Halder11_CharacteristicsSLE

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Dispersion Analysis in Hypersonic Flight During PlanetaryEntry Using Stochastic Liouville Equation

Abhishek Halder and Raktim Bhattacharya

Texas A&M University, College Station, Texas 77843-3141DOI:10.2514/1.51196

A framework is provided for the propagation of uncertainty in planetary entry, descent, and landing. The

traditionalMonteCarlobased dispersionanalysisis overly resource-expensive for such high-dimensional nonlinear

systems and does not provide any methodical way to analyze the effect of uncertainty for mission design. It is shown

that propagating the density function through Liouville equation is computationally attractive and suitable for

further statistical analysis. Comparative simulation results are provided to bring forth the efcacies of theproposed

method. Examples aregivenfrom theentry, descent, andlanding domainto illustrate howone canretrievestatistical

information of interest from an analysts perspective.

Nomenclature

Bc = ballistic coefcientCLCD = lift-to-drag ratioF = augmented dynamics vectorf = dynamics vectorg = acceleration due to gravity GMR0h2, where GM is the

Gravitational constant for Marsh = altitudens = number of statesnp = number of parametersp = parameter vectorR0 = mean equatorial radius of Marst = timeV = Mars-relative velocityvc = normalizing velocity constant

R0

q , where gR20

X = augmented state vector

x = state vector = ight-path angle = latitude = longitude = Martian atmospheric density0 = reference-level density = bank angle = probability density function = velocity azimuth angle measured from North = trace of the Jacobian of the dynamics = rotational angular velocity for Mars

Subscript

i = variable number

I. Introduction

W ITH the need to develop next generation entry, descent, andlanding (EDL) technologies for planetary exploration, quan-tication of uncertainties have become a major technical challengefor space system simulations. In particular, Mars EDL technologies

arewitnessing a paradigm shift as we strive for increasingthe landingmass capability at a higher surface elevation (lower atmosphericdensity) while improving the landing accuracy. Future missions

including Mars Science Laboratory (MSL), Mars Sample Return(MSR), and Astrobiology Field Laboratory (AFL) are pushing theEDL technology limits toward the aforesaid directions to pave theway for human exploration missions. As pointed out in [1,2], from1976 to 2008, all six NASA missions which landed successfully onMars, were based on Viking-era EDL technologies characterized bylanding mass of less than 600 kg, landing elevation of less than1:4 km Mars Orbiter Laser Altimeter (MOLA) reference withlanding footprint uncertainty in hundreds of km. In fact, they all hadsimilar design attributes, e.g., 70sphere-cone aeroshell, supersonicDisk-Gap-Band (DGB) parachutes with diameter less than theViking 16 m designand SLA-561V thermal protection system(TPS).In contrast, the MSL mission, scheduled to launch in 2011, has ahigher landing mass, higher surface elevationfor landing, an order of

magnitude improvement in landing footprint uncertainty, a largerDGB parachute and a new TPS. Braun and Manning [1] have listedthe probable future directions of EDL technological pursuits andchallenges thereof. To achieve such technical readiness level,development of these new generation of landers will require veryhigh-delity simulations enabling the assessment of risk andincorporating that knowledge for the purpose of robust missiondesign supported by statistical verication and validation. Moreprecisely, a probabilistic quantication of being inside or outsidesome predened operational safety marginis soughtthat can accountthe associated uncertainties in initial conditions and systemparameters resulting off-nominal trajectories.

Traditionally, a MonteCarlo (MC) based dispersion analysis iscarried out for this purpose where one simulates a large number oftrajectories for randomly sampled initial conditions and parameter

values. If most or all of such trajectories remain inside the safetymargin, one can at best hope for the system safety and reliabilitywithout any quantitative guarantee whatsoever. Usually theengineers responsible for subsystem models identify the uncertaintybounds and decide about the sampling strategy based on theirexperience. Clearly, brute force MC simulations are not the bestapproach for such mission critical uncertainty analysis. Moreover,for high-dimensional and nonlinear dynamics like spacecraft EDL,MC simulations are tremendously expensive as one strives tosimulate individual trajectories one by one for uncertainties inhundreds of states and parameters and their combinations. In spite ofallthese drawbacks, almostall space mission uncertainty analysis aredone with MC simulations including Mars Pathnder [3], METEOR[4] recovery module, Stardust [5] comet sample return capsule, MarsMicroprobe [6], MSP 2001 Orbiter and Lander [7] and to-be-launched MSL [8] mission. In fact, many important decisions inmission designhave beenhistorically drivenby MC based dispersionanalysis. Mars landing site selection [9], design of the Pathnder

Received 17 June 2010;revision received 24 September2010; accepted forpublication 27 September 2010. Copyright 2010 by Abhishek Halder.Published by the American Institute of Aeronautics and Astronautics, Inc.,with permission. Copies of this paper may be made for personal or internaluse, on conditionthat thecopier paythe $10.00per-copyfee to the CopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; includethe code 0731-5090/11 and $10.00 in correspondence with the CCC.

Graduate Student, Department of Aerospace Engineering; [email protected]. Member AIAA.Assistant Professor, Department of Aerospace Engineering; raktim@

tamu.edu. Member AIAA.

JOURNAL OFGUIDANCE, CONTROL, ANDDYNAMICSVol. 34, No. 2, MarchApril 2011

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aeroshell thermal protection system [10] and parachute deploymentalgorithm [3] areexamples for the same. This heavy bias toward MCsimulations among EDLanalysispractitioners is partlydue to itseaseof implementation and partly due to the scarcity of alternativeanalysis methods. Two primary EDL simulation frameworks whichare seeing extensive use at present, are NASA Langley ResearchCenters Program to Optimize Simulated Trajectories II (POST2)[11] and NASA Jet Propulsion Laboratorys Dynamics Simulator forEntry, Descent and SurfaceLanding (DESENDS) [12]. Both ofthese

two presently rely on MC based dispersion analysis for EDLsimulations.

The limitations of MC based dispersion analysis are well-known[13] (see Ch. 1), i.e., poor computational scalability and lack of anycoherent methodology to quantify the evolution of uncertainty in astatistically consistent manner. Consequently, researchers havepursued different methods for uncertainty propagation in dynamicalsystems in general while few of them have been attempted in EDLdomain. In this paper, we will classify these methods in two broadcategories:parametric(where one evolves the statistical moments)andnonparametric propagation of uncertainty (where one evolvesthe full probability density function, or PDF). There have been threemajor directions in parametric propagation of uncertainty as listedbelow.

1) The simplest method in this category assumes a linear statespace description of the form _x Ax, x2Rn. With the initialcondition uncertainty assumed to be Gaussian, one then sets forpropagating the mean ( x) and covariance (P) matrix by the well-

known equations _x A x and _P ATP PA. The computationalefciency of this method mainly depends on the efciency of thenumerical algorithm to integrate the covariancepropagationequationand insome cases, may evenperform worse thanMC [14]. Of course,themajordrawbackis that a typical EDLsystem with large numberofstates involving highly nonlinear dynamics with non-Gaussian initialjoint PDF is too far to t in this framewrok. Nevertheless, this hasbeen attempted in EDL problem (see [14]).

