreate the integrals ofpolynomials. Perhaps the most well known quadrature is the Gaussian QuadratureRule (GQR). Let us consider the integration of a polynomial function f(x) withrespect to the probability distribution function ρ(x).

∫ b

a

f(x)ρ(x)dx =N∑i=1

wif(xi) (2.48)

Substituting the dth order Taylor series expansion for f(x) given by (2.4) into(2.48):

∫ b

a

ρ(x)(c0 +c1x+c2x2 + . . .+cdxd)dx =

N∑i=1

wi(c0 +c1xi+c2x2i + . . .+cdxdi ) (2.49)

Recall that the coefficients ci represent the partial derivatives of x evaluated atx∗. Equating the coefficients gives a set of d+ 1 equations known as the Moment

18

Constraint Equations (MCE’s) (2.50):

∫ b

a

ρ(x)dx =N∑i=1

wi

∫ b

a

xρ(x)dx =N∑i=1

wixi

∫ b

a

x2ρ(x)dx =N∑i=1

wix2i

...∫ b

a

xdρ(x)dx =N∑i=1

wixdi

(2.50)

The left hand side of the MCE’s can be solved analytically for the given pdf, andthe right hand side can be solved for the points xi and weights wi. From (2.50), itis clear that there are 2N variables wi, xi available to solve the d+1 MCE’s. Thisimplies that N points are required to exactly solve a d = 2N − 1 order polynomial.The fundemental theorem for Gaussian quadrature, and a discussion of the methodsand algorithms needed to compute x,wi can be found in Ref. [13].

The points xi and weights wi are closely related to orthogonal polynomialswith respect to weighting function ρ(x). Take the polynomial φk(x) to be anorthogonal polynomial with respect to ρ(x) with coefficients ck and real rootsa < t1 < t2 < . . . < tk < b defined as

φk(x) = ck

k∏i=1

(x− ti) (2.51)

The quadrature points are simply the roots of this polynomial xi = ti, and theweights can be calculated using Eq. (2.52) [8].

wi =∫

Ω

φk(x)(x− xi)φ′k(xi)

ρ(x)dx (2.52)

To illustrate how this method works we will provide an example. Let us considerthe integration of the function f(x) over the interval [−1, 1], and that that thefunction f(x) can be reasonably approximated by a third-order polynomial p(x).

19

Take the pdf to be uniform ρ(x) = 12 .

12

∫ 1

−1f(x)dx ≈ 1

2

∫ 1

−1p(x)dx (2.53)

Take the 3rd order polynomial p(x) and it’s derivative p′(x) to be:

p(x) = a0 + a1x+ a2x2 + a3x

3

p′(x) = a1 + 2xa2 + 3x3a3(2.54)

In this case there are 4 unknown coefficients ai, so we need 4 equations to solve.Take 2 points x1 and x2, and plug them into p(x) and p′(x)

p(x1) = a3x31 + a2x

21 + a1x1 + a0

p(x2) = a3x32 + a2x

22 + a1x2 + a0

p′(x1) = 3a3x21 + 2a2x1 + a1

p′(x2) = 3a3x22 + 2a2x2 + a1

(2.55)

Solving for the coefficients in terms of p(x), and it’s derivatives we find:

a0 =p(x2)x21(x1 − 3x2) + x2(p(x1)(3x1 − x2)x2 − x1(x1 − x2)(x2p

′(x1) + x1p′(x2)))

(x1 − x2)3

a1 =6x1x2(p(x2)− p(x1)) + (x1 − x2)(x22p′(x1) + x2

1p′(x2) + 2x1x2(p′(x1) + p

′(x2)))(x1 − x2)3

a2 =3p(x1)(x1 + x2)− 3p(x2)(x1 + x2)(x1 − x2)3

− (x1 − x2)(x2(2p′(x1) + p′(x2) + x1(p′(x1) + 2p′(x2)))

(x1 − x2)3

a3 =−2p(x1) + 2p(x2) + (x1 − x2)(p′(x1) + p′(x2))

(x1 − x2)3

(2.56)

20

The integral of the polynomial is known to be given by equation (2.57)

12

∫ 1

−1p(x)dx = 1

2

∫ 1

−1(a3x

3 + a2x2 + a1x+ a0)dx

= 12

[a3x4

4 + a2x3

3 + a1x2

2 + a0x]1

−1

= a0 + a2

3

(2.57)

Substituting our expressions for a0 and a2 into (2.57) and regrouping similar termsof p(x) and p′(x) gives us equation (2.58)

12

∫ 1

−1p(x)dx =(x2 − x3

2 + x1(1 + 3x22))

(x1 − x2)3 p(x1) + (−x1 + x31 − x2 − 3x2

1x2)(x1 − x2)3 p(x2)

− 2x2 + x1(1 + 3x22)

3(x1 − x2)2 p′(x1)− 2x1 + x2 + 3x21x2

3(x1 − x2)2 p′(x2)

(2.58)

To find an expression for the integral of p(x) in the form of (??), we must set thecoefficients of the p′(x) terms equal to zero and set the coefficients of the p(x) termsequal to wi.

0 = −2x2+x1(1+3x22)

3(x1−x2)2 , 0 = −2x1+x2+3x21x2

3(x1−x2)2

w1 = (x2−x32+x1(1+3x2

2))(x1−x2)3 , w2 = (−x1+x3

1−x2−3x21x2)

(x1−x2)3

(2.59)

Solving the equations equal to zero in the first row of (2.4.2) provides the pointsx1 and x2 which can then be substituted into the second row of equations andsolved for w1 and w2. In the case of the 3rd order polynomial, the GQR points andweights are:

x1 = 1√3 , x2 = − 1√

3w1 = 1

2 , w2 = 12

(2.60)

Notably, these points correspond to the roots of the 2nd order Legendre Polynomial.To further emphasize this point, calculate the weights using Eq. (2.52) for ρ(x) = 1

2

and a second order Legendre Polynomial. The roots of the 2nd Legendre polynomialφ2(x) are easily found to be:

φ2(xi) = 0 = 12(3x2 − 1), xi =

[1√3,− 1√

3

](2.61)

21

Plugging these values into Eq. (2.52)

w1 =∫ 1

−1

φ2(x)(x− x1)φ′2(x1)ρ(x)dx = 1

2

∫ 1

−1

12√

3(3x+

√3)dx

= 14√

3

[3x2

2 +√

3x]1

−1= 1

2

(2.62)

The result is the same whether the points and weights are solved for long-handor using the orthogonal polynomial approach. In general, the points xi can becalculated for many different pdf’s by taking the roots of orthogonal polynomials(see Table 2.3), and the weights wi can be calculated using Eq (2.52). Next anexample is presented which will summarize many of the concepts presented in thischapter and show how they are utilized in the polynomial approximation method.

2.5 Example Higher Order Sensitivity ProblemIn this case, we wish to determine the least squares approximation for the functionf(x) over the interval [−1, 1].

f(x) = sin(5x)e−2x2 (2.63)

In this problem, the scaling factor S in Eq. (2.14) is not necessary because x is

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

x

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

f(x)

f(x)

d=1

d=2

d=3

d=4

d=5

d=6

d=7

(a) Best Polynomial Fit

1 2 3 4 5 6 7

Approximation Order (d)

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

RM

SE

(b) RMSE

Figure 2.4. Polynomial Approximation for N = 50 and Varying d

chosen such that x = ζ. We compute the inner products assuming the function to

22

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

x

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

f(x)

N=5

N=10

N=25

N=50

N=100

N=500

N=1000

Continuous

f(x)

(a) Best Polynomial Fit

1 1.5 2 2.5 3

log10(N)

-1.5

-1

-0.5

0

log10(R

MS

E)

Discrete Approximation

Continuous Approximation

(b) RMSE

Figure 2.5. Polynomial Approximation for d = 7 and Varying N

have a uniform distribution ρ(x) = 12 and the orthogonal basis functions φ(x) to

be the Legendre Polynomials. Starting with a discrete case, we take 50 uniformlyspaced samples of x and the function f(x), and compute the discrete inner products(Eq. (2.18)) required for the A and B matricies. The normal equation (2.23),can then be solved for the least squares coefficients. This process is repeated forvarying the approximation order between d = 1 and d = 7. After the coefficientsare calculated the best fit polynomials are plotted in Figure 2.4(a), and the RootMean Square Error (RMSE) for each of the approximations is plotted in Figure2.4(b) using 1000 points uniformly spaced points. RMSE error is defined as:

RMSE =

√∑1000i=1 [f(xi)− cTφ(xi)]2

1000 (2.64)

From inspection of Figure 2.4(a), it is readily apparent that as the approximationorder is increased the approximation becomes closer to the true function f(x). Thesame trend is quantitatively shown in Figure 2.4(b), which plots the RMSE betweenthe polynomial approximation and f(x) vs the order of polynomial approximation.

