Available online at www.sciencedirect.com

ScienceDirect

Procedía CIRP 10 (2013) 9-16

www.elsevier.com/locate/procedia

12th CIRP Conference on Computer Aided Tolerancing

Statistical tolerancing based on variation of point-set

Jianxin Yanga*, Junying Wangb, Zhaoqiang Wub, Nabil Anwerc

aBeijing University of Technology, Beijing 100124, China

bTsinghua University, Beijing 100084, China cLURPA, ENS de Cachan, Cachan Cedex 94235, France

Abstract

Functional tolerancing of mechanisms has now been well accepted in industry and become a major concern for academia. After a brief comparison of existing 3D functional tolerance analysis models, a statistical tolerancing approach based on variation of pointset is proposed in this paper. The toleranced surface is represented by a point-set which is consistent with its parametric equations, the semantics of geometrical tolerances are parameterized with respect to variation of point-set and associated mathematical interpretations of tolerance zone are formalized. Three methods are presented to extract individual points from point-set: characteristic points of MGRE, characteristic points of geometrical surface and discrete points of geometrical surface. The tolerance chains through key assembly features are analyzed and represented by homogenous transform matrix. The over-constrained degrees of freedom of a complex junction with multiple mating features are taken into account in accordance with datum precedence. The approach is applied to statistical tolerance analysis of a coordinate measuring machine to analyze deviation distribution of a geometrical functional requirement on the ending geometric feature.

© 2013 The Authors.PublishedbyElsevierB.V.

Selection and peer-review under responsibility of Professor Xiangqian (Jane) Jiang

Keywords: Statistical Tolerancing; Point-Set; Tolerance Analysis; Monte Carlo Simulation; Small Displacement Torsor

1. Introduction

Functional tolerancing of mechanisms has now been well accepted in industry since this approach contributes to a reduction of manufacturing cost while improving product quality [1]. Each functional requirement is translated to a geometrical characteristic of distances and angles defined between ending geometric features, and must be respected to allow for product manufacturing and assembly or to obtain successful product operations that match desired performance. In order to check respect of each geometrical requirement, the designer must determine the set of influential parts and calculate the influences of both deviations on individual parts and interfaces (gap, flush, clearance) between the parts, which is known tolerance analysis [2]. According to its

* Corresponding author. Tel.: +86-10-67391702; fax: +86-10-67391617 . E-mail address: yangjx@bjut.edu.cn.

objectives, tolerance analysis approaches can be classified as worst-case and statistical. Worst-case analysis determines the extreme displacement of each ending geometric feature resulting from the limits specified on the contributors while statistical analysis determines the full frequency distribution of the contributions. Since the worst-case tolerancing always generates over-quality, statistical tolerancing has been more and more widely used in industry.

Researches on 3D functional tolerance analysis have become a major concern for academia and several mathematical models have been developed. Ballot et al. [3] present a coherent model of geometric specification representation with the concept of deviation torsor, gap torsor and part torsor associated to undetermined components of small displacements, the functional requirement is calculated by two operations over the torsors: composition for serial link and aggregation for parallel link, however the number of configurations for complex mechanisms increases very quickly and the

2212-8271 © 2013 The Authors. Published by Elsevier B.V.

Selection and peer-review under responsibility of Professor Xiangqian (Jane) Jiang

doi: 10.1016/j .procir.2013.08.006

designer should manually select appropriate configurations, this model is adopted only for worst-case analysis. Giordano et al. [4] propose a deviation and clearance domain model to translate a tolerance zone or a clearance into a region of 6D configuration space, two topological operations (Minkowski sum and intersection) have been built over all configuration spaces, the generalization of this method to complex mechanisms requires an efficient numerical algorithm. Davidson et al. [5] propose the Tolerance-Map model for any combination of tolerances on a target feature, which is constructed from a basis-complex and described with areal coordinates, some special configurations are chosen to decouple the orientation variations in the tolerance zone, the operation for tolerance propagation is to undertake a succession of Minkowski sums of all contributing Tolerance-Maps which is similar to domain model, however the influence of parallel links has not been considered. Anselmetti [6] discretizes the boundary of the ending surfaces at many analysis points and calculates the functional requirement by associated analysis lines, the analytical deviation transfer equations have been derived taking into account both virtual material conditions and all possible contact configurations, it is possible to find explicit relationships between functional requirement and tolerances of influential parts, which offers a tolerance synthesis solution with different optimization criteria, however it requires determination of analytical transfer equations case by case. The existing methods mainly focus on worst-case tolerance analysis and are very time-consuming for a good accuracy.

