SIMULTANEOUS OPTIMAL UNCERTAINTY APPORTIONMENT AND … · SIMULTANEOUS OPTIMAL UNCERTAINTY...

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Int. J. Mech. Eng. & Rob. Res. 2014 Joe Hays et al., 2014

SIMULTANEOUS OPTIMAL UNCERTAINTYAPPORTIONMENT AND ROBUST DESIGN

OPTIMIZATION OF SYSTEMS GOVERNED BYORDINARY DIFFERENTIAL EQUATIONS

Joe Hays1,3*, Adrian Sandu2, Corina Sandu1 and Dennis Hong1

*Corresponding Author: Joe Hays, [email protected]

The inclusion of uncertainty in design is of paramount practical importance because all real-lifesystems are affected by it. Designs that ignore uncertainty often lead to poor robustness, suboptimalperformance, and higher build costs. This work proposes a novel framework to perform robustdesign optimization concurrently with optimal uncertainty apportionment for dynamical systems.The proposed framework considerably expands the capabilities of contemporary methods byenabling the treatment of both geometric and non-geometric uncertainties in a unified manner.Uncertainties may be large in magnitude and the governing constitutive relations may be highlynonlinear. Uncertainties are modeled using Generalized Polynomial Chaos and are solvedquantitatively using a least-square collocation method. The computational efficiency of this approachallows statistical moments to be explicitly included in the optimization-based design process. Theframework formulates design problems as constrained multi-objective optimization problems,thus enabling the characterization of a Pareto optimal trade-off curve that characterizes the entirefamily of systems within the probability space; consequently, designers are able to produce robustand optimally performing systems at an optimal manufacturing cost. A nonlinear vehicle suspensiondesign problem, subject to parametric uncertainty, illustrates the capability of the new frameworkto produce an optimal design at an optimal manufacturing cost.

Keywords: Uncertainty apportionment, tolerance allocation, robust design optimization, dynamicoptimization, nonlinear programming, multi-objective optimization, multibody dynamics,uncertainty quantification, generalized polynomial chaos

ISSN 2278 – 0149 www.ijmerr.comVol. 3, No. 1, January 2014

© 2013 IJMERR. All Rights Reserved

Int. J. Mech. Eng. & Rob. Res. 2014

1 Mechanical Engineering Department, Virginia Tech, Blacksburg, VA, USA.2 Computer Science Department, Virginia Tech, Blacksburg, VA, USA.3 US Naval Research Laboratory, Washington DC, USA.

Research Paper

INTRODUCTIONThe design for manufacturability community haslong studied the adverse effects of uncertainty

in kinematically assembled mechanicalsystems; these studies are generally referredto as tolerance analysis and tolerance

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allocation. Tolerance analysis approaches theproblem from the perspective of analyzinguncertainty accumulation in assembledsystems. Conversely, tolerance allocationdetermines the best distribution and maximizedmagnitudes of uncertainties such that the finalassembled system satisfies specifiedassembly constraints. Initially, these studies onlytreated rigid kinematic relations of thedimensional uncertainties (Chase, Greenwoodet al., 1990; Chase, 1999a and 1999b; andBarraja and Vallance, 2002), but have beenextended to include flexible (Merkley, 1998) anddynamical systems (Lee, Gilmore et al., 1993;Krishnaswami and Kelkar, 2003; and Arenbeck,Missoum et al., 2010).

Early works by Chase and co-workersinvestigated various techniques for theallocation of geometric tolerances subject tocost-tolerance tradeoff curves. Theirapproaches included both nonlinearprogramming (NLP) and analytic Lagrangemultiplier solutions for two-dimension (2D) andthree-dimension (3D) problems where anumber of cost-tolerance models wereanalyzed and compared (Chase, 1999a and1999b). Extensions of their initial workincluded the costs of different manufacturingprocesses for a given feature; thus, an optimalcost-to-build included the optimal set ofprocesses to complete the production of thevarious components (Chase, Greenwoodet al., 1990).

The conventional design methodologyprescribed a sequential approach whereoptimal dimensions were designed firstfollowed by a tolerance analysis or toleranceallocation (Krishnaswami and Kelkar, 2003).However, researchers quickly learned that

superior designs could be achieved bydesigning the optimal system parametersconcurrently with the allocation of thegeometric tolerances (Arenbeck, Missoumet al., 2010; and Rout and Mittal, 2010).

It is the understanding of the authors of thiswork that these studies have focused solelyon the affects of geometric relateduncertainties of relatively small magnitude.Conversely, this work presents a novelframework that extends the ability to apportionuncertainties including, but not limited to,geometric related tolerances. Specifically, thenew framework simultaneously performsRobust Design Optimization (RDO) andOptimal Uncertainty Apportionment (OUA) ofdynamical systems described by linear ornonlinear Ordinary Differential Equations(ODEs). Examples of non-geometricuncertainties include: Initial Conditions (ICs),sensor and actuator noise, external forcing,and non-geometric related systemparameters. Uncertainties from geometric andnon-geometric sources may be relatively largein magnitude and are addressed in a unifiedmanner; namely, they are modeled usinggeneralized Polynomial Chaos (gPC) and aresolved quantitatively using a non-intrusiveLeast-Square Collocation Method (LSCM).The computational efficiencies gained by gPCand LSCM enable the inclusion of uncertaintystatistics in the optimization process; thusenabling the concurrent OUA and RDO.

The new framework is initially presented ina constrained NLP-based formulation; thisformulation is general and independent to theconstrained NLP solver approach selected(i.e., gradient versus non-gradient basedsolvers); however, in the event that an

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unconstrained solver technique (such asGenetic Algorithms, Differential Evolution, orsome other Evolutionary Algorithm) isselected, an unconstrained penalty-basedformulation is also provided.

A simple kinematic tolerance analysis casestudy is presented to first illustrate howgeometric tolerances may be accounted forthrough gPC. The benefits of the newframework are then illustrated in an optimalvehicle suspension design case-study wherethe optimal apportionment of systemparameters minimizes the cost-to-build whilemaintaining optimal and robust performance.The suspension case-study was specificallyselected to illustrate the apportionment of non-geometric related uncertainties.

The author’s prior work related to RDO(Hays, Sandu et al., 2011e) and motionplanning (Hays, Sandu et al., 2011a, 2011b,2011c and 2011d) of uncertain dynamicalsystems may be found in these respectivereferences. These works contain an additionalreview of literature and information related touncertainty quantification, gPC, RDO, andmotion planning of uncertain systems.

