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ELSEVIER Procedia CIRP 43 (2016) 172 - 177

www.elsevier.corr/locate/procedia

14th CIRP Conference on Computer Aided Tolerancing (CAT)

A quantitative Comparison of Tolerance Analysis Approaches for Rigid Mechanical Assemblies

Benjamin Schleich®-*, Sandro Wartzacka

aChair of Engineering Design KTmfk, Friedrich-Alexander-University Erlangen-Nürnberg, Martensstraße 9, 91058 Erlangen, Germany * Corresponding author. Tel.: +49-(0)9131-85-23220; fax: +49-(0)9131-85-23222. E-mail address: schleich@mfk.fau.de

Abstract

Tolerance analysis is a key tool to predict the consequences of geometric variations on product quality. During the last decades, various approaches for the computer-aided tolerance analysis have been proposed, where each of them has specific advantages and disadvantages. In this contribution, three tolerance analysis approaches, namely tolerance stacks, vector loops, and the tolerance analysis based on the Small Displacement Torsor are quantitatively compared with the tolerance analysis based on Skin Model Shapes considering a typical case study. The novelty of the contribution lies in the profound assessment of these approaches and their results.

© 2016 The Authors.Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the organizing committee of the 14th CIRP Conference on Computer Aided Tolerancing

Keywords: Tolerance Analysis; Variation Simulation; Skin Model Shapes.

1. Introduction and Motivation

The presence of geometric part deviations is ubiquitous throughout the product life-cycle from manufacturing, to assembly, inspection, and product usage [1]. Since these geometric part deviations distinctly affect the function and quality of mechanical products, there is a strong necessity for companies to manage these geometric variations. In this regard, geometric dimensioning and tolerancing (GD&T) activities aim at specifying limits for such geometric part deviations in order to ensure the product function and to meet quality goals.

In this context, tolerance analysis is a key tool for product and process developers to predict the effects of inevitable part deviations on functional key characteristics of mechanical assemblies and to assess the consequences of variation on product quality [2,3]. During the last decades, various approaches for the computer-aided tolerance analysis have been proposed, where each of them has specific advantages and disadvantages. However, most of these approaches imply shortcomings, such as the missing consideration of form deviations and the incomplete conformance to international standards for the geometric product specification and verification (GPS). With the aim to overcome these shortcomings, the concept of Skin Model Shapes as a novel approach for modelling product shape variability and for the computer-aided tolerance analysis has been developed [4,5]. It is based on the Skin Model as a fundamental concept of modern GPS standards [6] and employs discrete ge-

ometry schemes, such as surface meshes and point clouds, for the virtual representation of deviated workpieces.

The aim of this contribution is the quantitative assessment of this novel approach for the tolerance analysis in comparison to established tolerance analysis methods, where the focus is laid upon rigid mechanical assemblies. This comparison is performed employing a typical tolerance analysis case study.

2. State of the Art and Related Work

Tolerancing aims at specifying allowable limits for geometric part deviations, which inevitably result from manufacturing imprecisions [7], to ensure the product assemblability and functional requirements [2,8]. In this context, tolerance analysis is a key tool to predict the effects of geometric part deviations on assembly characteristics without the need for physical mock-ups, where "the objective of tolerance analysis is to check the extent and nature of the variation of an analyzed dimension or geometric feature of interest for a given GD&T scheme" [9].

Three main issues in tolerancing research regarding the tolerance analysis are to establish mathematical models for the expression and representation of geometric deviations, geometric specifications, and geometric requirements, to model the effects of these geometric deviations on the assembly and the system behaviour, and to provide solution techniques for these models, such as worst-case or statistical evaluations [10]. During the

2212-8271 © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the organizing committee of the 14th CIRP Conference on Computer Aided Tolerancing doi: 10.1016/j.procir.2016.02.013

last decades, various approaches have been proposed in order to solve these issues, which can roughly be classified as deviation accumulation methods, where the functional key characteristic is expressed as a function of geometric part deviations, and tolerance accumulation approaches, where the tolerance zones to be analysed are expressed as subsets of multidimensional spaces, accumulated (using Minkowski sum and intersection), and compared to the functional subset in the multidimensional space [2]. For both of these categories, several approaches have been proposed, such as parametric tolerance analysis [11], simple tolerance stacks [9], solid offsets [12], vector loops [13], and based thereon the direct linearization method [14], which are deviation accumulation methods, and the tolerance analysis based on the Small Displacement Torsor [15,16], Tolerance-Maps'® [17], deviation domains [18] and their expression by polytopes [19] as well as their use for the formulation of the tolerance analysis issue in quantifier notion [2], which are tolerance accumulation approaches.