A statistical tolerancing approach based on variation of point-set is proposed in this paper. The toleranced surface is represented by a point-set with respect to its parametric equation. The semantics of geometrical tolerances are parameterized and expressed by variation of point-set. The relationship between variation of pointset and degrees of freedom of SDT model are established. Three methods are presented to extract characteristic points from infinite point-set. The tolerance chains through key assembly features are represented by homogeneous transform matrix. Considering displacement of point-set as random variables of Monte Carlo simulation and a statistical tolerance analysis is performed to analyze deviation distribution of a geometrical functional requirement on the ending geometric feature.

2. Tolerance modeling based on variation of point-set

2.1. Definition of variation ofpoint-set

A tolerance specification is defined on a non-ideal geometrical feature which is functionally equivalent to

the nominal feature and limits the variation between nominal feature and real feature. A nominal geometrical feature can be considered as a point-set P of Euclidean space E3, which is consistent with its parametric equationf as shown in Fig.1.

Fig. 1. Point-set of a nominal geometrical feature

The real feature is a non-ideal geometry of which the points deviate from the nominal feature. A tolerance zone limits all possible variations of integral feature or derived feature. The point-set P' of the real feature can be expressed by variation of the point-set P within the tolerance zone, as shown in Fig. 2.

fl ff2

Fig. 2. Point-set of a real geometrical feature within tolerance zone

An individual point on real feature can be considered as deviating from nominal feature in local normal direction. A local coordinate system is thus required for description of the point-set on a geometrical feature, which will simplify parametric expression of variation of point-set and define tolerance specification unambiguously. For canonical geometrical features, the rules for construction of a local coordinate system are as follows:

• If a geometrical feature or its derived feature is a point, the origin of the coordinate system should coincide with the point and the three axes of the coordinate system can be determined by part model. A spherical coordinate system can be also used.

• If a geometrical feature or its derived feature is a straight line, the origin of the coordinate system should coincide with midpoint of the line. The z axis of the coordinate system is identical to the line and the other axes can be determined by right-hand rule. A cylindrical coordinate system can be also used.

• If a geometrical feature or its derived feature is a plane, the origin of the coordinate system is located at the center of mass. The z axis of the coordinate system is parallel to the normal of the plane and the other axes are determined by right-hand rule.

2.2. Parametric expression of tolerance zone based on variation of point-set

A tolerance zone is a feasible variation region of a geometrical feature. The shape and size of the tolerance zone are inherent attributes of the tolerance specification; however its position and/or orientation are determined by a datum reference frame. A tolerance zone can be also expressed by a parametric variation region of point-set. For example, the mathematical definition of a planar circular tolerance zone based on variation of point-set is

P'|p'(xi+Ax, yi+Ayi) (1)

where Ax,-, Ay e [-t/2, t/2] and ^Axi2 + Ay2 < t/2 .

1 I / / / / /

M- D -►

// Tar A

Fig. 3. A planar geometrical feature with composite tolerances

For a geometrical feature with composite tolerances, each tolerance specification defines different constraints on variation of point-set. The co-existence of these specifications can be indicated by tolerance principles (independent principle, virtual material condition et al.). As shown in Fig. 3, for a planar geometrical feature with position, parallelism and flatness tolerances, the feasible variation region of point-set can be defined as

P | Pi ( x, y,, D + Az, ) where

Az( e [-Td/ 2, Td /2], Datum : z=D , Az( — Az. < Tpar, Datum : z = 0,

|d(A^Az. )| < yja2 + b1 + c

< Tlt, Datum : ax + by + cz + d = 0 ■

2.5. Relationship between variation of point-set and SDT tolerance model

The variation of point-set between real feature and nominal feature can be expressed by a homogenous transform matrix

"xf ■ i -y P u xi

P' = y z; = r -ß i a -a i v w X y,

i 0 0 0 1 1

= T,xP

where a, fl, y, u, v and w are six components of a small displacement torsor D [7], which is defined by the nature of related surface, some displacement components that leave the nominal surface invariant are noted Ind and the others are constrained by tolerance specification.

