Error Analysis for the In-Situ Fabrication of Mechanisms - Stanford ...

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Error Analysis for the In-Situ Fabrication of Mechanisms - Stanford ...

Fig. 2 In-situ mechanism prototypes fabricated via ShapeDeposition Manufacturing †3‡: „a… a polymer insect-leg prototypewith embedded pneumatic actuator, pressure sensor andleaf-spring joint „b… a hexapedal robot with integrated sensors,actuators and electronics „c… an ‘‘inchworm’’ mechanism, withintegrated clutch components, „d… a slider-crank mechanismmade from stainless steel. Images courtesy the Stanford Centerfor Design Research and Rapid Prototyping Laboratories.1.2 Introducing Mechanism Error Analysis. The modernscientific treatment of mechanism error estimation dates to theearly 1960’s 11 12. In the several decades since, many alternativeapproaches to error analysis for mechanisms have beenproposed—each with various simplifying assumptions and differentlevels of complexity 13 14 15 16 17. All approaches,however, attempt to solve the same basic problem—to predict thenature and amount of performance deterioration in mechanismsas a result of non-ideal synthesis, fabrication, materials orcomponentry.In this paper the focus is on kinematic performance. In otherwords, we assume that we are always able to describe the desiredtask in terms of an output equation of the form:y f , (1)Fig. 3 Micromechanisms and devices built using in-situ fabricationtechniques. Images courtesy Sandia National Laboratories,SUMMiT„tm… Technologies, www.mems.sandia.gov.Used with permission.Fig. 4 Comparing conventional and in-situ manufacturingmethods—process flow chart. Actions that impart accuracy tothe mechanism are specifically identified.where y denotes the (m1) vector of output end-effector locations,coupler-point positions or output link angles, isa(k1) vector of known driving inputs, and isa(n1) vector ofindependent mechanism variables—including deterministic orrandomly distributed geometric parameters and/or dimensions.The function f (•) is called the kinematic function of the mechanismand is, in general, assumed to be a continuous and differentiablei.e. smooth non-linear mapping from the mechanism parameterspace to an output space e.g. a Cartesian workspace. Inthe absence of higher-pairs i.e. joints that have line and pointcontact, as opposed to surface contact, between their memberlinks and multiple-contact kinematics, the smoothness assumptiongenerally holds true.1.3 Conventional Mechanism Error Analysis. Conventionalerror analysis deals with degradation in the performance ofa mechanism as a result of parametric or dimensional variations,and play in joints. The parameters typically considered are linklengths for planar linkages, or some form of the Denavit-Hartenberg 18 parameters for spatial linkages. Error in the performanceof known mechanisms can be estimated analytically ifcertain assumptions are made, rendering the underlying mathematicaltreatment more tractable. For example:• Mechanism dimensions and parameters have a known, givenvariability characteristic—either deterministic, or stochastic.• Dimensional/parametric variations and clearance values aresignificantly smaller than their nominal values.• Individual component variations are independent, uncorrelatedand identically distributed.• The output is, at most, a weak non-linear function of themechanism parameters at the operating configuration of interest.As a result of these assumptions, it becomes possible to approximatethe actual error by lower-order estimates. Other assumptionse.g. negligible variability of the clearance value itself,Normal or Uniform distribution of component parameters etc.,which either eliminate unnecessary model complexity or enableanalytical tractability, are also commonly made.1.3.1 Sensitivity Analysis. Sensitivity analysis is based onthe Taylor-series expansion of the output function. As stated in Eq.1, the end-effector position, coupler path or output angle of amechanism can be expressed as:y f , (2)810 Õ Vol. 125, DECEMBER 2003 Transactions of the ASME


where 1 , 2 , ¯ , k T are the k known driving inputs, and 1 , 2 , ¯ , n T are the n mechanism parameters or dimensionssubject to random, or worst-case deterministic, variability.Since is assumed static for a given mechanism configurationi.e. the driving inputs are held perfectly to their nominalvalues, it is dropped from the equation for notational simplicity.The previous equation is re-written as:y f (3)Expanding this function in Taylor-series around the nominalvalues of the mechanism parameters ( nom nom 1 , nom 2 , ¯ , nom n T ), we get:n fy f nom i1 i i inom nom 1 n2! 