a matrix approach to the representation of tolerance zones and

Int J Adv Manuf Technol (1997) 13:630-636 © 1997 Springer-Veflag London Limited

Tie In~bnel aeumal

Rdvanced Illanufacturing Technoi@

A Matrix Approach to the Representation of Tolerance Zones and Clearances

A. Desrochers* and A. Rivibret *Laboratoire d'Automatique et de M6catronique, l~cole de technologie supdrieure-Universit6 du Qu6bec, Canada and tLaboratoire d'Ing6nierie Intdgr6e des Syst~mes Industriels, Saint-Ouen, France

This paper presents a matrix approach coupled to the notion of constraints for the representation of tolerance zones within CAD~CAM (computer-aided design and manufacture) systems. The proposed theory, reproduces the measurable or non- invariant displacements associated with various types of' tolerance zone. This is done using the homogeneous transforms commonly associated with robotic modelling.

This matrix representation is completed by a set of inequalities defining the bounds of the tolerance zones. More- over, the generation of the model and its mathematical definition allows for its use in the representation of clearances and for the computation of tolerance transfer. An example illustrates the application of the model to a simplified gear pump mechanism.

Keywords: CAD/CAM; Clearance; Fit; Homogeneous transforms; Model; Tolerance zone

1. Introduction

Current CAD/CAM systems allow the generation of three- dimensional models of mechanical parts and the simulation of their machining on screen. Nevertheless, a large gap still exists between the "design" (CAD) and "manufacturing" (CAM) part of these software packages. It would seem appropriate to fill this gap through the addition and integration of technological information to the geometrical model. A method for representing and exporting geometrical tolerances within CAD/CAM systems is proposed.

The approach relies on the representation of tolerances using standardised zones constructed around the theoretical surfaces of the part.

Correspondence and offprint requests to: Professor A, Desrochers, Department of Automated Production Engineering, ~;cole de technologie sup&ieure, I100, rue Notre-Dame Ouest, Montreal, Quebec H3C 1K3, Canada.

From a mathematical point of view, these tolerance zones are represented by homogeneous transforms reproducing the degrees of invariance associated with the zone. As will be shown, this representation will prove particularly suitable for tolerance analysis and clearance computation.

At this point, it might be worth noting that many authors have already studied this problem. However, the literature reveals that all of them have favoured approaches which represent tolerances by a general model. These approaches include the offsetting representation [1], half space representation [2], variational [3] or parameterised model, etc. All of these approaches, derived from given geometrical modellers, prove unsatisfactory for properly integrating and representing the functional and technological features of tolerances, as pre- scribed by international standards.

To overcome these difficulties, a list of the various types of tolerance zone has been established and each one has been modelled using a suitable representation. Furthermore, the representation model adopted allows computation of tolerance transfer as well as clearances within assemblies. Finally, the model itself is based on the general concept of TTRS which defines high-level entities bearing technological contents [4,5].

2. Presentation of Matrix Tolerancing

2.1 Presentation

The positional tolerancing of mechanical parts describes the relative position uncertainty of a surface with respect to a set of data.

It is well known that position and displacement are math- ematically identical problems, the difference being only a matter of interpretation. In view of this, the displacements which led to the definition of seven types of elementary surface [4,6,7] have been studied first, as presented in Fig. 1.

2.1.1 Displacement Parameters

There are numerous methods for representing a displacement. One of them is to use homogeneous transforms as in robotics.

Tolerance Zones and Clearances 631

@ General surfaces

Prismatic surfaces

Surfaces of revolution

~ x

Hel ica l s u r f a c e s

Cylindrical surfaces

Planar surfaces

Spherical surfaces

i .