Now we hold the polynomial order to be fixed at d = 7, and vary the number ofdiscrete points N used to calculate the approximation. Figure 2.5(a) shows plotsof polynomial approximations for varying N , and Figure 2.5(b) shows a log-logplot of RMSE vs N . Notice that as N →∞ the discrete 7th order approximationasymptotically approaches the continuous polynomial approximation. The con-

23

tinuous least square approximation is found by computing the inner products inEq. (2.25) using the continuous definition in Eq. (2.19). A 10th order Gaussianquadrature rule is used to compute these integrals numerically. The least squaresapproximation coefficients found using continuous inner products represent themost accurate possible dth order polynomial approximation of f(x). The motivationbehind the higher order sensitivity approximation method is to calculate the TaylorPolynomial coefficients ci numerically using the continuous least squares method.This allows us to calculate the Taylor polynomial coefficients of a function withoutexplicitly taking Jacobians of f(x).

2.5.1 Alternatives to Gaussian Quadrature

Gaussian Quadrature is a powerful tool, that provides a minimal number of pointsneeded to compute polynomial integrals in 1D. Unfortunately, integrating poly-nomials in higher dimensions becomes significantly more difficult. To integrate apolynomial of the same degree in higher dimensions using Gaussian quadrature,one must construct a grid of quadrature points by taking the tensor product ofxi. Unfortunately, this leads to exponential growth in the necessary number ofpoints with increasing dimension. For a Gaussian quadrature rule that is able tointegrate an n dimensional, d = 2N − 1 degree polynomial, one needs Nn Gaussianquadrature points. This means that even a moderately high polynomial degree andsystem dimensionality, say, n = 6 and N = 5, 15, 625 points are needed to computethe integral. Due to the high computational load this creates, especially when theevaluation of a single point is computationally expensive, higher dimensions canquickly create restrictions on the degree of the polynomial approximation and byextension, on the accuracy of the solution.

Fortunately, there are other quadrature methods that exist which are able toreduce the number of points needed in higher dimensions. Sparse grid quadraturesare constructed by taking specific tensor product combinations of one dimensionalquadrature rules to reduce the effective dimension of the system [14]. Unfortunately,this process introduces negative weights which can introduce significant errorto the integrated function. In particular, the Unscented Transformation (UT)which can integrate up to 2nd order polynomials in n dimensions [15], and itsextension Conjugate Unscented Transformation (CUT) [16–20] will be utilized in

24

(a) CUT Points in 2D

𝝈

𝝈𝝈

𝒄𝟐

𝒄𝟐

𝒄𝟐 𝒄𝟑𝒔𝟑

𝒔𝟑𝒔𝟑

(b) CUT Points in 3D

Figure 2.6. A Schematic of CUT Axes

the implementation of the polynomial approximation method.The CUT approach can be considered an extension of the conventional UT

method that satisfies additional higher order moment constraints. Rather thanusing tensor products to determine quadrature points and weights as in Gaussquadrature, the CUT approach judiciously selects special structures to extractsymmetric quadrature points constrained to lie on specially defined axes as shownin Figure 2.6. For each cubature point, two unknown variables, a weight wi anda scaling parameter ri are assigned. The moment constraints equations for thedesired order are derived in terms of unknown variables ri and wi. Because of

Clenshaw−Curtis

Gauss−Legendre

Sparse Grid

CUT81

2

3

4

5

0

1

2

3

4

5

6

x 104

Num

ber

of P

oin

ts

20

5

59

161

314

5

5

6561

59049

729

81

925

53

125

165

385

625781

3125

Dimension of random variableApplied Quadrature Scheme

Figure 2.7. Comparison of Quadrature Schemes for Same Order of Accuracy

25

the symmetries of cubature points, the odd-order moment constraint equationsare automatically satisfied, so that wi and ri are found by solving just the evenorder equations. The order of these moment constraint equations dictates the setof cubature points, often referred to as CUT4, CUT6 and CUT8 points. For thisanalysis, the CUT8 points will be used because these points satisfy the most momentconstraints, which provides the highest accuracy. These new sets of so-called sigmapoints are guaranteed to exactly evaluate expectation integrals involving polynomialfunctions with significantly fewer points. Tabulated values for the CUT8 scalingparameters (ri) and weights (wi) can be found in [20].

Figure 2.7 shows a comparison of the number of points required for CUT andGauss-Legendre quadratures for similar accuracy, clearly illustrating the reducedgrowth exhibited by the CUT method. The CUT method has already beenimplemented sucessfully in cross-disciplinary fields of study, including: the study ofvolcanic ash plumes, conjunction analysis, control of glucose in diabetic patients, soilcharacterization for civil engineering purposes, and analysis of energy harvesters[21–25]. More details about the CUT methodology and its comparison withconventional quadrature rules can be found in Ref. [16–20,26]. With the applicationof the CUT approach, we can generate higher order sensitivity matrices in acomputationally efficient manner.

26

Chapter 3 |The Uncertain Lambert Prob-lem

3.1 Problem BackgroundThe classical Lambert Problem is a Two Point Bounday Value Problem (2PBVP),which connects two position vectors at known times to provide an initial velocityvector. This problem and its multiple variants have been well studied in litera-ture [27, 28]. In this chapter, we are particularly interested in characterizing theuncertainty associated with the Lambert Problem solution as a function of positionvector uncertainties at initial and final times. Due to limited measurement accuracyin sensor data, the characterization of uncertainty in the Lambert’s solution isan important operational concern for many Space Situational Awareness (SSA)problems including initial orbit determination, conjunction analysis, data associ-ation, and maneuver detection. The Uncertain Lambert Problem is introducedand discussed in detail in Schumacher et al. [29], where a state transition matrixformulation is used to characterize uncertainty. A similar analytic approach tocharacterize the uncertainty associated with the Lambert solution as a functionof the initial and final state uncertainties involves linearizing the lambert solutionusing first order partial derivatives about the nominal orbit [30, 31]. The linearvariational approximation to the lambert solution is a computationally efficientprocess, however these linear analyses are only valid if initial and final state uncer-tainties are relatively small and may not provide insight into the exact distributionof error associated with the Lambert solution.

27

There is a large base of existing literature dedicated to the characterizationand propagation of uncertainty in nonlinear dynamical systems [32–34], howeverthe application of these methods to the solution of the Lambert problem is anunexamined problem. Luo et al. [35] reviews a wide range of methods and algorithmsfor the characterization and propagation of uncertainty, noting that typicallyhigher order statistical moments are neglected due to exponentially increasingcomputational load for increasing state dimensionality.

The motivation of this work is to include higher order terms (HOT’s) of the Lam-bert solution to increase solution accuracy and the domain of position uncertaintiesunder which the solution remains valid. Computing these higher order sensitivityterms requires the evaluation of multi-dimensional expectation integrals whichtraditionally becomes quickly computationally expensive at higher dimensions. Tocircumvent this issue, the current analysis utilizes the non-product quadraturemethod known as Conjugate Unscented Transformation (CUT) to compute thehigher order sensitivity terms of the Lambert solution with respect to uncertaininitial and final position vector in a computationally tractable manner.

3.2 Simulation DescriptionThis section will discuss the uncertain Lambert problem formulation, and specificassumptions and parameters for two test case simulations. The two test cases arefor a Low Earth Orbit (LEO) and a Geosynchronous Transfer Orbit (GTO) andeach will be analyzed using varying degrees of higher order sensitivity matrices.Both test cases are assumed to have uncertainty in the initial and final positionand governed by deterministic two-body dynamics, however this method can beextended to apply to orbit dynamical models that include perturbing effects.

