a comprehensive study of three dimensional tolerance analysis methods

Computer-Aided Design 53 (2014) 113Keywords:3D tolerance analysisT-MapMatrixUnified JacobianTorsorDirect linearization methodComparison

tolerances only. This paper reviews fourmajormethods of 3D tolerance analysis and compares thembasedon the literature published over the last three decades or so. The methods studied are Tolerance-Map(T-Map), matrix model, unified JacobianTorsor model and direct linearization method (DLM). Eachof them has its advantages and disadvantages. The T-Map method can model all of tolerances andtheir interaction while the mathematic theory and operation may be challenging for users. The matrixmodel based on the homogeneous matrix which is classical and concise has been the foundation ofsome successful computer aided tolerancing software (CATs), but the solution of constraint relationscomposed of inequalities is complicated. The unified JacobianTorsor model combines the advantagesof the torsor model which is suitable for tolerance representation and the Jacobian matrix which issuitable for tolerance propagation. It is computationally efficient, but the constraint relations betweencomponents of torsor need to be considered to improve its accuracy and validity. The DLM is based on thefirst order Taylors series expansion of vector-loop-based assemblymodelswhich use vectors to representeither component dimensions or assembly dimensions. Geometric tolerances are operated as dimensionaltolerances in DLM, which is not fully consistent with tolerancing standards. The results of four modelswith respect to an example are also listed to make a comparison. Finally, a perspective overview of thefuture research about 3D tolerance analysis is presented.

2014 Elsevier Ltd. All rights reserved.

Contents

1. Introduction........................................................................................................................................................................................................................ 22. 3D tolerance analysis models ............................................................................................................................................................................................ 3

2.1. T-Map (Tolerance-Map) model ............................................................................................................................................................................. 3

Corresponding author at: Shanghai Key Laboratory of Digital Manufacture for Thin-walled Structures, Shanghai Jiao Tong University, Shanghai, China. Tel.: +86 02134206306; fax: +86 021 34206306.

E-mail address: [email protected] (S. Jin).

http://dx.doi.org/10.1016/j.cad.2014.02.014Contents lists available at ScienceDirect

Computer-Aided Design

journal homepage: www.elsevier.com/locate/cad

Review

A comprehensive study of three dimensional toleranceanalysis methodsHua Chen a, Sun Jin a,b,, Zhimin Li a, Xinmin Lai a,ba Shanghai Key Laboratory of Digital Manufacture for Thin-walled Structures, Shanghai Jiao Tong University, Shanghai, Chinab State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai, China

h i g h l i g h t s

Introduce four major 3D tolerance analysis models briefly. Make a comprehensive comparison and discussion between them. Expound the connotation of 3D tolerance analysis. Present a perspective overview of the future research about 3D tolerance analysis.

a r t i c l e i n f o

Article history:Received 29 October 2013Accepted 27 February 2014

a b s t r a c t

Three dimensional (3D) tolerance analysis is an innovative method which represents and transferstolerance in 3D space. The advantage of 3D method is taking both dimensional and geometric tolerancesinto consideration, compared with traditional 1/2D tolerance methods considering dimensional0010-4485/ 2014 Elsevier Ltd. All rights reserved.

nologically related surfaces (TTRS) [26], infinitesimal matrix [27],matrix [2830], small displacement torsor (SDT) [31,32], and pro-portioned assembly clearance volume (PACV) [33,34]. Similarly,for tolerance propagation, the approaches or methods consist ofthe linearization method [35], system moments [36,37], quadra-ture [3840], reliability index [41,42], the Taguchi method [43,44],Monte Carlo simulations [45,46], network of zones and datums[47], kinematic formulation [48], the direct linearization method(DLM) [49,50], Jacobianmatrix [51,52], state space [53,54], and the

placement matrix and transfer them with a homogeneous matrix.An SDT model introduced by Clment et al. [31] uses six small dis-placement vectors to represent the position and orientation of anideal surface in relation to another ideal surface in a kinematicway.Desrochers et al. [62] put forward a unified JacobianTorsor modelwhich combines the advantages of the torsor model and the Jaco-bian matrix. Chase et al. [50] introduce a DLM based on the firstorder Taylors series expansion of vector-loop-based assemblymodels which use vectors to represent either component dimen-2 H. Chen et al. / Computer-Ai

2.2. Matrix model.................................................................................2.3. Unified JacobianTorsor model....................................................2.4. DLM method..................................................................................

3. Discussion and comparison ......................................................................4. Conclusion .................................................................................................

Acknowledgments ....................................................................................References..................................................................................................

1. Introduction

The objective of tolerance analysis is to check the feasibilityand quality of assemblies or parts for a given GD&T scheme. Theresults of tolerance analysis include worst case variations andstatistical distribution of functional requirement, acceptance rates,contributors and their percent contributions, and the sensitivitycoefficients with respect to each contributor. Tolerance analysis isan essential part formechanical design andmanufacturing becauseit affects not only the performance of products but also the cost.

Tolerance analysis, including tolerance representation and tol-erance propagation (tolerance transfer), can be classified intomany categories based on the analysis objective and analysis ap-proach, as shown in Fig. 1. According to dimensionality, thereare one dimensional (1D), two dimensional (2D) and three di-mensional (3D) tolerance analyses. Three approaches are appliedfor 1/2/3D tolerance analysis, i.e., worst case (deterministic case),statistical case and Monte Carlo simulation. Rigid and flexible tol-erance analysis are two differentmodels in the light of analysis ob-jective. The former is surface-based and needs shape closure only,such as engines tolerance analysis; the latter is point-based andneeds shape and force closure simultaneously, such as auto-bodiestolerance analysis where the finite element method (FEM) is usedto take the deformation into consideration [14]. The division intopart level and assembly level is another classification. The stack-up effect of assembly can be described by virtue of assembly func-tion explicitly or implicitly, depending on the assembly methodand sequence, as well as the property of components [5]. Toler-ance analysis runs through the whole process of the product, in-cluding design, process planning, manufacturing, inspection, butthe objective may be different in each phase. For example, the tol-erance scheme, i.e., conventional (parametric) and geometric toler-ance will be selected and specified, and then tolerance analysis forfunctional requirement will be carried out in design phase. Mean-while, besides manual analysis, computer aided tolerancing soft-ware (CATs), such as VisVSA R, 3DCS R and CETOL R are applied totolerance analysis successfully [69]. To be sure, the classificationof tolerance analysis will be more and more complicated with thedevelopment of mechanical design and manufacturing.

Over the last thirty years, a large amount of fundamental re-search efforts has been given to explore the mathematical basisfor tolerance analysis. For tolerance representation, the models orconcepts include variational geometry [1012], variational class[13,14], virtual boundary [15,16], feasibility space [17,18], vecto-rial approach [19], virtual joints [20], degree of freedom (DOF)[2123], Tolerance-Map (T-Map) [24,25], topologically and tech-variational method [55]. It is worth noting that the partition of twocategories mentioned above is approximate and based mainly onded Design 53 (2014) 113

......................................................................................................................... 5

......................................................................................................................... 7

......................................................................................................................... 9

......................................................................................................................... 10

......................................................................................................................... 11

......................................................................................................................... 12

......................................................................................................................... 12

their strong suits, because there is no boundary between the toler-ance representation and propagation for these models, such as theTTRS [56].

As new generations of tolerancing standards, i.e., ASME Y14.5-2009 [57] and ISO 1101 [58] were released and popularized, ge-ometric tolerances are generally accepted as industry practices.The traditional 1/2D tolerance analysis models are insufficient tomeet the ever-tightening and increasingly complex requirementsof tolerance analysis in various fields [59]. More specifically, varia-tions of a feature caused by geometric tolerances are three dimen-sional, which cannot be considered by 1/2D methods. Researchersand engineers need a newmethod that can analyze how those geo-metric tolerances are represented and propagated in three dimen-sional space urgently. It is the 3D tolerance analysis method. Letus take a combustion engine as an example, as shown in Fig. 2.The translational and rotational variations of piston accumulatedby geometric and dimensional tolerances of crank-link parts have asignificant impact on the compression ratio. In addition, tolerancesof parts affect not only the dimensional quality of assembly, butalso other qualities such as frictional work [60,61] and sealing.Finding out the mapping relationship of tolerance between partsand functional requirements and performance indexes is impor-tant to engine design. 3D tolerance analysis methods will offer asignificant clue for understanding the role of every tolerance ofparts in the variation stream (gray boxes in Fig. 2).