2 fi1 i2 inom nom 2 fi 2 ij i j i nom i j nom j ¯nom(4)or, using a more concise notation:y f nom fnom nom 1 2 f2 nom 2 ¯2! nom(5)For small, independent variations about the nominal configuration,a linear approximation can be made—thereby rewriting theabove equation as:y f nom f nom (6)nomor y f (7)nomThe quantity f /] nom is known as the sensitivity Jacobian ofthe mechanism, evaluated at the nominal configuration. This Jacobianrelates the component variability ( ) in the mechanismparameter space to the output variation ( y ) in Cartesian space.This is classical sensitivity analysis, where all variational effectsare bundled into a simple parametric space, and all higher ordereffects are neglected.Equation 7 is used as the basis for error analysis and toleranceallocation. For error analysis, the component variability ( ) andsensitivity Jacobian ( f /) are known for a given mechanismconfiguration. The output error ( y ) is then a simple calculation.The component variability can either be expressed as worst-casevalues, or as stochastic variations in link parameters. Each ofthese approaches is discussed in the next sections.For tolerance allocation problems, the maximum permissibleoutput error ( max y ) and sensitivity Jacobian are known. Equation7 forms the basis for the constraint equations, and the objectiveis to maximize the overall variability i.e. ), given the constraints.Greater allowable variability typically means lowermanufacturing and inspection costs, and thus, is preferred. Onesimple formalization of the tolerance allocation problem is asfollows:subject to:minimize Z i1n1 (8)g max y f 0 andnomg i i 0; i1,2,...,n. (9)Here, an assumption is made that each component variabilityparameter is weighted equally in the cost function, which may notalways be true. Some manufacturing parameters may be easier tocontrol accurately than others e.g. hole size can typically be heldto tighter tolerances than center-distance between holes. Additionally,zero tolerance or close-to-zero tolerance for some parameters,which is permissible for the above formalization, is infeasiblefor real manufacturing processes. Non-homogeneousmanufacturing capability within the mechanism workspace is alsonot considered in this system.The optimization problem can be solved using standard methodsof parametric programming—Lagrange multipliers, or Powell’sconjugate direction method i.e. unconstrained optimizationof a penalty function 19. An example of these optimizationtechniques applied to mechanism tolerance allocation can befound in 20.1.3.2 Deterministic, Worst-Case Error Estimation. In worstcaseerror estimation, each parameter i is assumed to take exclusivelyone of two deterministic values min i and max i . Furthermore,it is assumed that min i nom i max i ;i1,2,...,n, where nom i is the nominal value of the ith parameter.The objective of this kind of error estimation is to determine theworst case envelope of the mechanism performance error. Exceptfor applications where performance within specified limits is absolutelycritical, the worst-case analysis results in conservativeestimates of error and thereby, over-design of components. Sincethe worst performance can occur for any combination of minimumand maximum component parameter values, the techniqueproceeds by exhaustive calculation of total error for each combinationof individual error values. For n parameters, this leads to asearch space of 2 n combinations for each mechanism configuration.If the objective is to find the worst-case performance withinthe entire workspace of the mechanism, then this calculation hasto be repeated at each incremental driver position.An alternative approach is to use dynamic programming 2122 to estimate the maximum error without computing the totalerror for every possible combination. The assumption made whileusing this technique is that the global optimization problem can bere-stated as a multi-stage optimization problem, with the nth stagesolution related to the (n1)th stage solution through a functionalequation. While this technique results in significant reductionof the computational burden involved, it is not guaranteed tofind the global optimum when the underlying monotonicity assumptionsdo not hold.1.3.3 Stochastic Error Estimation. Statistical error estimationproceeds by assigning a probability distribution functionPDF to each variable parameter i . The component dimensionunder consideration is assumed to be a random variable, distributedaccording to the characteristics of its underlying PDF, denotedas p i ( i ). The cumulative distribution function CDF ofthe output functions can then be estimated using standard techniquesfor stochastic analysis. If certain assumptions can be madee.g. linearity, independence, identical distribution etc., the estimationof the distribution and moments of the output function ishighly simplified.The error equation Eq. 7 can be replaced by an equivalentequation for the stochastic estimation of each output CDF, asfollows:P Y jy j 1... nyj .p 1 ... n 1 ,..., n d 1 ...d nwhere j1,2,...,m (10)and for independent and uncorrelated ip 1 ... n 1 ,..., n i1 p i i (11)nJournal of Mechanical Design DECEMBER 2003, Vol. 125 Õ 811