2) In polynomial chaos (PC) method, one derives a set ofdeterministic ordinary differential equations (ODEs) using eitherGalerkin projection [15] or the stochastic collocation [16] and then

solves that set of ODEs. Although this method can handle nonlineardynamics with non-Gaussian uncertainties, one ends up solving ahigher dimensional state space problem, which becomes intractablefor a realistic EDL simulation. Further, the method is difcult toapply for large nonlinearities [17] and computational performancedegrades (due tonite-dimensional approximation of the probabilityspace) if long-term statistics is desired. Recently, this method hasbeen applied to the EDL domain [18].

3) Another method in this category is called the direct quadraturemethod of moments (DQMOM) [19] where the PDFis approximatedas a sum of Dirac delta functions with evolving parameters. Thismethod suffers from the Hausdorff moment problem [20]. There aresome variations of this method where the PDF is expressed as aweighted sum of few constituent PDFs, referred to as partial PDFs,

and propagates them instead of the Dirac delta functions [21].Nonparametric propagation of uncertainties can be done in twoways: approximate method and direct method.

1) Approximate method is where one tries to estimate (in non-parametric sense) the underlying PDF. The method aims toapproximate the solution of thePDF transport equation.This methodis widely exercised in statistics community [22] under the name ofkernel density estimation, although most applications there concernwith static data. In a dynamical system, optimal values of theparameters must be determined at every instant of time. Many specialcases of this, may be constructed depending on the type of kernelfunction and the criteria for optimization. A least-square errorminimization set up is described in [23] and is shown to have goodcomputational performance. Further generalizations are possible byconsidering general basis functions which are not density functionsthemselves. However, this too can suffer from high computationalcost arising due to the explicit enforcement of normality constraintand moment closure constraint at each step of the optimization

procedure. Moreover, for high-dimensional state spaces likeplanetary EDL, recursively performing constrained optimizationbecomes extremely challenging.

2) Direct method is where one works with the PDF transportequation and instead of approximating its solution, strives to solvethat equation directly. In the absence of process noise (which is thecase we will restrict ourselves in this paper), this transport equationreduces to the stochastic Liouville equation (SLE) [24], which is aquasi-linear partial differential equation (PDE), rst order in both

space andtime.This equation describes thetime evolutionof thejointPDF over the state space, which itself is changing due to the knowndynamics. In this paper, we argue that the stochastic Liouvilleequation can be easily solved in such direct way using the method ofcharacteristics (MOC). Because all the statistics can be derived fromthe PDF, from an information point of view, it is denitely superiorthan parametric propagation methods. Further, since we will belooking to solve theSLE directly, thesolutions will satisfy thecriteriato be PDF. Hence the problems like moment closure or normalityconstraints are not required to be enforced explicitly.

Our objective in this paper is to demonstrate that an MOCimplementation in direct method can be computationally attractiveand it does provide the necessary rigor for a statistically consistentuncertainty quantication for the EDL problem. To see why it is thecase, one must realize that in solving the SLE, one propagates the

joint PDF prescribed at the initial time subject to the deterministicdynamics. In MC method, one randomly picks a single initialconditionand computes thetrajectory andthen repeatsthe process. Inthe SLE method, instead ofindividual realizations (initial conditionsand/or parameters), one propagates the ensemble of realizations.This, in essence, means that the number density of trajectories meetthe continuum hypothesis [24]. Just like the continuity equation inuid mechanics transports theuid mass in conguration space, theSLE transports the probability mass in phase space.

This paper is organized as follows. The SLE framework will bedescribed in the next section along with some illustrative examples.The nonlinear Hypersonicight dynamics model for planetary entrywill be mentioned in Sec.III. Before attempting to solve the SLE forthe full problem, we will describe some special cases in Sec. IV,which will provide some insight to the analysis. In Sec. V, thenumerical simulations will be described. Section VI will describehow to perform further EDL specic statisticalanalysis and it will beshown that many typical analysis problems of EDL interest, tsnicely in the SLE framework. SectionVIIwill conclude the paper.

II. Stochastic Liouville Equation

Consider the nonlinear autonomous state space model

_x fx; p; x2Rns ; p2Rnp (1)which can be put in an augmented form

_X FX; X2Rnsnp (2)

whereX x p T

is the extended state space.It is well known that under certain regularity conditions on the

functionf, the existence and uniqueness of the solution of Eq. (1)subject to the initial condition x0 x0, can be guaranteed (seetheorems 3.2 and 3.3 in [25]). With mild conditions, one can alsoensure that (theorems 3.4 and 3.5 in [25]) the unique solution of (1)depends continuously on x0, p and t. These propertiesare assumed tohold in all subsequent analysis.

Given such a deterministic function f (or equivalently F), thetransport equation that convects the probabilistic uncertainty in theinitial condition and parameters, is given by

@X; t@t

Xnsi1

@

@XiX; tFiX 0 (3)

which is a quasi-linearrst-order PDE with the joint PDF X; tbeing the dependent variable. This is the SLE which states that thespatio-temporal evolution of the joint PDF occurs in a way that

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preserves the total probability mass. Put differently, if we pick up acontrol volume in the (extended) state space, the net ux ofprobability mass must be zero provided no realizations are created ordestroyed (no source or sink), which holds true since all realizationsstart from the same initial condition [24]. SLE can be seen as thetransport equation associated with the Perron-Frobenius (PF)operator [26].

A. Method of Characteristics

In this section, we briey describe the MOC and show how thathelps in reducing a linear or quasi-linear PDE to an ODE along thecharacteristics. Application of MOC to nonlinear PDEs can be foundin [27].

Consider a PDE of the form

Xni1

aiz1; z2; . . . ; zn;U@U

@zi z1; z2; . . . ; zn;U (4)

with U being the dependent variable and z1; z2; . . . ; zn are the nindependent variables. The characteristic curves corresponding toEq. (4) are given by the LagrangeCharpit equations [28]

dz1a

1z

1; z

2; . . . ; z

n;U

dz2

a2

z1; z

2; . . . ; z

n;U

. . . dzn

anz1; z2; . . . ; zn;U dU

z1; z2; . . . ; zn;U (5)

Geometrically, this means that the (n 1) dimensional vectoreldF : a1z1; z2; . . . ; zn;U; a2z1; z2; . . . ; zn;U; . . . ;

anz1; z2; . . . ; zn;U; z1; z2; . . . ; zn;U

is tangent to the surface UUz1; z2; . . . ; zn 8 fz1; z2; . . . ; zngT 2Rn. In other words, the solution of the PDE (4) is an (n 1)dimensional surfaceUz1; z2; . . . ; zn and it can be constructed as theunion of the integral curves[or characteristic curvesgivenby Eq.(5)]of the vectoreld F.

To derive the characteristic curves for the SLE, we put Eq. (3) in aform similar to Eq. (4) (using product rule of differentiation)Xns

i1FiX

@X; t@Xi

@X; t

@t X; t

Xnsi1

@FiX@Xi

(6)

From Eq. (5), it readily follows that the characteristic curves forEq. (6) are given by

dX1

F1 dX2

F2 . . . dXns

Fns dt

1 dX; tX; tPnsi1 @Fi@Xi (7)

The equation above shows that the characteristic curves for the SLEare nothing but the trajectories of the dynamics given by Eq. ( 2).