Fig. 4. Variation of point-set on a planar geometrical feature

It is also possible to calculate the small displacement torsor with a certain number of points on a geometrical feature. As shown in Fig.4, a planar geometrical feature can be represented by three non-collinear points, which are extracted from the point-set. These three points move along the normal direction and the associated homogenous transform is

x, x j xk " 1 0 P 0" x, x j xk

y, yj yk 0 1 -a 0 X y, yj yk

Az, Az. Azk -P a 1 w 0 0 0

1 1 1 0 0 0 1 1 1 1

which yields

- xß + yta + w = Az,

— xß + y .a + w = Azj -xkß + yka + w = tek

where Az,-, Azj, Azke [-t/2, t/2], xi, xj, xke [-a, a] andyi, yj,

yk* [-b, b].

As shown in Eq. 5, the resultant interval of the small displacement torsor varies from the choice of selected points in the same tolerance zone, it reaches minimum when the points are placed at the vertices of the plane:

-t/ 2< aß-ba + w<,t/ 2 - tl 2 < -a^ -ba + w£t/ 2 -t/ 2< aß + ba + w<,t/ 2 -1/2 <, -aß + ba + w<t/ 2

which is identical to SDT tolerance model and ensures that all variations of point-set are within the tolerance zone.

3. Methods for extracting individual points from point-set

Because the point-set on a geometrical feature is infinite, tolerance analysis over the entire point-set is unpractical. Three methods are presented to extract individual analysis points from point-set of a geometrical feature: characteristic points of MGRE (Minimum Geometrical Reference Element) [8], characteristic points of geometrical surface and discrete points of geometrical surface. The deviation distribution of geometrical feature can be analyzed by Monte Carlo simulation [9] taking variation of extracted points as random variables.

3.1. Characteristic points of MGRE

Fig. 5. Characteristic points of MGRE of a cylindrical feature

A geometrical feature can be represented by its associated MGRE, which is a combination of a point, a line and a plane. The line and the plane of a MGRE can be also represented by several characteristic points: two endpoints for a line and three non-collinear points for a plane. For a tolerance specification applied on the MGRE of a geometrical feature, the geometric deviation simulated by characteristic points of MGRE is effective. As shown in Fig. 5, the MGRE of a cylindrical surface is its axis, which can be defined by the endpoints P\ and P2, the variation of these two points within a cylindrical tolerance zone is

where a=(Ay2-Ayi)/L e [-t/L, t/L], /?=(Ax2-AxO/Le [-t/L, t/L], u^A^+Ax^/2 e [-t/2, t/2] and v^Ay^Ay^/2 e [t/2, t/2], L is the length of the cylindrical surface.

3.2. Characteristic points of geometrical surface

Fig. 6. Characteristic points of a spherical surface

For a tolerance specification applied on the surface of a geometrical feature, such as cylindricity for a cylindrical surface, it is necessary to use the variation of characteristic points of geometrical surface. The number of characteristic points of geometrical surface is determined by parametric equation of the geometrical surface including an addition point for representing form deviation, such as four non-collinear points for a plane, five non-coplanar points for a spherical surface and six non-coplanar points for a cylindrical surface. The characteristic points should be located on the boundary of geometrical surface. As shown in Fig. 6, for a spherical surface located at the origin, the variation of five characteristic points is

Ap = (0,0, ArJ

AP2=(V3Ar2,0,-Ar2/2) (9)

■ AP3 = (V3Ar3 / 4,3Ar3 /4,-Ar3 /2) AP4 = (V3A r4 /4,-3Ar4 /4,-A r4 /2) AP5 =(0,A r5,0)

where Ar,e [-//2, //2].