Fig. 6The effects of link length variation in an assembled 4-bar mechanism2 Worst-Case Error Analysis for In-Situ FabricationThe conventional error models presented in Sec. 1.3 cannot bedirectly applied to in-situ fabrication since this fabrication techniquediffers from conventional sequential shape-and-assemblefabrication techniques in some fundamental ways. Primarily, thedifferences are:• In-situ fabrication is blind to conventional component boundaries.Consequently, the input to the analysis is not the dimensionalvariability in links, but the absolute position and orientationvariability in joints. As the assembly is built, joints are createddirectly or embedded within a surrounding matrix of part andsupport material. Links are formed around the joints. Parametricvariability is therefore a function of joint placement accuracy.• Tolerance stack-up due to dimensional/parametric errors incomponents is not an issue for in-situ fabrication. Instead, joints,and other features such as coupler points or end-effectors, areplaced in the workspace with a known absolute accuracy.• Gaps and clearances in joints are manifest directly in thegeometry of the support structure. In conventional fabrication, thegap geometry is a consequence of the interaction amongstcomplementary mating/fitting feature geometries.• Conventional error analysis does not explicitly allow for theconsideration of variable accuracy within the manufacturingworkspace. But when entire mechanisms are fabricated In-situ, thebuild configuration or pose can be chosen to make best use ofthe manufacturing error characteristics.These differences are accounted for in the general abstractmodel for in-situ fabrication and the associated error analysistechniques presented below.2.1 An Abstract Model for In-Situ Fabrication. The maindifference between conventional error analysis, and error analysisfor in-situ fabrication lies in the form of the inputs into the model.Conventional error analysis treats parametric variability i.e. variabilityin link-lengths etc. as a given constant input. In-situ erroranalysis estimates parametric variability for each build configurationfrom the location variability of the joints that make up thelinkage. The parametric variability is determined by the sensitivityof each parameter to the joint positions and orientations at a givenbuild pose. An important observation is that the mechanism parametersthat result from such fabrication are not independent, butpair-wise correlated. This is because multiple adjacent parametersdepend upon the same independent inputs i.e., the positionsand orientations of their shared joints. Although several parameterscan all be adjacent to each other if they share a commonjoint, their correlation is still taken pair-wise since covariance isdefined on random variable pairs. The degree of correlation dependsupon the configuration in which the mechanism is fabricatedalso called the build pose. The output variability, in turn, isdetermined by the sensitivity of the output function to the mechanismparameters at each operating configuration. Figures 6 and 7illustrate the fundamental differences between the two scenarios,for the simple case of a four-bar mechanism.2.2 Frames and Notation. We assign a global workspacedatum frame (OXYZ) and local datum frames (o i x i y i z i ) associatedwith each feature of interest, see Fig. 8. Without loss ofgenerality, it can be assumed that the z-axis of the global frame isaligned with the process growth direction e.g. vertical, or spindleaxis.If the feature of interest is a joint, then it is assumed that thelocal joint z-axis (z i for the ith joint is aligned with the jointfreedomaxis i.e. nominal pin/shaft axis for revolute joints, directionof translational motion for prismatic joints etc.. The directionof the x-axis of the ith frame (x i ) is taken as that of the commonnormal between the ith and adjacent i1th nominal joint axes.Typically, the position and orientation of each feature frame isspecified in the global frame, and the feature geometry is specifiedin the local frame. The nominal location of the origin in the ithlocal frame is represented as the position vector p i in the globalframe or alternately, as the homogeneous coordinatesx i ,y i ,z i ,1), and the nominal orientation of the ith frame is representedby the direction vector z i with direction numbersl i ,m i ,n i ). Alternately, the z-axis of the joint frame can beuniquely represented in a global frame in terms of its Plücker 27coordinates (Q i ,Q i ), where:Q i q 1i ,q 2i ,q 3i (17)are the direction numbers, and:Q i p i Q i q 1i ,q 2i ,q 3i (18)2is the moment vector of the line. Furthermore, we can let q 1iq 2 2i q 2 3i 1 without any loss of generality, making these coordinatesthe same as the direction cosines of the line.Fig. 7 The effects of joint location variation for an in-situ fabricated 4-barmechanismJournal of Mechanical Design DECEMBER 2003, Vol. 125 Õ 813


SLS, Stereolithography etc. The variability region is simply aworst-case or stochastic characterization of the variation in frameposition and orientation, given its nominal location and otherprocess-specific parameters. While this methodology extends tothe general spatial scenario, it is illustrated here with a simpleplanar example.2.3.1 Planar Example. In the planar case, the orientations ofthe joint axes i.e. z i ) are discarded, as all joint axes are assumedparallel. Given a nominal joint location p nom (x nom ,y nom ), theprecision function returns a region R as follows:Rx nom ,y nom , (21)In deterministic worst-case analysis, this function returns the extremalpositions of the region in which the actual joint lies, asfollows:Rworstcase,x min ,x max ,y min ,y max (22)Fig. 8 Frames and notation for the abstract model of in-situfabricationThus, using this representation, the