UNCERTAINTYQUANTIFICATIONGeneralized Polynomial ChaosGeneralized Polynomial Chaos (gPC), firstintroduced by Wiener (Wiener, 1938), is anefficient method for analyzing the effects ofuncertainties in second order randomprocesses (Xiu and Karniadakis, 2002). Thisis accomplished by approximating a sourceof uncertainty, , with an infinite series ofweighted orthogonal polynomial bases calledPolynomial Chaoses. Clearly, an infinite series

is impractical; therefore, a truncated set of po

+ 1 terms is used with Νop representing theorder of the approximation. Or,

op

j

jj

0...(1)

where Rj are known coefficients; andRj are individual single dimensional

orthogonal basis terms (or modes). The basesare orthogonal with respect to the ensembleaverage inner product,

0, dfjiji ...(2)

for ji

where R is a random variable that mapsthe random event , from the samplespace, , to the domain of the orthogonalpolynomial basis (e.g., ); f() is theweighting function that is equal to the jointprobability density function of the random

variable. Also, jjj ,1, when usingnormalized basis; standardized basis areconstant and may be computed off-line forefficiency using (2).

The choice of basis depends on the typeof statistical distribution that best models agiven uncertain parameter. In reference (Xiuand Karniadakis, 2002), a family oforthogonal polynomials and statisticaldistribution pairs was presented. Therefore,gPC allows a designer to pick anappropriate distribution and polynomial pairto model the uncertainty. For example, thetolerance analysis community generallymodels geometric uncertainties with eithera Normal, or Gaussian, distribution, denotedby N(, ), or with a Uniform distribution,

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denoted by U(a, b); where is the mean, is the standard deviation, and a and b arethe lower and upper bounds of the distributionrange, respectively. When modeling anuncertainty with N(, ) the correspondingexpansion basis is the probabilist’s Hermitepolynomials, Hn(), and the expansion withknown coefficients is,

opHHHH 00 210

...(3)

where the domain is . Similarly,,when the uncertainty is better modeled withU(a, b), then Legendre polynomials are used,Ln(), and the expansion with knowncoefficients is,

02Laba

opLLLab 00

2 21

...(4)

where the domain is . (For moreinformation regarding possible distribution/polynomial pairs the interested reader shouldrefer to (Xiu and Karniadakis, 2002).

Any quantity dependent on a source ofuncertainty becomes uncertain and can beapproximated in a similar fashion as (1),

bn

j

jj

0

...(5)

where () is an approximated dependentquantity; j are the unknown gPC expansioncoefficients; and Nbn is the number of basisterms in the approximation. The orthogonalbasis may be multidimensional in the eventthat there are multiple sources of uncertainty.The multidimensional basis functions are

represented by bnj R . Additionally, becomes a vector of random variables,

pp

nn R ,,1 , and maps the sample

space, , to an np dimensional cuboid, pn (as in the example of Legendre

chaoses).

The multidimensional basis is constructedfrom a product of the single dimensional basisin the following manner,

pokin

iij nkpipn

p 1,0,21

21

...(6)

where subscripts represent the uncertaintysource and superscripts represent theassociated basis term (or mode). A completeset of basis may be determined from a fulltensor product of the single dimensionalbases. This results in an excessive set of pn

op 1 basis terms. Fortunately, themultidimensional sample space can bespanned with a minimal set of

!!/! opopb pnpnn basis terms. The minimalbasis set can be determined by the productsresulting from these index ranges,

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12

1

0

00

pp non

o

o

iiipi

ipipi

The number of multidimensional terms, nb,grows quickly with the number of uncertainparameters, np, and polynomial order, po.Sandu et al. showed that gPC is mostappropriate for modeling systems with arelatively low number of uncertainties (Sandu,Sandu et al., 2006a and 2006b) but canhandle large uncertainty magnitudes linear andnonlinear constitutive relations.

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Once all sources of uncertainty, (), anddependent quantities, (), have beenexpanded then the constitutive relationsdefining a given problem may be updated. Forexample, constitutive relations may bealgebraic or differential equations,

0, Α ...(7)

,;; tt D ...(8)

Equation (7) is an implicit algebraicconstitutive relation and (8) is a differentialconstitutive relation. It is important to note thatthe dependent quantities are functions of timein (8), therefore, (5) is modified to,

bn

j

jj ttt0

;, ...(9)

It is instructive to notice how time andrandomness are decoupled within a singleterm of the gPC expansion. Only the expansioncoefficients are dependent on time, and onlythe basis terms are dependent on the nb

random variables, . If any sources ofuncertainty are also functions of time then (1)needs to be updated in a similar fashion as(9) and then all dependent quantities will haveto be expanded using (9). Without loss ofgenerality, the proceeding presentation willassume that all sources of uncertainty are time-independent to simplify the notation.

Substituting the appropriate expansionsfrom (1), (5), or (9) into the constitutive relationsresults in uncertain constitutive equations,

0,0 0

b on

j

p

j

jjjj t Α ...(10)

bn

j

jj t0

;

b on

j

p

j

jjjj t0 0

, D

...(11)

where the nb expansion coefficients, j or j(t),from the dependent quantities are theunknowns to be solved for.

There are a number of methods in theliterature for solving equations such as (10)-(11). The Galerkin Projection Method (GPM)is a commonly used method; however, this isa very intrusive technique and requires acustom reformulation of (10)-(11). As analternative, samplebased collocationtechniques can be used without the need tomodify the base equations.