, x

, w v" x

• w v x

v , x

w

w v x

C~13 -SrCa+ CrSl3Sct S,/Sa+CrSpCa u ] i

SyC[3 CTCa+SySl3Sa -CTSa+STSI3Ca v [ I

-sp cps~ cpca w l L o o o ~J C7cI3 -s~tca+cTsl3sa sTsa+eTSl3Cct o ] s7cI3 cyccc+s~sl]sa -c?sa+sTsl]ca v -Sp CpS~ CpCa w l

L o o o ] J

-cvcp -s r cvsP u3 i

S~Cl3 C~ S~Sl3 v [ I

I - sp o c~ w l L o o o 1J

[C~Cp -sT CvSP u7 v 1

I -S13 o CP w l ~ u *

Lo o o 1A [ c ~ c ~ - s r c~s~ o3

svcI3 Cr S~SI3 v I -s13 o cp w l L o o o ~J -crc~ -s~ crsp uq S~Cl3 cy SrSl3 o

I - s p o cp o l L o o o ,J

[1 0 0 u-]

0 1 0 v

Io o 1 wl k o o o l J

V X

Fig. 1. Description of the seven classes of elementary surfaces.

As a reminder, an homogeneous transform matrix is composed of a (3 × 3) rotation matrix, and a translation (3 x 1) column vector.

A displacement will depend on six independent parameters; three rotation parameters and three translation parameters. Each of these parameters defines a displacement in itself which will be referred to as elementary displacement. The rotation displacements will be expressed as angles of rotation around the coordinate system axis; angles cq [3, -y around the x-, y- and z-axes. The Roll-Pitch and Yaw convention has been adopted for the definition of these angles. Finally, the translation displacements will be expressed by components u, v and w representing, respectively, translations along the x-, y- and z-axis.

The corresponding homogeneous transform matrix describing the six elementary displacements can be written as follows:

foo / l ooo 1 0 1 0 v T(u) = T(v) =

0 1 0 1 0

0 0 1 0 0 1

(1)

f 0 0 0 1 0 0 T(w) =

0 1 w

O O 1

I °° C~ - S a

R(a) = Sc~ Cc~

0 0

F~S~ -S'y 0 C-/ 0

R(~/) = 0 1

L0 0 0

ol i os o 0 1 0 0

R([3) = -S[3 0 C[3 0

1 - 0 0 0 1

0

Every displacement D1 can be described as the product of six elementary displacements T(uO, T(vO, T(wl), R(eq), R([3~) and R(~) . The order of these components is important since rotations are not commutative. For the Roll-Pitch and Yaw convention the multiplication order is as follows: T(w) x T(v) x T(u) x R(7) x R(f3) × R(a).

632 A. Desrochers and A. Rivikre

2.1.2 Displacements Applied to the Representation of Surfaces

By definition, the position of a geometrical element with respect to a complete reference frame is modified only by displacements which do not leave it globally invariant. Through studies of these displacements, seven types of elementary surface were established, as presented in Iqg. 1.

For instance, in the case of a general surface, no displacements can leave it invariant with respect to a complete reference frame. Consequently, for this surface, the set of non- invariant displacements will be represented by a homogeneous transform matrix which includes all six elementary displacements.

Moreover, a cylindrical surface is left invariant when con- sidering translation and rotation along its own axis. If the cylinder axis is regarded as the X-axis of the surface, then the global displacement matrix (Fig. 1) can be used providing its u and e~ parameters (translation and rotation about the X-axis) are set to zero. In this way, the matrix presented in Fig. 1 is obtained.

A similar reasoning can be applied to each one of the seven classes of surface.

When it comes to defining the position of a surface with respect to an incomplete reference frame, the problem increases in complexity. Indeed, in such an event, one must then take into account the displacements which either leave or do not leave the reference frame invariant. This leads to the definition of 44 cases of positional tolerancing [5].

2.2 Representation of Standard Tolerance Zones

A tolerance zone is a surface or volume space limited by one or several surface or line definition elements [7-10]. Nine tolerance zones have been reported:

Four are surface elements:

1. Surface defined by a planar disk.

2. Surface defined by a planar annulus.

3. Surface defined by a straight planar strip.

4. Surface resulting from the offsetting of a line.

Five are volume elements:

1. Inside volume of a square section prism.

2. Inside volume of a cylinder.

3. Inside volume of a cylindrical annulus.

4. Volume resulting from the offsetting of a plane.

5. Volume resulting from the offsetting of a surface.

2.2.1 Representation of the tolerance zone defined by the inside volume of a cylinder

Here, the previously established concepts will be applied to the case of a tolerance zone bounded by the inside volume of a cylinder.