B. Solution Methodology

It can be noted from Eq. (7) that using MOC, along the trajectory,one can reduce the SLE to an ODE of the form

dX; tdt

X; tX (8)

where X: Pnsi1 @Fi@Xi is the trace of the Jacobian of theunderlying dynamics and hence, evolves with time. At this point, it isapparent that if the intial state and parametric uncertainties arespecied in terms of a joint PDF 0 : X0; 0, then one canwrite the solution of Eq. (8) as

X; t

0exp Z

t

0

X

d (9)

The exponential in Eq. (9) is formally known as the orderedexponential [29] and is analogous to the Dyson operator of the

quantum Liouville equation[30] in statistical quantum mechanics.Because the ordered exponential is a ratio of the instantaneous andinitial PDFs, one may interpret it as a likelihood ratio [ 31].

It canbe observed thatXt is thedivergence of thevectoreldand hence, is a measure of the rate of change of the phase space(Lebesgue) volume. For example, ifF is linear time invariant, then must be a constant. Depending on the sign of this constant, the phasespace volume can expand (expanding ow) or contract (contractiveow) exponentially fast or may remain constant (volume-preserving

ow). The Liouville Theorem [32] tells us that the case ofdivergence-free vectoreld ensures that the system is Hamiltonian.Notice that a nonlinear vectoreld can be Hamiltonian too.

In general, it is hardly possible to evaluate the integral in Eq. (9)analyticallyand thus mandates numerical solution. Once the solutionfor the joint PDF X; t is obtained, one cannd the marginal PDFsby integrating out the other states over their respective domains,namely

Xi; t ZD1

. . .

ZDi1

ZDi1

. . .

ZDns

X; t dX1. . . dXi1dXi1. . . dXns (10)

whereD

i is the domain of the ith state variable at time t. Here it isimportant to realize that since the domain in the state space isdeforming with time, one must know the instantaneous domain tocarry out the integration in Eq. (10). This will be explained in moredetails later in the paper.

C. Illustrative Examples

We now provide some examples to clarify the ideas presentedabove. Specically, we want to illustrate how MOC enables thesolution of SLE (which is a PDE) by solving ODE initial valueproblem along the trajectories.

1. 1-D Example

Let us consider the simple 1-D dynamics [31] given by _x x2

with initial condition x0 x0. Then the solution of this initial valueproblem is given by

xx0; t x0

1 tx0(11)

Consequently, we have

xt 2xt 2x01 tx0

)exp

Z

t

0

x d

exp

2x0

Z t

0

d

1 x0

1 tx02 (12)

which, from Eq. (9), leads to

x; t 0x01 tx02 (13)Now we can ndx0 x0x; t using Eq. (11) as

x0 x

1 tx (14)

and substitute this to Eq. (13) to yield

x; t 0

x

1 tx

1 tx

1 tx

2

0 x

1tx1 tx2 (15)

Figure 1 shows the spatio-temporal evolution of the PDF x; taccording to Eq. (15), when the initial PDF is chosen to be a standardnormal distribution. The surface plot in Fig.1aillustrates the rise ofthePDF peak with time, accompanied with a shrinkage of itssupport.As Fig.1bshows, ast! 1, the PDF tends to become a dirac deltadistribution. This is not surprising since the origin being the unique

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equilibrium of this dynamics, in the asymptotic limit, all probabilitymass gets in that sink. Thus, the support of the stationary distributionhas Lebesgue measure zero.

2. 2-D Example

Next, consider a planar vectoreld

_x x 2ylogx2 y2 _y y

2x

logx2 y2 (16)

with given initial conditions x0 x0and y0 y0. Looking at theform of the dynamics, we convert Eq. (16) from Cartesian to polarcoordinates using the standard transformation r _r x _x y _y and_ x _yy _x

x2y2 to obtain

_r r _ 1log r

(17)

purely as a matter of working convenience. The initial conditions forEq. (17) are r0 :r0

x20 y20

p and 0 :0 arctan y0x0.

From the polar equations, it immediately follows that as t! 1,rt !0 andjtj ! 1 implying that the origin is a globallyasymptotically stable spiralfor this nonlinear system (Fig.2). Notice,however, that a linear stability analysis predicts the origin to be astable star. Infact, one caneasily solve Eq. (17) toget the trajectory inclosed form

rr0; t r0et; 0; t 0 log

log r0log r0 t

(18)

which corroborates the asymptotic behavior mentioned above.Further, one can compute

@@r

r @@

1

r

1

)exp Z t

0 r; d et (19)

From Eq. (18), we also get

r0r; t ret; 0; t log

log r tlog r

(20)

Thus, Eqs. (19) and (20) result

r ; ; t 0r0; 0et 0

ret; log

log r tlog r

et (21)

If the initial conditions are sampled from a uniform distribution,

the transient PDFs resemble the phase portrait of Fig.2, convergingtoward a dirac delta at the origin. To examine the case for non-uniformly sampled initial conditions, an initial PDF is taken whichhasa high probability around 0 andis symmetric about thesame.The polar plots of Fig.3shows the PDF contours att 0, 0.2, 0.5,1.0, 1.4 and 2.0, respectively, for the dynamics given by Eq. ( 16). Itcan be observed that the support of the transient PDFs shrinkprogressively and spirally converge toward the origin. The center(periphery) has high (low) probability.

Remark 1: The two simple examples given above illustrate howMOC solves the SLE. In MOC, the initial value problem (IVP) givenby Eq. (8) is solved along the characteristics (which in case of SLE,are the integral curves or trajectories of theow). Thus the integral inEq. (9) is a path integral computed along each trajectoy (see Fig. 4).

a) With time, the PDF support shrinks and tends toward

the asymptotic limit of impulse function.

2

0

2

2

4

6

0

5

b) Stacked PDF snapshots at the initial and five consecutive times,

starting with a standard normal distribution.

Fig. 1 Evolution of theN0; 1initial PDF according to Eq. (15).

3 2 1 0 1 2 3

3

2

1

0

1

2

3

3 2 1 0 1 2 3

3

2

1

0

1

2

3

Fig. 2 Vectoreld (left) and an ensemble of trajectories in the phase space (right) for the nonlinear system given by Eq. (16).



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To further clarify this, consider a divergence-free vectoreld. ThenEq. (8) tells us that the joint PDF remains constant as long as we areridinga particular trajectory. The value of this constant is differentalong a different trajectory. Thus, a volume-preserving ow, ingeneral, does admit a spatio-temporally evolving PDF. For the same

reason, in the above examples, initial condition was computed as afunction of the current state and time to substitute for X0 inX0; 0 [see Eqs. (14), (15), and (21)].

Remark 2: For the SLE, since the trajectories are same as thecharacteristic curves and trajectories cannot intersect (due touniqueness), the solutions of the MOC are no where discontinuous.Further, notice that, to compute theinverse mapofx xx0; p; t, ofthe form x0 x0x; p; t, our earlier stated assumption of thecontinuity ofx on x0,p, andt, comes into play.

III. Nonlinear Flight Dynamics for Hypersonic Entry

In this paper, we will concentrate on applying the theoreticalframework described above to the problem of hypersonic EDLmodeled through Vinhs equations [33]. We will work with twodifferent versions of the model, a three state model where the

dynamics is assumed to be purely longitudinal, and a more generalsix-state model with lateral-longitudianl coupling. Both thesemodels describe the trajectory of the center-of-mass of the spacecraftentering into the Mars atmosphere.

A. Three-State Model

Assuming the entire trajectory is contained in the longitudinalplane, one can write the following nondimensionalized three-stateh ; V ; model for nonrotating spherical Mars with zero bank angleight.