The 3D tolerance analysis is an innovativemethodwhich repre-sents and transfers tolerance in 3D space. Geometric tolerances anddimensional tolerances, as well as the interaction between themin the tolerance zone can be taken into consideration by 3D toler-ance analysis methods. Moreover, abundant results, i.e., the trans-lational and rotational variations of target feature are obtained inthese methods. Many models have been developed for 3D toler-ance representation and propagation since 1990s. Portman [27]introduces a spatial dimensional chain where the individualerror is represented as an infinitesimal matrix to model the tol-erance propagation. Fleming [47] illustrates the geometric rela-tionships by a network of zones and datums connected by arcs towhich constraints are assigned. The effects of these constraints arecalculated through the network between nodes. Rivest et al. [48]propose a kinematic formulation which exploits the kinematiccharacter of a toleranced feature relative to its datum. These threemethods are preliminary explorations of 3D methods. Laperrireand Lafond [20,51] use virtual joints for tolerance representa-tion and the Jacobian matrix for tolerance propagation. Davidsonet al. [24] present a T-Map representing all possible variations ofsize, position, form, and orientation for a target feature. Desrochersand Rivire [29] represent the variations of a feature with a dis-sions or assembly dimensions. Some models mentioned abovehave been applied extensively by virtue of CATs.

fThe purpose of this work is to discuss four typical methods of3D tolerance analysis, i.e., the T-Map model, the matrix model, theunified JacobianTorsor model and DLM, all of which are researchhotspots recently. The TTRS theory is also introduced because it isthe basis of the matrix and torsor models. Although several reviewpapers have introduced thesemethods briefly [6370], it is the firsttime to put them together to make a comparison in 3D context. Itshould be noted that the concept of 3D here emphasizes tolerancerepresentation and propagation in three dimensional space, othermodels, such as the state space model which focuses on tolerancetransfer with a state space method, the variational method whichpays close attention to variation propagation of fixtures with totaldeterministic locating (321 scheme), are not discussed becausethey are point-based tolerance schemes where variations causedby geometric tolerances are not three dimensional.

The rest of the paper is organized as follows: Section 2 givesan overall introduction of these 3D tolerance analysis models.Section 3 makes a comprehensive discussion and comparisonbetween them. Finally, conclusions with a perspective overview

In order to illustrate the modeling process of T-Map, here weconsider a cross-section of a round bar with a size tolerance t onits length, as shown in Fig. 3. Fig. 3(a) shows the tolerance zoneABCDwhere all points of the end facemust lie in, and the Cartesiancoordinate system. Assuming a perfect round plane with diameterd and no thickness, the possible displacement of this plane is thetolerance zone, i.e., the volume limited by the planes 1 and 2, andthe cylindrical diameter of the bar. The planewould translate alongthe z axis and rotate around the x axis freely on two dimensionaloccasions where the plane is always parallel to the x axis. Theplane 3 in Fig. 3(a) represents the greatest clockwise tilt of theperfect plane in tolerance zone. After that, the two dimensionalset of planes in Fig. 3(a) defined by three basis planes 1, 2, 3are mapped to the areal coordinates, as shown in Fig. 3(b). Anyend plane of the round bar that satisfies the size tolerance and isparallel to the x axis will be represented by expressed as Eq.(1). Especially, the points on the line-segment 12 in Fig. 1(b)represent the parallel planes that are perpendicular to the z axis inFig. 1(a) and lie between 1 and 2. The points on the line-segment13 in Fig. 1(b) correspond to these planes in Fig. 1(a) that areparallel to the x axis, pass through point B, and lie in the toleranceH. Chen et al. / Computer-Ai

Fig. 1. Categories o

Fig. 2. The mapping relationship of tolerances between parts and functionalrequirements and performance indexes in engines.of the future research about 3D tolerance analysis are given inSection 4.ded Design 53 (2014) 113 3

tolerance analysis.

2. 3D tolerance analysis models

2.1. T-Map (Tolerance-Map) model

The T-Map R (Patent No. 6963824) model developed by David-son et al. [24] is a hypothetical Euclidean volume of points, theshape, size, and internal subsets of which represent all possiblevariations in size, position, form, and orientation of a target fea-ture. A T-Map is a convex set resulting from a one-to-one mappingfrom all the variational possibilities of a feature within its toler-ance zone, constructed from a basis-simplex and described withareal coordinates.

The areal coordinates use areal parameters to describe theposition of a point in a reference triangle. Given three fixed points1, 2, 3, called basis points that are chosen in Euclidean space,the position of any point is uniquely determined by the linearequation:

= 11 + 22 + 33 (1)where 1, 2, 3 are areal coordinates about , and have therelation 1 + 2 + 3 = 1. Either one or two of these coordinateswill be negative when is chosen outside 123.zone between 1 and 3. Similarly, the points on the line-segment23 in Fig. 1(b) correspond to these planes in Fig. 1(a) that are

4 H. Chen et al. / Computer-Ai

Fig. 3. Modeling process of the T-Map for a round bar with a size tolerance.

Fig. 4. A plane in the Cartesian frame of a tolerance zone.

parallel to the x axis, pass through point C , and lie in the tolerancezone between 2 and 3.

The three dimensional tolerance zone at the end of the roundbar is obtained by a full sweeping operation around z axis with therectangleABCD in Fig. 3(a). Therefore, the three dimensional T-Mapis modeled by revolving the triangle in Fig. 3(b) a full turn aroundthe line 12, it is a right-circular dicone shown in Fig. 3(c). Itshould bementioned that the areal coordinates can also be used toidentify points in three dimensional space where four basis pointsare non-coplanar. The points 1, 2, 3, 4 in Fig. 3(c) are selectedto establish a tetrahedron of reference for three dimensional areacoordinates in the T-Map. Setting

i = 1, any end plane of the

round bar that satisfies the size tolerance will be represented bythe linear equation:

= 11 + 22 + 33 + 44. (2)The transformation from T-Map in Fig. 3(c) to Cartesian coordi-nates in Fig. 3(a) is worth mentioning. Presuming a plane in theCartesian coordinates, as shown in Fig. 4, its position can be de-scribed by the equation px + qy + rz + s = 0. p, q, r are the di-rection cosines in which r is approximately equal to 1 because therotation displacement of the plane shown in Fig. 3(a) around thez axis is smaller than other two rotation displacements in the tol-erance zone, and s is the absolute distance from the plane to theorigin of coordinates. Therefore, the planes in the tolerance zoneare distinguished by the coordinates p, q and s only. q in Fig. 3(b) isobtained by assigning dimension or length on q in Fig. 3(a) becauseq is dimensionless. The lateral dimension t of q axis in Fig. 3(b) ist = d(t/d) = d tan() = dq = q. p is obtained in the same wayas q. Consequently, the transformation of any plane in the toler-ance zone of Fig. 3(a) from areal coordinates to Cartesian coordi-nates is:pqr

s

= 0 0 0 t/d0 0 t/d 01 1 1 1t/2 t/2 0 0

1234

. (3)It should be stressed that a T-Map for a single part is always

convex. This property allows the usage of fundamental principlesof convex sets, such as Minkowski sums and differences.

We continue to use the round bar shown in Fig. 3 as an objectiveto illustrate themodeling process of formandorientation tolerancefor T-Map, but now a flatness tolerance t1 and a parallelism

tolerance t2 which is relative to a datum plane (another end ofbar) are specified besides the size tolerance t . According to theded Design 53 (2014) 113

Fig. 5. (a) A half section of the T-Map for a round bar with a form tolerance and asize tolerance. (b) A half section of the T-Map for a round bar with an orientationtolerance and a size tolerance.