These approaches are also the basis for tolerance analysis software [9,20], such as MECAMaster®, which is based on the SDT, 3DCS®, VSA®, and CeTol®, which use parametric approaches (CeTol® used vector loops in former versions), and PolitoCAT®, which employs polytopes.

Furthermore, a considerable number of review papers highlighting the similarities and differences of the aforementioned approaches exist, such as [9,11,17,21-24]. Based on these works, it can be found, that most of the proposed tolerance analysis methods are not capable of holistically considering form deviations, are consequently not fully conform to tolerancing standards and imply shortcomings regarding the combination of 3D tolerance zones, envelope and independence principles, material condition modifiers, and datum precedence.

In contrast to these established approaches, the concept of Skin Model Shapes [4,5] employs discrete geometry representation schemes, such as point clouds and surface meshes, for the representation of parts and assemblies considering all kinds of geometric deviations and grounds on the Skin Model as a fundamental concept of modern GPS standards.

3. Overview of considered Tolerance Analysis Approaches

As it has been argued, various approaches for the computer-aided tolerance analysis have been proposed during the last decades, where three major approaches are tolerance stacks, vector loops, and the tolerance analysis based on the Small Displacement Torsor (SDT). In contrast to these approaches, the tolerance analysis employing discrete geometry representations of deviated workpieces considering form deviations is a novel method. In the following, these four approaches are briefly highlighted before they are applied to a typical case study.

Tolerance Stacks. Tolerance stacks are a simple and straightforward approach to model the effects of part deviations on distances between different part features in an assembly. In this regard, tolerance stacks include most often only dimensional tolerances, though modern modifications of this method also consider geometric tolerances [9]. The procedure of performing the tolerance analysis based on tolerance stacks comprises firstly the definition of a stack coordinate system, secondly the identification of the stack path and the formulation of the stack

equation, and finally the evaluation of the stack equation using worst-case or Monte-Carlo methods [9].

Vector Loops. In contrast to tolerance stacks, where traditionally only dimensional part deviations are modelled, vector loops also consider geometric part deviations and kinematic variations [13]. The mechanical assembly is modelled as a loop of vectors, where each of these vectors represents an assembly dimension, which in turn may be either a dimensional part deviation, a geometric part deviation, or a kinematic variation. In this regard, geometric part deviations are only considered as the effect they may have on mating points between parts and kinematic variations denote the kinematic effects of dimensional and geometric part deviations on the mating parts [24].

The different steps for performing a tolerance analysis using the vector loop model involve the creation of the assembly graph, the definition of datum reference frames for the different parts, the definition of kinematic joints and the creation of datum paths, the identification of vector loops, and the derivation of stack-up equations [24].

Small Displacement Torsor. The Small Displacement Torsor (SDT) describes the displacement of a geometric element by a translation vector and a linearised rotation matrix written as a three-element rotation vector [15], i. e. the SDT t is given as t = (t w) with t, w e K,3x1. With this, the displacement Ap of a point p is expressed as:

A p = t + w x p (1)

where t is the vector of translations, i. e. t = [tx ty tzJ, w is the vector of rotations, i. e. w = [a/3y], and x is the cross-product.

Thus, the SDT can be used to express the displacement of each part in an assembly, leading to the part SDT, the displacement of points on a toleranced feature, leading to the deviation SDT, and the relative displacement between two parts, leading to the gap SDT [15]. As the allowable displacements of each point on a toleranced feature are constrained by the respective tolerance zone(s), inequations between the entries of the deviation SDT and the respective tolerances can be formulated, which leads to the concept of deviation domains [18]. These constraints and consequently the boundaries of the deviation domains are in general not linear [19]. However, there are approaches to express the deviation domains by polytopes and thus to replace the non-linear constraints by sets of linear constraints. For more details, the reader is referred to [18,19].