As depicted in Fig. 2, a cylindrical zone of diameter t is constructed around the line element [A,B] representing, for instance, the axis of a cylindrical surface. The reference frame

~ t

["iz il ........... ; # ' " . ) ,,'

Fig. 2. Cylindrical tolerance zone.

(0, x, y, z) is defined with 0 as the middle point of the [A,B] segment and X oriented along the AB vector.

The application of this tolerance zone will now be shown for the case of the positional tolerancing of a cylindrical surface with respect to a complete set of data. The set of displacements which do not leave segment [A,B] globally invariant is represented by the homogeneous transform matrix D(v, w, ~, y) expressed in the (0, x, y, z) reference frame:

I C.yC[~ -Sy C,tS[~ 0 = ]SyCf3 C~ S~Sf3 v

D(v,w,~,y) L-~ ~ O0 C~O Wl (2)

Additional constraints are also needed to ensure that the segment [A,B] remains within the bounds of the cylindrical tolerance zone:

--< ~ t with Y,~ D x A

d A

ea d B - a

(3)

0--< [3--<w and 0 ~ y ~ w

Therefore complete modelling of the tolerance zone is achieved through a matrix (2) describing the possible deviations of axis [A, B] along with a set of constraints (3) representing the geometry and dimensions of the tolerance zone.

It is worth noting that this approach provides a global model of the tolerance zone. Indeed, it allows the representation of position uncertainties for any point M associated with segment [A,B]. Moreover, the actual choice of invariant displacements reflects the geometry of the tolerance zone rather than its function. For instance, in the case of a parallelism, the translation of the axis within the tolerance zone itself loses all technological meaning. The displacement v and w should then be withdrawn from matrix (2) to properly reflect the orientation tolerance (parallelism).

The representation of the eight other tolerance zones follows the same principles and logic that has just been shown for the cylindrical tolerance zone [5]. Furthermore, the example presented in the following section uses exclusively cylindrical tolerance zones.

(+ +

+

1~+,,., /

v

7" x o.

//~17H? [ ]

• %

Fig. 3. Pump body (part 1).


IA,I@I,,, ++- I";I tc, l@l,, ,,,. Io.,I

iB,I+{. +,~ +,I

x

3. Application to a Simple Mechanism

The example that has been chosen is a simplified gear pump assembly. It includes three functional parts; the pump body (Fig. 3) and two shafts supporting the gears (Figs 4 and 5). The plate cover has been left out for the sake of simplicity.

3.1. Representation of the Mechanism with Tolerances

3.2 Clearance Computation Between Both Gears

As an example, the computation of the clearance between parts 2 and 3, the two shafts and gears, will be attempted. To achieve this, the mechanism must be considered as a whole. The matrix tolerancing techniques presented earlier will be applied to each part of the assembly.

The complete gear pump assembly can be represented by a binding graph. On this graph, parts are illustrated by sets of functional surfaces while contacts are symbolised by links. Moreover, each link carries a tolerance specification characterising the associated clearance. For instance, a concentricity specification of 0.037 defines the shaft clearances within their respective bore. Indeed, according to the small displacement model, a cylindrical clearance zone is equivalent to a concentricity tolerance since, in both cases, they result from the association of two cylindrical, coaxial surfaces. The geometric tolerancing specifications of the drawing are also represented

+++ .......... ii .... l 25. r, q

on the graph by additional links within each set of functional surfaces.

In this example, clearance computation is simplified by this particular type of mechanism featuring essentially parallel shafts and bores. Finally, as can be seen (Fig. 6), datum surface D~ is hatched in order to designate it as the starting point for the relative displacement calculation of the surfaces involved in the clearance computation between parts 2 and 3.

Eight tolerancing surfaces have been defined for the simplified gear pump assembly. However, only six of them are actually relevant for the computation of the clearance between the two gears. As shown on the binding graph, these six surfaces are the ones involved in a loop passing through the link between both gears. For the sake of simplicity, all of the surfaces including the gears are considered to be cylindrical.