3.3. Discrete points of geometrical surface

[APKAXÍ.Ak.O) |AP2 =(Ax2,Ay2,0)

where Ax¡, Ay,e [-//2, t/2] and ^Axf + Ay2 < t/2-The corresponding small displacement torsor is

Cylinder

a u P V

The real geometrical surface of a part is a non-ideal

(7) geometrical surface with form deviation. A finite set of discrete points can be extracted from all point-set of geometrical surface within a certain density. The density of discrete points can be determined by local radius of curvature of nominal geometrical surface. The nominal geometrical surface is covered by a polygonal mesh of which the nodes are chosen as discrete points P¡ (i =

(8) 1, ... , n) of this surface. An iterative algorithm is developed to generate variation of discrete points: 1) For each discrete point Pi, it moves randomly along local normal direction to a variation point p' within the

position tolerance zone; 2) If a orientation tolerance and/or form tolerance are applied on the geometrical surface, it is necessary to check whether p' is within corresponding floating tolerance zone with a substitute surface generated by a least-squares association operation, if not, the point will be regenerated. 3) Repeat Step 1) and (2) until variation of all discrete point are generated. For example, variation of discrete points of a planar geometrical surface and the substitute surface are shown in Fig. 7.

where Ti_,i+ is the internal transform matrix between two functional features of the same part, Tii+1 is the kinematic transform between two functional features of different parts if there is a physical contact between them, and K is the local homogenous coordinates of key geometrical feature.

In real condition, interface between the parts is a real geometrical surface which can be represented by variation of point-set, the homogenous transform matrix from the base part to the real ending functional features

P + AP =

^Tt1- X T1_,1+ X Tt1+ X T1,2 X Tt2- X T2-,2+ x!t 2+x- X T„_1_„ X T~[ X T„_,„+ x Ttn+ J

where Tti-, Tti+ is homogenous transform matrix of the real geometrical surface relative to the nominal geometrical surface of previous and preceding part interface. Therefore the deviation of key geometrical feature can be calculated by

AP = ( P + AP)-P

^Tt1-X T1-,1+ X Tt1+ X T1,2 X Tt21 X T2—,2+ xT„^x---xT , xT-'xT xT, ^

12+ n—1,n tn- n—,n+ tn+

T1-,1+ X T1,2X T2-,2+X X Tn-1,n X Tn-,n+

Fig. 7. Variation of discrete points of a planar geometrical surface and the substitute surface

4. Statistical tolerance analysis model based on variation of point-set

Fig. 8. Tolerance accumulation chain in nominal and real condition

As shown in Fig. 8, for an assembly stacked by a series of parts without geometrical deviation in nominal condition, the homogenous transform matrix from the base part to the nominal ending functional feature is

P = Tw+ xT,,2 xxr„-1,n xT„_,„+ xK (10)

Ti_,i+ and Ti-1,i are determined during product design stage and can be retrieved from CAD models. However Tti-, Tti+ are generated during actual machining operations and can be simulated by variation of point-set. The actual distribution of part deviations from metrology can be also used for more reliable result. The deviation is a function of small displacement torsor of part interface, which is also a function of variation of pointset on each mating geometrical feature:

-FT{T^a ,ß,y, u, v, w)) ~ F^l, ^

where j = 1, ..., n and 1,j is displacement variation of the ith individual point on the jth mating geometrical feature.

Fig. 9. Assembly junction with three planar mating features

For a complex assembly junction with multiple mating features [10], there are redundant constraints on

some degrees of freedom between parts. The precedence order of the constraints on different features can be indicated by a datum reference frame composed of primary, secondary and tertiary datum. As shown in Fig. 9, the priority of a junction with three planar mating features is A > B > C, a local coordinate system is established on plane A, the combined small displacement torsor for this junction can be defined in accordance with the datum precedence as

R*(0 T\t3) Ry{ti) TVk/sinö)

yRXtjsne) TXti) ,

Thus the hybrid tolerance chain with several links in parallel is transformed into a serial tolerance chain. It will simplify tolerance analysis of complex assembly.

5. Application to a coordinate measuring machine

Fig. 10. Structure decomposition of a coordinate measuring machine

A simplified mechanism of a coordinate measuring machine (CMM) is shown in Fig. 10. It is composed by workbench 1, sliding rail 2, spinner rack 3 and feeler lever 4. According to technical specifications of the coordinate measuring machine, the position deviation of the measuring probe on the feeler lever relative to the workbench is key criteria for measuring accuracy. It should be validated under different configurations of the mechanism, where L e [10, 25], Q1 e [jt/4, tt/2] and 02e[;r/3, 2jt/3]. A geometrical functional requirement is defined between the base of the workbench and the measuring probe. The tolerance specifications of the parts are annotated in Fig. 11.