Fig. 6. An assembly of two round bars for the T-Map.

tolerancing standards, the flatness tolerance zone is defined by twoparallel planes with the distance t1. Its position and orientationare not constrained within the size tolerance zone. The internaltriangular sub-set drawn with a dashed line in Fig. 5(a) representsthe flatness tolerance while the hatched triangle correspondsto positions that the flatness tolerance zone can occupy. TheMinkowski sum of the sub-set of the dashed triangle and thehatched triangle is the tolerance t . Similarly, the parallelismtolerance zone is defined by two parallel planes which are parallelto planes 1 and 2 with the distance t2, represented by thedashed triangle in Fig. 5(b). Because the parallelism tolerance zonecan only translate up or down within the tolerance zone t , theMinkowski sum of sub-set of dashed triangle and its translationalzone shown in Fig. 5(b) is a truncated map along the q axis.

In order to illustrate the tolerance propagation of T-Map, asimple assembly composed of two parts is shown in Fig. 6 as anexample. Part 1 is a round bar. Part 2 consists of two round barsseparated by an offset b. They are assembled end to end coaxially.The functional surface is the upper face of the bar with diameter d2in part 2. The dimensions and tolerances of assembly can be seenin Fig. 6. The coordinate frames are similar to Fig. 3.

The variations of orientation in the tolerance zone of part 1 areamplified and cause positional variations of the functional surfacebecause of the offset b. The accumulation T-Map depends on theoffset b and the diameters d1 and d2 since functional requirements

for the target face in the assembly control the assignment oftolerances to the individual parts. Here we only consider the

H. Chen et al. / Computer-Ai

Fig. 7. (a) q-s sections of the accumulation and functional requirement T-Maps forthe assembly when b = 0. (b) q-s section of two operand dicones for the assemblywhen b = 0.

Fig. 8. A planar feature in a tolerance zone.

condition where d1 > d2. The T-Maps of part 2 and functionalrequirements in which the aspect ratios are unity are selectedto be constructed on the basis tetrahedron 1234 in Fig. 3(c).When b = 0, the half-section of the accumulation (solid line)and functional requirement (dashed line) T-Maps are shown inFig. 7(a) in which the accumulation T-Map is inscribed in thefunctional T-Map in order to avoid specifying excessively tighttolerances on individual parts. ta and tf which are equal to t1 +t2 represent accumulation tolerance and functional requirementtolerance along the length direction. ta which is equal to t11 (d2/d1)+ t2 represents orientation tolerance of functional surface,which is tighter than functional requirement.

However, it is somewhat complicated if b = 0. Assuming a ro-tational angle q1 in the tolerance zone of part 1, s of T-Map ofpart 1 becomes s+ bq1 which results in point 3 in Fig. 3(c) movesdownward by the amount bt11/d1 and the opposite point 7 movesupward by the same amount. That is an oblique elliptic diconetruncated by the orientation tolerance t11, shown by the dashedline in Fig. 7(b).When b (d1d2)/2, the point3 in T-Map of part1 is still within the functional T-Map, and the accumulation toler-ance in the axial direction is t1+t2.When b > (d1d2)/2, the point3 in T-Map of part 1 touches the line 1f 3f , and the accumulationtolerance along the axial direction is t11 ((2b+d2)/d11)+t1+t2.

Besides round bars, T-Maps have been developed for other fea-tures, such as polygonal bars [25], axes [7173], angled faces [74],pointline clusters [75], and planar and radial clearancewith a sta-tistical way [7678]. T-Map can also be used for tolerance alloca-tion or synthesis, the detailed discussion is not within the scope ofthis paper.

2.2. Matrix model

The matrix model introduced by Desrochers and Rivire [29]uses a displacement matrix D to describe the small displacementsof a feature within the tolerance zone and the clearance between

two features. Using the theory of the set of displacements byHerv [79], Clment et al. [26,31,32] have proven that there areded Design 53 (2014) 113 5

only seven elementary surface types, as shown in Table 1. The com-binations of surfaces are called TTRS when two surfaces belong tothe samepart or Pseudo-TTRSwhen two surfaces belong to twodif-ferent parts [80]. The reference elements are classified by the con-cept of minimum geometric datum element (MGDE) [81]. Thirteenrelative positioning constrains of basic components of the MGDE,i.e., point, line and plane are defined, as shown in Table 1. The con-cept of functional requirements is also declared in the wake of theTTRS [82].

The 4 4 homogeneous matrix D including a 3 3 rotationalmatrix and a 3 1 translational matrix is chosen here to representthe relative displacement of a feature within tolerance zone:C C S C + C SS S S + C SC uS C C C + S SS C S + S SC vS CS CC w

0 0 0 1

(4)where , , are rotational displacements around the x, y, z axesrespectively; u, v, w are translational displacements along thex, y, z axes respectively; C is the abbreviation of cos() and S is sin().

Tolerance is only meaningful in directions other than thosethat leave a surface invariant with respect to itself [83], so everyparameter in the matrix can be seen as a micro DOF that leavesthe feature non-invariant. Let us take a planar surface shown inFig. 8 as an example, the ideal plane without form tolerance andthickness can translate along the x axis and rotate around the y andz axes in the tolerance zone. In other words, a plane has three non-invariant degrees, i.e.,, and uwithin its tolerance zone. Anotherthree displacements (, v and w) are invariant and set to zero.Therefore, Eq. (4) can be simplified for the planar non-invariantmatrix D (see Eq. (5)). Non-invariant displacement matrix of theother five surfaces can be seen in Table 1.C C S C S uS C C S S 0S 0 C 0

0 0 0 1

. (5)The matrix representation is completed by a set of inequalities

defining the bounds of every component in the matrix. Theseinequalities depend on the type of surface and tolerance. If morethan one tolerance is specified on the same feature, their effectsare calculated through the principle of effects overlapping. Takingthe four vertexes (A, B, C,D) into consideration, the bounds of theplanar surface with t shown in Fig. 8 can be expressed by:tSL u(A,B,C,D) tSU

t/a t/at/b t/b

with

uAuBuCuD

= DCACBCCCD

(6)where CA,B,C,D are coordinates for points A, B, C , and D.

Two types of datum reference frame (DRF) need to be identifiedfor tolerance transfer in the matrix model when the assemblygraph is created, i.e., global DRF (R) which is the evaluationreference of the functional feature and local DRF (Ri) of each partfeature. The homogeneous matrix P which depends only on thenominal geometry represents the transformation of the R to the Ri.Given two points in Ri, the theoretical pointM and the same pointM after a displacement in the tolerance zone, [MM ] representsthe total displacement of theM because of matrix Di, which can beexpressed by:MM

R = P1RRi

MM

Ri

= P1RRi

M Ri [M]Ri

= P1RRi (Di I) [M]Ri . (7)

6 H. Chen et al. / Computer-Aided Design 53 (2014) 113

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The equation above gives the range of displacements of the ithfeature in the global DRF that is allowed by the tolerance zone.More points are necessary to specify the constraints that keep thefeature inside the bounds of the tolerance zone. Taking the planarsurface shown in Fig. 8 as an example, the additional constraints ofpoints A, B, C and D are:

tSL (Di I)MA,B,C,D

Rix tSU (8)

wherex represents the directional vector of tolerance zone.It should be mentioned that other geometric tolerances, such

as orientation tolerance, also can be modeled by Eq. (8) throughthe arithmetic operation of some vertex points according to thetolerancing standards. Moreover, a cylindrical clearance zone canbe viewed as an equivalent concentricity tolerance in the matrixmodel.