3.2. 1 Part 1 (Pump-Body)

Tolerancing surfaces A~, B~, C1 and D1 have been defined on part 1. However, only surfaces Bj and D~ will be taken into account in the computations. They are the bores into which the shafts will be assembled and turn. Additionally, the tolerance zones being similar for surfaces B~ and D~, the demon- stration will be restricted to the tolerance zone associated to surface B~.

Prior to any tolerance zone modelling, a mathematical formu- lation must be defined for the expression of a given reference frame relative to another one.

A matrix PR--R+ is therefore established to express relative to a local reference frame R~ (associated to the surface D~),

+ ............. 31;i+ .......... ........... x

Fig. 4. Shaft and gear (part 2). Fig. 5. Leading shaft and gear (part 3).

634 A. Desrochers and A. Rivikre

~rt 3

Fig. 6. Schematic representation for the tolerancing of the three parts composing the mechanism. Dotted boxes indicate displacement associated with the zone.

any point or displacement initially defined in the mechanism coordinate system, R.

PR~RI: Transform matrix mapping R into R~, R being the coordinate system of the mechanism and R~ the coordinate system of the surface D~

(o,.~,.y,.,~ = [Pn~R,] X (O,x,y,z) (4)

Determination of the Constraints on a Point M Associated to the Dl l Tolerance Zone. In the R~ coordinate system, the general homogeneous transform matrix D~ representing the tolerance zone characterising the location of a cylinder whose axis is oriented along X, can be expressed as follows:

[c c13 = I s~C13 (5)

D,l(v,w,[3,~/) l-~13

However, in this case the B~ bore is subject to a location tolerance specification bearing a cylindrical zone modifier and using D~ as datum. This implies a zero value for the w parameter of the matrix acting perpendicularly to the plane created by both the B~ and D~ axis. The v value expresses any variation in the distance between the axes while the " /and [3 parameters represent any angular deviation of the axes within the cylindrical tolerance zone.

-s'~ c~s13 o

c.y s~s13

o C13 w

0 0 1

To establish the constraints which will be acting upon any point within this tolerance zone, it will be necessary to define the displacement vector of a point M in the tolerance zone expressed in the R~ frame.

Given M, a point located on the theoretical axis of bore B~, and M', the same point M after a displacement (v, 13, 7). Then,

[MM'] is the total displacement of the theoretical point M after the application of the D~ transform.

Consequently, in the R~ reference frame,

[ M M ' I R , = [M']R, -- [M]R 1 = D , , X [M]R, -- [M]R,

[MM']R, = Dll X [PR--R1] X [M]e- I X [PR--R,] X [M]R

hence,

[MM']R, = (D1L - ]) x [PR--J X [M]R (6)

Additionally, this vector must comply with several constraints related to the geometry and size of the tolerance zone B~:

[Ms~M'B~o]R~ • y~ <-- ~bt3/2 [MB~bM'8~b]R~ • Y~ --< ~bt3/2(7)

0 -- "¢ --< "rr 0 --< 13 <-- 7r and naturally v <-- qbt3/2

and w <-- +t3/2

Determination of the Constraints on a Point M Associated to the 972 Tolerance Zone. The Dl2 transform matrix describing, in the R2 frame, the uncertainty zone associated with the clearance between surfaces A2 and D~ which are cylinders with the same axis, is identical to Oil (v, w, 13, '~/) as presented in equation (5). Indeed the A2 and D~ bore axis are subject to a concentricity tolerance specification which, in turn, implies a cylindrical zone [7].

This matrix translates the smNt relative displacements possible between parts 1 and 2 given the specified H7f6 fit.

Again, the displacement vector of a point M within this uncertainty zone expressed in the current reference frame R2 may be established.:

[MM']R 2 = (D12 - I) X [PR_.R2 ] X [M]R (8)

Appropriate geometrical and dimensional constraints are once again specified accordingly.

[MDI~, M'D~]R2 " Y2 <- 0.037/2 (9)

[Mo~h M'D~b]R 2 " Y2 -< 0.037/2

0 ----- 13 <-- "rr and 0 --< y ~ 'rr with v --< 0.037/2

and w -< 0.037/2

3.2.2 Parts 2 and 3 (Gear and Pinion)

For parts 2 and 3, the tolerancing surfaces A2, B2, A3 and B3 have been defined. The surface A2 will be the datum surface for the definition of the tolerance zone associated with B2, while the tolerance zone associated with B 3 will be established with respect to datum surface A3.