nominal configuration(C nom ) of a mechanism can be represented in terms of the localframe positions and orientations as:C nom p nom i ,z nom i ; i1,2,...n (19)or alternately, in terms of the joint-axis Plücker coordinates as:C nom Q i ,Q i ; i1,2,...n (20)Fabrication proceeds by constructing or embedding non-idealjoints at the given nominal locations. By quantifying the extent ofthese errors, it is possible to predict overall performance errors inmechanisms fabricated in-situ. The complete procedure is describedin later sections Secs. 2.4 and 3.2.3 Heterogeneous Workspace Modeling. For modelingvariable fabrication accuracy within the process workspace, weassume that we have a precision function that returns the variabilityregion R of a joint in the build space, given the nominalposition and orientation, and other process parameters . Notethat the precision function is process-specific and needs to beempirically determined for each process, such as SDM, FDM,Similarly, in stochastic analysis, the function returns a probabilitydistribution that describes the position of the point as a randomvariable, as follows:Rnormal, x , x 2 , y , y 2 (23)In the most general case, R is a closed region of arbitrary geometrywithin which the actual joint position (x,y) lies with a knownprobability distribution. By applying the precision function to allthe joint and coupler points (x i nom ,y i nom ) in a planar mechanism,we get joint variability regions R i as:R i x i nom ,y i nom , (24)In other words, the regions R i determine the characteristics of theinterval or random values that represent the variable nature of thejoint locations. The mechanism parameters i are functions e.g.distance function of the form i (x i x j ) 2 1/2 ) of the positionsand orientations, and the parametric variability is a functionof the joint variability regions (R i ), all at the given build configuration(C b ): i i R 1 ,R 2 ,...R n ; i1,2,...n (25)Error analysis involves estimating the variability in the linkparameters i using the above equation, and then applying sensitivityanalysis techniques to determine the error in the outputfunction at various operating configurations for a mechanismFig. 9 Actual and schematic diagrams of the planar 4-bar crank-rocker mechanism used as anexample in this paper. The parameter values are: L 1 Ä15 cm, L 2 Ä5 cm, L 3 Ä25 cm, L 4Ä20 cm, L 5 Ä7.5 cm, L 6 Ä20 cm „after Mallick and Dhande, 1987…. Stochastic simulations onthe example are performed with a positional variance 2 xkÄ0.01 cm 2 . Worst case simulationsare performed with a positional variability of 0.3 cm, equivalent to the 3 stochastic error.814 Õ Vol. 125, DECEMBER 2003 Transactions of the ASME


Fig. 10 Multiple positions of the example 4-bar mechanism, correspondingto 30 deg increments of the input angle, that is fabricated in-situ. In the following sections, this process isdescribed, and illustrated using the specific planar 4-bar mechanismshown in Fig. 9. The mechanism parameter values werechosen to allow checking of results with earlier published work28. However, in that case, the authors consider a clearance errorof 0.05 cm in the joints, along with a 0.5 percent error in linklength. Since neither link variability or clearance are variablesin our analysis, the comparison is qualitative. To aid with discussionof the results, the mechanism is also shown in variousconfigurations i.e. specific values of the driving angle in Fig.10. In Sec. 4, the analysis is extended to cover general spatialmechanisms.2.4 Error Estimation. Worst case error estimation for insitufabrication proceeds in two stages. First, at a candidate buildpose C b , all the worst case parameter values ( i WC ) are evaluatedby choosing, in sequence, all possible combinations of the worstcasefabrication input values (p i WC ,z i WC ).The precision function Eq. 24 returns these extremal valuesof the position of each joint, given the mechanism nominal buildpose. For k fabrication input variables, this process generates 2 kcandidate mechanisms at each pose C b . Figure 11 shows the exampleof a mechanism with 3 mm square precision regions and acandidate build configuration.In the second stage, the error in the output function y is evaluatedfor each one of the candidate mechanisms produced in thefirst stage. This calculation is repeated for all operating angles, forevery build pose. Overall, if c operating and build positions areconsidered for a mechanism with m independent degrees of freedom,and k independent fabrication variables, the determinationof worst-case error boundaries for the output has computationalcomplexity O(2 k mc 2 ). Dynamic programming approaches 22can significantly improve upon the computational complexity, butneed to be re-stated appropriately for each specific problem.Figure 12 illustrates the results of the worst-case error estimationfor the example 4-bar mechanism for a few candidate buildposes. The coupler-point location is shown as a cloud of points inthe vicinity of the nominal coupler-point, with each point correspondingto one combination of worst-case joint locations. Figure13 plots the worst-case variability of the coupler-point locationi.e. half the perimeter of the bounding box for each cloud in Fig.12 as a function of the build configuration. Of the four buildconfigurations evaluated, the one corresponding to 180 deg isevidently best for minimizing the worst-case errors in couplerposition.Fig. 11 An example build configuration and worst-case variationsin joint and coupler point locations3 Stochastic Error Analysis for In-Situ FabricationThe worst-case method presented in the previous section is bothoverly conservative, and computationally expensive for most applications.By contrast, a stochastic approach results in superiorJournal of Mechanical Design DECEMBER 2003, Vol. 125 Õ 815