Sandu et al. (2006a) and Cheng and Sandu(2009a) showed that the collocation methodsolves equations such as (10)-(11) by solving(7)-(8) at a set of points, cp

nk nkp 1, R ,

selected from the np dimensional domain ofthe random variables pnR . Meaning, at anygiven instance in time, the random variables’domain is sampled and solved ncp times with

k (updating the approximations of allsources of uncertainty for each solve), then theuncertain coefficients of the dependentquantities can be determined. This can beaccomplished by defining intermediatevariables such as,

bn

jcpk

jjk nk

0

0, ...(12)

Substituting the appropriate intermediatevariables into (10) and (11) respectively yields,

0, krkkΑ ...(13)

krkkk ,D ...(14)

where k = 0 … ncp, r = 1 … np, and eachuncertainty’s intermediate variable is,

op

jk

jjrkrk

0

...(15)

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Equations (13)-(14) provide a set of ncp

independent equations whose solutionsdetermine the uncertain expansioncoefficients of the dependent quantities.Since (13) is implicitly defined, there are twooptions in determining k : use anumerical nonlinear system solver such asNewton-Raphson, or, solve (7) for a newrelation that defines k explicitly. Theuncertain expansion coefficients of thedependent quantities are determined byrecalling the relationship of the expansioncoefficients to the solutions as in (12). Inmatrix notation, (12) can be expressed as,

Tkk ...(16)

where the matrix,

cpbkj

jk nknjA 0,0,, ...(17)

is defined as the collocation matrix. It’simportant to note that cpb nn . Theexpansion coefficients can now be solvedfor using (16),

...(18)

where A# is the pseudo inverse of A if nb < ncp.If nb = ncp, then (18) is simply a linear solve.However, (Cheng and Sandu, 2007; Chengand Sandu, 2009a; 2009b and 2009c; andCheng and Sandu, 2010) presented the Least-Squares Collocation Method (LSCM) wherethe stochastic dependent variable coefficientsare solved for, in a least squares sense, using(18) when nb < ncp. Reference (Cheng andSandu, 2009a) also showed that as cpn

the LSCM approaches the GPM solution; byselecting bcpb nnn 43 the greatestconvergence benefit is achieved with minimalcomputational cost. LSCM also enjoys the

same exponential convergence rate asop .

The nonintrusive nature of the LSCMsampling approach is arguably its greatestbenefit; (7) or (8) may be repeatedly solvedwithout modification. Also, there are a numberof methods for selecting the collocation pointsand the interested reader is recommended toconsult (Xiu and Hesthaven, 2005; Sandu,Sandu et al., 2006a; Xiu, 2007; Cheng andSandu, 2009a; and Xiu, 2009) for moreinformation.

Once the expansion coefficients of thedependent quantities are determined thenstatistical moments, such as the expectedvalue and variance, can be efficientlycalculated. Arguably the greatest benefit ofmodeling uncertainties with gPC is thecomputational efficiencies gained whencalculating the various statistical moments ofthe dependent quantities. For example,(Papoulis, Pillai et al., 2002) defines thestatistical expected value as,

dwE ...(19)

and the variance,

dwVar 22

2 E ...(20)

with the standard deviation, Var .Given these definitions and leveraging theorthogonality of the gPC basis, these momentsmay be efficiently computed by a reduced setof arithmetic operations of the expansioncoefficients,

000 , E ...(21)

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bn

j

jjjE

1

22 ,

...(22)

Also, recall that jjj ,1, . when usingnormalized basis; standardized basis areconstant and may be computed off-line forefficiency using (2). A number of efficientstatistical moments may be determined fromthe expansion coefficients. The authorspresented a number of gPC based measuresusing efficient moments such as (21)-(22) in(Hays, Sandu et al., 2011a, 2011b, 2011c,2011d and 2011e).

To summarize, the following basic steps aretaken in order to model uncertainty and solvefor statistical moments of quantities dependenton uncertainties:

• Model all sources of uncertainty byassociating an appropriate ProbabilityDensity Function (PDF).

• Expand all sources of uncertainty with anappropriate single dimensional orthogonalpolynomial basis. The known expansioncoefficients are determined from the PDFmodeling the uncertainty.

• Expand all dependent quantities with anappropriately constructed multi-dimensionalbasis.

• Update constitutive relations with theexpansions from steps 2-3. The newunknowns are now the expansioncoeff icients f rom the dependentquantities.

• Solve the uncertain constitutive relations forthe unknown expansion coefficients; thiswork uses the LSCM technique.

• Calculate appropriate statistical momentsfrom the expansion coefficients.

The following two sections summarize thismaterial in the context of: uncertain kinematicassemblies, and uncertain dynamical systems.

Uncertain Kinematic AssembliesGeneralized Polynomial Chaos may be usedfor tolerance analysis, where the effects ofgeometric uncertainties in kinematicallyassembled systems are quantified. Theassembly features to be analyzed may bedefined through an explicit algebraicconstitutive relation such as,

lP 0 ...(23)

where P represents assembly relations suchas gaps, positions, and orientations ofsubcomponent features (Law, 1996). Thedependent assembly variables, , must satisfyclosed kinematic constraints, if any; these maybe implicitly defined as was shown in (7),

0 cl ...(24)

After solving (24) then any assembly featuredefined by (23) may be evaluated.

Once the independent kinematic features,or contributing dimensions, havemanufacturing tolerances prescribed, i()then (23)-(24) become uncertain constitutiverelations. Solving the uncertain versions of (24)yields the uncertain dependent assemblyvariables,(), and (23) may then be solvedfor the uncertain assembly features, P().

The procedure defined in Section 2.1 foralgebraic constitutive relations allowsdesigners to calculate statistical moments ofthe dependent assembly features and quantifythe effects of the prescribed dimension

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tolerances. A simple tolerance analysisexample based on gPC is presented inprevious section.

Uncertain Dynamical SystemsGeneralized Polynomial Chaos haspreviously been shown to be an effective toolin quantifying uncertainty within RDO (Hays,Sandu et al., 2011e) and robust motioncontrol (Hays, Sandu et al., 2011a, 2011b,2011c and 2011d) settings. This workpresents a new framework that enables RDOconcurrently with OUA of dynamical systemsdescribed by ODEs. The new framework isnot dependent on a specific formulation of theEquations-Of-Motion (EOMs); formulationssuch as Newtonian, Lagrangian, Hamiltonian,and Geometric methodologies are allapplicable. An Euler-Lagrange ODEformulation may describe a dynamical system(Murray, Li et al., 1994; and Greenwood,2003) by,

,,,,, ttqNtttqCttqM

ttttq ,,, F ...(25)

where gcntq R are the generalizedcoordinates with dn

dgc tnn R ; are thegeneralized velocities and—using Newton’sdot notation—HF contains their timederivatives; pnR includes systemparameters of interest; dd nntqM R, is thesquare inertia matrix; dd nnttqC R ,,includes centrifugal, gyroscopic, and Corioliseffects; dnttqN R ,, are the generalizedgravitational and joint forces; and int R arethe ni applied wrenches (e.g., forces ortorques). (For notational brevity, all futureequations will drop the explicit timedependence).

