Vectorial Method of Minimum Zone Tolerance for Flatness, Straightness, And Their Uncertainty...

INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 15, No. 1, pp. 31-44 JANUARY 2014 / 31

© KSPE and Springer 2014

Vectorial Method of Minimum Zone Tolerance forFlatness, Straightness, and their Uncertainty Estimation

Roque Calvo1,#, Emilio Gómez2, and Rosario Domingo3

1 Department of Mechanical Engineering and Construction, Universidad Politécnica de Madrid, Ronda de Valencia, 3; 28013 Madrid, Spain2 Department of Mechanical Engineering and Construction, Universidad Politécnica de Madrid, Ronda de Valencia, 3; 28013 Madrid, Spain

3 Department Construction and Manufacturing Engineering, Universidad Nacional de Educación a Distancia (UNED), Juan del Rosal, 12; 28040 Madrid, Spain# Corresponding Author / E-mail: [email protected], TEL: +34 913367465. FAX: +34 913367676

KEYWORDS: Flatness, Form tolerance, Minimax problem, Minimum zone, Planar straightness, Measurement uncertainty

Flatness and planar straightness are fundamental form tolerances in engineering design and its materialization through

manufacturing processes. Minimum zone tolerance is a preferred approach of flatness and straightness for widely accepted ISO and

ANSI standards. In this paper, we propose a novel accurate method of minimum zone tolerance based on vectorial calculus of point

coordinates. The non-linear minimax formulation of the original flatness or straightness problem is transformed into a set of linear

problems. Next, the optimal solution of the envelop planes or lines is reached through vectorial calculus for both flatness and planar

straightness. Then, the developed algorithms are compared to a selection of methods with published tests in recent and classic

literature on the topic, reaching the best attained accuracies or outperforming them in the trials. Finally, we propose a new

decomposition of the uncertainty contributions for analysis and the improvement of sampling strategy. We conclude remarking the

practical contributions of the proposals.

Manuscript received: July 29, 2013 / Accepted: November 27, 2013

1. Introduction

Flatness and straightness are both fundamental tolerances of form in

precision design and manufacturing engineering, for product

dimensioning and its verification through direct measurement, or as a

support to verify other specifications. The tolerances of prismatic parts

are ordinary referred to a datum, plane or line, idealization of a physical

plane or its orthogonal projection. The measurement of angular

magnitudes or squareness is also subject to the determination of planes

or lines and their tolerances. In machining, not only the tolerance of the

parts, but those of the supporting tooling and the machine tool itself are

involved in the manufacturing process capability and its control for

specification compliance.1

Relative deviations of physical planes or lines are often controlled

in the job shop with analogical dial instruments, displacing its probe on

the surface, but today digital measurement on coordinate measuring

machines (CMM) are widespread for product verification. The CMM

uses a set of discrete points coordinates that is evaluated to determine

a plane or a line. The standards ISO 11012 or ANSI Y14.53 establish

the minimum zone (MZ) as a preferred criterion of tolerance of form,

before other used methods like the least-squares. The standards consider

flatness (planar straightness) the minimum distance between two parallel

planes (lines) with all the set of points between them -envelop planes

or lines-. Even so, the standards do not establish explicitly how to

determine the minimum zone. Conversely, the least squares criterion is

univocally defined by its mathematical formulation and direct solution.

In addition, the least-squares criterion conveys the statistical power of

maximum likelihood in the optimization, when the measured point set

is treated statistically. The interval of confidence of the model estimators

-plane or straight line coefficients- complements the analysis of

uncertainty in its statistical approach. Noteworthy, the widely accepted

approach to express measurements and their uncertainties lays on the

statistical treatment of the variables, where the least-squares criterion

finds also its own roots. Therefore, due to several reasons including its

easy calculation, least-squares algorithms are generally in the routines

of CMM machines. Nevertheless, the physical interference and contact

of surfaces are determined by the outstanding points of the plane, so the

minimum zone criteria responds better to functional tolerancing4 and

fitting.

The MZ criterion is referenced in early works by Chetwynd5 in the

research of dual conditions of linear programming models. Antony et

al.6 confirmed that at least 4 measured points must be included for

DOI: 10.1007/s12541-013-0303-8

32 / JANUARY 2014 INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 15, No. 1

flatness assessment: In two different possible configurations, 2-2 and 3-

1, depending on the minimum number of points lying on each of the

two envelop planes. In order to ensure an exact MZ solution, these

necessary conditions would require evaluating all possible configurations

from a given data set in a combinatorial hard problem. These

configurations will be revised later in the construction of the proposed

algorithm.