Fig. 12Worst case coupler-point positional error, plotted on the coupler patherror estimates in constant-time as opposed to exponential or lineartime for worst-case methods. However, the conventional approachto stochastic error estimation needs modification in orderto be applicable to in-situ fabrication.In this analysis, we assume that the joint coordinates positionsand orientations are independent random variables with knowndistributions. Given the nominal location of a joint i, the precisionfunction Eq. 24 returns the appropriate distribution for its actuallocation. Mechanism parameters like link-lengths, jointangles, joint offsets and skew angles are functions of the independent,random joint coordinates. This, in turn, makes the parametersthemselves random variables which are pairwise correlatedbeing jointly dependent on the same independent variables. Theoutput, then, is a complex function of correlated random variables.The probability distribution i.e. PDF of a known function ofrandom variables can, in principle, be derived exactly from thegiven, analytically specified, distributions of the original randomvariables. However, in practice, the exact derivation is intractablein the absence of certain simplifying assumptions, due to the complexityof the algebra involved. For a weakly non-linear functionof independent and uncorrelated random variables, the mean andvariance of the function can be approximated directly from themean and variance of the underlying random variables, as illustratedin Eqs. 15 and 16. When the simplifying assumptionsi.e. independent and uncorrelated do not hold, the function propertiesneed to be determined analytically by integrating the joint-PDF see Eq. 28, by modifying the approximation techniques toinclude the effects of correlation, or by using Monte Carlo simulationtechniques. In general, the analytical technique is not tractablefor all but the simplest of cases. In the following section, animproved approximation technique for the estimation of the momentsof a weakly-nonlinear function of correlated random vari-Fig. 13 Total worst-case coupler-point positional errors, plotted against operating angle for fourdifferent build configurations, corresponding to different values of the input angle. The error valuescan be compared with 3 stochastic errors „see Fig. 17….816 Õ Vol. 125, DECEMBER 2003 Transactions of the ASME


Fig. 14 First order estimates of the link-length variance compared to the results of a Monte Carlosimulationables is developed and applied to the problem of stochastic errorestimation for mechanisms that are fabricated in-situ. The resultsare compared to those obtained by Monte Carlo simulation.3.1 Estimating the Parametric Variance. Equation 15can be applied directly to the mechanism parameters ( i ), giventhe stochastic properties i.e. mean and variance of the joint variables(x k ). The parameters are simple functions i.e. sums, productsand differences of the joint variables, which are assumedindependent and uncorrelated. Moreover, the variance in any jointvariable can be assumed to be much smaller than its mean formacro-scale devices, since the precision of fabrication equipmentis typically several orders-of-magnitude smaller than the partdimensions. This implies that the variability in the mechanismparameters can be approximated as a linear function weightedby the sensitivity coefficients of the variability in the input, asfollows: 2 i k i2x xkk2 ; i1,2,...n (26)where 2 iis the variance of the ith mechanism parameter, and x krepresents the kth joint variable, and 2 xkrepresents the varianceof the kth joint variable. If the joint variables follow Normal distributionstypical for most physical random processes involvingmany noise factors, then the parameters too will follow a Normaldistribution.The parameters i , however, are correlated random variables.The correlation coefficients ( ij ) of each parameter pair ( i , j )can be approximated using the sensitivity coefficients as follows: ij k ix k jx k2xk i j(27)Figure 14 compares the first order estimate of link-length variabilityagainst that obtained by Monte Carlo simulation, for thelinks in the example 4-bar in Fig. 9. Since the mechanism is builtin-situ, the link length variation is a consequence of variations injoint location.Figure 15 compares the pairwise correlation coefficients obtainedfor the approximation in Eq. 27 against those obtained byMonte Carlo simulation, for the same four links of the example4-bar. In both cases, the approximation yields results that are veryclose to the simulation—illustrating the validity of the assumptionof independence. Note also