The tolerance analysis employing the SDT concept is performed by propagating the different SDTs using Minkowski sums and intersections to obtain the possible deviations of a feature or point of interest [16,19].

Skin Model Shapes. In contrast to the aforementioned approaches, the tolerance analysis based on Skin Model Shapes (SMS) is a novel method, which allows the consideration of form deviations and is conform to international standards for the GPS. Skin Model Shapes are specific outcomes of the Skin Model [6] as a basic concept of modern standards for the geometric product specification and verification employing discrete geometry representation schemes, such as point clouds and surface meshes [4]. The tolerance analysis based on Skin Model Shapes [25] comprises the generation and scaling of deviated workpiece representatives according to specified toler-

ances [26], their processing using computational geometry algorithms for the relative positioning and assembly simulation [27], as well as the measurement of functional key characteristics (FKC) from the simulated assemblies (see Fig. 1).

Fig. 1. The Tolerance Analysis based on Skin Model Shapes

4. Case Study

In order to provide a quantitative comparison between the different tolerance analysis approaches, a typical case study consisting of four parts as shown in Fig. 2 is analysed. The cubes are subsequently assembled on the clip, where a three-point-move is applied in negative z-direction and a two-point-move is performed in negative x-direction resulting in three contact points between the respective cube and the clip and two contact points between the respective cube and the previous cube or clip, respectively. The functional key characteristics are the position variation pos of the feature of interest with reference to the datums A and B on the clip and its parallelism variation par with reference to A and B. In this regard, par comprises only the orientation defects of the feature of interest with reference to the datum system and pos comprises also the location defects (see Fig. 3).

Tolerance Analysis using Tolerance Stacks. The tolerance analysis based on tolerance stacks starts with the definition of a stack coordinate system and the formulation of the stack path, which can be seen from Fig. 4. The stack equation yields to:

l = lo + l1 + l2 + l3 (2)

The position deviation pos of every assembly can then be calculated from its actual length l and the nominal length l by:

= 2 • l - l

Since tolerance stacks traditionally do not consider geometric part deviations, the position tolerances given in Fig. 2 ((3) and (7)) are converted to dimensional part deviations, where each of the position tolerances is interpreted as a dimensional tolerance lying symmetrically around the nominal part dimension, leading to l0 e 10.0 ± 1.0/2, l1 e 50.0 ± 1.0/2, l2 e 70.0 ± 1.0/2, and l3 e 30.0 ± 1.0/2. The stack equation is then used to determine the worst-case and it is evaluated statistically, where the results are discussed in the last paragraph.

Tolerance Analysis using Vector Loops. In contrast to the tolerance analysis based on tolerance stacks, gaps between the parts due to their geometric deviations are considered in the vector loop approach. Thus, with the gaps g;^;+1 between the parts i and i + 1, the vector loop (1D) is obtained as (see Fig. 5):

l* = gStart^0 + l0 + g0^1 + l\ + g1^2 + l2 +g2^3 + l3 + g3^End

The gaps g;^;+1 depend on the geometric deviations of the mating parts, which result from the perpendicularity tolerances ((1), (5)) and the parallelism tolerances ((2), (6)). In this regard, each of these tolerances lead to a rotation around the y-axis of the corresponding feature. From these rotations around the y-axis of the mating features and the part heights , the gap between each two parts can be computed by:

gi^i+1 = 0.5 •|(hi+1 • A+1 - hi • A)|, (5)

where 6 e [-t(2'6)/h;; t(2'6)/h;] is the orientation defect of part i around the y-axis due to the parallelism tolerance t(2,6) and yS;+1 e [-t(1'5)/hi+1; t(1'5)/hi+1] the orientation defect around the y-axis of part i+1 due to the perpendicularity tolerance t(1'5)(see Fig. 6). Thus, depending on the rotations of the mating features, the gap between two parts takes a value between min(g;^;+1) = 0 (when 6 and yS;+1 balance out) and max(g;^;+1) = 0.5 • (t(2'6) + t(1,5)), whereas the gap gSt^ = t(1,5)/2 and g3^End = t(2,6)/2.