For convenience, the assembly in Fig. 6 is also used to demon-strate the tolerance analysis of thematrixmodel, as shown in Fig. 9where three local DRFs (R1, R2, R3) and one global DRF (R0) arespecified. The assembly graph is shown in Fig. 10. The functionalrequirement is the displacement of point a along the z axis in theglobal DRF, which is equal to za/D1 + za/D2. Two matrices D1 andD2 which are similar to Eq. (5) are established for tolerance repre-sentation of the upper surface relative to the under surface of tworound bars. With Eq. (7), we can obtain the two displacements ofpoint a:

za/D1 = u1 + b cos(1) sin (1)+ l2 (cos(1) cos(1) 1) (9)

za/D2 = u2. (10)The constraints of the components ofD1 under the tolerance t1 andt11 with respect to four vertexes of part 1 are:

t12

u1 + (d1 cos(1) sin(1))/2

u1 d1 sin(1)/2u1 (d1 cos(1) sin(1))/2

u1 + d1 sin(1)/2

t12 (11)t11

d1 cos(1) sin(1)

d1 sin(1) t11. (12)

The constraints of the components of D2 under the tolerance t2with respect to four vertexes of part 2 are:

t22

u2 + (d2 cos (2) sin(2))/2

u2 d2 sin(2)/2u2 (d2 cos(2) sin(2))/2

u2 + d2 sin(2)/2

t22 (13)where t1/2 u1 t1/2, t2/2 u2 t2/2, t11/d1 1 t11/d1, t11/d1 1 t11/d1, t2/d2 2 t2/d2, t2/d2 2 t2/d2.

Standard optimization algorithms, such as simplex, can beapplied to solve this problem.

The statistical method of the matrix model for tolerance anal-ysis can be found in [30]. It is worth mentioning that some CATsbased on the matrix model have been developed and applied suc-cessfully, such as CATIA.3D FDT R [6] and FROOM [68,69].

2.3. Unified JacobianTorsor model

The unified JacobianTorsor model introduced by Desrocherset al. [62] combines the advantages of the torsor model whichis suitable for tolerance representation and the Jacobian matrixwhich is suitable for tolerance propagation. Tolerance analysis of

this model is in a kinematic way because of the concept of torsorwhich is also on the basis of TTRS.ded Design 53 (2014) 113 7

Fig. 9. The assembly of two round bars for the matrix model.

Fig. 10. Assembly graph of the matrix model.

Fig. 11. Small displacement torsor of a surface.

The torsor, also known as small displacement torsor (SDT), isa small displacement screw used to represent the position andorientation of an ideal surface or its feature (axis, centerline, plane)in relation to another ideal surface with a kinematic way [34]. Asshown in Fig. 11, at a given point O on nominal surface S0, the SDTof variational surface S1 from S0 can be expressed as:

T = 1/0 1/0 = u v w

(14)

where 1/0 is the rotational vector around the axis including threespecific vectors, i.e., , and indicate the vectors around theaxes x, y and z in the local reference system respectively; likewise,1/0 is the translational vector and u, v, w are three specific vectorsalong the axes x, y and z respectively.

The relative translation between two surfaces at any pointM inEuclidean space can be obtained by a linearization rule in terms of1/0 and a cross product of 1/0 and vector

OM .

dM = 1/0 + 1/0 OM =

uvw

+ y+ z x zx+ y

. (15)

The SDT is well suited to not only 3D tolerance representationbut also 3D metrology [84,85]. It should be noted that the SDTfor tolerance representation is the first order approximation ofthe matrix model introduced in the previous subsection [86]. The

rotation angles in Eq. (4) can be simplified by sin() andcos() 1 because the angles are very small in the tolerance zone.


Then the 3 3 rotational matrix can be written as: 1 1 1

. (16)

And the small displacement of any pointM can be obtained by:

dM =

uvw

+ 1 1 1

xyz

=uvw

+ y+ z x zx+ y

. (17)

Generally, the constraints of components of SDT depend onthe type of tolerance and feature. The SDT used for tolerancerepresentation of planar and cylindrical surfaces, as well as theirjoints (clearance) has been studied deeply. Seven types of surfaceand their screws are listed in [86]. The concept of non-invariant ofthe matrix model is also effective in the SDT model. , v and w inEq. (14) that haveno effect on tolerance analysis of the plane shownin Fig. 8 can be set to zero to reduce the scale of computationalwork.

Tolerance representation with SDT is concise and intuitionistic,but it is difficult for tolerance transfer [64,65]. Therefore, Jacobianmatrix is introduced into tolerance analysis.

The Jacobian method is a linear arithmetic formulation appliedin series robot system, mapping the displacement or velocity ofjoints to end joint. Laperrire and Lafond [20,51] bring it intotolerance analysis by introducing the concept of functional pairsexpressed by a set of virtual joints.

There are two types of functional pairs in the assembly, i.e., in-ternal pair and contact or kinematic pair. The former is composedof two functional elements (FEs) on the same part; the latter ismade up of two FEs on different parts if there is a physical or po-tential contact between them.

The Jacobian matrix for ith FE can be expressed as:

[J]FEi =

[Ri0]33 [RPti]33

... [W ni ]33 ([Ri0]33 [RPti]33) ...

[0]33... ([Ri0]33 [RPti]33)

(18)

whereRi0represents the local orientation of ith frame with

respect to 0th frame that is the global reference system; [RPTi] isa projection matrix designating the unit vectors along local axesrespectively for tolerance zone tilted according to the directionof tolerance analysis;

W niis a skew-symmetric matrix allowing

the representation of the vector among the ith and nth frame (endpoint), defined in Eq. (19);

Ri0 W ni reflects the leverage effect

when the small rotations of FE are beingmultiplied by terms of theJacobian matrix.

W ni33 =

0 dzni dynidzni 0 dxnidyni dxni 0

(19)

where dzni = dzn dzi, dyni = dyn dyi, dxni = dxn dxi.The SDTmodel is suitable for tolerance representationwhile theJacobian matrix is suitable for tolerance propagation. The unifiedJacobianTorsor model combines the advantages of both methods,ded Design 53 (2014) 113

the expression of which can be written as follows:

u, u

v, v

w,w

,

,

,

FR

= [J]FE1 [J]FEn

u, u

v, v

w,w

,

,

,

FE1

...

u, u

v, v

w,w

,

,

,

FEn

(20)

where FR represents the functional requirement; (, ) is thetolerance interval where must lie in, other vectors follow thesame way. The interval arithmetic is incorporated into Eq. (20)to allow tolerance analysis to be performed on a tolerance zonebasis rather than on a point basis.

The assembly graph of Fig. 9 with the unified JacobianTorsormodel is shown in Fig. 12, including two internal FEs and onecontact FE. The target feature is the upper face of the bar withdiameter d2 in part 2. All coordinate frames are in the middle ofthe tolerance zone or contact zone. The contact pair between part1 and part 2 is considered as zero because there is no clearancebetween two contacting planes. With Eq. (20), the final expressionof the unified JacobianTorsor formulation about the assembly inFig. 9 is:

(u, u)

(v, v)

(w,w)

(, )

(, )

( , )

FR

=

1 0 0 0 l2 b0 1 0 l2 0 00 0 1 b 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

FE1

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

FE2

(0, 0)

(0, 0)

(t1/2, t1/2)(t11/d1, t11/d1)(t11/d1, t11/d1)

(0, 0)

FE1

(0, 0)

(0, 0)

(t2/2, t2/2)(t11/d2, t11/d2)(t11/d2, t11/d2)

(0, 0)

FE2

. (21)

As can be seen, the orientation tolerance mainly limits therotational displacements of the upper surface of part 1. We onlyfocus on the value along the z axis of FR in this paper. The resultshowswmust lie in the interval of [((t1+t2)/2+bt11/d1), ((t1+t2)/2 + bt11/d1)]. It should be pointed out that the bt11/d1 of theresult is the so called leverage effect caused by the small rotationaldisplacement t11/d1 and the offset b.

The statistical method of the unified JacobianTorsor model

where Monte Carlo simulation is applied has been developed[8789]. Moreover, the unified JacobianTorsor model can also be


Fig. 12. Assembly graph of the unified JacobianTorsor model.

Fig. 13. An example of kinematic joints.

used for redesign of assembly tolerance where the contributionof each FE can be calculated [90], and geometrical variationsmanagement in a multi-disciplinary environment [91].

2.4. DLM method

The DLM (Direct Linearization Method) proposed by Chaseet al. [49,50] is based on the first order Taylors series expansionof the assembly kinematic constraint equations with respect toboth the assembly variables and the manufactured variables inassembly. The assembly equations expressed by the vector-loop-based assembly models which use vectors to represent eithercomponent dimensions or assembly dimensions take three mainsources of variation into account in a mechanical assembly. Theyare dimensional variations and geometric feature variations whichare the results of the natural variations inmanufacturing processes,and kinematic variations are small adjustments between matingparts that occurred at assembly due to the dimensional andgeometric variations in manufacturing phrase.