that these results hold for an examplewith tolerances that are looser than is common in macroscopicdevices. As a percentage of the link lengths, the tolerances aremore characteristic of MEMS devices.3.2 Estimating the Output Variance. In the previous section,we have established a method for efficiently estimating thevariance and correlation coefficients of the parameters of amechanism that has been fabricated in-situ. Our real interest inthis treatment, however, is in the behavior of the output functiony during operation. As indicated earlier, the output is a functionof the mechanism parameters which, being dependent functions ofthe given independent random variables i.e. the joint variables,are themselves correlated random variables. Thus, the simplifyingassumptions which could be made for the estimation of parametricvariability are not applicable for the estimation of output variability.No simple analytical technique exists for the determinationof the distribution of a general function of correlated randomvariables. In theory, the cumulative distribution function of theoutput can be evaluated as follows:P Y y 1¯ nf 1 ,... n .p 1 ... n 1 ,... n d 1 ...d n (28)However, the joint distribution function p 1 ... n( 1 ,... n )isnot easy to determine when the random variables i are correlated.Furthermore, the upper limits of the multiple integral needJournal of Mechanical Design DECEMBER 2003, Vol. 125 Õ 817


Fig. 15 Pairwise correlation coefficients of the link lengths—first-order results compared to the MonteCarlo simulationto be expressed in terms of the output variables, which is notanalytically feasible except for the simplest of cases.The assumption that makes this problem tractable, once again,is that of weak-nonlinearity in the output function. In other words,if we can assume that the second and higher-order terms in theTaylor Series expansion of the output function can be discarded,then it is possible to derive an expression that directly produces anapproximate estimate for the output variance, given the variance( 2 i) and correlation coefficients ( ij ) of the mechanism parameters.Furthermore, if the total number of parameters is large i.e.n5), then, according to the Central Limit Theorem, the outputfunction will follow an approximately Normal distribution, regardlessof the individual parameter distributions 23. Thus, bymaking the linear approximation, we completely side-step theevaluation of the extremely problematic multiple integral in Eq.28. The derivation of the approximation equation is given inAppendix A, and the final result is summarized below:n2 2 fy i1 i 2 f fi2 i j i j ij i j(29)where i1,2,...n and ji. In the special case where only adjacentparameters share a joint variable, ij 0 for non-adjacentparameters, and the above equation needs to be evaluated only forthe cases where ji1. Note that all the sensitivity coefficientsin the above equation are evaluated at the nominal operating configuration of the mechanism. Comparison of Eq. 29 and Eq.15 reveals that they differ only in the second term on the RHS.This term, then, is the adjustment term that accounts for the correlationeffect that results from the co-dependence of the mechanismparameters on the same joint coordinates.Summarizing, the first order approximations are the only tractable,general purpose estimates of the output function variability.Equation 29 indicates that the output error depends upon theoutput function sensitivity coefficients evaluated at the nominaloperating configuration, the parametric variances, and the pairwisecorrelation coefficients of the parameters. The parametricvariances and the correlation coefficients are functions of themechanism build pose, during in-situ fabrication. Equation 29succinctly relates the fabrication workspace to the operationalworkspace, thereby presenting us with a method for evaluating theoptimal build pose, given an operational tolerance specification.This issue is explored in more detail in 29.Figure 16 compares the first order estimated coupler-point errorfor the example 4-bar fabricated in-situ against the Monte Carlosimulations of the same quantity. Also included are the estimatesusing the conventional approach, which does not include the considerationof correlation effects. Comparisons can also be madebetween these results, and those of the worst case error estimatepresented earlier see Fig. 13. The worst-case and stochastic estimatesfor a specific build angle are compared in Fig. 17. It isclear from the comparison that the worst-case method is significantlymore conservative in its estimation of output error.Figure 18 plots the simulated coupler-point variance against thenumber of random trials. This helps with the estimation of theminimum number of trials needed in order for the random estimatesto converge to a steady value between 4000 and 10,000 inthis case.4 Extension to Spatial ParametersWhile the detailed treatment of spatial error analysis, with supportingnumerical results, is beyond the scope of this paper, thetheoretical extension of the error analysis techniques presentedabove to spatial systems is straightforward once the essential conceptshave been established. Spatial systems are traditionally describedin terms of the Denavit-Hartenberg parameters see Section1.3.4, or modifications thereof. Spatial error analysis is theprocess of relating variability in the spatial parameters to errors inthe output function.For in-situ fabrication, parametric variability is not directlyavailable, but is a function of the position and orientation variabilityin joint placement. Earlier sections in this paper have dealtwith the issue of estimating the output variance, given the stochasticcharacteristics of the joint variables. The approach has been818 Õ Vol. 125, DECEMBER 2003 Transactions of the ASME