In contrast to the orientation tolerances, the position tolerances ((3) and (7)) are considered to have no effect on the feature rotations and hence on the gaps, but only on the part dimensions l*. However, it has to be noticed, that the parallelism tolerances ((2),(6)) are covered by the position tolerances ((3),(7)) and that the perpendicularity tolerances ((1),(5)) have an effect on the effective part length. This can be seen from Fig. 6, where it can be found, that the part length l* results from the total part length li as l* e (l; - t(1'5)/2) ± (t(3'7) - t(2'6)) with t(3'7) being the value of the position tolerance ((3),(7)). Hence, the l* result to r0 e 9.75 ± 0.5/2, lj e 49.75 ± 0.5/2, l*2 e 69.75 ± (0.5/2, and

l30 e 29.75 ± 0.5/2. 1 2

Similarly to the tolerance analysis based on tolerance stacks, the length of the vector loop of an assembly in equation (4) leads to the position deviation pos = 2 -|l* - l|.

Tolerance Analysis employing the SDT. As it has been mentioned, the tolerance analysis employing the SDT requires firstly the expression of the specified tolerances as constraints on the components of the deviation SDT for toleranced features and then their propagation using Minkowksi sums and intersections. By doing so, the domain of possible deviations of a feature or point of interest is obtained. For this case study, a focus is laid upon the deviations of the feature of interest regarding the rotations around the y- and z-axis (6 and y) to finally calculate the parallelism deviation par. In order to perform this, the tolerances leading to rotational feature defects are expressed as deviation domains considering the part positioning scheme.

In this regard, the perpendicularity tolerances ((1) and (5)) influence the feature rotations of the respective features around the y-axis (6), whereas the parallelism tolerances ((2) and (6)) influence the feature rotations around the y- and the z-axis (6 and y). However, according to the positioning scheme, the feature rotations around the y-axis (6) of the preceding parts and features manifest in gaps between the parts without having an effect on the rotational feature deviations of the feature of interest (the feature rotation of the feature of interest around the y-axis is solely influenced by the parallelism tolerance (4)). Thus, the perpendicularity tolerances ((1) and (5)) have no effect on the rotational feature deviations of the feature of interest around the analysed axes. In contrast to that, the parallelism tolerances ((2) and (6)) affect the rotation of the feature of interest around the z-direction, where the deviation domains of the parallelism tolerances ((2) and (6)) can be simplified as can seen from Fig.

Z7 0.1

_L 0.5

H3 [10]

Œ3 G30]

// par A B

0 pos A B

Feature of In 10

// 0.5 A B (2) D 0. 1

0 1.0 A B (3) // 0. 5 A |(4)

HP- /

// 0.5 A B

0 1.0 A B

Fig. 2. Case Study

7. Moreover, the rotation of the feature of interest around the y-axis is solely influenced by the parallelism tolerance (4). However, as the effect of the parallelism tolerance (4) on the feature rotation around the x-axis is not analysed, the deviation domain of tolerance (4) can be simplified as can be seen from Fig. 7.

As the deviation domains of all relevant tolerances have been identified, they can be propagated to obtain the deviation domain of the feature of interest with respect to the 3 and y deviations, which can be seen from Fig. 8.

Fig. 5. Vector Loop (1D) of the Assembly

Fig. 6. Conversion of the Position Tolerances to Part Lengths for the Vector Loop Approach

Fig. 3. Explanation of the Functional Key Characteristics of the Case Study

lo h l2 l3

Fig. 4. Tolerance Stack of the Assembly

Tolerance Analysis by Skin Model Shapes. In order to analyse the effects of part tolerances on the assembly behaviour employing Skin Model Shapes, deviated surface meshes of the single parts are generated, which are then scaled to be conform to the specified tolerances. Thereafter, these part representatives are assembled according to the positioning scheme. Finally, the key characteristics are measured from the obtained assemblies. Fig. 9 shows an exemplary assembly with coarsened mesh and

magnified form deviations. The nominal surface meshes of the parts have been generated using proprietary finite-element software, where mesh refinement has been applied to the mating surfaces (see Fig. 10).