_h Vsin (22a)

_V R02Bc V

2

gR0v2c sin (22b)

_ R02Bc

CLCD

V gR0v2c

cos

V

1 h 1

V

(22c)

Here the model for Martian atmospheric density variation [34] istaken as

0exp

h2 hR0h1

(23)

where h2 20 kmand h1 9:8 km. The mean equatorial radius ofMars will be taken asR0 3397 km.

B. Six-State Model

Here we present the more generalformof Vinhs equations, whichis a nondimensionalized six-state h ; ; ; V ; ; model. This

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

Fig. 3 The PDF contours computed from Eq. (21) att0, 0.2, 0.5, 1.0, 1.4, and 2.0, respectively.

Fig. 4 In MOC based SLE method, along each sample trajectory, theprobability weights are updated during dynamics propagation. In MCmethod, one tries to reconstruct a histogram toapproximatesuch weight

distribution, as a postprocessing step. So the main advantage of SLE

compared with MC is the ability to update exactprobability weights onthe y and hence, the samples are colored, so to speak, with the color-

value being proportional to the value of the instantaneous joint PDF.



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modeltakes the self-rotation rate of the planet and the bankangle into account.

_h Vsin (24a)

_ Vcos sin 1 h (24b)

_ Vcos cos 1 h cos (24c)

_V R02Bc

V2 gR0v2c

sin

R20

2

v2c1 h cos sin cos cos sin sin (24d)

_ R02Bc

CLCD

Vcos gR0v2c

cos

V

1 h 1

V

(24e)

_ R02Bc

CLCD

Vsin

cos Vcos 1 h tan cos

2R0vc

tan cos sin sin

R20

2

v2c

1 hVcos

sin cos cos (24f)

was calculated from the rotational time period of Mars, which is24 h, 39 min, and 35.24 s. The density variation is taken identical tothe three-state model.

IV. Application of SLE to Some Specic Cases

Before solving SLE for the models described in the precedingsection, we will examine certain restricted cases of the same. Sincethe three and six-state models, in general, require numerical solutionfor the PDF, considering some specic cases will give us somephysical understanding of the problem. Many case studies of thisnature can be found in [14] (see Ch. 7).

A. Horizontal Flight

For horizontal ight, 0and h constant. Therefore, only thesecond equation remains to be considered in Eq. (18), whichbecomes

_V R02Bc

V2

)Z

V

V0

dV

V2 R0

2Bc

Z t

0

dt since h is constant; so is

)V V01 R0

2BcV0t

(25)

which implies that V decreases monotonically with time. In this

case,V; t 0V0 exp Rt0V d, whereV R0Bc V.Therefore

V; t 0V0

1 R02Bc

V0t

2

0V0

V0V

2

0

V

1 R02Bc

Vt

1

1 R02Bc

Vt2 (26)

Thus, given an initial PDF describing the intial condition uncertainty,Eq.(26) provides an algebraic expressionfor determining the PDFatany current time and velocity.

B. Vertical Flight

This special case concerns the vertical descent ( 2) in anonlifting trajectory. Consequently, we eliminate Eq. (22c) as allterms in it are identically zero. Hence, we are left with Eqs.(22a),(22b), and (23). Substituting in Eq. (22b) as a function of h,we get two rst-order coupled nonlinear ODEs in h and V, shownbelow

_h V (27a)

_V K1V2eh K2 (27b)

withK1 0R02Bc eh2=h1 , R0h1

andK2 gR0v2c . With 2K1Veh,

the SLE needs to be solved numerically along with the abovedynamics.

With nominal initial altitude h0 80 km and nominal initialvelocity V0 3:5 km=s, and assuming 5% uniform dispersion inboth h0and V0, the SLE was solved numerically. The simulation wasrepeated for 15% uniform dispersion in both h0and V0. In both thecases, 1000 samples were taken to represent the trajectory ensemble.Figure5shows the color-coded scatter plots att 19:13seconds inthehVplane with the color-value being proportional to the value ofthe bivariate joint PDF at that instant. Notice that, a larger dispersionin the initial conditions results in more spread in thepointcloud at thesame instant of time.

2.5 2.6 2.7 2.8 2.9 3 3.1 3.28

10

12

14

16

18

20

22

V (Km/s) V (Km/s)

h(Km)

a) Scatterplot at t= 19.13 sec with 5% uniform initialcondition uncertainties in h0 and V0

b) Scatterplot at t= 19.13 sec with 15% uniform initialcondition uncertainties in h0 and V0

1.5 2 2.5 3 3.50

5

10

15

20

25

30

35

h(Km)

Fig. 5 Scatterplot snapshots for uncertainty evolution of systems (27a) and (27b) with uniform dispersions in the initial conditions.

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V. Numerical Simulations

In this section, the numerical simulation set up is described forsolving the SLE for the three- and six-state models described inSec.III. The simulation framework is provided in detail followed byresults and discussions.

A. Simulation Setup

The nominal initial conditions were taken to be h0 80 km,0 24:01N, 0 341:03E, V0 3:5 km=s, 0 2 and0 0:0573. The nominal values of the parameters were taken asBc 72:8 kg=m2, 0 0:0019 kg=m3 and CLCD 0:3. Because themodels described in Sec. III are nondimensionalized, numericalintegration was performed in nondimensional time ~t 0to 0.7 withthe nondimensional step-size of~t 0:01. One caneasily convertitback to the physical time by multiplying the nondimensional time

with a factorR 0vc

. In this paper, results are presented for two kinds of

initial uncertainties, viz. 5% uniform dispersion in each variable andGaussian dispersion about the nominals with 10% variance alongeach dimension.

B. Simulation Framework

The simulation framework comprises of three main modules as

described below.

1. Sampling Initial Distribution

The initial uncertainties are specied by an initial joint PDF. Oncethe initial joint PDF is known, one needs to generate a prespeciednumber of samples such that they best represent that joint PDF. Forthe case of uniform initial distribution, one may do a grid-baseddiscretization or for high dimensions, opt for a pseudorandomnumber generator using low-discrepancy sequences like Haltonsequence [13] to avoidthe curse of dimensionality. Some preliminarycomparative simulation results along these lines were reported in[35]. In this paper, samples from uniform initial PDF were generatedusing multidimensional Halton sequence. For nonuniform initialPDFs, one need to use probability integral transform (e.g., BoxMuller transform in case of normal distribution) methods [36].However, one must resort to the Markov Chain MC [3739](MCMC) techniques to achieve better computational performancefor sampling any general initial PDF in high dimensions [40].

2. Uncertainty Propagation

The samples from the initial uncertainty polytope are propagatedaccording to the dynamics given in Sec.IIIand the SLE is solved ateach time step. Four libraries are required to achieve this. Thedynamics library species the nonlinear model and the atmospheremodel is provided in a separate library, which is used by thedynamics. Another library does gradient computation (analyticallyfor thepresent caseor usingnite differencing for a black boxmodel)needed to solve the SLE. A fourth-order RungeKutta (RK4) basedintegrator was used to propagate the dynamics and for solving theSLE. All results reported below are from the MATLABimplementation using variable step-size ode45integrator.