Assume that the nonlinear function f(x∗) expresses the deterministic Lambertsolution in terms of mean (n× 1) input vector x∗ and mean (m× 1) output vectory∗.

y∗ = f(x∗) (3.1)

In this analysis f(x∗) represents the deterministic lambert solver, and is based onthe universal variable formulation of the Battin-Vaughn algorithm [27,28]. Detailson this algorithm can be found in Ref. [36]. We wish to formulate a stochastic

28

solution to this problem by substituting x intof(x∗)where x is a (n× 1) continuousrandom vector with the joint pdf ρ(x).

y = f(x) (3.2)

In the uncertain Lambert Problem, f(x) denotes the statistical mapping betweenthe uncertain position vectors the solution for the initial velocity:

x =

r1

r2

, y = v1 (3.3)

The relationship between x and x∗ is facilitated by the normalized (n×1) continuousrandom vector ζ, where each element ζj is associated with a zero mean unit varianceGaussian pdf:

x =x∗ +√

Σζ = x∗ + δx

ρ(ζj) = 1√2πe−

ζ2j2

(3.4)

Where x∗ is the mean vector of x, and Σ is the covariance matrix associated withx. δx represents the deviation in input due to uncertainty. Note that becuase ζis a normally distributed vector, the orthogonal basis functions φ(ζ) used for thepolynomial approximation will be the Hermite Polynomials (see Table 2.3). Figure3.1 illustrates the setup for the Uncertain Lambert Problem.

To solve the uncertain Lambert problem, we use the least squares solutiondeveloped in Chapter 2 to compute higher order sensitivity matrices of the deter-ministic Lambert Problem f(x∗) which approximates the solution to the stochasticmapping function f(x). For each test case, the uncertain Lambert solution willbe approximated as a dth order polynomial in terms of the standardized randomvariable ζ

yj ≈ cj,0 + c(1)Tj φ(ζ)(1) + . . . c(d)T

j φ(ζ)(d) (3.5)

To compute the least squares coefficients, the CUT8 points and weights for n = 6

29

Figure 3.1. Uncertain Lambert Problem Diagram

are used to numerically evaluate the normal equations

A = CB

Aj,k = 〈yj(x),φ(ζ)k〉ρ , Bi,k = 〈φ(ζ)i,φ(ζ)k〉ρ(3.6)

Recall that because the basis functions (Hermite Polynomials) are orthogonal〈φi, φj〉 = 0 for i 6= j, so the coefficients can be evaluated as:

cj,i =〈yj(x),φ(ζ)i〉ρ〈φ(ζ)i,φ(ζ)i〉ρ

(3.7)

Figure 3.2 illustrates the overall structure of the polynomial approximation method,and the process involved in solving for the approximation coefficients. For moredetails on computing the approximation coefficients refer to Chapter 2.

Test case 1 is a near-circular LEO orbit with a radius of |r1| ≈ 7000km andinclination i = 2. The nominal initial and final state vectors for case 1 correspondto a time of flight dt = 1200s and are shown in Table 3.1. This test case is replicatedfrom Ref. [29] for comparison purposes. The pdf prescribed to both the initial and

30

Figure 3.2. Polynomial Approximation Method Summary

final position vectors are Gaussian distributions with equal σ2 = 0.01km2 variancesin the x, y, and z-directions. This corresponds to a position uncertainty standarddeviation of 100m in each dimension.

Σ =

0.01 0 00 0.01 00 0 0.01

km2 (3.8)

The second test case simulation is for a sample Geostationary Transfer Orbit(GTO) to demonstrate the utility of this method in orbits other than LEO. Theassumed GTO orbit is an equatorial transfer (i = 0) between the LEO altitude intest case 1 (610.6 km) and GEO altitude (35786 km). The final position vector isat a true anomaly of 5 before the apoapsis (θ = 175). The time of flight for testcase 2 is dt = 16936.7s. The nominal state vectors for test case 2 are summarizedin Table 3.2. The pdf perscribed to both the initial and final position vectors is thesame as in test case 1; Gaussian with 100m standard deviations (Σ = 0.01× I6×6).

3.2.1 Simulation Results

This section will present the results of each simulation and discuss their significance.First, we will consider the LEO case, and look into the results of using Monte CarloMethods to calculate the covariance matrix Σ and the mean vector µ for the joint

31

Parameter State 1 State 2Time (s) 0 1200x (km) -2039.8845 -6995.7285y (km) 6672.88669 -166.39802z (km) 232.675383 -7.0380479x (km/s) -7.236669 0.15969047y (km/s) -2.2063637 -7.5422634z (km/s) -0.0783 -0.2633659

Table 3.1. Test Case 1 LEO State Vec-tor

Parameter State 1 State 2Time (s) 0 16936.7x (km) 6981.5960 -41605.0402y (km) 0 3639.9694z (km) 0 0x (km/s) 0 -0.5027y (km/s) 9.8977 -1.6169z (km/s) 0 0

Table 3.2. Test Case 2: GTO StateVector

distribution ρ(ζ), and the extent to which this methods can help us validate oursolution. The covariance matrix, statistical moments, and mean state vector for theLEO case will then be calculated using the CUT method and directly comparedto the results for this test case in Ref. [29]. For each test case, the higher ordersensitivity matrices for orders 1-3 will be determined and used to calculate theerror ε at randomly sampled points in the distribution ρ(ζ).

Monte Carlo sampling is often used as a method to approximate the solution tonon-linear dynamical problems. In Ref. [29], the authors use Monte Carlo methodsto validate their solution to test case 1 using both a differential correction and alinear variational analysis, noting that the convergence of the Monte Carlo methodis restricted due to sampling limitations. For the current LEO test case, we willdemonstrate the difficulties associated with Monte Carlo sampling to reproduceexpectation values of the initial position and velocity vectors. To determine theseexpectation values numerically, points in the initial and final position space aresampled and a Lambert Problem is solved for v1 at each point.

We seek to compare both the mean µ and covariance Σ of the input spacecalculated using a Monte Carlo method and using the CUT8 method. Considerthe mean Eq. (3.9a) and variance Eq. (3.9b) approximated using N Monte CarloPoints, xi, with r1 sampled from the true distribution ρ(ζ), and v1 calculated bysolving the corresponding Lambert Problem:

µ =∫ ∞−∞

xρ(ζ)dζ ≈ µMC = 1N

N∑i=1

xi (3.9a)

32

10 2 10 3 10 4 10 5 10 6 10 7 10 8

Number of MC Points

10 -9

10 -8

10 -7

10 -6

10 -5

10 -4

10 -3

10 -2

Me

an

Err

or

()

Monte Carlo Method

CUT8 Method

(a) Mean Error (εµ)

10 2 10 3 10 4 10 5 10 6 10 7 10 8

Number of MC Points

10 -14

10 -12

10 -10

10 -8

10 -6

10 -4

10 -2

Covaria

nce E

rror

()

Monte Carlo Method

CUT8 Method

(b) Covariance Error (εΣ)

Figure 3.3. Error in Expectation Values of x vs. N

Σ =∫ ∞−∞

(x− µ)(x− µ)Tρ(ζ)dζ ≈ ΣMC = 1N

N∑i=1

(xi − µ)(xi − µ)T (3.9b)

Figures 3.3(a) and 3.3(b) show the RMS error in mean εµ = RMS(µ− µMC)and the matrix norm of the error in covariance: εΣ = ‖Σ−ΣMC‖ . The RMS errorin mean and covariance is plotted for varying values of N , as well as the error foreach computed using CUT8 method. Observing Figures 3.3(a) and 3.3(b)we seean approximate trend of one decimal place of error reduction for every additionalfactor of 100 sample points. This steep computational cost for improved accuracyillustrates the limitations of the Monte Carlo method and how the need for highaccuracy solutions can quickly conflict with the practical limitations of on-boardcomputational capabilities. Contrasting the error in Monte Carlo approximationwith the error computed using the CUT8 method (shown as a large red dot inFigures 3.3(a) and 3.3(b)) , there is a very clear advantage to CUT8 which givesboth greater accuracy and drastically reduced computational load, requiring only745 points. Due to the computationally expensive nature of solving a lambertproblem to get a single point for the solution space and the relative inaccuracyof the Monte Carlo method, only the CUT8 method will be considered from thispoint on.