Results of the Tolerance Analysis Approaches. The tolerance analysis approaches have been used to determine the FKCs of the study case, where worst-case as well as statistical evaluations of the tolerance analysis models have been performed. Table 1 highlights the considered tolerances in each approach.

Table 1. Considered Tolerances in the different Tolerance Analysis Approaches.

Approach Tolerances Explanation

Stacks (3),(7) Conversion to Dim. Tolerances

Vector Loop (1)-(3),(5)-(7) Conversion to Gaps & Dim. Tolerances

SDT (1),(2),(4)-(6) Evaluation of Orientation Defects

SMS (1)-(7) (+ Form) With & without Form Tolerances

-0.010 0.00 0.010 -0.005 0 0.005 -0.005 0 0.005

Fig. 7. Simplification of the Deviation Domains: Left: Simplification of the Parallelism Tolerances ((2),(6)) - as the Feature Rotations around the y-axis (0) manifest in Gaps between the Parts, only the Rotations around the ^-axis (7) have to be considered. Right: Resulting Effect of the Parallelism Tolerance of the Clip (4) - since only the Feature Rotations around the y- and ^-axis are considered, the Deviation Domain of the Parallelism Tolerance (4) is simplified.

Cube 1

Resulting Domain

ß ß ß Cube 2 (6) Cube 3

Fig. 8. Relevant Feature Deviation Domains and resulting Deviation Domain (0 and y) for the Orientation Defects of the Feature of Interest

In this regard, the worst-case results for the tolerance analysis using tolerance stacks can be calculated as min(l) = 23=oimnft) = 158;max(l) = £3=0max(l;) = 162 and consequently max(pos) = 2 • max(|l-l|) = 4 with l = 160. In contrast to that, the worst-case limits for the length of the vector loop in equation (4) result as min(l') = 158.5 and max(l') = 162.0 and hence again max(pos) = 4.0. The difference in the minimum length between the tolerance stack and the vector loop arises from the joint consideration of orientation and location tolerances. In contrast to that, the tolerance analysis based on Skin Model Shapes without consideration of form deviations gives max(pos) = 4.26. This is because also the effect of the parallelism tolerance (4) is considered, which leads to a rotation of the assembly around the y-axis and hence to an increased position deviation of the feature of interest. In contrast to that, the consideration of form deviations leads up to max(pos) = 5.35, which can be explained by irregular contact points between the parts due to form deviations, that accumulate through the assembly and lead to additional position deviations of the feature of interest. Furthermore, based on the results of the SDT approach for the worst possible feature rotations, the maximum parallelism deviation can be calculated as max(par) = 2.07, where it results from the tolerance analysis based on Skin Model Shapes as max(par) = 2.07 without and as max(par) = 1.76 with consideration of form deviations.

Beside the worst-case analysis, statistical evaluations have

Fig. 9. Resulting Assembly for the Tolerance Analysis based on Skin Model Shapes with coarsened Mesh and magnified Form Deviations: Initial Part Deviations (left), Accumulated Deviations through the Assembly (right)

Fig. 10. Surface Meshes of the Parts: coarse Mesh for Visualization (left), dense Mesh with Mesh Refinement on Mating Surfaces for Computation (right).

been performed, where Gaussian input probability densities have been chosen with u = 0.05,a = 0.1/6 for the form tolerances, u = 0.3, a = 0.4/6 for the orientation tolerances, and U = 0.75, a = 0.5/6 for the location tolerances. These Gaussian distributions have been chosen due to their broad application in industry and implementation in proprietary CAT tools. However, other probability distributions, for example based on observations from manufacturing processes, could also have been used. The results of the statistical tolerance analysis are shown in Fig. 11, where pos*G & para*G denote the results of the tolerance analysis based on Skin Model Shapes with and posG & paraG without consideration of form deviations, posG are the position deviations calculated by tolerance stacks, and posG using the vector loop approach. It can be found, that tolerance stacks underestimate the position deviation pos, since gaps between the parts as a result of the orientation defects are not considered. Furthermore, it can be seen, that the consideration of form deviations leads to a slight negative shift of the probability densities for the parallelism deviation par. This is because the form deviations decrease the possible feature rotations and hence result in a decreased parallelism deviation of the feature of interest.