Fig. 16 First order estimates of coupler-point variance for the an in-situ fabricatedcrank-rocker mechanism „Fig. 9… using conventional stochastic analysis, analysismodified to for in-situ fabrication, and direct Monte Carlo simulationillustrated using a planar example, and the technique is extendedhere to cover general spatial mechanisms. The basic issue thatremains to be addressed for the spatial case is that of explicitlyexpressing the spatial parameters illustrated in Fig. 5 in terms ofthe joint-frame positions illustrated in 8. This is a fairly simpleproblem in the analytical geometry of three dimensions 30.Given the origin coordinates (p i ,p j ,p k ) and the direction numbers(z i ,z j ,z k ) of the axes of three adjacent spatially locatedjoints, the modified Denavit-Hartenberg parameters of the jthjoint can be expressed in terms of the joint Plücker coordinatessee Section 2.1 of the three joint axes (Q i ,Q i ), (Q j ,Q j ) and(Q k ,Q k ), and those of the two common normals (Q ij ,Q ij ) and(Q jk ,Q jk ). This notation is illustrated in Fig. 19. The directioncoordinates of the common normal are given as:Q ij q 1ij ,q 2ij ,q 3ij whereq 1ij q 2i q 3 j q 3i q 2 jq 2ij q 3i q 1 j q 1i q 3 jq 3ij q 2i q 3 j q 3i q 2 j (30)and the moments of the common normal between axes i and j aregiven as follows this can be extended to j and k by symmetry:Q ij q 1ij ,q 2ij ,q 3ij whereq 1ij Q ij•q 2ij q 3 j q 3ij q 2 j Q i q 2ij q 3i q 3ij q 2i Q j Q ij 2q 2ij Q ij•q 3ij q 1 j q 1ij q 3 j Q i q 3ij q 1i q 1ij q 3i Q j Q ij 2q 3ij Q ij•q 1ij q 2 j q 2ij q 1 j Q i q 1ij q 2i q 2ij q 1i Q j Q ij 2 (31)The modified Denavit-Hartenberg parameters for link j cannow be written as:Journal of Mechanical Design DECEMBER 2003, Vol. 125 Õ 819


Fig. 17 Comparison of 3 stochastic and worst-case „deterministic… error estimatesfor crank-rocker mechanism j arcsinQ jk a j Q j•Q k Q k •Q jsin i d j p k p j • Q jkQ j Q jk 2l j p j p k • Q jkQ j Q jk 2 j arcsinQ jk Q ij (32)Since the mechanism parameters are now known in terms of thejoint positions and orientations, it is possible to estimate the errorin output function given the variability in joint location usingtechniques similar to those outlined for the planar case earlier. Theprocess proceeds by writing the product of homogeneous transformationmatrices three translations and two rotations, that transformone local coordinate frame to the adjacent frame the jthframe to the kth frame in this case, as follows:A k j T0,0,d j Rz j , j Ta j ,0,0Rx j , j T0,0,l j (33)Next, the first-order Taylor Series approximation of the transformationmatrix is written as follows: Ak A k jd j d j A k j j j A k ja j a j A k j j j A k jl j l j(34)The parameter variabilities i.e. d j, j, a j, jand l j) arenow either interval for worst-case analysis or random for sto-Fig. 18anglesConvergence rates of Monte Carlo simulations for different operational820 Õ Vol. 125, DECEMBER 2003 Transactions of the ASME


• There is the freedom to choose a build configuration that willminimize the output variability when the mechanism is in itsoperating configuration.• Important functional gaps and clearances can be controlleddirectly, by controlling the dimensions of sacrificial supportmaterial between mating parts, rather than being a consequenceof the mating of independently fabricated parts.We surmise that it will be particularly important to take advantageof these characteristics in fabricating MEMS and meso-scalemechanisms, for which the process variability is typically a largerpercentage of the feature size than for macroscopic devices.These topics are the subject of ongoing investigation. Someresults on the treatment of clearances and on build pose optimizationare provided in 29.Fig. 19 Notation for the derivation of modified Denavit-Hartenberg parameters from joint Plücker coordinates.chastic analysis parameters, the variances and correlation coefficientsof which can be obtained using the relationships derived inEq. 32.5 Conclusions and Future WorkA framework has been presented for reasoning about errors inthe performance of mechanisms that are slated to be built usingthe increasingly popular ‘‘freeform’’ fabrication techniques. Thisis achieved by formulating an abstract model for the in-situ fabricationof mechanisms, and solving the problem of analyticalestimation of the variance of the kinematic function, in the presenceof correlated random parameters. The fundamental assumptionsin this treatment of error analysis are:• The desired performance of the mechanism is specified interms of a kinematic output function, which is a continuous anddifferentiable mapping from a parameter space to the operationalworkspace usually a Cartesian space. This assumption limits theapplication of the methods presented to linkages with lower pairsand ‘‘well-behaved’’ higher pairs only.