In addition to the mean and covariance, we will also calculate higher orderstatistical central moments of the initial position and initial velocity space. We willonly concern ourselves with the diagonal terms of these central moments, which

33

can be calculated with Eq. 3.10:

E[(xj − E[xj])q] =∫

Ω(xj − x∗j)qρ(ζ)dζ =

N∑i=1

wi(xi,j − x∗j)q (3.10)

Where E[(xj−E[xj ])q] is the qth central moment. The results of the 1st raw momentx∗ calculated using Eq. 3.9a, and the 2nd − 4th central moments calculated usingEq 3.10 are shown in Table 3.3. Looking at the first three rows of Table 3.3 we cansee that the mean and variance exactly replicate the mean and variance prescribedto the initial position vector r1. We can also see that the skewness of both r1 and v1

are numerically very close to zero, indicating that both variables are symmetricallydistributed about the mean vector. The variance of the v1 vector is relativelysmall for the given initial and final position distribution (10−8km/s), however it islarger than the 3rd and 4th central moments by at least 7 orders of magnitude. Thefact that the first and second moments of v1 are the only central moments withsignificant magnitude demonstrates that the solution for initial velocity is Gaussianin nature. To directly compare the CUT8 results with the results in Ref. [29], we

Table 3.3. Statistical Moments Calculated Using CUT8Mean Variance Skewness Kurtosis

x1 -2.0399E+03 1.0000E-02 -3.9085E-12 3.0000E-04y1 6.6729E+03 1.0000E-02 1.3861E-11 3.0000E-04z1 2.3267E+02 1.0000E-02 4.7152E-13 3.0000E-04x1 -7.2367E+00 1.4447E-08 -1.3931E-19 6.2613E-16y1 -2.2064E+00 2.4970E-08 7.8989E-17 1.8705E-15z1 -7.8321E-02 1.3534E-08 4.1393E-20 5.4952E-16

must calculate the covariance matrix for r1 and v1. This covariance matrix can becalculated by evaluating the deterministic Lambert problem at the CUT points,and using Eq. (3.11) for x = [r1, v1]T .

Σ =∫

(x− x∗)(x− x∗)Tρ(ζ)dζ =N∑i=1

wi(xi − x∗)(xi − x∗)T (3.11)

As expected, the covariance matrix given in Ref. [29] Table 3 matches the Covariancecalculated using the CUT8 method in Table 3.4 almost exactly to the precisiongiven. Notably, in Ref. [29] the off-diagonal terms are assumed to be uncorrelated

34

Table 3.4. Test Case 1: Covariance Σ Using CUT8 Methodx1 (km) y1 (km) z1 (km) x1(km/s) y1(km/s) z1(km/s)1.000E-02 -1.084E-19 -6.776E-21 -7.615E-06 5.365E-06 1.866E-07-6.776E-21 1.000E-02 6.098E-20 5.365E-06 -1.182E-05 -3.063E-07-6.776E-21 -4.065E-20 1.000E-02 1.865E-07 -3.062E-07 -3.037E-06-7.615E-06 5.365E-06 1.865E-07 1.444E-08 -6.507E-09 -2.271E-105.365E-06 -1.182E-05 -3.062E-07 -6.507E-09 2.497E-08 3.987E-101.865E-07 -3.062E-07 -3.037E-06 -2.271E-10 3.987E-10 1.353E-08

Table 3.5. Test Case 1: 1st Order Approximation Coefficientsx1 (km) y1 (km) z1 (km) x1(km/s) y1(km/s) z1(km/s)1.499E+00 -1.468E+00 -5.104E-02 1.603E+03 -5.019E+02 -1.737E+01-9.527E-01 1.676E+00 4.894E-02 -3.735E+02 1.250E+03 1.278E+01-3.307E-02 4.888E-02 2.708E-01 -1.294E+01 1.275E+01 8.835E+021.899E-03 -3.013E-03 -1.045E-04 2.598E+00 -1.126E+00 -3.888E-02-9.430E-04 1.012E-04 3.916E-05 -4.412E-01 6.281E-01 1.438E-02-3.205E-05 3.846E-05 -1.028E-03 -1.516E-02 1.377E-02 2.343E-01

and set to zero, however, in evaluating the covariance matrix numerically we makeno assumptions about correlation. In light of this, it is reassuring to see that theoff-diagonal terms of the covariance matrix are indeed very small, further assuringus that the CUT8 method works properly.

Now that the CUT8 method has been shown to be effective in reproducing theresults of both Monte Carlo and linear variational methods given in Ref. [29], theCUT method can be used to compute the inner products required to determine theleast squares coefficients. The coefficients for the first order sensitivity coefficientsare in many ways analogous to the State Transition Matrix used in linear variationalmethods. The State Transition Matrix given in Ref. [29] Table 2 matches the 1st

order least squares coefficients given in Table 3.5 to a precision of roughly threedecimal places. This demonstrates an approximate equivalency of the two methods.The results for the polynomial approximation to the Lambert Problem using higherorder sensitivity matrices will now be presented for both test cases 1 and 2.

3.2.2 Test Case 1: LEO

Coefficients for the initial velocity approximation are calculated for polynomialorders 1-3, and the magnitudes of the coefficient for polynomial order approximation

35

1st Order 2nd Order 3rd Order

Polynomial Approximation Order

-15

-10

-5

0

Co

eff

icie

nt

Ord

er

of

Ma

gn

itu

de

CV

x

CV

y

CV

z

Figure 3.4. Plot of Coefficient Order of Magnitude for Varying Approximation Orders

are summarized in Figure 3.4. CVx , CVy , and CVz are the RMS values for all elementsin the 1st, 2nd, and 3rd row of coefficients in each c(d)

j matrix (see Eq. (2.10)). Asthe order of the polynomial approximation increases the order of magnitude of thecoefficients decreases, indicating that the higher orders are contributing less to theapproximation of y.

10,000 random points xi were sampled from the normalized distribution ρ(ζ),and used to compute the polynomial basis vector φ(ζ). The approximation for v1

can then be calculated at each sample point, and the error can be found using Eq.(2.16). The same 10,000 points are used for each polynomial approximation ordershown. Figure 3.5 shows 3D scatter plots of the initial velocity with ECI positioncoordinates on the x,y, and z axes. Figure 3.5(a) shows the true velocity magnitudeon the color axis, and Figures 3.5(b)-3.5(d) show initial velocity error on the coloraxis.

The same data is presented in figure 3.6 as a 2D contour plot where the x and yaxes are the Mahalanobis Distance of initial and final position vectors respectively,and the colorbar is the same for that in Figure 3.5(a). The Mahalanobis DistanceDi is a the multi-dimensional analog to the standard deviation, and describes howfar a point is from the mean vector x∗. D1 and D2 are measurements of how far asample point is from r∗1 and r∗2 respectively, and are calculated as follows:

Di =√

(ri − r∗i )TΣ−1(ri − r∗i ) (3.12)

36

232.4

232.5

6673.2

232.6

232.7

Z P

ositio

n (

km

)

232.8

6673 -2039.6

232.9

Y Position (km)

233

-2039.8

X Position (km)

6672.8-2040

6672.6-2040.2

7.5656

7.5657

7.5658

7.5659

7.566

7.5661

7.5662

7.5663

|V1

| (k

m/s

)

(a) True Velocity Magnitude

232.4

232.5

6673.2

232.6

232.7

Z P

ositio

n (

km

)

232.8

6673 -2039.6

232.9

Y Position (km)

233

-2039.8

X Position (km)

6672.8-2040

6672.6-2040.2

-15

-14

-13

-12

-11

-10

-9

-8

-7

log

10

(|V

1

| (k

m/s

)

(b) 1st Order Approximation Error

232.4

232.5

6673.2

232.6

232.7

Z P

ositio

n (

km

)

232.8

6673 -2039.6

232.9

Y Position (km)

233

-2039.8

X Position (km)

6672.8-2040

6672.6-2040.2

-15

-14

-13

-12

-11

-10

-9

-8

-7

log

10

(|V

1

| (k

m/s

)

(c) 2nd Order Approximation Error

232.4

232.5

6673.2

232.6

232.7Z

Positio

n (

km

)232.8

6673 -2039.6

232.9

Y Position (km)

233

-2039.8

X Position (km)

6672.8-2040

6672.6-2040.2

-15

-14

-13

-12

-11

-10

-9

-8

-7

log

10

(|V

1

| (k

m/s

)

(d) 3rd Order Approximation Error

Figure 3.5. Test Case 1: V1 Spatial Representation

Observing Figures 3.5 and 3.6, there is an obvious trend of decreasing error aspolynomial approximation order increases, perhaps most drastically between 1st

and 2nd order. In 3.6(b) it is notable that the error contours tend to monotonicallyincrease as mahalanobis distance increases. Intuitively it makes sense that thelinear 1st order approximation drops in accuracy as the point moves farther awayfrom the mean. Contrasting this trend with higher order approximations, theerror decreases and then increases again as mahalanobis distance increases (seeFigure 3.6(d)). The pattern shown in both 2nd and 3rd order approximationscan be interpreted as the solution for v1 having a near Gaussian distribution,because it is very accurately approximated by a second order approximation. Thismakes intuitive sense, because the input was Gaussian and due to a relatively shortpropagation time, there was not ample time for non-linearities to be introduced.