ty 1 1

1 / A - / / paraG paraG

// \ _ if T i \ i

0 0.5 1

Parallelism

Position

Fig. 11. Results of the Case Study: Probability Densities of the FKCs

Moreover, the 0 and y deviations of the feature of interest calculated by the approach based on Skin Model Shapes can be seen from Fig. 12, where WCSDT indicates the deviation domain calculated by the SDT approach, devWC indicates the re-

3 - ................• S 1 t • devwc

• • dev^

2 - I t • devg

-2 -1.5 -1 -0.5 0 0.5 1 1.5

Fig. 12. Results of the Case Study: Rotation Defects ofthe Feature of Interest

suits of the worst-case analysis employing Skin Model Shapes, dev¿ the results of the statistical tolerance analysis with, and devG without consideration of form deviations. It can be seen, that the tolerance analysis based on Skin Model Shapes also allows the worst-case analysis of orientation deviations and that the consideration of form deviations in the statistical tolerance analysis results in a slightly decreased spread of the orientation defects of the feature of interest due to irregular contact points compared to the case where the form deviations are nil.

In summary, it can be found, that the tolerance analysis based on Skin Model Shapes leads to comparable results as the three established tolerance analysis approaches for the case where the form deviations are considered to be nil. However, this novel approach also allows the consideration of form deviations in conformance to international GPS standards, which have, as it has been shown, distinct effects on the quality of mechanical assemblies. Thus, the tolerance analysis based on Skin Model Shapes allows a more realistic prediction of assembly characteristics in virtual product development and a hence supports holistic geometric variations management.

5. Conclusion and Outlook

Tolerance analysis is a key tool to predict the effects of inevitable part deviations on assembly characteristics. In this contribution, a novel approach for the computer-aided tolerance analysis, namely the tolerance analysis based on Skin Model Shapes, has been compared to three established tolerance analysis methods considering a typical case study of tolerancing research. Based on the obtained results, it can be found, that the consideration of form tolerances in conformance to GPS standards, which is enabled by Skin Model Shapes, leads to more realistic predictions of assembly characteristics.

However, works covering related tolerance analysis problems, such as over-constrained assemblies or case studies considering thermal expansion and part deformations, are to be performed in the future.

Acknowledgements

The authors thankfully acknowledge a funding of the German Research Foundation (DFG, grant number WA 2913/15-1).

References

[1] Mathieu, L., Ballu, A.. A model for a coherent and complete tolerancing process. In: Davidson, J., editor. Models for Computer Aided Tolerancing in Design and Manufacturing. Springer Netherlands. 2007, p. 35-44.

[2] Dantan, J.Y., Qureshi, A.J.. Worst-case and statistical tolerance analysis based on quantified constraint satisfaction problems and monte carlo simulation. Comput.-Aided Des. 2009;41(1):1-12.

[3] Soderberg, R., Lindkvist, L., Dahlstrom, S.. Computer-aided robustness analysis for compliant assemblies. J. Eng. Des. 2006;17(5):411-428.

[4] Schleich, B., Anwer, N., Mathieu, L., Wartzack, S.. Skin model shapes: A new paradigm shift for geometric variations modelling in mechanical engineering. Comput.-Aided Des. 2014;50:1-15.

[5] Anwer, N., Schleich, B., Mathieu, L., Wartzack, S.. From solid modelling to skin model shapes: Shifting paradigms in computer-aided tolerancing. CIRP Ann. 2014;63(1):137-140.

[6] Anwer, N., Ballu, A., Mathieu, L.. The skin model, a comprehensive geometric model for engineering design. CIRP Ann. 2013;62(1):143-146.

[7] Srinivasan, V.. Computational metrology for the design and manufacture of product geometry: A classification and synthesis. J. Comput. Inf. Sci. Eng. 2006;7(1):3-9.