3. Postprocessing of the Joint PDF Data

As the samples from the initial PDF are propagated according todynamics, the joint PDF at any given time is represented by theinstantaneous distribution of those evolved samples. Because of thenonlinear dynamics, such a distribution, in general, is a scattered dataset residing over the extended state space. These evolved joint PDFsare required to compute the marginal distributions that correspond tothose typically used in EDL analysis. For doing this, one needs toisolate a snapshot of interest and integrate out the dimensions otherthan whose marginal is sought. In Sec. II.A, webriey touched uponthe fact that because of dynamics, the domain or the support of thejoint PDF deforms with time and the integration for marginalcomputation needs to be carried out over few dimensions of this

instantaneous domain. This brings forth the problem of integrationover high-dimensional scattered data.

One way to tackle this problem is to interpolate these scattereddata, which itself is numerically challenging. We mention here thatsince the joint PDFvalues were computed directly by solving SLE, itis an interpolation problem as opposed to function approximation.Alternatively, one may attempt the numerical integration withoutinterpolation. For this, one can sprinkle a new set of Halton points(prefarably more than the number of samples) inside the bounding

boxof this static/time-frozen data and then use these newly sprinkledpoints as the quadrature points to carry out quasi-MC (QMC)integration [13] (see Ch. 2). The computational cost associated withthis approach comes from the evaluation of the joint PDF values atthis new set of points, which can be determined by rst back-integrating the dynamics and then forward integrating the SLE forthese quadrature points.

Notice, however, that for computing marginals from MC simu-lations, one takes a frequentist approach and counts samples in thebins lying on the requisite slices. Because we are interested tocompare the SLE derived marginals with those obtained from MC, asimilar (and computationally less heavy than described above)method can be employed to approximate the marginals from SLE. Inthe PF-derived SLE method, since one has a probability weightassociated with each sample, one can do a binning similar to MC

histograms. Only this time, instead of counting the number ofsamples in each bin, the bin weight can be assigned as the average ofthe joint PDF values of the samples in that bin (see Fig. 6).In both cases, the individual bin weights need to be normalized withrespect to the bin size and total probability weight. All marginals ofPF-derived SLE method presented here, are computed in thisfashion.

Fig. 6 A schematic for computing marginals from scatteredinstantaneous joint PDF data. For example, after solving the SLE for

three-state Vinhs equations with initial condition uncertainties, tocompute univariate marginal inh, one would take Vslices at differentsample levels of altitude. At each such slice (the one with solid edges),one can average out the Vand directions using the joint PDF valuesas explained in Sec. V.B.3and obtain a scalar value. One would then

shift this slice at the nexth sample level (dotted slice) and repeat the

exercise. This results the h-marginal vectorhh as shown. By takingslices in other orthogonal directions, one can similarly getVV and. The idea can be easily extended to compute higher dimensional

marginals.



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C. Results and Discussion

For the three-state model, it is possible to visualize the joint PDFusing three dimensional color-coded scatter plots similar to Fig. 5.Such plots are shown in Fig. 7 for 1000 samples at~t 0:05, 0.20,0.30, and 0.50 with both uniform (top row) and Gaussian (bottomrow) initial condition uncertainties. It can be observed that at~t 0:05, the joint PDFs are slightly perturbed from the respectiveinitials. As time progresses, the probability mass accumulates nearzero altitude and zero velocity and the ight-path angle (FPA)

assumes a steep value. This is in agreement with the physicalintuition as the vehicle, with high probability, slows down throughthe lower part of the atmosphere.

Starting from the uniform initial joint PDF, the evolution of theunivariate and bivariate marginals for the three-state Vinhsequations, are shown in Fig. 8. The same for the Gaussian initial PDFare plotted in Fig. 9. Theunivariate MC (dashed) andPF (solid) PDFsare in good match. Thebivariate maginals show thegeneral trend thatPF-derived marginals (bottom row) capturethe concentration of the probability mass well (by virtue of theprobability weights obtained by solving SLE) while theMC bivariatemarginals (top row) tend to smear it out (because of the histogramapproximation). This can be seen, for example, in Vbivariate plots.Similar trends can be observed for the six-state model. For brevity, inFig. 10, we only show thesnapshotof univariate PDFs at~t

0:30 for

the six-state model with uniform initial PDF.Thesimulation resultsshown above bears testimonyto thefact that

with same number of simulations, PF operator based approach canbetter resolve the PDF compared with MC method. This is notsurprising since the former assigns explicit probability weightscomputed by solving theSLE while the latter tries to construct a PDFusing crude histogram approximation. The success of the latter (interms of good approximation of thePDF) is heavily dependent on thenumber of sample trajectories being evolved. Hence, SLE-based PFmethods can be computationally attractive over MC, particularly inhigh-dimensional nonlinear problems like spacecraft EDL, as itevolves less number of high initial probability samples to achieve anaccurate resolution of the PDF.

To quantify the closeness of the MC and PF based marginal PDFs

shown before, two information-theoretic quantities were used, viz.KullbackLeibler (KL) divergence KL and Hellinger distance HL.The KL divergence measures the distance between two PDFs Pxand Qx, and is dened as

KLPkQ:Z1

1Px logPx

Qx dx (28)

It canbe interpreted as therelativeentropy between twoPDFsand is apseudometric (nonsymmetric and does not obey triangle inequality).A similar quantity, Hellinger distance, is dened as

2HLPkQ:1

2

Z11

Px

p

Qx

p 2 dx (29)

and it does obey the triangle inequality. Further, since Hellingerdistance lies between 0 and 1, it can be interpreted as the percentage

distance between two densities. In Fig. 11, KL divergence andHellinger distance between the respective univariate marginals of thethree-state Vinhs equations are plotted for~t 0:30 for the case withuniform initial PDF. The superscriptsdenote the respectivevariables.The PF marginals are taken as the reference densities. We observethat MC marginals arein close match with the PFonesat this instant.However, as the PDF evolves further, the difference between thehistogram-based MC and SLE-based PF marginals become moreprominent, as was qualitatively observed in Fig. 8. The twoinformation-theoretic metrics provide a quantitative measure of thesame. Figure 12shows the variation of these information metricswith sample size for ~t 0:50. Notice the increase in the ordinatevalues by at least an order of magnitude, compared with the same inFig. 11. Theexercise can be repeated for various time instances to get

an idea of the computational performance.

VI. Further Statistical Analysis

In this section, we will demonstrate few case studies pertaining toEDL specic analysis in the SLE framework.

A. Case I: Tracking Uncertainty

It is of interest to compute the probability that the ight-path angle(FPA) will be within a specied interval, i.e.,min max. Thisproblem is important in the context of tracking the spacecraft by aspace-based antenna. Univariate FPA marginals (like those shown inFigs. 810) can be computed at different times to calculate thetracking probabilities. Such information can be crucial from missiondesign perspective.