It is the kinematic variations which result in implicit assemblyfunctions. As shown in Fig. 13, the kinematic variable F depends onthe variables , R, t and H , and the position of contact points a andbwhich are called kinematic joints (dashed rectangles). Kinematicjoints describe motion constraints at the contact points betweenmating parts. There are six common joints in 2D assemblies andtwelve common joints in 3D assemblies.

The vectors in a matrix form are arranged in chains or loopsrepresenting the accumulation of variations mentioned above invector-loop-based assembly models. Firstly, ignoring the geomet-ric tolerances, the assembly constraints with the vector-loop-based assembly models can be expressed as a concatenation ofhomogeneous transformation matrices:R1 T1 R2 T2 Ri Ti Rn Tn Rf = H (22)

where Ri is the rotational transformation matrix between the vec-tors at node i; Ti is the translational matrix of vector i;Rf is theded Design 53 (2014) 113 9

Fig. 14. The effects of flatness tolerance t at the kinematic joint b of Fig. 13.

final closure rotational transformationmatrix with the global DRF;H is the resultant matrix which is equal to the identity matrix for aclosed loop, or the final gap or clearance and its orientation for anopen vector loop.

Eq. (22) describes a series of rotations and translations to trans-form the local coordinates from vector-to-vector until it has tra-versed the entire vector loop and returned to the starting point. Itis important to note that the rotational value ofRi is always relativeto the prior vector. It is a positive angle when the rotational direc-tion is same as the prior vector. It is a negative angle, otherwise.

Although smaller than dimensional variations, the accumula-tion and propagation of geometric feature variations are similar todimensional variations. In the vector-loop-based assembly mod-els, geometric tolerances are considered by placing at the contactpoint between mating surfaces with zero length vectors havingspecified variations or tolerances. In other words, the geometrictolerance associated with each joint may result in an independenttranslational variation or rotational variation or both. It should bepointed out that the effect of feature variations in 3Ddepends uponthe joint types and which joint axis you are looking down. Fig. 14shows the effects of flatness tolerance t at the kinematic joint bof Fig. 13. Fig. 14(a) represents a translational variation in the xyplane while Fig. 14(b) represents a rotational variation in the xzplane. There are three variations in all at this joint where anotherrotational variation around the z axis is not shown, which impliesthe DOF of kinematicmotions and the DOF of feature variations aremutually exclusive.

All the possible combinations of geometric feature toleranceswith kinematic joint types in 2D space and 3D space can be seenin [92].

Assuming a geometric feature tolerance is added to joint i,assembly constraint equation (22) can be rewritten as:

R1 T1 R2 T2 Rig Tig

Ri Ti Rn Tn Rf = H (23)where

Rig Tig

is the transformationmatrix caused by geometric

tolerance of ith feature.It is complex to solve Eqs. (22) and (23) because they are

nonlinear equations. But the approximate solution with the DLMmethod is accurate enough for tolerance analysis. The first orderTaylors series expansion of assembly constraint equations for aclosed loop can be written as:

1HC = A 1X + B 1U + F 1 = [0] . (24)And for an open loop is:

1HO = C 1X + D 1U + G 1 (25)where 1HC is the vector of clearance variations in a closed loopand 1HO is the vector of assembly variations in an open loop;1X is the vector of variations of dimensional variables; 1U isthe vector of variations of assembly variables; 1 is the vectorof variations of geometric feature variables; A and C are the firstorder partial derivatives of the dimensional variables in the closed

loop and open loop respectively; B and D are the first order partialderivatives of the assembly variables in the closed loop and open


loop respectively; F and G are the first order partial derivatives ofthe geometric feature variables in the closed loop and open looprespectively.

Among Eqs. (24) and (25),1U is obtained by solving these twoequations. For the closed loop,1U is given in Eq. (26) if B is a full-ranked matrix and in Eq. (27) if B is a singular matrix.

1U = B1 A 1X B1 F 1 (26)1U = (BT B)1 BT A 1X (BT B)1 BT F 1. (27)

From Eqs. (25) to (27), we can obtain the 1U in the open loopas:

1U = (C D B1 A) 1X + (G D B1 F) 1 (28)1U =

C D BT B1 BT A 1X+G D BT B1 BT F 1. (29)

Tolerance accumulation of DLM can be estimated with a worstcase way and a statistical way, as shown in Eqs. (30) and (31).

TW =mj=1

|Sdij | T dij +n

j=1|Sij | Tij (30)

TS = m

j=1|Sdij | T dij

2+

nj=1

|Sij | Tij2

(31)

where Sdij and Sij are sensitivity matrices of dimensional variables

and geometric variables respectively, which are the coefficientsof the 1X and 1 in Eqs. (26)(29); m and n are the number ofdimensional and geometric variables respectively.

Let us continue to take the assembly shown in Fig. 9 as theexample to demonstrate the computational process of DLM. That isan open loop where the assembly constraint equations are explicitbecause no adjustable elements exist. The geometric tolerance t11involves a rotational variable around the x axis in part 1 whichis equal to t11/d1. According to Eq. (23), the resultant vector alongthe z axis at point a is l1 + l2 cos() + b sin(). From Eq. (25),the stack-up variation along the z axis is (t1 + t2 cos())/2 +(b cos() l2 sin()) .

The second order tolerance analysis (SOTA) method wherethe second order Taylors series expansion of the assemblykinematic constraint equations is taken into account by MonteCarlo simulation has been developed to enhance the accuracy ofDLM [93].

Because the derivatives of the assembly function with respectto both the assembly andmanufactured variables are more readilyfrom the vector model, the DLM is more computationally efficientover other models for tolerance analysis. Owing to the long-termresearch by association for the development of computer-aidedtolerancing software (ADCATS), DLM has been applied on CATssuccessfully, such as CETOL 6 Sigma R.

3. Discussion and comparison

So far, we have listed four 3D tolerance analysismethods partic-ularly based on the literature published over the last three decadesor so. And a simple example has beenused todemonstrate the anal-ysis process of these models. Each of them has its advantages anddisadvantages. This section discusses and compares them in detail.

The T-Map based on area coordinates cannot only model all3D variations of a feature, such as size, orientation and form, butalso model completely and precisely the interactions of them. It is

completely compatiblewith the ASME/ISO standards for geometrictolerance and suitable for tolerance synthesis. The size and shapeded Design 53 (2014) 113

of the accumulation map are controlled by the dimensions andshapes of target surfaces in assembly, which gives expression tothe connotation of 3D tolerance analysis. However, the Minkowskioperation for tolerance propagation is not straightforward andnot suitable for computation by hand, and the calculation ofclearance for two planar surfaces or the pin-hole assembly hasbeen developed in 1D situation only. In otherwords, T-Map has notyet been fully developed. A mass of efforts is still needed to studythe algorithms of sensitivities of contributors and their percentcontributions, as well as the statistical arithmetic. Moreover, theT-Map for axes is four dimensional,which is difficult for illustrationin 3D situation. A better method for the visualization of higherdimensional maps is needed.

The matrix model uses a displacement matrix to describe thesmall displacements of a feature within the tolerance zone and theclearance between two features. This model, completed by a set ofinequalities defining the bounds of the tolerance zones, reproducesthe measurable or non-invariant displacements associated withvarious types of tolerance. It is very efficient for computationand can be integrated into CAD systems easily. The statisticalmethod for thematrixmodel has also beendevelopedwhich bringsthe constraint relations between the translational displacementsand rotational displacements into computation by Monte Carlosimulation. But the analysis objective and constraint objective arepoints, which lead to different results with respect to differentpoints. The optimization may be difficult when lots of inequalitiesare obtained. In addition, it is unable to take the form tolerance intoaccount.