[8] Weill, R., Clement, A., Hocken, R., Farmer, L., Gladman, C., Wirtz, A., et al. Tolerancing for function. CIRP Ann. 1988;37(2):603-610.

[9] Shen, Z., Ameta, G.,Shah, J.J., Davidson, J.K.. A comparative study of tolerance analysis methods. J. Comput. Inf. Sci. Eng. 2005;5(3):247-256.

[10] Dantan, J.Y., Gayton, N., Dumas, A., Etienne, A., Qureshi, A.J.. Mathematical issues in mechanical tolerance analysis. In: Proceedings of the 13th Colloque National AIP PRIMECA. 2012.

[11] Shah, J.J., Ameta, G., Shen, Z., Davidson, J.. Navigating the tolerance analysis maze. Comput.-Aided Des. Applic. 2007;4(5):705-718.

[12] Requicha, A.A.G.. Toward a theory of geometric tolerancing. The International Journal of Robotics Research 1983;2(45):45-60.

[13] Gao, J., Chase, K.W., Magleby, S.P.. Generalized 3-d tolerance analysis of mechanical assemblies with small kinematic adjustments. IIE Transactions 1998;30(4):367-377.

[14] Wittwer, J.W., Chase, K.W., Howell, L.L.. The direct linearization method applied to position error in kinematic linkages. Mech. Mach. Theory 2004;39(7):681-693.

[15] Bourdet, P., Mathieu, L., Lartigue, C., Ballu, A.. The concept ofthe small displacement torsor in metrology. In: Ciarlini, P., Cox, M.G., Pavese, F., Richter, D., editors. Advanced Mathematical Tools in Metrology II. World Scientific Publishing Company; 1996, p. 110-122.

[16] Li, H., Zhu, H., Li, P., He, F.. Tolerance analysis of mechanical assemblies based on small displacement torsor and deviation propagation theories. Int. J. Adv. Manuf. Technol. 2014;72(1-4):89-99.

[17] Ameta, G., Serge, S., Giordano, M.. Comparison of spatial math models for tolerance analysis: Tolerance-maps, deviation domain, and ttrs. J. Comput. Inf. Sci. Eng. 2011;11(2):021004-021004.

[18] Giordano, M., Samper, S., Petit, J.. Tolerance analysis and synthesis by means of deviation domains, axi-symmetric cases. In: Davidson, J., editor. Models for Computer Aided Tolerancing in Design and Manufacturing. Springer Netherlands. 2007, p. 85-94.

[19] Homri, L., Teissandier, D., Ballu, A.. Tolerance analysis by polytopes: Taking into account degrees of freedom with cap half-spaces. Comput.-Aided Des. 2015;62:112- 130.

[20] Prisco, U., Giorleo, G.. Overview of current cat systems. Integrated Computer-Aided Engineering 2002;9:373-387.

[21] Farmer, L., Gladman, C.. Tolerance technology — computer-based analysis. CIRP Ann. 1986;35(1):7 - 10.

[22] Chase, K., Parkinson, A.. A survey of research in the application of tolerance analysis to the design of mechanical assemblies. Res. Eng. Des. 1991;3(1):23-37.

[23] Nigam, S.D., Turner, J.U.. Review of statistical approaches to tolerance analysis. Comput.-Aided Des. 1995;27(1):6 - 15.

[24] Polini, W.. Geometric tolerance analysis. In: Colosimo, B.M., Senin, N., editors. Geometric Tolerances. Springer London. 2011, p. 39-68.

[25] Schleich, B., Wartzack, S.. A discrete geometry approach for tolerance analysis of mechanism. Mech. Mach. Theory 2014;77:148 - 163.

[26] Schleich, B., Wartzack, S.. Evaluation of geometric tolerances and generation of variational part representatives for tolerance analysis. Int. J. Adv. Manuf. Technol. 2015;79(5-8):959-983.

[27] Schleich, B., Wartzack, S.. Approaches for the assembly simulation of skin model shapes. Comput.-Aided Des. 2015;65:18 - 33.