B. Case II: Landing Footprint Uncertainty

Computing the landing footprint uncertainty has been one of thekey aspects of EDL analysis. Important decisions like landing riskevaluation and trajectory correction maneuver (TCM) design dependon it. A list of factors contributing toward landing footprint un-certainty, can be found in [9]. Almost all EDL analysis has been

6570

7580

3

3.5

42.6

2.4

2.2

2

1.8

1.6

h (Km)V (Km/s)

FPA(d

egrees)

4050

6070

2.5

3

3.5

43

2.5

2

1.5

1

0.5

h (Km)V (Km/s)

FPA(d

egrees)

4050

6070

2

2.5

3

3.52

1.5

1

0.5

0

h (Km)V (Km/s)

FPA(d

egrees)

20

40

60

0

1

2

38

6

4

2

0

h (Km)V (Km/s)

FPA(d

egrees)

50

100

150

2

3

4

55

4

3

2

1

0

h (Km)V (Km/s)

FPA(degrees)

0 50

100150

0

2

4

610

5

0

5

h (Km)V (Km/s)

FPA(degrees)

0

100

200

0

2

4

610

5

0

5

10

h (Km)V (Km/s)

FPA(degrees)

200 0

200400

0

2

4

660

40

20

0

20

h (Km)V (Km/s)

FPA(degrees)

Fig. 7 Scatter plots of the joint PDF with three dimensional supporth;V; at~t0:05, 0.20, 0.30, and 0.50, respectively. Columns show differenttimes, rows signify different initial PDFs (uniform at top and Gaussian at bottom row).



9/16

based on evolving a bivariate Gaussian in latitude and longitude andthereby characterizing a3landing ellipse representing the landingfootprint uncertainty. Historically, the landing ellipses have spannedhundreds of kilometers (see Fig.13).

However, depending on the initial uncertainty and systemdynamics, the latitudelongitude bivariate marginal can be far fromGaussian, resulting the3estimates unrealistic. On the other hand,computing this marginal using MC method is not only computa-tionally expensive but can be inaccurate, for reasons discussed inSec. V.C. Figure 14 compares the latitudelongitude bivariate marginal at the nal time, computed for the six-state modelusing MC (left) and SLE-based PF (right) method. Notice that, theSLE-based PFmethod (right in Fig. 14) predicts thelanding footprintto be at approximately 377 deg east and 21.3 deg north, withmaximum probability and a very small dispersion around it. Itascertains that the landing probability everywhere else is zero. In

contrast, MC method (left in Fig. 14) can at best predict a highprobability around 357381 deg east and 20.521.4 deg north and isunable to do any further renement of the landing footprintuncertainty. Not surprisingly, such huge MC dispersion in latitudelongitude results 3- landing ellipse spanning hundreds of kilo-meters. It is evident that SLE-based PF method outperforms MC.

Looking at such dramatic localization of uncertainty computedthrough SLE (Fig.14), one should not get confused by thinking thatSLE underpredicts uncertainties. SLE simply computes exactprobability density values whereas MCapproximatelyreconstructsthese values through histogram. Based on the approximationparameters (e.g., number of bins, overlapping or nonoverlappingbins, equal or unequal bin-size, etc.), the MC method, as a piecewiseconstant approximation algorithm, may underpredict or overpredictuncertainty. In either case, SLE stands superior as exact value,irrespective of the relative size of the PDF support.

70 75 80

0.08

0.09

0.1

0.11

0.12

0.13

h (Km)

3 3.5 42.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3

V (Km/sec)

2.5 2 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

FPA (degrees)

h (Km)727476

3.4

3.5

3.6

h (Km)727476

3.4

3.5

3.6

h (Km)

FPA(degrees)

7274762.4

2.2

2

1.8

h (Km)

FPA(degrees)

7274762.4

2.2

2

1.8

V (Km/sec)

FPA(degrees)

3.43.53.62.4

2.2

2

1.8

V (Km/sec)

FPA(degrees)

3.43.53.62.4

2.2

2

1.8

40 60 800

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

h (Km)

2 3 40

0.2

0.4

0.6

0.8

1

1.2

1.4

V (Km/sec)

2 1 00

0.2

0.4

0.6

0.8

1

1.2

1.4

FPA (degrees)

h (Km)

50 602.4

2.6

2.8

3

3.2

h (Km)50 60

2.4

2.6

2.8

3

3.2

h (Km)

FPA(degrees)

50 60

1.5

1

0.5

h (Km)

FPA(degrees)

50 60

1.5

1

0.5

V (Km/sec)

FPA(degrees)

2.5 3

1.5

1

0.5

V (Km/sec)

FPA(degrees)

2.5 3

1.5

1

0.5

20 40 600

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

h (Km)

1 2 30

0.2

0.4

0.6

0.8

1

1.2

1.4

V (Km/sec)

10 5 00

0.05

0.1

0.15

0.2

0.25

0.3

0.35

FPA (degrees)

h (Km)30 40 50

1.5

2

2.5

h (Km)

V(Km/sec)

V

(Km/sec)

V(Km/sec)

V(Km/sec)

V(Km/se

c)

V(Km/sec)

30 40 50

1.5

2

2.5

h (Km)

FPA(degrees)

30 40 50

6

4

2

h (Km)

FPA(degrees)

30 40 50

6

4

2

V (Km/sec)

FPA(degrees)

1.5 2 2.5

6

4

2

V (Km/sec)

FPA(degrees)

1.5 2 2.5

6

4

2

Fig. 8 The univariate and bivariate marginals for the case of uniform initial condition uncertainty at ~t0:05, 0.30 and 0.50, respectively. Thesimulation is for three-state Vinhs equations with 5000 samples. For univariate marginals, PF results are in solid and MC results are in dashed. For

bivariate marginals, PF results are in the bottom row and MC results are in the top row.



10/16

C. Case III: Heating Rate Uncertainty

It is of interest to compute the univariate density of heating rate _Q(j=s) given by

_Q 14

CfV3S 1

4Cf0exp

h2 hR0

h1

V3S

V3eh V; h say

where 14

Cf0Seh2h1 and R0

h1. Here Cfrefers to the skin-friction

coefcient of the exterior surface area of the entry capsule. Let us

introduce an auxiliary random variable & V V; h. Thefunctions and dene a mappingV; h7! _Q; &. Jacobian (andits determinant) of this transformation can be calculated as

Root of theinverse mappingis found to be

V; h

&;1

log

&3

_Q

(31)

From the equation before Eq. (31), one can evaluate the determinantof the Jacobian atthe rootof the inverse mapping given byEq. (31) as

detJ

&3

_Q

&3 _Q (32)

Because ;;V > 0, it is easy to verify that the mapping

V; h7! _Q; & dened by the functions and , is bijective.

0 100 2000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

h (Km)

2 4 60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

V (Km/sec)

4 2 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

FPA (degrees)

h (Km)

V(Km/sec)

50 1002.5

3

3.5

4

V(Km/sec)

50 1002.5

3

3.5

4

h (Km)50 100

4

3

2

1

50 100

4

3

2

1

V (Km/sec)

h (Km) h (Km) V (Km/sec)





FPA(degrees)

2.5 3 3.5 4

4

3

2

1

FPA(degrees)

2.5 3 3.5 4

4

3

2

1

0 100 2000

0.005

0.01

0.015

0.02

0.025

0.03

h (Km)

0 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

V (Km/sec)

20 0 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

FPA (degrees)

V(Km/sec)

40 80 120

1

2

3

4

V(Km/sec)

40 80 120

1

2

3

4

40 80 12010

5

0

5

40 80 12010

5

0

5

FPA(degrees)

2 410

5

0

5

FPA(degrees)

2 410

5

0

5

0 200 4000

0.005

0.01

0.015

0.02

0.025

h (Km)

0 50

0.2

0.4

0.6

0.8

1

1.2

1.4

V (Km/sec)

100 0 1000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

FPA (degrees)

V(Km/sec)

50 150 250

1

2

3

4

V(Km/sec)

50 150 250

1

2

3

4

50 150 250

40

20

0

50 150 250

40

20

0

FPA

(degrees)

2 4

40

20

0

FPA(degrees)

FPA(degrees)

FPA(degrees)

FPA(degrees)

FPA(degrees)

FPA

(degrees)

FPA(degrees)

2 4

40

20

0

Fig. 9 The univariate and bivariate marginals for the case of Gaussian initial condition uncertainty at~t0:05, 0.30 and 0.50, respectively. Thesimulation is for three-state Vinhs equations with 5000 samples. Conventions for the MC and PF plots are same as in the previous gure.