The unified JacobianTorsor model combines the advantages ofthe torsormodelwhich is suitable for tolerance representation andthe Jacobian matrix which is suitable for tolerance propagation.To overcome the limitations and difficulties of point-basedapproaches, the interval arithmetic is brought into the model toallow tolerance analysis to be performed on a tolerance zonebasis rather than on a point basis. Tolerance analysis of thismodel is in a kinematic way because of the concept of torsor.It is more suitable for the representation and propagation ofclearance in 3D situation. The statistical method and toleranceallocation for this model have also been studied. Nevertheless, theconstraint relations between the components of torsor need to beconsidered to improve the accuracy of results and conform to thetolerancing standards better. Aswith thematrixmodel, the unifiedJacobianTorsor model cannot deal with the form tolerance too.

The DLM is based on the first order Taylors series expansionof the assembly kinematic constraint equations with respect toboth the assembly variables and the manufactured variables inan assembly. Three main sources of variations, i.e., dimensionalvariations and geometric feature variations, as well as kinematicvariations, are distinguished and represented by the vector-loop-based assembly models. Although all types of tolerance can bemodeled, and the results of statistical case and worst case canbe calculated efficiently with the sensitivity matrix, this methodheavily depends on the users expertise and experience to obtaincorrect results. More specifically, how to define the joint types andthe effects of geometric variations are dependent of users choices.Meanwhile, the relationship between the geometric toleranceand the dimensional tolerance needs continuous optimization tocoincide with the tolerancing standards better.

The difference and comparison between four models with sixitems are listed in Table 2. The symbol represents unknownor unable to calculate based on the published literature. and represent point-based and surface-based respectively.w and represent the variation of target surface along the z axis and aroundthe x axis in the assembly depicted in Fig. 6 respectively.The results of four models corresponding to the assemblyshown in Fig. 6 are listed in Table 3 where only the condition of

d/details about these models, as well as the differences betweenthem are left to the readers. Generally, each of these toleranceanalysis models has its own strengths and weaknesses, and it isup to the users to make the wise choice according to the analysisobjective and condition.

We would like to conclude this paper by presenting a perspec-tive overview of the future research about 3D tolerance analysiswhich is considered to be challenging but promising.

(1) Working conditions of the assembly, such as deformation be-cause of force and temperature, and variation of joint due to

gap between 3D models and reality. F shown in Fig. 13 de-pends on the variables , R, t and H , and the transfer routewhich passes point a or point b or both. Most often, a se-ries route passing one of two points is selected for tolerancepropagation. However, there are two routes that participatein tolerance propagation actually. It is complicated for parallelconnections because there are interactions between them.

(5) The sensitivity and percent contribution of tolerance are veryuseful for tolerance optimization, especially in a statisticalcase. Some of 3D tolerance analysis methods, such as T-Mapposition but also orientation of the analytic objective clearly in 3Dspace; (2) the variations caused by selection of different points areavoided.

4. Conclusion

The discussion and comparison have been given in the previoussection where our subjective judgment comes into play. More

propagation. Meanwhile, the role of form tolerance heavily de-pends on the users expertise and experience, which is illus-trated in DLM. Approximately, a runout tolerance can be seenas the combination of form tolerance and positional tolerance,which is more complex than form tolerance.

(4) The existing 3D models focus on connections in series mainly.The solution of tolerance representation and propagation forparallel connections in the assembly will greatly reduce theH. Chen et al. / Computer-Ai

Table 2Comparisons of four models.

Worst case Statistical case Sensitivity andpercent contribu

w w

T-Map

Matrix

Unified JacobianTorsor

DLM

Table 3Comparisons of the results of four models with a worst case.

Results

T-Map t1 + t2 + t11 ((2b+ d2)/d1 1)Matrix t1 + t2 + 2b cos(t11/d1) sin(t11/Unified JacobianTorsor t1 + t2 + 2bt11/d1DLM t1 + t2 cos(t11/d1)+ 2(b cos(t11

b > (d1 d2)/2 is considered. It should be noted that the result ofthe matrix model is obtained by ignoring the constraint inequal-ities because the symbolic parameters impede the optimization.Residual gaps between 3D and 2D which is t1 + t2 are approxi-mated and simplified. As can be seen, the residuals reflect the prop-erty of 3D tolerance which takes the structure of assembly and thegeometric tolerance into consideration. This is a significant differ-ence of tolerance analysis between 3D and traditional 1/2D meth-ods. The result of T-Map depends on not only the shapes of targetsurface but also the diameters of bars in the assembly. Because therotational variation aroused by the orientation tolerance t11 is verysmall, the results of the matrix model and DLM can be approxi-mated and simplified as((t1+t2)/2+bt11/d1), which is the resultof the unified JacobianTorsor model.

The rotational displacement can be obtained by the T-Map model and the unified JacobianTorsor model because thesetwo models are surface-based approaches. More specifically, theaccumulation maps illustrated in Fig. 7 express both translationaland rotational variations of functional surface. Similarly, Eq. (21)of the unified JacobianTorsor model includes all of translationaland rotational variations of functional surface. However, the resultof the matrix model is the variation of point a on functionalsurface. The pose of a point in space is described by positionrather than orientation. If another point of functional surface isselected as the target objective, the resultmay be different. So doesthe DLM. Compared with point-based approaches, the advantagesof surface-based models include: (1) the results reflect not onlylubricating medium, need to be taken into account to improvethe reliability. The combustion engine is a typical example,ded Design 53 (2014) 113 11

tionGeometric tolerance Objective Application (CATs)

Form Orientation Position

Residual between 3D and 2D

2bt11/d1 + (d2/d1 1)t111)+ 2l2 (cos(t11/d1) sin(t11/d1) 1) 2bt11/d1

2bt11/d1d1) l2 sin(t11/d1)) t11/d1 2bt11/d1

which endures high temperature and high pressure and forcedliquid lubrication when it works. There is no doubt that the ge-ometric feature and joint (clearance) are different to the staticcondition. Pierre et al. [94] integrate the thermomechanicalstrains into the SDTmodel and Zhang et al. [95] take the work-ing condition into the unified JacobianTorsor model, whichcan be seen as the preliminary work. The interaction betweenthe temperature and force and lubrication aggravates the dif-ficulties of the analysis process.

(2) The constraints between translational and rotational vectors of3D tolerance analysis models should be optimized further inorder to conform to the tolerancing standards better and en-hance the accuracy, because they depend on each other in thetolerance zone. According to the envelope principle, and must shrink to zero when u arrives at its limited position for aplanar feature in the tolerance zone, as shown in Fig. 8. There-fore, it should bewrong to take both the limited values of trans-lation and rotation into computation simultaneously. The uni-fied JacobianTorsor model and DLM cannot deal with theseconstraints. Other 3Dmodels which are not discussed in detailin this paper also need to consider the constraints.

(3) The mathematic models of tolerance analysis, especially thetolerance propagation for form tolerance and runout tolerancewhich includes circular runout and total runout, still need alarge deal of research. The position and orientation of form tol-erance are random in a tolerance zone (see Fig. 5(a)), whichmeans the form tolerance is not deterministic in toleranceand the matrix model, still lack suitable algorithm to calcu-late them. Moreover, for a feature specified by more than one


tolerance, the algorithm for separating and calculating each ofthem has not been presented by far.

Acknowledgments

The work described in this paper is supported in part bygrants from the National Natural Science Foundation of China(Grant Nos. 51121063, 51175340) and the National Science &Technology Pillar Program during the 12th Five-year Plan Period(Grant No. 2012BAF06B03). The authors are grateful for thesefinancial supports.

References

[1] Dizioglu B, Lakshiminarayana K. Mechanics of form closure. Acta Mech 1984;52:10718.

[2] Takezawa N. An improved method for establishing the process wise qualitystandard. Rep Stat Appl Res Union Japan Sci Eng (JUSE) 1980;27(3):6376.

[3] Liu SC, Hu SJ. An offset element and its application in predicting sheet metalassembly variation. Int J Mach Tools Manuf 1995;35(11):154557.

[4] Liu SC, Hu SJ. Variation simulation for deformable sheet metal assembly usingfinite element methods. Trans ASME, J Manuf Sci Eng 1997;119:36874.