The relationship between the timederivatives of the independent generalizedcoordinates and the generalized velocities is,

,qHq ...(26)

where ,qH is a function of the selectedkinematic representation (e.g., Euler Angles,Tait-Bryan Angles, Axis-Angles, EulerParameters, etc.) (Haug, 1989; Nikravesh,2004; and Hays, Sandu et al., 2011d).However, if (25) is formulated with independentgeneralized coordinates and the system hasa fixed base then (26) becomes q .

The trajectory of the system is determinedby solving (25)-(26) as an initial value problem,where 00 qq and 00 . Also, the systemmeasured outputs are defined by,

,, qqy O ...(27)

where ony R with no equal to the number ofoutputs. One helpful observation is that thedynamic outputs defined in (27) are analogousto the kinematic assembly features in (23); theyare both functions of the dependent quantitiesof their respective systems defined in (24) and(25)-(26), respectively.

The EOMs defined in (25)-(26) have the formof the differential constitutive relations definedin (8). Any uncertainties in ICs, actuator inputs,sensor outputs, or system parameters areaccounted for by (). All system states {q, },and associated outputs, y, are dependentquantities represented by ().

The procedure defined in previous fordifferential constitutive relations allowsdesigners to calculate statistical moments ofthe dynamic states and outputs, thusquantifying the effects of the systemuncertainties over the trajectory of the system.

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list of optimization variables, xnx R , includesselect design parameter means as well asvariances, or standard deviations, of theuncertainties to be apportioned. ConcurrentRDO and OUA is possible by converting therobust performance objectives of RDO toconstraints and adding them to the list ofproblem constraints itemized in 0,,, ty C .The authors’ work in (Hays, Sandu et al.,2011e) illustrates how robust performanceobjectives may be defined within a gPCsetting. Therefore, the solution of (28) yields asystem design that minimizes themanufacturing cost-to-build subject tospecified robust performance criterion definedthrough constraints. Equation (28) isformulated as a constrained Multi-ObjectiveOptimization (cMOO) problem, meaning, aslong as at least two performance constraintshave opposing influences on the optimum, thena Pareto optimal set may be determined asthe constraint boundaries are adjusted. Forexample, if each robust performanceconstraint, , is bounded in the followingmanner . A Pareto set will beobtained for unique values of and/or aslong as the associated constraint is active.Once a given constraint becomes inactive, thatconstraint has no influence on the optimalvalue.

The NLP defined in (28) may beapproached from either a sequential nonlinearprogramming (SeqNLP), or from asimultaneous nonlinear programming(SimNLP) perspective (Biegler, 2003; andDiehl, Ferreau et al., 2009). (The literaturesoccasionally refers to the SeqNLP approachas partial discretization and to the SimNLP asfull discretization (Biegler and Grossmann,

The new framework presented in previoussection will build upon this formulation to enableRDO concurrently with OUA of dynamicalsystems described by ODEs.

OPTIMAL DESIGN ANDUNCERTAINTYAPPORTIONMENTThe new framework for simultaneous RDO andOUA of dynamical systems described by ODEsis now presented. This formulation builds uponthe gPC-based uncertainty quantificationtechniques presented in previous sections.Sources of uncertainty may come from ICs,actuator inputs, sensor outputs, and systemparameters; where parametric uncertaintiesmay include both geometric and non-geometricsources. The NLP-base formulation of the newframework is,

wJx

min

s.t ;;,;,;,; tttttq F

;,;,;; tttqHtq

;,;,;; ttqtqty O

0,;,;,; tttty C

ftf qtqqq ;,;0 0

ftf qtqqq ;,;0 0 ...(28)

where the problem objective, J, is a weightedvector function, anRw , defining na

costuncertainty trade-off curves for themanufacturing cost associated with eachuncertainty being apportioned. Equation (28)is subject to the dynamic constitutive relationsdefined in (25)-(27), and their associated ICsand optional Terminal Conditions (TCs). Whenperforming simultaneous RDO and OUA, the

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2004)). In the SeqNLP approach, thedynamical Equations (25)-(27) remain ascontinuous functions that may be integratedwith standard off-the-shelf ODE solvers (suchas Runge-Kutta). This directly leverages theLSCM-based gPC techniques described inprevious section and yields a smalleroptimization problem as only the optimizationvariables, x, are discretized. On the contrary,the SimNLP approach discretizes (25)-(27)over the trajectory of the system and treats thecomplete set of equations as equalityconstraints for the NLP. The discretized statevariables are added to x to complete the fulldiscretization. As such, the SimNLP approachrequires a slight modification in the formulationto account for the full discretization of (25)-(27)in light of the LSCM technique:

wJx

min

s.t 11111 ,,, qF

1111 ,,qHq

1111 ,, qqy O

1,011,01 ;0,;0 qqqq

cpcpcpcpcp nnnnnq ,,, F

cpcpcpcp nnnn qHq ,,

cpcpcpcp nnnn qqy ,,

cpcpcpcp nnnn qqqq ,0,0 ;0,;0

0,,, ty C ...(29)

Equation (29) duplicates the deterministicdynamical Equations (25)-(27) ncp times whereeach set has a unique collocation point, k.Each unique set of dynamical equations is thenfully discretized and x is updated appropriately.However, the system constraints,

0,,, ty C are calculated using thestatistical properties determined by the LSCMand the ncp sets of dynamical equations. Thus,the SimNLP approach has a much larger setof constraints and optimization variables thanthe SeqNLP approach, but, enjoys a morestructured NLP that typically experiences fasterconvergence.

The Direct Search (DS) class ofoptimization solvers—techniques such asGenetic Algorithms, Differential Evolution, andParticle Swarm—typically only treatunconstrained optimization problems. To usethis kind of solver, all the inequality constraintsin (28) need to be converted from hardconstraints to soft constraints; where hardconstraints are explicitly defined as shown in(28), and constraints that are added to thedefinition of the objective function, J, arereferred to as soft constraints. This isaccomplished by additional objective penaltyterms of the form,

21

;,0max; ttJ i

n

iconstc C

...(30)

where nc represents the number of systemconstraints, and is a large constant. With alarge , this relationship is analogous to aninequality like penalizing term, meaning, if theconstraint Ci is outside of its bounds—oroutside the feasible region—then it is heavilypenalized. When it is within the feasible regionthere is no penalty. (Also, by squaring the maxfunction its discontinuity is smoothed out;however, this is an optional feature and is onlynecessary for a solver that uses gradientinformation).