The determination of the constraints on the tolerance zone of surface B2 and B3 will not be performed here since it is similar to that of surface B1 shown in the previous section.


DII

DI2

I M-g~], 4,t3 ' - - 2

Cfl~r * x#t +CY * YM +,.WSY * d d - yM + v

[ M~M~ 1 0 .037 - 2

C,~y *.~+Cr *.u~d+S~ * , ~ - ~ + v

r-ls- [151 0 I01 , * t 3 _

I °, . [:J, :~-' r-1251 Io l ° 1

• R~ t

F1zsl lol ~ . O l S S y

D13 -------~] 0 .037 M.,~G, 'Y~ = < - -

CBSr*xM +Cr*N'/ +S~r*zM-yM +v

r-~sl Io I [o

1 • ?'1

F157 Io ~°,L _+o.olss y

D22

D33

[ - - ] <¢t4 ,%=a¢,= .y,=

L -IR~ 2

c4~r*~+Cr*~+S~r*~-~+v

[ - 1 , <°" m . M , , "Y' = - 2

C~r*~t+Cr *#4+SlUr *z~=yv/+v

F-sl tol t~L [:! [o

1

Fsl Io It R3

[:i ~° L, 4

1:¢ t4 - - y y

0 t 5 _

5-y

Fig. 7. Summary of the computational results.

Figure 7 illustrates the summary of the computational results so far.

3.2.3 Clearance Computation Between Gears B2 and B3

The computation of the clearance between both gears will result from the combined effects of the minimum and maximum distances separating their axes. These minimum and maximum distances will in turn be computed from the displacement vector e 2 and e3 on parts 2 and 3 as depicted in Fig. 8.

The displacement vector ez will be computed from the toterancing chain linking surface D~ to surface B2 and involving the sum of displacements D~z and D22 as presented in equation (t0). Similarly, the displacement vector e3 will be obtained from the tolerancing chain connecting datum surface D~ to the cylindrical surface B 3 through the sum of displacements D~ ~, Oi3 and D33 (Fig. 8) as shown in equation (11).

e2 = {(PheLR2 x ([DI2]R2 - / ) x PR~R2 X [MD1]R ) (10)

+ (P~RIR3 × ( [ D 2 2 ] R 3 - - / ) X PR--% X [MB2]R)} " Y

e3 p~D 1 = {( R--R, X ( [Dl l ]R , - / ) × PR-.R,

p;1 ( R--R, × ([D13]R, -- / ) X PR_.RI

(~LR~ × ([D3dR, - / ) × PR--R~

x [ M < k ) +

x [MB,]R) +

x [MB3]R)} " Y

(11)

L ..... L M B3a M B3b

Maximum distance

M B2e M B2b

b : ..... I

M B3a M B3b

15 15

M B2a M B2b

Fig. 8. Maximum and minimum distances between the axes of the gears for the computation of the corresponding clearance.

636 A. Desrochers and A. RiviOre

The determination of the clearance between the gears will imply the optimisation of two distance functions corresponding to the worst case scenarios illustrated in Fig. 8.

Therefore, the maximum and the minimum distances between the gear axes will be obtained from the maximisation of the following functions:

for the maximum distance : ]e3a - e2~]

subject to the parallelism constraints :

le3,, - e~l >- Ie~o - e=~l (12)

and for the minimum distance : I-e~° + e2~l

subject to the parallelism constraints :

l e3b - - e2b I <-- l e 3 a - e2a I ( 1 3 )

The maximisation is performed using a standard optimisation algorithm such as simplex.

The numerical value obtained from these computations may be cross-checked using classical tolerancing chains. Using the values presented in Fig. 7, both approaches yield the same result:

Clearance = _+[(@3/2 + 0.037/2 + qbt5/2) (14) + (0.037/2 + (bt4/2)]

Consequently the resulting dimension separating the gear axes will be: 15 x:~+tv2 + <bt4/2 + cbt5/2 + o.o37)

result would have been very difficult to compute using conven- tional approaches.