According with the ISO or ANSI standards, the determination of the

MZ for flatness involves not only the definition of the envelop planes

from the data points measured, but also its associated uncertainty. Since

the set of points is a sample from the whole surface, the strategy of

sampling includes the path of measurement and the number of points.

Previous research shows7 that the number of points has a higher

influence than the path of sampling in the results uncertainty. Due to

the close relationship of the number of measured points with the

measuring cost, several works8,9 have sought the reduction of the

sample size for a desired precision.

This paper is primary dedicated to the process of calculating the

flatness or planar straightness and its uncertainty after proper sampling.

Looking for an accurate calculation of flatness, several methods have

been developed since the 1980’s that can be classified in a first instance

in two main categories:10 Based and non-based on computational

geometry. We can find other classifications11 where computational

geometry methods are at the same level of least-squares methods or

recent meta-heuristics. The first taxonomy by Lee10 is preferred and his

review is followed and complemented in the next paragraphs, because

the research line associated with computational geometry has provided

reference values to other new or improved methods.

We can mention among the non-computational geometry methods:

• Direct least squares12 or weighted least squares.13 Its MZ interval

is ordinary overestimated, when the data points are not well aligned

with a CMM coordinate axis. Shunmugam14 proposed a method based

on the median, more robust estimator that the mean.

• Non-linear optimization techniques. Its goal is minimizing the

maximum distance from an ideal reference plane or point. Some works

in the field are those by Shunmugam,15 Wang,16 Kaiser and Krishnan,17

Damodarasamy and Anand18 or Cheraghi et al..19 A characteristic point

of these non-linear search methods is the non-convexity of the

optimization problem and the need of several trials to look for a global

optimum.

• Approximation methods. Based on linear programming, in some

cases they sacrifice accuracy for an easy implementation. We find in

this area the already mentioned work of Chetwynd.5 In addition,

Prakasvudhisarn,20 Car and Ferreira,21 Weber et al.22 or Zhu and Ding.23

• Exchange methods. They construct the solution in a sequential search

of a better solution replacing a current set of points by a new one. The

works of Fukuda and Shimokohbe,24 Huang et al.,25 Deng et al.,26

Danish and Shunmugam27 or Burdekin and Pahk28 are between them.

• Meta-heuristic methods. They have been applied to the minimum

zone problem after successful use in difficult combinatory problems. In

this category we find the genetic algorithm (GA) of Sharma et al.,29 Cui

et al.30 or Liu et al.31 combining GA with geometric calculation, particle

swarm optimization by Kovvur,32 and the gradient ascent approach of

the evolution algorithm by Malyscheff et al..33 Some weakness of these

methods could be the non-convergence to the exact solution or the

computation difficulties facing big data sets.

As a main second type of methods, computational geometry tackles

the problem through the construction of a convex hull and the

identification of the smallest convex domain including all the points.

Works in this field includes Houle and Toussaint,34 Anthony et al.,6

Traband et al.,35 Samuel and Shunmugam,36 Hermann37 and Lee.10 These

methods based on convex hull enumerate all the possible solutions

looking for an accurate result.

For planar straightness many authors have tried to adapt the same

spatial algorithm used for flatness, but others have tried to get benefit

directly from the 2D geometry, like the iterative method by Danish and

Mathew.38

In this context, our research in the tolerance of form and fitting is

looking for analytic solutions.39 We propose a new method that is

developed in Section 2, including its application for planar straightness

in Section 3. Then, we apply it in Section 4 for performance evaluation

to data sets from the literature, focused on the most accurate methods

referenced above. Across Section 5 we estimate the uncertainty

associated to flatness evaluation, in a breakdown useful for sampling

strategy improvement. Finally, in Section 6 we evaluate the results with

concluding remarks.

2. Vectorial Formulation of the Minimum Zone for Flatness

We consider the minimum zone for flatness where the envelop

planes must comply with:

• At least one point of the data set must lie on each envelop plane.