11/16

Consequently, the joint density in the transformed variables &; _Q isgiven by

&; _Q V; h

j detJj (33)

where :; : is the bivariate marginal in V and h. Further, _Q

R1

0 &; _Q d&. Thus, from Eqs. (3133), we have

_Q Z10

1 _Q

&; 1

log

&3

_Q

d& (34)

Figure 15 shows the univariate PDFs for _Q per unit area (inW=cm2) for the three-state model (left) and for the six-state model(right), both with uniform initial condition uncertainty. The solidlines show the SLE result and the dashed lines show the MC basedhistogram approximation.

D. Case IV: Chute Deployment Uncertainty

Quantication of uncertainty for the purpose of chute deploymentis of growing interest as future Mars missions are expected to deployat higher Mach numbers [1]. It is important to schedule thedeployment in an Mach numberMand dynamic pressureq regime

40 60 800

0.05

0.1

0.15

0.2

h (Km)

20 250.1

0.2

0.3

0.4

0.5

Latitude (degrees N)

340 360 3800.015

0.02

0.025

0.03

Longitude (degrees E)

2 3 40

1

2

3

V (Km/sec)

2 1 00

0.5

1

1.5

2

FPA (degrees)

10 8 60

0.5

1

1.5

Azimuth (degrees)

Fig. 10 The univariate marginals for the case of uniform initial

condition uncertainty at~t0:30. The simulation is for 6 state Vinhsequationswith 10,000samples.PF results arein solid and MCresults are

in dashed.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

3.98

3.99

4x 10

4

KLh

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

0.35

0.352

0.354

KL

V

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 10

4

0.0235

0.0236

0.0237

Size of the initial condition ensemble

KL

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

8.71

8.72

8.73x 10

3

HLh

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

0.246

0.248

0.25

HLV

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 10

4

0.0676

0.0678

0.068


HL

Fig. 12 Variation of the KL divergence (left) and Hellinger distance (right) with number of samples at the instant~t0:50.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

2.07

2.08

2.09x 10

6

KLh

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

2.1

2.12

2.14x 10

3

KLV

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

8.26

8.28

8.3x 10

5


KL

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

7.2

7.22

7.24x 10

4

HLh

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10

4

0.023

0.0231

0.0231

HLV

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

4.52

4.54

4.56x 10

3


HL

Fig. 11 Variation of the KL divergence (left) and Hellinger distance (right) with number of samples. Snapshots are compared at~t0:30.

http://-/?-http://-/?-


12/16

such that the parachute can bear the stress and provide the requisiteaerodynamic deceleration performance. The question consideredhere is whether the bivariate density in Mand q is contained in aprespeciedM-qbox. If signicant amount of probability mass liesoutside this M-q box at the moment of chute deployment, then itwould be probabilistically unsafe to deploy the supersonic parachuteat that moment. From a mission design perspective, one can repeatthis analysis to nd the best time to deploy the chute for robustperformance. Historically, for DGB parachutes, the Viking qual-ication program has guided the designM-q box dimension to beM 1:12:2and q 239850 Pa[12].

To characterize the Mach-q uncertainty, one needs to nd the

transformed bivariate density M; q from the bivariate marginaldensity V; h. The mapping consideredhere is V; h7!M; q andis dened by

M VR?Th

p V; h (35)

q 12

V2 12

0exp

h2 hR0

h1

V2 V2eh V; h

(36)

where 12

0eh2h1 and R0

h1, as dened in Case III. Here is the

ratio of specic heats and R? is the difference between them(assuming ideal gas). As before, one mustnd the Jacobian of thistransformation and compute the determinant as

J@@V

@@h

@

@V

@

@h

" #

1

R?Th

p V2

R?

p Th3=2dTdh

2Veh

V2eh

" # (37)

)detJ V2eh

R?Thp 1

ThdT

dh

(38)

Now one can proceed to compute the roots of the inverse mapping,i.e.,h and Vas functions ofMand q. To do this, one can substituteV2 M2R?Th[from (28)] in Eq. (36) to get

Theh qM2R?

C say (39)

At this point, let us assumeTh A Bhfor Mars atmosphere.Then Eq. (39) leads to

A Bh Ceh

0 (40)This transcendental equation in h can be solved in closed form interms of theLambertWfunction

hAB

1

W0

C

B e

AB

(41)

provided B 0, which holds true for the case under consideration.Notice that, since ;B;C > 0, the associated LambertWfunction issinglevalued (zerothbranch(usually referredas theprincipal branch)of W, denoted as W0) and consequently Eq. (41) is the uniquesolution of Eq. (40). It should be emphasized here that the constantAis a negative scalar. This is because the temperature is given by T(in

Viking 1,2 (1976)Pathfinder (1997)

Phoenix (2008)

MER A,B (2004)

MSL (2011)

150 100 50 50

40

20

20

40

100 150

Fig. 13 Schematic comparison of landing footprints of Mars missions. To make a comparison between their sizes, all ellipses are drawn with the same

center and same orientation. The scale on each axis is in Km (data taken from [ 1]). The ellipse for the upcoming MSL mission is anticipated.

20 30 40 50 60 700

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Q dot (W/cm2) Q dot (W/cm2)

20 30 40 50 60 700

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Fig. 15 Heating rate PDF snapshot at~t0:30 for the three-state model with 5000 samples (left) and for the six-state model with 10,000 samples, bothwith uniform IC uncertainty (right). Solid lines show SLE derived PDFs, dashed lines show MC based histogram approximation.


Longitude(d

egreesE)

19 20 21 22350

355

360

365

370

375

380

385

390


Longitude(d

egreesE)

19 20 21 22350

355

360

365

370

375

380

385

390

Fig. 14 Comparison of the latitudelongitude bivariatemarginal PDF at the nal time, from MC (left) and SLE-based PF

(right) method.



13/16

C) a Bhwhere 273:15> a > 0, B > 0 and h is in meters.Hence, the absolute temperature T (in Kelvin) a Bh 273:15 A Bhwith A a 273:15< 0. This will avoid anyconfusion about the sign ofh.

Substituting Eq. (41) in Eq. (28), one can obtain

V M

R?Thp

M

R?B

W0

C

B e

AB

s (42)

Thus theroot forthe inverse mapping is thedoubletV; h given byEqs. (41) and (42), where it is reemphasized thatC is a function ofbothMand q, as given in Eq. (39).