37

(a) True Velocity Magnitude (b) 1st Order Approximation Error

0 1 2 3 4 5

d1 (Mahalanobis Distance)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

d2 (

Mahala

nobis

Dis

tance)

-15

-14

-13

-12

-11

-10

-9

-8

-7

Log10(|

V1

|) (

km

/s)

(c) 2nd Order Approximation Error

0 1 2 3 4 5

d1 (Mahalanobis Distance)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

d2 (

Mahala

nobis

Dis

tance)

-15

-14

-13

-12

-11

-10

-9

-8

-7

Log10(|

V1

|) (

km

/s)

(d) 3rd Order Approximation Error

Figure 3.6. Test Case 1: V1 Mahalanobis Distance Representation

Table 3.6. Test Case 1: Initial Velocity ErrorPolynomial Order RMS(εV1) (km/s))

1st Order 6.4256E-92nd Order 3.1885E-123rd Order 3.1881E-12

The RMS value for initial velocity error at the random samples for each polynomialapproximation order are summarized in Table 3.6. Inclusion of HOT’s in the2nd and 3rd order approximations shows a clear advantage over the 1st orderapproximation by lowering the RMS error by more than 3 orders of magnitude,and smoothing the error distribution contours (see Figures 3.6(c) and 3.6(d)).

38

1st Order 2nd Order 3rd Order

Polynomial Approximation Order

-14

-12

-10

-8

-6

-4

-2

0

Co

eff

icie

nt

Ord

er

of

Ma

gn

itu

de

CV

x

CV

y

CV

z

Figure 3.7. Plot of Coefficient Order of Magnitude for Varying Approximation Orders

3.2.3 Test Case 2: GTO

The results of test case 2 will now be presented. The least squares coefficients fororders 1-3 are computed in the same manner as in test case 1. Figure 3.7 depicts themagnitude of coefficients CVx , CVy , and CVz for approximation orders 1-3. Noticethat the decreasing trend in coefficient magnitude as approximation order increasesis similar in this test case as in test case 1.

A different 10,000 random samples are taken from ρ(ζ) and used to computethe approximated v1 as well as error in v1. The results are plotted in Figures 3.8as a 3D scatter plot and in Figure 3.9 as a 2D Mahalanobis Distance contour plot.Additionally, the RMS error values for polynomial approximation orders 1-3 aregiven in Table 3.7.

Comparing Figure 3.8(b) with it’s counterpart in test case 1, Figure 3.5(b), it isnoticeable that there is larger error in the 1st order approximation for test case2 than for test case 1. Despite the larger error for the 1st order approximationrelative to the LEO case, adding higher order terms still significantly decreasesthe approximation error to the point where the 3rd order approximation errors forboth cases are the same order of magnitude. Table 3.7 shows the RMS error forvarious polynomial approximations of test case 2. This result shows that regardlessof what the error is in the first order approximation, the error can always bereduced by including HOT’s. Another interesting insight of this result is that when

39

0.4

-0.2

0

0.2

Z P

ositio

n (

km

) 0.2

Y Position (km)

6981.80

0.4

X Position (km)

6981.6-0.2

6981.4

9.8975

9.8976

9.8977

9.8978

9.8979

9.898

|V1

| (k

m/s

)

(a) True Velocity Magnitude

232.4

232.5

6673.2

232.6

232.7

Z P

ositio

n (

km

)

232.8

6673 -2039.6

232.9

Y Position (km)

233

-2039.8

X Position (km)

6672.8-2040

6672.6-2040.2

-15

-14

-13

-12

-11

-10

-9

-8

-7

log

10

(|V

1

| (k

m/s

)

(b) 1st Order Approximation Error

232.4

232.5

6673.2

232.6

232.7

Z P

ositio

n (

km

)

232.8

6673 -2039.6

232.9

Y Position (km)

233

-2039.8

X Position (km)

6672.8-2040

6672.6-2040.2

-15

-14

-13

-12

-11

-10

-9

-8

-7

log

10

(|V

1

| (k

m/s

)

(c) 2nd Order Approximation Error

232.4

232.5

6673.2

232.6

232.7Z

Positio

n (

km

)232.8

6673 -2039.6

232.9

Y Position (km)

233

-2039.8

X Position (km)

6672.8-2040

6672.6-2040.2

-15

-14

-13

-12

-11

-10

-9

-8

-7

log

10

(|V

1

| (k

m/s

)

(d) 3rd Order Approximation Error

Figure 3.8. Test Case 2: V1 Spatial Representation

Table 3.7. Test Case 2: Initial Velocity ErrorPolynomial Order RMS(εV1) (km/s))

1st Order 3.3502E-072nd Order 1.0538E-103rd Order 4.1705E-12

the time of flight for the Lambert Problem is increased the non-linear dynamicshave a larger influence on the solution. For example, in test case one the time offlight is dt = 1200s and in test case two the time of flight is dt = 16936.7s. Afterincluding the 2nd order approximation term in test case 1, the solution does notreally improve by much because the non-linearities are not as influential as in testcase two, where the solution needs to include the third order term to achieve thesame level of accuracy.

40

(a) True Velocity Magnitude (b) 1st Order Approximation Error

(c) 2nd Order Approximation Error

0 1 2 3 4 5

d1 (Mahalanobis Distance)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

d2 (

Mahala

nobis

Dis

tance)

-15

-14

-13

-12

-11

-10

-9

-8

-7

Lo

g1

0(|

V1

|) (

km

/s)

(d) 3rd Order Approximation Error

Figure 3.9. Test Case 2: V1 Mahalanobis Distance Representation

41

Chapter 4 |The Reachability Set Problem

4.1 Problem BackgroundUncertainty in all aspects of satellite operations including state measurements,dynamic modeling, physical parameters, and maneuver capabilities is a persistentproblem that raises issues in many areas of Astrodynamics and Space SituationalAwareness. The concept of reachability is well studied in mathematics, dynamics,optimal control theory, and game theory [37–43] and involves determination of allpossible states of a dynamic system at some future time based on a set of initialconditions, constraints and uncertainties. Despite the rich history of reachabilitytheory in these fields, there are few investigations into the application of reachabilitysets to SSA problems.

Reachability sets have important implications on SSA, and can provide us insighton creating efficient and accurate methods for sensor tasking, collision assessment,and data association. Imagine, for instance, the scenario of an orbit insertionmaneuver. There are uncertainties associated not only with the initial positionand velocity, but also with the attitude and magnitude of the maneuver itself. Ifthe uncertainty is high enough that we cannot track the satellite post-maneuver,then knowledge of the reachability set and probability distribution of the futurepositions of the satellite would be very useful for finding an appropriate searchspace.

The use of reachability sets for SSA applications was first introduced by Holzingerand Scheeres in 2009 [44] where reachability theory was applied proximity operationssituations using the linearized Clohessy-Wiltshire relative equations of motion to

42

provide insight into proximity operations. More recently, reachability sets formaneuevering spacecraft [45–47] as well as trajectory design for non-kepleriansystems [48] have been investigated.

Due to the computational burden associated with propagating non-linear orbitdynamics and the sheer number of propagation points required to accuratelycapture higher order statistical moments, computing accurate reachability setshas traditionally been a practical impossibility. The CUT approach to computinghigher order sensitivity matrices proposed in this thesis provides the tool necessaryto efficiently compute the matrices, enabling previously unavailable speed andaccuracy.

4.2 Simulation DescriptionThe objective of the reachability set problem is to find the set of all possible outputsY that can be reached at the final time tf given uncertain initial state x and controlu in the admissible control set U. Thus, the reachability set can be defined as:

Y (x0,U, tf ) = ∀x ∈ <n|u ∈ U, s.t. x(tf ) = x (4.1)

The underlying mathematical framework of this problem is the same as that in theprevious chapter, with the main difference being the inputs x, outputs y and thenonlinear function f(x). In the Reachability Set Problem, x is considered to bethe initial position and velocity of the satellite, y is the final position and velocity,and f(x) is the dynamic model.

x =

r1

v1

, y =

r2

v2

, u =

∆vθ

ψ

(4.2)

Each simulation discussed is comprised of periods of coasting (u = 0) separated byimpulsive ∆v maneuvers (where θ and ψ are the attitude angles). For the examplesin this chapter, the function f(x∗,u∗) represents the deterministic propagation of

43

two body dynamics in between impulsive thrust maneuvers.

r = −µr|r|3

, f(x,u) =∫ t2

t1

x + g(uk, tk)dt (4.3)

Where µ represents the gravitational parameter of the central body, and g(uk, tk)represents the impulsive maneuvers. To compute f(x∗,u∗) for deterministic vari-ables x∗, u∗, the Runge-Kutta45 method is used to numerically integrate x∗ viaMatlab’s in-built ode45 command. We seek to compute the statistical mapping forthe uncertain state vector x which gives the reachability set.