Once the inequality constraints have beenconverted to objective penalty terms, Equation(28) can be reformulated as,

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;min tJJ constx w

s.t ;;,;,;,; tttttq F

;,;,;; tttqHtq

;,;,;; ttqtqty O

00 ;,;0 qtqqq f ...(31)

where the equality constraints from thecontinuous dynamics are implicit in thecalculation of the objective function. ThisSeqNLP approach enables (31) to be solvedby the DS class of unconstrained solvers.

The new framework presented in (28), (29)or (31) allows designers to directly treat theeffects of modeled uncertainties during aconcurrent RDO and OUA design process. Theformulations are independent of theoptimization solver selected; meaning, if aconstrained NLP solver—such as SequentialQuadratic Programming (SQP) or InteriorPoint (IP)—is selected, then any of the threeformulations presented is appropriate,depending upon the designer’s preferencesregarding hard/soft constraint definitions andpartial/full discretization. On the contrary, if anunconstrained solver is selected, then (31) isthe formulation of choice.

The computational efficiencies of gPCenable the inclusion of statistical moments inthe OUA objective function definition as wellas in the RDO constraint specifications; thesestatistical measures are available at areduced computational cost as compared tocontemporary techniques. However, theframework does introduce an additional layerof modeling and computation. In (Hays, Sanduet al., 2011e), the authors presented generalguidelines of when to apply the framework for

RDO problems. From an OUA perspective, thefollowing general guidelines can helpdetermine if a given design will benefit fromthe new framework:

Non-Geometric Uncertainties: Traditionaltolerance allocation techniques have beendeveloped for the apportionment of geometricrelated uncertainties. The new frameworkprovides a unified framework that enables thesimultaneous apportionment of both geometricand non-geometric related uncertaintiessimultaneously in dynamical systems.

Large Magnitude Uncertainties: Again,traditional tolerance allocation techniques havebeen developed under the assumption that theuncertainty magnitudes are sufficiently small.This assumption is generally valid for geometricmanufacturing related uncertainties, however,it may not be valid for non-geometric relateduncertainties. This point is illustrated in thecase study presented in previous section.

Simultaneous RDO/OUA Design: Asmentioned in Section 0, the researchcommunity has already found that concurrentoptimal design and tolerance allocation yieldsa superior design than the traditionalsequential optimization approach. However,the concurrent design studies to date have onlytreated geometric uncertainties; the newframework in (28), (29) or (31) enables RDOconcurrently with OUA and treats non-geometric uncertainties in addition to thegeometric.

In what follows, previous section presentsa tolerance analysis example for a simpleassembly described by kinematic algebraicconstraints. The purpose of previous sectionis to show that the gPC framework is

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immediately applicable to tolerance analysisproblems. Previous section show-cases thenew framework for simultaneous RDO andOUA when applied to a vehicle suspensionsubject to non-geometric uncertainties.

TOLERANCE ANALYSISBASED ON GPCThis section illustrates gPC-based toleranceanalysis using a one-way mechanical clutchfound in a lawn mower or some other smallmachinery; the problem definition wasborrowed from (ADCATS, 2006) and will beused for comparison purposes. The clutchassembly, shown in Figure 1, is comprised ofan outer ring, a hub, and a roller bearing in aclose-loop kinematic relation. Leveraging thesystem symmetry, the problem considersexplicitly only a quarter of the mechanism; itsindependent assembly variables are edca ,,, and dependent variables are ,,b . The basic goal of the tolerance

analysis is to ensure that the pressure angleremains within the specified range of

oo 86 .

The assembly feature being analyzed, thepressure angle , is a dependent variable,therefore, there is no need for an explicitassembly feature relation as defined in (23),only the closed-loop kinematic constraintEquation (24) is necessary. This twodimensional vector relation is:

,,,,,, edcba

00

edcbaedcba

...(32)resulting in three equations for the threedependent unknowns. The independentvariables, or dimensions, are assignedtolerances where each is assumed to have anormal distribution; their respective mean andstandard deviation values are presented inTable 1. The analysis proceeds by applyingthe procedure outlined in previous section,where each independent variable i isapproximated as shown in (1); uncertaindependent variables are expanded using(5); and all approximations are substituted into(32) resulting in uncertain algebraicconstitutive relations. The LSCM methodsamples the probability space ncp times andsolves for the dependent variable expansioncoefficients through least-squares using (18).

Figure 1: One Quarter of a ClutchAssembly. The Independent Variables

are Dimensions {a, c, d, e} and theDependent Variables are {b, , }.

Proper Operation of the Clutch Requires6° < < 8°

Parameter Mean () Std. () Units (SI)

a 27.645 0.0167 m m

c, d 11.430 0.0033 m m

e 50.800 0.0042 m m

Table 1: Independent Variable Meanand Standard Deviations

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The respective mean and standard deviationstatistical moments are then efficientlydetermined by (21)-(22). The results of thisanalysis for various gPC approximation ordersare shown in Table 2.

Table 2 also contains the results of theanalysis when applying the Direct LinearizationMethod (DLM), as used in (Chase, 1999a; andADCATS, 2006), as well as a Monte Carlo(MC) based analysis using 2.5 million samples.Additionally, Table 2 reports the absolute valueof the errors between the solution of thedifferent methods when compared to the MCresults, MC , in the third column; the numberof samples used are shown in the fourthcolumn; and the associated computation timesare shown the fifth column. All methods validatethat the pressure angle remains within thespecified range; for example, the DLM methodhad the largest reported variation and its 3x

solution had a range of oo 6763.73605.6 x ,which is within specification.

Using the high sample MC solution as abaseline for comparison shows that the 2nd

order gPC solution results in comparableresults to the DLM. The 3rd order gPC solutionseems to be the best approximation point whenconsidering both computational cost andaccuracy. The reported computation timeswere taken from an unoptimized Mathematica

code running on an HP Pavilion with the Intel i7processor and 6 GB of RAM. Clearly the gPCapproach is more computationally burdensomethan the DLM, however, when considering themore general nature of the gPC approach—asdiscussed in previous sections—and therelatively cheap cost to increase the accuracyof the solution beyond what the DLM canprovide, a designer may find that this trade-offis worth the expense. In previous section, thebenefits of the gPC approach become moreapparent when large magnitude variations andnon-geometric uncertainties are included withina problem’s scope.