[5] Chase KW, Magleby SP, Gao J. Tolerance analysis of 2-D and 3-D mechanicalassemblies with small kinematic adjustments. In: Zhang HC, editor. Advancedtolerancing techniques. New York: John Wiley & Sons; 1997. p. 10337.

[6] Prisco U, Giorleo G. Overview of current CAT systems. Integr Comput-AidedEng 2002;9:37387.

[7] Salomons OW, Houten FJAM, Kals HJJ. Current status of CAT systems.In: ElMaraghy HA, editor. Geometric design tolerancing: theories, standardsand applications. London: Chapman & Hall; 1998. p. 43852.

[8] Shen Z. Tolerance analysis with EDS/VisVSA. ASME J Comput Inf Sci Eng 2003;3:959.

[9] Islam MN. Functional dimensioning and tolerancing software for concurrentengineering applications. Comput Ind 2004;54:16990.

[10] Hillyard RC, Braid IC. Analysis of dimensions and tolerances in computer-aidedmechanical design. Comput-Aided Des 1978;10(3):1616.

[11] Lin VC, Light RA, Gossard DC. Variational geometry in computer-aided design.Comput Graph 1981;15(3):1717.

[12] Light RA, Gossard DC. Modification of geometric models through variationalgeometry. Comput-Aided Des 1982;14(4):20914.

[13] Requicha AAG. Toward a theory of geometric tolerancing. Int J Robot Res 1983;2(4):4560.

[14] Requicha AAG. Representation of tolerances in solid modeling: issues andalternative approaches. In: Pickett MS, Boyse WJ, editors. Solid modelingby computers: from theory to applications. New York: Plenum Press; 1984.p. 322.

[15] Jayaraman R, Srinivasan V. Geometric tolerancing: I. Virtual boundaryrequirement. IBM J Res Dev 1989;33(2):90104.

[16] Srinivasan V, Jayaraman R. Geometric tolerancing: II. Conditional tolerance.IBM J Res Dev 1989;33(2):10522.

[17] Turner JU, Wozny MJ. A mathematical theory of tolerance. In: Wozny MJ,Mclaughlim JL, editors. Geometric modeling for CAD applications. ElsevierScience Publishers, IFIP; 1988. p. 16387.

[18] Turner JU. A feasibility space approach for automated tolerancing. Trans ASME,J Eng Ind 1993;115(3):3416.

[19] Wirtz A. Vectorial tolerancing a basic element for quality control. In:Proceedings of the 3rd CIRP seminar on computer aided tolerancing. 1993.p. 11528.

[20] Laperrire L, Lafond P. Modeling tolerances and dispersions of mechanicalassemblies using virtual joints. In: CD-ROM proceedings of 25th ASME designautomation conference. 1999.

[21] Bernstein N, Preiss K. Representation of tolerance information in solidmodels.In: Proceedings of 15th ASME design automation conference, September 1989.DE-Vol. 19-1. p. 478.

[22] Zhang BC. Geometric modeling of dimensioning and tolerancing. Ph.D. Thesis.Arizona State University. 1992.

[23] Kramer GA. Solving geometric constraint systems: a case study in kinematics.MIT Press; 1992.

[24] Davidson JK, Mujezinovi A, Shah JJ. A newmathematical model for geometrictolerances as applied to round faces. ASME J Mech Des 2002;124:60922.

[25] Mujezinovi A, Davidson JK, Shah JJ. A newmathematical model for geometrictolerances as applied to polygonal faces. ASME J Mech Des 2004;126:50418.

[26] Desrochers A, Clment A. A dimensioning and tolerancing assistancemodel forCAD/CAM systems. Int J Adv Manuf Technol 1994;9:35261.

[27] Portman VT. Modelling spatial dimensional chains for CAD/CAM applications.In: F. Kimura, editor. Proceedings of the 4th CIRP design seminar on computer-aided tolerancing. 1995. p. 7185.

[28] Cardew-Hall MJ, Labans T, West G, Dench P. A method of representingdimensions and tolerances on solid based freeform surfaces. Robot Comput-

Aided Manuf 1993;10:22334.

[29] Desrochers A, Rivire A. A matrix approach to the representation of tolerancezones and clearances. Int J Adv Manuf Technol 1997;13:6306.ded Design 53 (2014) 113

[30] Whitney DE, Gilbert OL, Jastrzebski M. Representation of geometric variationsusing matrix transforms for statistical tolerance analysis in assemblies. ResEng Des 1994;6(4):191210.

[31] Clment A, Desrochers A, Rivire A. Theory and practice of 3D tolerancing forassembly. In: Proceedings of the CIRP seminar on computer aided tolerancing.USA: Penn State University; 1991.

[32] Clment A, Rivire A. Tolerancing versus nominal modeling in next generationCAD/CAM system. In: Proceedings of the CIRP seminar on computer aidedtolerancing. 1993. p. 97113.

[33] Teissandier D, Coutard Y, Grard A. Three-dimensional functional tolerancingwith proportioned assemblies clearance volume (U.P.E.L: Unions pondresdspaces de libert). In: Proceedings of the 1996 engineering systems designand analysis conference presented at the 3rd Biennial joint conference onengineering systems design and analysis, ESDA (ASME, Petroleum Division),PD-Vol. 80.8. P. 12936.

[34] Teissandier D, Coutard Y, Grard A. A computer aided tolerancing model:proportioned assembly clearance volume. Comput-Aided Des 1999;31(3):80517.

[35] Fortini ET. Dimensioning for interchangeable manufacture. New York:Industrial Press; 1967.

[36] Evans DH. Statistical tolerancing: state of the art, part 2.Methods of estimatingmoments. J Qual Technol 1975;7(1):112.

[37] Evans DH. Statistical tolerancing: state of the art, part 3. Shift and drifts. J QualTechnol 1975;7(2):716.

[38] Evans DH. An application of numerical integration techniques to statisticaltolerancing. Technimetrics 1967;9(3).

[39] Evans DH. An application of numerical integration techniques to statisticaltolerancing, IIa note on the error. Technimetrics 1971;13(2).

[40] Evans DH. An application of numerical integration techniques to statisticaltolerancing, IIIgeneral distributions. Technimetrics 1972;14(1).

[41] Parkinson DB. The application of reliability method to tolerancing. ASME JMech Des 1982;104:6128.

[42] Parkinson DB. Reliability indices employingmeasures of curvature. Reliab Eng1983;15379.

[43] Taguchi G. Performance analysis design. Int J Prod Des 1978;16:52130.[44] DErrico JR, Zaino NA. Statistical tolerancing using a modification of Taguchis

method. Technometrics 1988;30(4):397405.[45] DeDoncker D, Spencer A. Assembly tolerance analysis with simulation and

optimization techniques. SAE Trans 1987;96(1):10627.[46] Grossman DD. Monte Carlo simulation of tolerancing in discrete parts

manufacturing and assembly. Technical report STAN-CS-76-555. ComputerScience Dep., Stanford University, USA; 1976.

[47] Fleming A. Geometric relationships between toleranced features. Artif Intell1988;37:40312.

[48] Rivest L, Fortin C, Morel C. Tolerancing a solid model with a kinematicformulation. Comput-Aided Des 1994;26:46576.

[49] Chase KW, Gao J, Magleby SP. General 2-D tolerance analysis of mechanicalassemblies with small kinematic adjustments. J DesManuf 1995;5(4):26374.

[50] Gao J, Chase KW, Magleby SP. General 3-D tolerance analysis of mechanicalassemblies with small kinematic adjustments. IIE Trans 1998;30:36777.

[51] Lafond P, Laperrire L. Jacobian-based modeling of dispersions affectingpredefined functional requirements ofmechanical assemblies. In: Proceedingsof IEEE international symposium on assembly and task planning. 1999.

[52] Laperrire L, EIMaraghy HA. Tolerance analysis and synthesis using Jacobiantransforms. CIRP AnnManuf Technol 2000;49(1):35962.

[53] Jin J, Shi J. State space modeling of sheet metal assembly for dimensionalcontrol. Trans ASME, J Manuf Sci Eng 1999;121:75662.