• The output is a weakly non-linear function of the inputs. Thisenables a first-order Taylor Series approximation of the error atthe points of interest.• In-situ fabrication is abstracted as a process of independentinsertions of joints which could have internal clearances into afabrication workspace, with a known accuracy. The inaccuracy isspecified as worst-case limits on position and orientation for deterministicerror analysis or variances with known distributionsfor stochastic error analysis.Note that no assumptions of planarity or of homogeneity in workspacecharacteristics are made anywhere in the methodology.Analysis of parametric errors in spatial mechanisms has also beencovered in the theoretical formulation.This paper demonstrates that differences in the manufacturingprocess flow for in-situ fabrication leads to fundamental differencesin how process input variability is manifested in the kinematicoutput of a mechanism.For stochastic analysis, the essential result is that we must accountfor correlations among adjacent links. In this paper we havepresented a modified stochastic analysis that accounts for the correlationsand shown that it compares favorably with numericalMonte Carlo simulations.Although the need to consider correlations in the variabilities oflink parameters somewhat complicates the analysis, in-situ fabricationalso affords some important advantages over conventionalfabrication for reducing output variability, notably:• Tolerances do not accumulate along serial chains.AcknowledgmentsWe gratefully acknowledge the support of the National ScienceFoundation MIP 9617994 and the Alliance for Innovative ManufacturingAIM at Stanford. We are also grateful for the assistanceprovided by members of the Stanford Center for DesignResearch and Rapid Prototyping Laboratories. Sanjay Rajagopalanalso thanks Jisha Menon for her generous support during theformulation of this work—some of which constitutes the basis forhis Ph.D. thesis at Stanford University.Appendix: Estimation of Mean and VarianceHere, we are concerned with the approximate estimation of themean and variance of an output y, described in terms of its outputfunction f (•) and a set of n random parameters 1 , 2 ,..., n as follows:y f (35)where f (•) is, in general, a continuous and differentiable nonlinearmapping, and the parameters are random variables withno assumptions made about their distributions, correlations or independence.It is assumed, however, that the function f (•) is onlyweakly non-linear i.e. high-order terms in it’s Taylor Series expansioncan be neglected and that the mean and variance of theparameters i are known, and denoted as ( i , 2 i).We begin by expanding the output function in its Taylor Series,about the mean values of the parameters, as follows:n fy f i ;i1,2,...,n i1 i i i n 2 f 1 2! i1 i2 2 f i i 2 ij i j i i j j¯ (36)With a little bit of rearrangement, the above equation can be rewrittenin terms of proxy variables i as: fy f i ;i1,2,...,n i1 i in i 2 fj i j i jO 3 (37)where i i i are zero-mean random variables, with allhigher order moments identical with i . The term O 3 stands forall terms in the Taylor Series expansion that are of third degree ormore, and are usually negligible.We now go about the task of estimating the mean and varianceof the output the LHS term, using the above equation. In thisJournal of Mechanical Design DECEMBER 2003, Vol. 125 Õ 821


egard, we make use of the following results, which are based onelementary applications of theorems in the area of MathematicalStatistics 23:E f f i E i 0VaryEy y 2 Cov i , jE i jE i E j (38)where E• stands for the expected value, Var• stands for thevariance and Cov• stands for the covariance. For notationalsimplicity, we denote the expected value, or mean, by the symbol with the appropriate subscript, and the variance by the symbol 2 . In addition, we use the covariance coefficient ( ij ), which isdefined as follows:Cov i , j ij i j(39)Note that 1 ij 1, and that ij 1 when i j and ij 0 forindependent or uncorrelated i and j . From the above equations,it is also apparent that:Cov i , jE i j, and E i j ij i j(40)Returning to the output expansion in Eq. 37, and using the resultsdetailed above, we are able to write the expression for theexpected value of the output function as follows:Ey y f i 0 i 2 fj i j E i j (41)or, using Eq. 40:Ey y f i i 2 fj i j ij i j(42)Equation 42 is a general expression for the approximation of themean of a function f (•) of random variables, which are—ingeneral—correlated.In a manner similar to the earlier analysis, we can use Eq. 37to write an expression for the output variance as follows:Vary 2 fy Ey y E 2 i1 i i i j 2 f2 i j i j f f i j i j E i jO 3 (43)Combining Eq. 43 with Eq. 40,n2 2 fy i1 i 2 f fi2 i j i j ij i j, i j(44)Equation 44 is a general expression for the approximation ofthe variance of a function of correlated random variables. The firstterm in the RHS expression is the variance assuming independentand uncorrelated parameters. 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