AN ILLUSTRATIVE CASE-STUDY OF CONCURRENTRDO AND OUAThis section illustrates the benefits of the newframework presented in previous sectionthrough a vehicle suspension design case-study where RDO and OUA are carried outconcurrently. The case-study showcases OUAfor a nonlinear system subject to largemagnitude non-geometric uncertainties.

A brief overview of the system dynamicsfollows to help clarify the concurrent RDO/OUAdesign presented thereafter; however, theinterested reader may consult (Hays, Sanduet al., 2011e) where more detailed informationis presented to explain the cMOO formulation

Parameter Variation (3) MC # of Samples, or Collocation Points Computation Time (s)

DLM 0.65788 0.00135 n/a 0.06

gPC, po = 2 0.65822 0.00101 30 0.34

gPC, po = 3 0.65900 0.00023 60 0.51

gPC, po = 5 0.65901 0.00022 168 2.73

MC 0.65923 0 2,500,000 217.42

Table 2: Various Results for the 3x One-Way Clutch Pressure Angle Variation

229


that is used to carry out both a deterministicand RDO of the suspension. This work focuseson illustrating OUA when performedsimultaneously with RDO.

Vehicle Suspension ModelThe case-study uses an idealized two Degree-Of-Freedom (DOF) nonlinear quarter-carsuspension model which is shown in Figure 2.When all the system parameters are knownexactly, the system is described by thefollowing deterministic nonlinear dynamicalEOMs,

uss

uss

ss zz

mzz

mkz ,1

D

guu

uus

uus

s

su zz

mkzz

mzz

mkz ,1

D

...(33)

The model has sprung and unsprungmasses, ms, and mu; vertical mass positionsabout the equilibrium, {zs, zu}, and velocities us zz , ; suspension spring and dampingcoefficients, ks, and bs; tire spring coefficient,

ku; and ground input position, zg. The systemis nonlinear due to the asymmetric dampingforce that is dependent on the velocity direction.

0,0,

,ususs

usussus zzzzb

zzzzbzz

D ...(34)

The ratio of damping forces is determinedby the scalar .

The ground input position, zg, is modeledby a series of isolated road bumps definedby,

tAzg sin ...(35)

where A represents the amplitude of the bump; / is the frequency of the irregularity

determined by the vehicle velocity and baselength of the irregularity ; and t representstime. Each bump was uniquely spaced withno overlap with one another and theiramplitudes were A = 0.15 meters. Thefrequencies of the speed bumps were selectedto be = [1, 5, 10, 15] Hertz. FilteredGaussian noise with a maximum amplitude ofA = 0.03 meters was superimposed over theseries of speed bumps. The cut-off frequencyof the filtered Gaussian noise was 35 Hertz.

Varying passenger and cargo loads,fatiguing/deteriorating suspensioncomponents, and variations in tire air pressureare all very practical sources of uncertainty ina vehicle. Therefore, five uncertain systemparameters were selected for this study, usss kbkm ,,,, . Each

uncertain parameter is assumed to have auniform distribution and is therefore modeledwith a Legendre polynomial expansion. Thistakes the form of,

prqrrrr nr ,,1,0 ...(36)

Figure 2: An Uncertain 2-DOFQuarter-Car Suspension Model with a

Nonlinear Asymmetric Damper. The FiveUncertain Parameters are, () = {ms(),

ks(), bs(), (), ku()}

230


This system results in the following set ofuncertain nonlinear EOMs,

uss

uss

ss zz

mzz

mkz ,1

D

usu

uss

su zz

mzz

mkz ,1

D

guu

u zzmk

...(37)

The ultimate goal for an optimal vehiclesuspension design is to characterize the trade-off effects between three conflicting objectives:the passenger comfort (ride), which is modeledas the acceleration of the sprung mass; thesuspension displacement (rattle); and the tireroad holding forces (holding). Once the optimaltrade-off relationship of these opposingobjectives is determined, the engineers areable to select a parameter set that optimallysatisfies the design’s requirements.

Concurrent OUA/RDOIt is assumed that the standard deviations ofthe uncertain sprung mass, 1

sm ms , and tire

spring constant, 1uk k

s , cannot be

manipulated by the design; therefore, they aretreated as fixed uncertainties. It is alsoassumed that the mean nonlinear dampingcoefficient, 0

ss is fixed. Therefore, the

search variables to carry out the OUA are theassociated variances 111 ,, sssOUA bkx . Thesame search variables used in (Hays, Sanduet al., 2011e) for RDO are reused here; theyare 00, ssRDO bkx . The final list of searchvariables is the union of the two sets, or

RDOOUA xx .

There are a number of methods presentedin the literature for defining a cost-uncertaintytrade-off curve (Chase, Greenwood et al.,

1990). This case-study assumes that thereciprocal power cost-uncertainty trade-offcurve used by the manufacturing communityis a reasonable definition for the selected non-geometric uncertainties in this case study. Thereciprocal power curve is defined as,

aki

iii niBA ,,1,

w ...(38)

where Ai is a bias cost associated with the ith

source of uncertainty being apportioned; Bi isthe cost that is scaled by the reciprocal powerof the selected variation magnitude, k

i , with

Rk defining the exponential power. Table 3shows the A, B, and k values used for theuncertainties associated with OUA

k x .

With this selected set of search variables,defined cost-uncertainty trade-off curves, anduncertain vehicle suspension dynamicsdescribed in (37), the correspondingconcurrent OUA/RDO problem may bedefined as,

a

ssss

n

iki

iii

bkkx

BAw1,,, 1110

min

uss

uss

ss zz

mzz

mkz ,1

D

usu

usu

su zz

mzz

mkz ,1

D

guu

u zzm

k

gu

t

s

u

u

s

s

zzrmsdtz

zzzz

yf

0

2

231


0 rideride JJ

extensionJJ rattlerattle ,0

ncompressioJJ rattlerattle ,0

0 holdingholding JJ

sss kkk 0

sss bbb 0

00 0,0 zzzz ...(39)

where wi is a scalarization weighting factor forthe i th apportionment cost and theuncertainasymmetric damping force is,

us zz ,D

0,

0,

ususs

ususs

zzzzbzzzzb

...(40)

Therefore, (39) simultaneously performsRDO and OUA subject to the uncertain systemdynamics defined in (37) and opposingperformance constraints for vehicle ride, rattle,and holding. In other words, (39) determinesthe optimal apportionment of the uncertaintiesin OUAx that satisfy the performanceconstraints; this is accomplished bysimultaneously determining optimal nominalsuspension values in RDOx . Equation (39) is arobust cMOO formulation in that a Paretooptimal set accounting for the system’s

uncertainties may be constructed by sweepingthrough a range of the performance constraintbounds, holdingrattlerattleride JJJJ ,,, . Recalling themean and standard deviation definitionsprovided in (21)-(22), the performanceconstraint definitions are,

01 rideyy Jass

032 rattlezzzz Jaauuss

054 uuss zzzzrattle aaJ

066 6 holdingyy Ja ...(41)

Therefore, the performance constraintsbound the mean values plus or minus aweighted standard deviation. The constants ai

are weighting factors of the standarddeviations; setting ai > 1.0 yields a moreconservative design. The interested reader isreferred to (Hays, Sandu et al., 2011e) for moredetails regarding the definition of the uncertainperformance constraints and this approach torobust cMOO.