y = f(x,u) (4.4)

To solve the reachability set problem, we use the least squares solution developedin Chapter 2 to compute higher order sensitivity matrices of each simulation withrespect to the uncertain variables. For each test case, the reachability set willbe approximated as a dth order polynomial in terms of the standardized randomvariable ζ.

yj ≈ cj,0 + c(1)Tj φ(ζ)(1) + . . . c(d)T

j φ(ζ)(d) (4.5)

To compute the least squares coefficients for the n dimensional uncertain inputs,the CUT8 points and weights are used to numerically evaluate the normal equations

A = CB

Aj,k = 〈yj(x), φk〉ρ , Bi,k = 〈φi, φk〉ρ(4.6)

Recall that because the basis functions (Hermite Polynomials) are orthogonal〈φi, φj〉 = 0 for i 6= j, so the coefficients can be evaluated as:

cj,i =〈yj(x), φi〉ρ〈φi, φi〉ρ

(4.7)

Figure 3.1 illustrates the problem setup for the Satellite Reachability Problem.Two test cases taken from literature will be considered in the chapter. The first

test case is a two burn maneuver based on the trajectory in Ref. [49] which takes asatellite from a LEO parking orbit into a specified orbit. The second test case isbassed on the Lunar Orbit Insertion (LOI) trajectory of the Korea Pathfinder Lunar

44

Figure 4.1. Satellite Reachability Problem Diagram

Orbiter mission described in Song et al [50]. Both test cases assume impulsiveuncertain maneuvers.

4.2.1 Test Case 1: Two Burn Maneuver

This test case is based on Betts [49] which determines the optimal three burntransfer orbit from a set LEO parking orbit to a final operations orbit of a specificinclination. Betts’ paper considers three burn transfers to several target orbitswith varying inclinations, however this test case will only look at the first examplewith target i = 63.4. Additionally test case 1, only considers the first two burnsfor the nominal optimal transfer orbit given [49], and the end of the simulationis taken to be the position where the third burn would occur. The parking orbitis a circular LEO with i = 37.4, and the first burn is applied at an argument oflatitude u = 255. The RAAN is not important for this analysis and is set to zerofor simplicity. The nominal orbital elements for the coasting transfer orbits aregiven in Table 4.1.

The nominal maneuvers are modelled as impulsive ∆v∗ burns applied at thenominal yaw and pitch angles ψ∗ and θ∗ respectively. To compute the post burnvelocity, ∆v is computed with respect to the velocity, cross-track, orbit-normal(VCN) frame and transformed to the inertial frame. A diagram of pitch and yaw

45

Parameter Parking Orbit 1st Transfer 2nd Transfera∗(km) 6667.32 22,835.4 29,243.5e∗ 0 0.7080 0.6785i∗() 37.40 35.78 57.88ω∗() N/A 254.0 259.6

Table 4.1. Test Case 1: Nominal Transfer Orbit Parameters

angles in the VCN frame is shown in Figure 4.2. The ∆v in the VCN frame can be

Figure 4.2. Diagram of Maneuver Geometry VNC Frame

computed as

∆vvcn = |∆v|

cos θ cosψcos θ sinψ− sin θ

(4.8)

The post maneuver velocity in the ECI frame v1 is then calculated to be

v1 = T

|v0|

00

+ ∆vvcn

(4.9)

Where T is the rotation matrix which maps the VCN frame to the ECI frame.The parameters of the first and second burn as well as the location that each burnoccurs is summarized in Table 4.2.

It is assumed that the initial position and velocity are deterministic and knownbased on the given parking orbital elements and argument of latitude at the 1stburn. The uncertain variables are the maneuver magnitudes and attitude angles. Itis assumed that the pitch and yaw angles as well as the ∆V magnitude are normally

46

Parameter 1st Burn 2nd Burn Final Position|∆v∗|(m/s) 2,383.5 1,235.7 N/A

θ∗() -1.36 7.21 N/Aψ∗() 5.99 -74.7 N/Au∗() 255 37.7 121.7t(s) 0 6,078.2 35,609.0

Table 4.2. Test Case 1: Nominal Burn Parameters

-1

3

0

-2

1

2

LEO Orbit

1st Burn

2

×107Z

Positio

n (

m)

X Position (m)

×107

Y Position (m)

×1070 1

3

2nd Burn

2nd Transfer Orbit

1st Transfer Orbit

4

02 -1

Figure 4.3. Test Case 1: Nominal Orbit

distributed random variables with mean given by the nominal parameters given inTable 4.2, and standard deviations given by Table 4.3.

Variable Standard Deviationσ∆V1 5 (m/s)σθ1 2σψ1 2σ∆V2 5 (m/s)σθ2 2σψ2 2

Table 4.3. Nominal Maneuver Uncertainties

47

4.2.2 Test Case 2: Lunar Orbit Insertion

This section will provide details on the various parameters and assumptions madefor the 2nd reachability test case. The current test case simulates a Lunar OrbitInsertion (LOI) maneuver based on 1st orbit insertion maneuver of the KoreaPathfinder Lunar Orbiter mission described in Song et al [50]. A similar examplecan be found in Houghton et al [51]. It is assumed that the orbiter approaches themoon on a hyperbolic arrival trajectory with i = 90 such that the satellite canenter into a polar orbit. The nominal LOI maneuver is planned to be executed atthe periapsis of the arrival orbit such that the target orbit has a period of 12 hours.A diagram showing the geometry of the arrival trajectory and insertion maneuveris shown in Figure 4.4. The nominal maneuver is planned to occur in the direction

Figure 4.4. Test Case 2: Diagram of Arrival Geometry

opposite the velocity vector at periapse and follows the relationship:

v∗1 = v∗hyp + ∆v∗

r∗1 = r∗hyp(4.10)

48

Table 4.4. Test Case 2: Nominal Orbit ElementsOrbital Element ValueSemi-Major Axis (a0) 6143.4 (km)Eccentricity (e0) 0.6847Inclination (i0) 90Argument of Periapse (ω0) 0RAAN (Ω0) 0True Anomaly (ν0) 0Period (P ) 12 hr

The semi-major axis (a0) and eccentricity (e0) of the target orbit are defined bythe relationships:

a0 =(

(P/2π)2

µ

)(4.11)

e0 = 1− Rm + hpa0

(4.12)

Where P is the orbital period, µ is the standard gravitational parameter of theMoon, Rm is the radius of the Moon, and hp is the altitude above the lunar surfaceat periapse. It is assumed that the parameters i0, ω0,Ω0, ν0, hp and hyperbolicarrival velocity at periapsis vhyp are design parameters that can be specified basedon mission objectives. The current analysis will use the same target orbit as inRef. [50], with hp = 200km, |v∗hyp| = 2.4km/s, time of maneuver ti = 0s and allother targeted orbit parameters shown in Table 4.4. The values given representthe nominal orbit of the spacecraft immediately after the burn. For test case 1,uncertainty will be accounted for in the (3× 1) Cartesian pre-maneuver positionand velocity vectors rhyp and vhyp, as well as in the (2×1) attitude of the maneuverθ, ψ and the scalar magnitude of the maneuver vector |∆v|. Figure 4.2 illustratesthe geometry of the orbit insertion maneuver. Note that all input variables areassumed to be normally distributed random variables and are represented in theVCN reference frame. This allows us to specify more realistic uncertainties (greaterin the velocity direction and less in the orbit normal and cross track directions).The input variables are listed in Table 4.5, and the nominal orbit is shown in Figure4.5.

To reduce the number of variables, we combine the arrival velocity and maneuverusing Eq. (4.10). We must now determine the post maneuver velocity v∗vcn1 mean

49

5

Incoming Hyperbolic Orbit

×106

Y Position (m)

0

Target Orbit

LOI

Maneuver

-8

-6

-4

×106

X Position (m)

-5

-2

Z P

ositio

n (

m)

×106

-10

0

-5

2

4

0

Figure 4.5. Test Case 2: Nominal LOI Maneuver

Table 4.5. Test Case 2: Input UncertaintyVariable Direction Standard DeviationPosition ev σ

(pos)v

Position ec σ(pos)c

Position en σ(pos)n

Velocity ev σ(vel)v

Velocity ec σ(vel)c

Velocity en σ(vel)n

Attitude N/A σθAttitude N/A σψ

Burn Magnitude N/A σ|∆v|

and covariance (Σ) in the VCN frame.

v∗vcn1 = E[vvcn1 ] = E[vvcnhyp + ∆vvcn] = v∗vcnhyp + ∆v∗vcn

Σvcnv1 = E[(vvcn1 − v∗vcn1 )(vvcn1 − v∗vcn1 )T ] = Σvcn

vhyp+ Σvcn

∆v

(4.13)

The mean hyperbolic arrival velocity v∗vcnhyp = 2.4km/s, and the mean maneuvervector ∆v∗vcn can be calculated using the orbital elements for the target orbit. Thecovariance matrix for the hyperbolic arrival velocity Σvcn

vhypis specified, however the

covariance matrix for the maneuver in VCN frame Σvcn∆v is not directly specified

50

from the burn magnitude and attitude uncertainties so we must calculate it. Theattitude model for this test case is the same as the one used for test case 1 Eq.(4.8) and (4.9). Using this relationship we can use the CUT8 numerical integrationscheme to calculate the covariance matrix in the VCN frame.