4. Conclusion

In this paper a methodology allowing the representation of tolerance zones and clearances within CAD/CAM systems has been developed. With the proposed approach, the degrees of freedom associated with the tolerance zones are translated in terms of matrices. This mathematical representation can then be used easily, as has been shown, for the computation of clearances and angular deviations.

Other applications can naturally be derived from this modelling procedure. Indeed, tolerance transfer and tolerance analysis computations could also be achieved in three dimensions using this method. As a matter of fact, the principles set forth for these types of computation are the same as those commonly employed for the determination of clearances.

Finally, the method is very general, which makes it suitable for any three-dimensional part and any type of surface. It should also be mentioned that the aim of this tolerancing. model is not to replace geometric dimensioning and tolerancing standards but rather to translate their corresponding tolerance zones into minute displacements, which provides an effective computational tool for tolerance related operations in three dimensions.

3.3. Computation of the Parallelism Error Between the Gears

The clearance computation that has just been presented could have been performed more easily using the classical tolerance stack up approach with the results of Fig. 7. However, this is not the case for the computation of the parallelism error between the gears. Here, the method of resolution remains the same as in the previous section, but the extreme position of the gears will be considered in terms of orientation rather than distance. Graphically, this involves two alternative combinations of the maximum and minimum distance shown in Fig. 8.

The displacement vectors e2 and e3 along with the y distance components are obtained, as in the previous example, with equations (10) and (11). The functions to be optimised, however, do reflect the computation of the c~ uncertainty on the parallelism:

I(-e3~ + e2,) + (e3b - e2b)l, I(e3~ - e2~) + (-e3b + e~b)I 0 5 )

Obviously, there are no constraints to be imposed on the gear orientations. The result of the optimisation process yields the following c~ uncertainty on the parallelism between the gears,

0/. . . . . . = (O/'DI2 -}" {Y~D22 ) "}- (O/'D I [ J¢" ~DI3 -I- 0~.D33)

or area× = 0.037125 + qbt4/16 + ~t3/30

+ 0.037/30 + ~t5/16 (16)

Since the angular deviations considered are very small, the approximation, tan (cx) = ot (rad) has been used. Therefore, the parallelism uncertainty between the gears will be 0±~m~. This

Acknowledgment

The authors wish to thank Doctor Dominique Gaunet, research and development engineer at Dassault-systames and Mr Inouk Dub6, Eng., for their contribution to this work.

References

1. A. A. G. Requicha, "Toward a theory of geometric tolerancing", Journal of Robotic Research, 2 (40), winter 1983.

2. R. Jayaraman and V. Srinivasan, "Virtual boundary requirements and conditional tolerances in mechanical design", RC 12942 (58050), IBM T. J. Watson Research Center, October 7, 1987.

3. J. U. Turner, "Exploiting solid models for tolerance computations", Geometric Modeling for Product Engineering, 1FIP t990 pp. 237- 258.

4. A. Desrochers, "Modble conceptuet du dimensionnement et du tol~rancement des m~canismes. Reprrsentation dans les syst~mes CFAO", Doctoral Thesis, Fv~cole Centrale Paris, September 24, 1991.

5. A. Rivi~re, "La g~omrtrie du groupe des drptacements appliqude la modrlisation du tol4rancement", Doctoral Thesis, Ecole Cen-

trale Paris, September 22, 1993. 6. J. M. Herve, "La grom6trie du groupe des d4ptacements appliqu4e

l'analyse cinrmatique des mdcanismes", State Doctoral Thesis, Paris VI, May 1976.

7. D. Gaunet, "ModUle formel de tolrrancement de position: contribution ~a l'aide au tol4rancement en CFAO', Doctoral Thesis, Ecole Centrale Paris, 1993.

8. "Dessins techniques - Tolrrancement g4omrtrique - Tol&ancement de forme, orientation, position et battement-Grndralitrs, drfi- nitions, symbotes, indications sur tes dessins", ISO, 1983.

9. "Dessins techniques- Cotation et tol~rancement des profils", tSO, 1987.

10. "Dessins techniques- Tolrrancement gromrtrique- Toldrancement de localisation', ISO, 1987.

a matrix approach to the representation of tolerance zones and

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