Combining Eqs. (38), (41), and (42) and noting that ThB

W0CB eA

B , one can derive

j detJj M2eAB

R?Bp

0BB@

C

Be

AB

eW0

CBe

AB

W0

C

B e

AB

r1CCA (43)

where the denitionof Lambert W function x WxeWx, has beenused (see Appendix for details). Finally, the joint bivariate density inM; qis given by

M; q V; h

j detJj (44)

where the right-hand side needs to be evaluated using Eqs. (4143).Notice that, instead of using Lambert W function, one may also

proceed by numerically solving Eq. (40).Appendix lists the temperature model and associated numerics

used in the simulation for computing the Mach number. The Mach-qbivariate marginals computed from Eq. (44) are plotted in Fig.16at~t 0:30and 0.50. The rst row shows the plots for the three-statemodel with uniform initial condition uncertainty, the second rowcorresponds to the same model with Gaussian initial conditionuncertainty. Plots in the last row are for the six-state model withuniform initial condition uncertainty and follow similar trend as thecorresponding three-state case. With the Mach-q box dimensionspecied earlier, one can notice that for a three-state dynamics,

12 13 14 15 16 17 18

200

250

300

350

400

450

500

Mach number Mach number

DynamicPressure(Pa)

5 10 15

250

300

350

400

450

500

DynamicPressure(Pa)

5 10 15 20 25 30 35

50

100

150

200

250

300

350

400

Mach number

DynamicPressure(Pa)

2 4 6 8

50

100

150

200

250

300

Mach number

DynamicPressure(Pa)

12 13 14 15 16 17 18

200

250

300

350

400

450

500

Mach number

DynamicPressure(Pa)

5 6 7 8 9 10 11 12400

420

440

460

480

500

520

540

Mach number

DynamicPressure(Pa)

Fig. 16 Bivariate PDF of the Mach number and dynamic pressure (Pa) at~t0:30 and 0.50. First and second rows are three-state model with uniformand Gaussian initial PDF, respectively. The third row corresponds to the six-state model with uniform initial PDF.



14/16

deploying a parachute at ~t 0:50 will have signicant reliabilitywith Gaussian initial uncertainty compared with uniform initialuncertainty.

VII. Conclusions

A framework based on the SLEis provided for dispersion analysisin planetary EDL. It was argued that in this framework, one can do asystematic initial condition and parametric uncertainty analysis byspatio-temporally evolving the joint PDF through the SLE. Variousanalytical and numerical examples are given to illustrate how theMOC can be used to directly solve SLE, thereby making thisframework not only computationally more tractable than traditionalMC analysis, but also more accurate. For the EDL problem, resultsare provided for both three-state and six-state Vinhs equations forhypersonic entry in Mars atmosphere. Further, some case studieswere presented to demonstrate howthis framework naturally enablesEDL specic statistical analysis. An initial implementation of themethodology described here, is now available in NASA JetPropulsion Laboratorys DSENDSsimulator systemfor high-delitynumerical experiments [41].

Appendix

I. LambertWFunction

The present section is intended to provide an overview of theLambertW function. This function W is dened as the inverse ofthe function fW WeW, where W is any complex number. Thefunction satises the equation

WzeWz z (A1)for every complex numberz. This equation has innite number ofsolutions, most of them being complex. Thus, W is a multivaluedfunction. The different solutions of Eq. (A1) are called differentbranchesofW, and are denoted as Wk. The indexktakes its valuesfrom an index setZ (the set of integers). Thus, different solutions orbranches of Eq. (A1) are W0z,W1z,W2z, etc. The case ofparticular interest is when the solution is real. Depending on the

domain ofW, one may have zero, one or two real solutions; in eachcase, the remaining solutions are complex. It is easy to see fromEq. (A1) thatWx is real, provided x2 1

e; 1 R. FigureA1

shows how Wx varies with x. From the gure, it follows that whenx2 1

e; 0,Wis double valued (two branchesW0 (solid) andW1

(dashed)) and forx2 0; 1, Wis singlevalued (only W0branch). Inother words, unique real root occurs only when the domain isrestricted to the non-negative reals.

The readers areencouraged to refer the paper by Corless et al. [42]foran excellentaccount on LambertWfunction from both theoreticaland applied point of view. Two more signicant references gearedtoward applications are [43,44].

II. Computing Mach Number

Case IV of Sec. VI discusses the quantication of chutedeployment uncertainty by computing the Mach-q bivariate PDF.Equation (35) modeled the Mach number as M V

R?Thp . The

denominator

R?Thp represents the speed of sound. In thissection, we describe how to compute the quantities , R?,and Th inthe present context.

A. Modeling

The ratio of specic heatsis modeled as

f 2f

(A2)

withfbeing the total number of degrees-of-freedom (DOF) of themolecule, which is further givenbyf 3n, where n is the number ofatoms in a molecule. The contribution to total number of DOFscomes from three types, viz. translational DOFs, rotational DOFs,

and vibrational DOFs; the amount of contribution from each typedepends on whether the geometric arrangement of the atoms in themolecule are linear or nonlinear (see Table A1).

At low temperature, the vibrational DOFs are not excited. Forexample, terrestial dry air, being primarily a mixture of diatomicgases (approx. 78% nitrogen and 21% oxygen), has only threetranslational and two rotational DOFs at low temperature. Hencelowair 525 1:4. But at high temperature, all f 3 2 6DOFsare excited and

highair 626 1:33.

Atmosphere in Mars, being largelyconstituted of (approx. 95%byvolume) triatomic carbon dioxide (a linear molecule), has ve totalmolecular DOFs at low temperature andnine molecular DOFs at hightemperature. In this paper, for the problem of hypersonic entry atupper Martian atmosphere (typically more than 7 km altitude),

1:4 is considered while for supersonic descent through thicker

(and warmer) atmosphere (below 7 km altitude), 119 1:22 isassumed.

B. ModelingR?

The difference between specic heats (R?), sometimes calledspecic gas constant, is given by

R? R

M (A3)

where R 8:3145 J mol1 K1 is the universal gas constantandMis the molar mass of the gas or gas mixture (in kgmol1). For carbondioxide, M 44:01 103 kg mol1, which leads to R?188:9230 J kg1 K1 [using Eq. (A3)].

C. ModelingThIn Mars atmosphere, thevariation of temperature Twith altitude h,

usually termed as the lapse rate, can be modeled as the followingpiecewise linear function

Th 23:4 0:002220h for h > 7000

31:0 0:000998h for h < 7000 (A4)

whereTis in deg Celcius andhis in meters.

Acknowledgments

This research work was supported by NASA Jet PropulsionLaboratory Directors Research and Development Fund DRDF-

0.5 1.0 1.5 2.0 2.5

6

5

4

3

2

1

1

3.0

Fig. A1 Real-valued LambertWfunction.

Table A1 Different types of molecular DOFs

Type Number of translational

DOF

Number ofrotational

DOF

Number ofvibrational

DOF

Linear molecule 3 2 3n 5Nonlinear molecule 3 3 3n 6

Data available at http://www.grc.nasa.gov/WWW/K-12/airplane/atmosmrm.html[accessed June 2010].

http://www.grc.nasa.gov/WWW/K-12/airplane/atmosmrm.htmlhttp://www.grc.nasa.gov/WWW/K-12/airplane/atmosmrm.htmlhttp://www.grc.nasa.gov/WWW/K-12/airplane/atmosmrm.htmlhttp://www.grc.nasa.gov/WWW/K-12/airplane/atmosmrm.html


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FY09 with J. (Bob) Balaram (section 347) as Jet PropulsionLaboratory principal investigator and Aron Wolf (section 343) as JetPropulsion Laboratory coinvestigator. In particular, we are thankfulto Bob for many insightful discussions.

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