Σ∆v =∫

(∆vvcn −∆v∗vcn)(∆vvcn −∆v∗vcn)Tρ(ζ)dζ

= wi(∆vvcni −∆v∗vcn)(∆vvcni −∆v∗vcn)T(4.14)

Where wi are the CUT8 weights and ∆vvcni isThe post maneuver velocity covariancematrix Σ(vcn)

v1 can now be calculated using Eq. (4.13). Assume that matrix T is therotation matrix which maps the VCN frame to the Moon Centered Inertial (MCI)frame. To determine the mean velocity as well as position and velocity covariancesin the MCI frame, the following relations are used:

v∗mci = Tv∗vcn1

Σ(mci)v1 = TΣ(vcn)

v1 TT

Σ(mci)r1 = TΣ(vcn)

r1 TT

(4.15)

Now we have complete representations for the mean and covariance of the statevector in MCI frame:

x =

r1

v1

, Σx =

[Σ(mci)

r1 03×3

03×3 Σ(mci)v1

], x∗ =

r∗mci1

v∗mci1

(4.16)

Using this framework we will examine 2 different scenarios: 1) Low Non-Linearity(LNL) Scenario, and 2) a High Non-Linearity (HNL) Scenario. For LNL, the finaltime of the simulation will be 30 minutes after the maneuver is performed, and forHNL, the final time will be 6 hours after the maneuver. The uncertainty conditionsfor low and high non-linearity cases are shown below in Table 4.6.

4.3 ResultsThe following section will discuss the results of both the two burn transfer orbit, andthe LOI maneuver test cases. Both test cases use the CUT8 method to calculatethe coefficients for polynomial approximation orders 1 through 4, then use the

51

Table 4.6. Simulation UncertaintiesVariable LNL HNLσ

(pos)v (km) 1 20σ

(pos)c (km) 0.1 2σ

(pos)n (km) 0.1 2

σ(vel)v (m/s) 5 15σ

(vel)c (m/s) 0.5 7.5σ

(vel)n (m/s) 0.5 7.5σθ (deg) 0.1 5σψ (deg) 0.1 5σ|∆v| (m/s) 5 15

coefficients to calculate a reachability set of 100,000 randomly sampled points. Aftercomputing the approximation coefficients, 5000 Monte Carlo points are randomlysampled from the distribution and propagated fully throughout the simulation.The error between the polynomial approximation and the fully propagated MCsamples is computed to determine the accuracy of the approximation.

4.3.1 Test Case 1: Two Burn Maneuver

The following plots depict the results of the for test case 1. Figure 4.6 shows theevolution of the reachability set over time using the most accurate (4th order)sensitivity matrix. Each color in Figure 4.6 represents all possible positions ofthe satellite at a given time. It is apparent that over time and with the addi-tion of uncertainty in the 2nd burn, the reachability set expands considerably insize. Figures 4.7 and 4.8 depicts the error at final time for varying polynomialapproximation orders for 5000 random samples. Figure 4.7 is a scatter plot withthe final satellite position r2 plotted on the X,Y,Z axes and the final positionerror ε is shown on a log scale in the color bar. Figure 4.8 plots the MahalanobisDistance of the initial samples for r1 on the x axis, v1 on the y axis, and showsthe final position error. Table 4.7 shows the RMSE for position and velocity atvarying polynomial approximation orders. Even at a 4th order approximation, theRMSE in the reachability set is quite large ≈ 14km. Although this seems like alarge error, considering the error relative to the 1st order error ≈ 63km and thespatial volume of the reachability set (on the order of 10, 000km3), this error is

52

Figure 4.6. Test Case 1: 4th Order Reachability Set Over Time

Approximation Order Position RMSE (m) Velocity RMSE (m/s)1 6.294E5 6.294E12 5.251E4 9.432E03 1.619E4 2.682E04 1.454E4 9.810E-1

Table 4.7. Test Case 1: Error Summary

pretty reasonable. The Mahalanobis distance plots show a similar trend whereincreasing the approximation order decreases the error, even if the improvment isonly marginal. The uncertainty in this simulation is relatively large compared tosome of the other cases that will be presented in this chapter, and the roughly 10hour propagation time allows non-linearities to accumulate in the solution, so theapproximation error is correspondingly large.

4.3.2 Test Case 2: Lunar Orbit Insertion Maneuver

This section will present and discuss the results of the LOI maneuver reachabilityset computation for both HNL and LNL scenarios. The parameters for each canbe found in Table 4.6. First the low uncertainty case will be given and then thecorresponding results for the high uncertainty case. The same metrics will be usedto assess the polynomial approximation as in test case 1.

53

2.5

3

2

Z P

ositio

n (

m)

×107

3.5

Y Position (m)

×107

1.5

4

-1.4

×107

X Position (m)

-1.61 -1.8-5

0

5

Log10(|ǫ|)

(a) 1st Order

2.5

3

2

Z P

ositio

n (

m)

×107

3.5

Y Position (m)

×107

1.5

4

-1.4

X Position (m)×107

-1.61 -1.8-5

0

5

Log10(|ǫ|)

(b) 2nd Order

2.5

2.5

3

2

×107

Z P

ositio

n (

m) 3.5

Y Position (m)

×1071.5

4

-1.4

X Position (m)×107

1 -1.6-1.8

-5

0

5

Lo

g1

0(|ǫ|)

(c) 3rd Order

2

2.5

2.5

2

3

×107

Z P

ositio

n (

m)

Y Position (m)

×107

3.5

1.5

4

1 -1.4

X Position (m)×107

-1.6-1.8

-5

0

5

Log10(|ǫ|)

(d) 4th Order

Figure 4.7. Test Case 1: Spatial Position Error Plot

Figures 4.10 and 4.11 depict the error in position between 5000 fully propagatedMC points and sensitivity matrix approximation using scatter plot and Mahalanobisdistance representations respectively. Table 4.11 summarizes the position andvelocity RMSE at all MC points. Figure 4.9 shows the evolution of the LNL case 4thorder reachability set over time. From inspection, we can tell that the accuracy of thepolynomial approximation improves as polynomial approximation order increases.For the low uncertainty test case, even the 1st order approximation providesreasonable accuracy. From Table 4.8 we can see that a first order approximationgives us an RMSE of 15m in position and 3.7cm/s is velocity. This is fairlygood accuracy considering the relatively large initial position uncertainties of1km. Depending on the accuracy requirements of the reachability set, a 1st order

54

(a) 1st Order (b) 2nd Order

(c) 3rd Order (d) 4th Order

Figure 4.8. Test Case 1 Mahalanobis Distance Position Error

approximation may be sufficient. However if greater accuracy is required, increasingthe approximation to 4th order gives us a near exact solution with error of about 10micrometers. Observing Figure 4.11 it can be seen that as polynomial approximationorder is increased, the variation of error with respect to Mahalanobis distance isdecreased. That is to say, the distribution of error becomes smoothed such thatthe error becomes uniform over the entire domain of the approximation. Thisphenomenon is similar to that which was observed for the uncertain Lambertproblem. It is also interesting to note that from Figure 4.11 it is clear that ingeneral the gradient of the error is more dependant of the velocity Mahalanobisdistance than on the position. Intuitively this makes sense, becasue a variationin velocity has larger effects on the resulting orbit than a variation in position.Propagated over time, a small change in velocity can greatly expand the possiblereachable positions.

Next, the results for the high uncertainty test case are presented and discussed.