ResultsThe following results show the progression ofthe suspension design from a deterministicoptimal performance design, to an RDOdesign, and finally the concurrent OUA/RDOdesign based on the new framework. Figure3 best helps visualize the differences betweenthe solutions obtained from the three designs;in the following figures the deterministic designoptimization is referenced as dOpt; RDO isreferenced as uOpt; and the simultaneousOUA/RDA is referenced as aOpt.

Figure 3 presents a specific 3D optimalsolution in the constraint space as projectionsonto the three orthogonal 2D planes, holding

Parameter Bias (Ai) Scaled Cost (Bi) Power (k)1sk 0.5 1.40 e+6 1.5

1sb 0.75 3.31 e+5 2

1s 0.85 7.33 e-1 1

Table 3: Independent Variable Meansand Standard Deviations

232


– rattle, holding – ride, rattle – ride; this solutionis obtained when the bounding constraints areset to mJJsmJ rattlerattleride 203.0,/220 32 ,

and mJholding 034.0 . The dOpt solution isrepresented by a solid dot and results in anactive holding constraint. The mean of the uOptsolution is represented by an asterisk. Whenall weighting factors are set to ai = 1, the meansolution is enclosed in a 3D cuboid whosedimensions are determined by the originalnon-optimal uncertainty standard deviations.The projections of the uOpt cuboid are denotedby the dotted lines. Upon close inspection ofFigure 3 it is apparent that the uOpt solutionhas an active rattle constraint. Since the uOpt

rattle

is so large, the uOpt solution was pushed to asignificantly lower uOpt

holding value when compared

to the deterministic dOptholdingx value.

Finally, the optimal mean solution obtainedfrom the new framework for a simultaneousOUA/RDO design is represented by adiamond. Again, when all weighting factors areset to ai = 1, the mean solution is enclosed ina 3D cuboid whose dimensions aredetermined by the optimally apportioneduncertainty standard deviations, OUAx . Theprojections of the aOpt cuboid are denoted bythe dashed lines. Notice how the apportioneduncertainties for the aOpt solution aresignificantly larger than the uOpt solution. Also,the result of the uncertainty apportionmentyields active ride and holding constraints.Figure 3 confirms that OUAx and RDOx arecoupled in that the aOpt solution is shifted whencompared to that found by uOpt; therefore, thesimultaneous search finds the true optimalsolution.

Figure 3: Projection of the 3D dOpt, uOpt, uOpt, Optimal Solutions Onto theThree Orthogonal 2D Planes. These Optimal Solutions were Determined When:

,203.0,/220 32 mJJsmJ rattlerattleride and mJholding 034.0 . Notice How Each Solution has aDifferent Set of Active Constraints

233


A Pareto optimal trade-off curve may beobtained by sweeping through a range ofvalues of the bounding constraints. Since thereare four performance constraints the actualPareto trade-off is a 5D surface. However, toillustrate, the ride and holding constraints arefixed and the OUA/RDO optimal solution isdetermined for a range of values of the rattlebound (where rattlerattle JJ ). The resultingPareto curve may be illustrated as a 2Dprojection into the objective/rattle plane; thisis illustrated in Figure 4.

One interesting observation is that the OUA/RDO framework determined the first twodesign points shown in Figure 4 to beinfeasible points; therefore, their values wereset to the objective value that results if nooptimization was performed. The reason forthese infeasible constraints is best illustratedby Figure 5, which shows the 2D projection ofthe uOpt ride objective trade-off with the rattleconstraint resulting from an RDO, or uOpt,design (Hays, Sandu et al., 2011e); the holding

bound is fixed mJholding 034.0 . The diamondcurve is obtained from a dOpt optimal designand the square curve is obtained from the RDOdesign, however, the triangle line shows thebounding constraint value assigned for theOUA/RDO, or aOpt, design 32 /220 smJride .When the ride bound (triangle line) is belowthe uOpt curve (square curve) the OUA/RDOdesign requires the uncertainties to bereduced, or tightened, to satisfy the constraints.However, since the variations of the uncertainmass, 1

sm , and tire spring constant, 1uk , were

assumed to be uncontrollable, the requireduncertainty reductions for the first two pointsended up being infeasible.

The resulting optimally allocateduncertainties for the various design points

Figure 4: A Single 2D Plane from the 5DPareto Optimal Set Showing the Trade-OffBetween the Cost-to-Build Objective and

Rattle Constraint Bound; the OtherConstraints are Held Constant at

32 /220 smJride , and mJholding 03.0

Figure 5: A Single 2D Plane Showing theTrade-Off Between the Ride Objective andRattle Constraint Bound When Performing

an RDO Design; where mJholding 034.0 isHeld Fixed. The Triangle Curve Represents

the Bounding Value for rideJ WhenPerforming the OUA/RDO Designs

Associated with Figure 3 and Figure 4. TheFirst Two Points on the Triangle Curve

Require Too Large of a Reduction inUncertainties and Therefore Result in

Infeasible aOpt Designs

234


illustrated in Figure 4 are shown in Figure 6.The optimal distribution of the allocateduncertainties is a function of the selected cost-uncertainty trade-off curves defined by (38) andeach associated weighting factor s; Table 4documents the apportionment results obtainedfrom the specific case when ,/220 32 smJride

,203.0 mJJ rattlerattle , and mJholding 034.0 .The final apportionment of optimal standarddeviations resulted in changes as large as300% relative to their respective initial valuesand as large as 64% relative to their respective

mean values; this clearly show-cases the newconcurrent OUA/RDO framework’s ability totreat uncertainties with large magnitudes.