Workbench

Sliding rail

Spinner rack

Feeler lever Fig. 11. Tolerance specification of the parts

The tolerance accumulation chain for the functional geometrical requirement is E-T1-M12-T2-M23-T3-M34-T4-K. Position deviation of the measuring probe AP = (APx

APy APz 1)T relative to its nominal position can be 6. Conclusion calculated by

rTxT xT xTT xT xT ^

± i ± i^ ± 12 ^ J- 2_ ^ 2 2+ 23

xT3-> T3 x T3+ x T34X T;!x T4

—TxT xT xT xTxT xT

"0 -1 0 L

-1 0 00

00 -1 0

0 0 0 1

- sin 61 -cos$ 0 0

-cos^ sin^ 0 0

0 0 -1 0

0 0 0 1

cos#2 sin#2 0 0"

- sin 02 cos#2 0 0

0 0 1 0

0 0 0 1

A statistical tolerance analysis software for the coordinate measuring machine has been developed to calculate position deviation distribution of the measuring probe, as shown in Fig.12. User can define simulation times and input kinematic variables for different nominal mechanism configurations. The normal distribution (in the first row) and uniform distribution (in the second row) assumptions are adopted for variation of individual points. The resultant deviation distribution obtained from characteristic points of MGRE, characteristic point of geometrical surface, discrete points of geometrical surface are denoted in dotted line, thick solid line and thin solid line respectively.

Fig. 12. Statistical tolerance analysis of the measuring probe with normal and uniform distribution assumptions

The associated object of a tolerance specification is extended to point-set of a geometrical feature. The tolerance model based on variation of point-set is accurate for interpretation of tolerance semantics and it is effective for statistical tolerance analysis of complex mechanisms. Among three methods for point-set extraction, characteristic points of MGRE can simulate deviation of a geometrical feature with minimum computation, however it is not applicable for some form tolerances; characteristic points of geometrical surface can reflect both MGRE deviation and form deviation of a surface, it is applicable for expressing deviation occurred on boundary of the surface; discrete points of geometrical surface can more accurately simulate variations of a continuous geometric surface, however iterative generation of the points and computation of association operation is time-consuming. Designers should choose an appropriate method in consideration of both simulation accuracy and computation complexity.

Acknowledgements

This work is supported by National Natural Science Foundation of China (Grant No. 51005002) and Scientific Research Common Program of Beijing Municipal Commission of Education (Grant No. KM201110005012).

References

[1] Islam MN. Functional dimensioning and tolerancing software for

concurrent engineering applications. Computers in Industry 2004; 54: 169-190.

[2] Hong YS, Chang TC. Chang. A comprehensive review of

tolerancing research. International Journal of Production Research 2002; 40: 2425-2459.

[3] Ballot E, Bourdet P, Thiebaut F. Determiniation of relative

situations of parts for tolerance computation. Proc 7th CIRP International Seminar on Compute-Aided Tolerancing 2001: 6372.

[4] Giordano M, Samper S, Petit JP. Tolerance analysis and synthesis

by means of deviation domains, axis-symmetric cases. Proc 9th CIRP International Seminar on Computer-Aided Tolerancing 2005: 85-94.

[5] Singh G, Ameta G, Davidson JK, Shah JJ. Worst-case tolerance of a

self-aligning coupling assembly using Tolerance-Maps. Proc 11th CIRP International Conference on Computer-Aided Tolerancing 2009: CDROM paper.

[6] Anselmetti B. Generation of functional tolerancing based on

positioning features. Computer-Aided Design 2006 ; 38 : 902-918.

[7] Bourdet P, Mathieu L, Lartigue C, Ballu A. The concept of small

displacement torsor in metrology. Advanced Mathematical Tool in Metrology II, Series Advances in Mathematics for Applied Sciences 1996: 110-122.

[8] Clement A, Valade C, Riviere A. The TTRSs : 13 oriented

constraints for dimensions, tolerancing & inspection. Advanced Mathematical Tool in Metrology III 1997: 24-42.

[9] Robert C, Casella G.. Monte Carlo Statistical Methods, Springer

[10] Anselmetti B. Cotation fonctionnelle tridimensionnelle et statistique, Hermes Lavoisier Edition 2008.