55

Figure 4.9. Test Case 2 LNL: 4th Order Reachability Set Over Time

Approximation Order Position RMSE (m) Velocity RMSE (m/s)1 1.566E1 3.764E-22 8.555E-2 2.997E-43 7.735E-4 3.468E-64 1.044E-5 5.564E-8

Table 4.8. Test Case 2 LNL: Error Summary

The evolution of the 4th order reachability set over time for the high uncertainty testcase is given by Figure 4.12. Figure 4.13 plots the spatial distribution of final positionerror for varying polynomial approximation, and Figure 4.14 plots these errors vsinitial position and velocity Mahalanobis distances. The RMSE for final positionand velocity are summarized in Table 4.9. Unsurprisingly, the error in the HNL caseare much higher than in the LNL case. In this case, adding higher order sensitivitymatrices up to 4th order gains almost 2 orders of magnitude in position RMSEwhen compared with 1st order. Although this is a significant improvement, the 4thorder approximation still gives a position RMSE of around 3.8km. Depending onthe application, this might still be too much error. Unfortunately, with the currenttheory of the conjugate unscented transform method and a 6 dimensional system,we are limited to 4th order approximations. If the desired accuracy is still not metusing the 4th order approximation, then either the uncertainty or propagation timemust be reduced, or another quadrature method is required to compute the 5th

56

4.82.86

X Position (m)×105

4.7

2.87

Y Position (m)

0 4.6

2.88

2.89

2.9

2.91

×106

Z P

ositio

n (

m) 2.92

2.93

2.94

2.95

-5

0

5

Lo

g1

0(|ǫ|)

(a) 1st Order

4.8

X Position (m)×105

2.86

4.7

Y Position (m)

2.87

0 4.6

2.88

2.89

2.9

2.91

×106

Z P

ositio

n (

m) 2.92

2.93

2.94

2.95

-5

0

5

Log10(|ǫ|)

(b) 2nd Order

4.8

X Position (m)×105

2.86

4.7

Y Position (m)

2.87

0 4.6

2.88

2.89

2.9

2.91

Z P

ositio

n (

m)

×106

2.92

2.93

2.94

2.95

-5

0

5

Log10(|ǫ|)

(c) 3rd Order

4.8

X Position (m)×105

2.86

4.7

Y Position (m)

0

2.87

4.6

2.88

2.89

2.9

2.91

×106

Z P

ositio

n (

m)

2.92

2.93

2.94

2.95

-5

0

5

Lo

g1

0(|ǫ|)

(d) 4th Order

Figure 4.10. Test Case 2 LNL: Spatial Position Error Plot

order and higher sensitivity matrices. However, computing sensitivity matricesof orders greater than 4 with methods other than CUT can be computationallyinfeasible.

Another interesting trend to notice in the higher order sensitivity method is thatthe amount that accuracy improves by adding higher order sensitivity matrices,is largely dependant on the magnitude of the error to begin with. For example,test case 1 had the lowest accuracy out of any of the test cases analyzed, with aposition RMSE of approximately 630km for the 1st order approximation. Addinghigher order matrices up to 4th order, did decrease the error but only by about 1.5orders of magnitude. Contrasting this fact with the LNL case of test case 2, wesee that the 1st order approximation gives a position RMSE of 1.6m whereas the4th order approximation has a position RMSE of 1E − 5. This means that adding

57

(a) 1st Order (b) 2nd Order

(c) 3rd Order (d) 4th Order

Figure 4.11. Test Case 2 LNL: Mahalanobis Distance Position Error Plot

Approximation Order Position RMSE (m) Velocity RMSE (m/s)1 2.321E5 3.873E12 4.132E4 1.023E13 1.238E4 4.639E04 3.766E3 1.718E0

Table 4.9. Test Case 2 HNL: Error Summary

higher order sensitivity matrices to the LNL case of test case 2 decreased the errorby roughly 6 orders of magnitude. This is a somewhat undesirable trend whichessentially means that the approximations with the highest error gain the least byadding higher order sensitivity matrices.

58

Figure 4.12. Test Case 2 HNL: 4th Order Reachability Set Over Time

-4

-6

×106

X Position (m)

-8

-10

-12

×105Y Position (m)

0

-2

0

×106

Z P

ositio

n (

m) 2

4

-5

0

5

Log10(|ǫ|)

(a) 1st Order

-4

-6

-8

X Position (m)

×106

-10

-12

-3

Y Position (m)×105

0

-2

-14

-1

0

1

×106

Z P

ositio

n (

m)

2

3

4

5

-5

0

5

Lo

g1

0(|ǫ|)

(b) 2nd Order

-2

-4

-6×106

X Position (m)

-8

-10

-12

Y Position (m)×105

0

-2

0

Z P

ositio

n (

m)

×106

2

4

-5

0

5

Lo

g1

0(|ǫ|)

(c) 3rd Order

-4

-6

-8×106

X Position (m)

-10

-12

Y Position (m)×105

-140

-2

0

2

Z P

ositio

n (

m)

×106

4

6

-5

0

5

Log10(|ǫ|)

(d) 4th Order

Figure 4.13. Test Case 1 HNL: Spatial Position Error Plot

59

(a) 1st Order (b) 2nd Order

(c) 3rd Order (d) 4th Order

Figure 4.14. Test Case 1 HNL: Mahalanobis Distance Position Error Plot

60

Chapter 5 |Conclusions

5.1 Summary of Current WorkThe goal of this thesis was to provide a probabilistic framework for solving theuncertain Lambert problem and the reachability set problem. This objective wasaccomplished by computing sensitivity matrices, which represent the higher orderTaylor series expansion terms of the solution to each problem. Necessarily, theTaylor series must be truncated to a finite order, which provides an approximatesolution. In both the uncertain Lambert problem and the reachability set problem,it has been demonstrated that increasing the order of sensitivity matrices decreasesthe approximation error and expands the domain over which the approximationis valid. The main innovation is derived from the use of non-product quadraturemethod Conjugate Unscented Transform (CUT) to compute higher order sensitivitymatrices in a non-intrusive and computationally efficient manner. The CUT methodhelps in curtailing the curse of dimensionality associated with computation of thesesensitivity matrices.

Although the developed approach is valid for a generic nonlinear system, theaccuracy of the developed approach is dependent upon the size of uncertain domainas well as the inherent nonlinearity of the problem. One will need to judiciouslyselect the order of polynomial approximation depending upon the initial uncertaintyand nonlinearity of the problem. The use of orthogonal polynomial is beneficialas calculation of higher order coefficients is independent of calculations of lowerorder coefficients and hence, these coefficients can be computed recursively toachieve the desired accuracy. Off course, the computational burden associated with

61

computing higher order sensitivity matrices will increase with increase in the orderof polynomial approximation. The CUT method which provides computationalefficiency provides 8th-order quadrature points only up to six dimensions. One mayneed to generalize the CUT method for higher order approximations or resort toother methods with increased computational burden to sensitivity matrices forpolynomial higher than fourth order.

5.2 Furture Research DirectionThe mathematical framework laid out in this thesis is a valid approach to ap-proximate any stochastic non-linear function; however, the focus was placed onSSA applications due to the inherent uncertainty present in space operations. Thesolution to these problems is not meant to be a final product in SSA, but as atool used in solving other problems. There are two sensible directions in whichfuture research can go: 1) "Upward", in which this method is implemented intomore macroscopic application, and 2) "Downward" towards refining the buildingblocks of the method itself.

First, consider the upward direction. The higher order sensitivity method hasfar-reaching applications in many fields, however, there are several clear applicationsin aerospace engineering, and in particular, in SSA. One possibility is to implementthis method for collision assessment of two bodies that are predicted to have aclose encounter. The reachability problem can be used to provide highly accurateprobability distribution functions of each object during the encounter which wouldenable far more accurate numerical calculation of the probability that these objectswill collide. Other possible applications of these problems include implementing thisframework in sensor tasking, maneuver detection, and data association algorithms.All of these applications must account for uncertainty, and could greatly benefitfrom an efficient approach to computing the stochastic solution to the lambertproblem and the reachability problem.

The second direction is to go inward. The CUT method is a very powerfultool, but like any numerical integration method, it has limitations. One possibleresearch direction is to expand the CUT method to satisfy even more momentconstraints or to determine CUT weights and points for higher dimensional spaces.This would enable even better accuracy in highly non-linear problems. There is

62

also potential to improve accuracy by looking into the possibility of combiningseveral localized approximations to approximate the entire domain. Using multiplelocal approximations could have the benefit of increasing accuracy over the entiredomain using relatively low order local approximations. Another possibility is toextend the reachability set problem to include continuous control variables. In theanalyses provided in this thesis all control inputs were assumed to be impulsive;however, developing the framework to compute reachability sets for finite burns orcontinuous thrust would greatly expand the applicability of the method.

63

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