The final resulting cost-to-manufacture forthe dOpt and uOpt designs was $17.70; thisis based on the initial non-optimal uncertaintylevels which apply to both designs. However,the aOpt solutions result in significant cost-to-manufacture reductions; Figure 4 shows thecost savings for the various design points,where savings as high as 74% wereexperienced. Clearly, the actual cost-savingsachievable by applying the new simultaneousOUA/RDO framework defined in (28), (29) or(31) is very dependent upon the definition ofthe respective costuncertainty trade-off curves,however, this case-study illustrates the powerof treating uncertainty up-front during thedesign process where robust optimallyperforming systems are designed at anoptimal cost-to-manufacture.

CONCLUSIONThis work presents a novel framework thatenables RDO concurrently with OUA fordynamical systems described by ODEs.Current state-of-the-art methodologies arelimited to only treating geometric uncertaintiesof small magnitude. The new frameworkremoves these limitations by treating bothgeometric and non-geometric uncertainties ina unified manner within a concurrent RDO/OUAsetting. Additionally, uncertainties may berelatively large in magnitude and the systemconstitutive relations may be highly nonlinear.The vehicle suspension design case-studysupports this message where uncertaintyallocations on the order of 300% of the initialvalue were obtained.

Figure 6: Optimally ApportionedUncertainties Determined from the NewSimultaneous OUA/RDO Framework. TheFirst Two Points were Infeasible Designs

and Therefore the Values were Setto their Initial Values

Table 4: Apportionment Results Obtainedfrom the New Concurrent OUA/RDO

Framework Corresponding to the Case

When ,/220 32 smJride

,203.0 mJJ rattlerattle and mJholding 034.0

1sk 11250 300.00% 64.27%

1sb 423.0763 20.19% 29.70%

1s 0.5213 300.12% 50.00%

fromNonoptimal

% fromNonoptimal % from

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The computational efficiency of the selectedgPC approach allows statistical moments ofthe uncertain system to be explicitly includedin the optimization-based design process. Theframework, presented in a cMOO formulation,enables a Pareto optimal trade-off surfacecharacterization for the entire family of systemswithin the probability space. The Paretotradeoff surface from the new framework isshown to be off-set from the traditionaldeterministic optimal trade-off surface as aresult of the additional statistical momentinformation formulated into the objective andconstraint relations. As such, the vehiclesuspension case-study Pareto trade-offsurface from the new framework enables amore robust and optimally performing designat an optimal manufacturing cost.

In future work, the authors will expand thenew framework to treat constrained dynamicalsystems described by Differential AlgebraicEquations (DAEs).

FUNDINGThis work was partially supported by theAutomotive Research Center (ARC), ThrustArea 1; the National Science Foundation(NSF); and the Computational ScienceLaboratory (CSL), Advanced VehicleDynamics Laboratory (AVDL) and theRobotics and Mechanisms Laboratory(RoMeLa) at Virginia Tech.

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Independent Variables

t Time

Random event

General

Random variable

xi Bottom right index generally indicates a state (with occasional exceptions).

xj Top right index generally indicates a stochastic coefficient, or mode.

kx Bottom left index generally associates x to a specific collocation point.

jik x The major variable annotations

( )T Transpose

qq , Partial derivative notations

#

1

Matrix inverse and pseudo inverse

( )° The quantity represents rotations in degrees

xx, Lower and upper bounds on x

E[x], x Expected value, or mean, of x

2, xxVar Variance of x

std[x], x Standard Deviation of x

Indexes and Dimensions

Ndn Number of degrees-of-freedom (DOF)

Ngcn Number of generalized coordinates, where dgc nn , dependent on kinematic representationof rotation.

Nsn Number of states, dgcs nnn

Non Number of outputs, ony R

Nin Number of input wrenches, inR

Npn Number of uncertain parameters

Nop Polynomial order

APPENDIX

List of Variables (Nomenclature)

239


Nbn Number of multidimensional basis terms

Ncpn Number of collocation points

Nxn Number of optimization variables (‘manipulated variables’)

Nan Number of apportionment variables (‘allocated variables’), where pa nn

Ncn Number of constraint equations

Nin Number of independent variables

Ndn Number of dependent variables

Tolerance Analysis

0 l Open-loop algebraic kinematic constraints defining P

c l Closed-loop algebraic kinematic constraints

P Algebraic relations describing kinematic assembly features (e.g., gaps, clearances, positions,etc.)

Dynamics

gcnq R Independent generalized coordinates

qq , Rates and accelerations of generalized coordinates

dnR Generalized velocities

Generalized accelerations

00 0,0 qq Initial conditions

ss nnH R Kinematic mapping matrix relating rates of generalized coordinates to generalized velocities

inR Input wrenches (e.g., forces and/or torques)

ss nnM R Square inertia matrix

snC R Centrifugal, gyroscopic and Coriolis terms

snN R Generalized gravitational and joint forces

F Differential operator

ony R System outputs

onO R Output operator

A Road irregularity amplitude

Road irregularity frequency

APPENDIX (CONT.)

240


Road irregularity length

Longitudinal vehicle speed

Uncertainty Quantification

Random event sample space

f() Joint probability density function

inR Vector of uncertain independent variables

dnR Vector of uncertain dependent variables

1 opR Single dimensional basis terms

bnR Multidimensional basis terms

Matrix of multidimensional basis

Hn(x) Hermite polynomial basis

Ln(x) Legendre polynomial basis

N(, ) Normal, or Gaussian, distribution with mean and standard deviation of [,]

U(a, b) Uniform distribution with range of [a, b]

A Algebraic operator

D Differential operator

cpnk R, Kth collocation point (confusing notation with mean, consider changing...)

cpniik R , Kth intermediate variable of the ith uncertain dependent variable, i, representing expanded

quantity

cpb nnA R Collocation matrix

Dynamic Optimization

xmin Optimization objective through manipulation of x

x List of manipulated variables

J Scalar objective function

wi Scalarization weights for the individual objective function terms

cnRC Inequality constraints (typically bounding constraints)

a i Standard deviation scaling parameters

Soft constraint penalty weight

APPENDIX (CONT.)

SIMULTANEOUS OPTIMAL UNCERTAINTY APPORTIONMENT AND … · SIMULTANEOUS OPTIMAL UNCERTAINTY...

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