[54] Mantripragada M, Whitney DE. Modeling and controlling variation propaga-tion in mechanical assemblies using state transition models. IEEE Trans RobotAutom 1999;15(1):12440.

[55] Cai W, Hu SJ, Yuan JX. A variational method of robust fixture configurationdesign for 3D workpieces. Trans ASME, J Manuf Sci Eng 1997;119(4A):593602.

[56] Desrochers A. A CAD/CAM representation model applied to tolerance transfermethods. ASME J Mech Des 2003;125:1422.

[57] ASME Y14.5-2009 dimensioning and tolerancing. New York: ASME; 2009.[58] ISO 1101: Geometric Product Specifications (GPS)-Geometrical toleranc-

ingtolerancing of form, orientation, location and run-out. 2004.[59] Hong YS, Chang TC. A comprehensive review of tolerancing research. Int J Prod

Res 2002;40(11):242559.[60] Zhu D, Cheng HS, Arai T, Hamai K. A numerical analysis for piston skirts in

mixed lubrication-part I: basic modeling. ASME J Tribol 1992;114:55362.[61] Zhu D, Hu YZ, Cheng HS, Arai T, Hamai K. A numerical analysis for piston skirts

in mixed lubrication-part II: deformation considerations. ASME J Tribol 1993;115:12533.

[62] Desrochers A, Ghie W, Laperrire L. Application of a unified JacobianTorsormodel for tolerance analysis. ASME J Comput Inf Sci Eng 2003;3:113.

[63] Marziale M, Polini W. A review of two models for tolerance analysis of anassembly: vector loop and matrix. Int J Adv Manuf Technol 2009;43:110623.

[64] Laperrire L, Desrochers A. Modeling assembly quality requirements usingJacobian or screw transforms: a comparison. In: Proceedings of the 4th IEEEinternational symposium on assembly and task planning, Soft Research Park.May 2011. p. 3306.

[65] Marziale M, Polini W. A review of two models for tolerance analysis of anassembly: Jacobian and Torsor. Int J Comput Integr Manuf 2011;24(1):7486.[66] Ameta G, Serge S, Giordano M. Comparison of spatial math models fortolerance analysis: tolerance-maps, deviation, domain, and TTRS. ASME JComput Inf Sci Eng 2011;11: 012004-18.


[67] Shah JJ, Ameta G, Shen Z, Davidson J. Navigating the tolerance analysis maze.Comput-Aided Des Appl 2007;4(5):70518.

[68] Salomons OW, Poerink HJJ, Haalboom FJ, Slooten FV, et al. A computer aidedtolerancing tool I: tolerance specification. Comput Ind 1996;31:16174.

[69] Salomons OW, Haalboom FJ, Poerink HJJ, Slooten FV, et al. A computer aidedtolerancing tool II: tolerance analysis. Comput Ind 1996;31:17586.

[70] Shen Z, Ameta G, Shah JJ, Davidson JK. A comparative study of toleranceanalysis method. ASME J Comput Inf Sci Eng 2005;5:24756.

[71] Davidson JK, Shah JJ. Geometric tolerances: a new application for linegeometry and screws. In: Proceedings of A symposium commemorating thelegacy, works, and life of Sir Robert Stawell ball upon the 100th anniversary ofa treatise on the theory of screws. University of Cambridge; 2000.

[72] Bhide S, Davidson JK, Shah JJ. A new mathematical model for geometrictolerances as applied to axes. In: Proceedings of DETC03, ASME 2003design engineering technical conference and computers and information inengineering conference. September 2003. p. 19.

[73] Bhide S, Ameta G, Davidson JK, Shah JJ. Tolerance-maps applied to thestraightness and orientation of an axis. In: Models for computer aidedtolerancing in design and manufacturing. 2007. p. 4554.

[74] Ameta G, Davidson JK, Shah JJ. The effects of different specifications onthe tolerance-maps for an angled face. In: Proceedings of ASME/DETC-2004.September, 57199.

[75] Ameta G, Davidson JK, Shah JJ. Tolerance-maps applied to a point-line clusterof features. ASME J Mech Des 2007;129:78292.

[76] Ameta G, Davidson JK, Shah JJ. Generate frequency distributions of clearanceand allocate tolerances for pin-hole assemblies. ASME J Comput Inf Sci Eng2007;7:34759.

[77] Ameta G, Davidson JK, Shah JJ. Influence of form on tolerance-map-generatedfrequency distributions for 1D clearance in design. Precis Eng 2010;34:227.

[78] Ameta G, Davidson JK, Shah JJ. Form tolerance on the frequency distributionsof clearance between two planar faces. ASME J Comput Inf Sci Eng 2011;11:011002-110.

[79] Herv JM. Analysis structurelle desmcanismes par groupe des dplacements.Mech Mach Theory 1978;13:43750.

[80] Desrochers A, Delbart O. Determination of part position uncertainty withinmechanical assembly using screw parameters. In: EIMaraghy HA, editor.Geometric design tolerancing: theories, standards and applications. 1998.p. 18796.ded Design 53 (2014) 113 13

[81] Clment A, Rivire A, Serr P, Valade C. The TTRSs: 13 constraints fordimensioning and tolerancing. In: EIMaraghy HA, editor. Geometric designtolerancing: theories, standards and applications. 1998. p. 12231.

[82] Clment A, Rivire A, Serr P. A declarative information model for functionalrequirements. In: Computer-aided tolerancing. 1996. p. 316.

[83] Salomons OW. Computer support in the design of mechanical products. Ph.D.Thesis. University of Twente, The Netherlands; 1995. p. 21142.

[84] Clment A, Bourdet P. A study of optimal-criteria identification based on thesmall-displacement screw model. CIRP Ann 1988;37(1):5036.

[85] Bourdet P, Marhieu L, Lartigue C, Ballu A. The concept of the small displace-ments torsor in metrology. In: International euroconference, advanced math-ematical tools in metrology. 1995.

[86] Desrochers A. Modeling three dimensional tolerance zones using screwparameters. In: Proceedings DETC: 25th design automation conference.September 1999. p. 895903.

[87] Laperrire L, Ghie W, Desrochers A. Statistical and deterministic toleranceanalysis and synthesis using a unified JacobianTorsor model. CIRP AnnManuf Technol 2002;51(1):41720.

[88] Ghie W. Statistical analysis tolerance using Jacobian Torsor model based onuncertainty propagation method. Int J Multiphys 2009;3(1):1130.

[89] Ghie W, Laperrire L, Desrochers A. Statistical tolerance analysis using theunified JacobianTorsor model. Int J Prod Res 2010;48(15):460930.

[90] Ghie W, Laperrire L, Desrochers A. Re-design of mechanical assembly usingthe unified JacobianTorsor model for tolerance analysis. In: Models forcomputer aided tolerancing in design and manufacturing. 2007. p. 95104.

[91] Desrochers A. Geometrical variations management in a multi-disciplinaryenvironment with the JacobianTorsor model. In: Models for computer aidedtolerancing in design and manufacturing. 2007. p. 7584.

[92] Chase KW, Gao J, Magleby SP, Sorensen CD. Including geometric featurevariations in tolerance analysis ofmechanical assembly. IIE Trans 1996;28(10):795807.

[93] Glancy CG, Chase KW. A second order method for assembly tolerance analysis.In: Proceedings of 1999 ASME design engineering technical conferences. 1999.DETC99/DAC-8707.

[94] Pierre L, Teissandier D, Nadeau JP. Integration of thermomechanical strainsinto tolerancing analysis. Int J Interact Des Manuf 2009;3:24763.

[95] Zhang W, Chen C, Li P, et al. Tolerance modelling in actual working conditionbased on JacobianTorsor theory. Comput-Integr Manuf Syst 2011;17(1):7783.

A comprehensive study of three dimensional tolerance analysis methodsIntroduction3D tolerance analysis modelsT-Map (Tolerance-Map) modelMatrix modelUnified Jacobian--Torsor modelDLM method

Discussion and comparisonConclusionAcknowledgmentsReferences

a comprehensive study of three dimensional tolerance analysis methods

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