Suppose by the moment that p1 and p2 are the envelop planes of minimum

zone at a minimum distance d and they do not contain at least one point

of the data set. In this case, we can always find a pair p’1 and p’2 of

parallel planes to p1 and p2 at a distance d’ < d, by supporting p’1 and

p’2 on the two points between p1 and p2 of minimum distance to p1 and

p2 respectively. Thus, p1 and p2 would not be the envelop planes of

minimum zone, in contradiction with the hypothesis. Therefore, p1 and

p2 must contain at least a point of the data set each one.

• The envelop planes contain 1, 2 or 3 points of the data set under

the configuration 2-2 or 3-1. The identification of these configurations

appears in the early works in the field.5,34 A plane is a vectorial subspace

of dimension two, so it can be defined by two non-co-linear vectors,

configuration 2-2. Otherwise the configuration 1-3 corresponds to the

definition of a set of two planes determined by one point on each one

and the common normal direction to the planes. This direction is

determined directly by three points, equivalent to the normal vector

result of the cross product of two vectors defined by those three non-

co-linear points. In the case of co-linearity, the three points of the

configuration just determine a line, so adding a fourth non co-linear

point will univocally define the plane.

Once we have approached to the envelop planes from a geometric

point of view, we can formulate the optimization problem. Noting by fp

the distance from the origin to each parallel plane that contains the point

p, the minimum zone problem for the data set of n points is the solution

of (1), formulation equivalent to (2).

, (1)min max fi fj–( ) i∀ j, 1 … n, ,=

INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 15, No. 1 JANUARY 2014 / 33

/ ; (2)

From (2) we derive (3), see Appendix A, as considering the function

of the distance continuous and at least C2. Note that the ordinary general

consideration of the measured points as a sample from a normal

distribution or other continuous distribution includes an even harder

hypothesis of C∞. This expression incorporates in the formulation a

supporting plane between the two envelop planes of the solution, in fact

its middle plane.

, (3)

⇔

Thus, we can substitute the problem (2) by its equivalent (4)

/ and ; (4)

The problem (4) is equivalent to (5), deduced in (6)

, ; (5)

subject to / is minimum

(4) ⇐ (5) because (2) ≡ (4)

(4) ⇒ (5) ; (6)

with

that is

but (4) ≡ (2), so is min.

⇒ is min. for (m, s) = (k, l)

Noteworthy, the condition (m, s) = (k, l) in (6) departs from the

solution of (4), but in the proposed algorithm we will resolve the

problem (5), where in general (m, s) ≠ (k, l).

We are proposing passing from one L∞ minimax problem (1) to n(n-

1)/2 minisum problems of L2 by (5). The underlying idea of the problem

reformulation is as follows: In the least-squares problem we look for

the solution of a supporting plane f* between the points, so the sum of

residuals fi- f* is minimum. The solution of f* is the mean plane. We

do not minimize any pair of residuals, but the sum of all of them (7).

⇔ (7)

In the minimum zone problem we try to minimize the distance

between two parallel envelop planes. The first step of the problem

consists on determining a pair of points and the minimum distance on

the envelop planes. It is accomplished creating a supporting plane that

links both conditions. The middle plane of those two planes is the

minimum optimum of the unconstrained formulation (3). Then, in a

second step we incorporate the rest of the points -constraint in (5)-

optimizing the residuals about this plane. We obtain from the n data set

points up to n(n-1)/2 pairs that enumerate all possible (m, s) non-

ordered pairs. We take by (5) the minimum distance solution /fk-fl/ for

the envelop planes, out of the n(n-1)/2 linear problems expressed by the

constraint. This condition contains the optimization of residuals around

the optimum solution of a pair and we will inspect every possible pair

looking for the minimum envelop.

In plain terms, we construct the solution around the unconstrained

solution of the minimum zone defined only by two points and a

supporting plane. The solution to the supporting plane is their middle

plane. So, let us enforced the rest of the points close around this plane,

minimizing its residuals around it, in order to obtain the minimum

distance between the extreme envelop planes. Nevertheless, it is not

evident that this supporting middle plane is not necessarily established

by the extremes k,l that define the envelop planes, but in general by

a different pair of points m,s of the data set. The demonstration passing

from (1) to (5) formalizes it.

The first part of the whole proposed algorithm is the n(n-1)/2 linear

problems expressed in (5), looking for the minimum distance between

the envelop planes. It allows identifying one point on each envelop plane.

The formulation of the constraint of (5) in the case of flatness is expressed

in (8), where the solution (a, b, c) is a normalized unitary vector and the

constraint is the relationship between the direction cosines of the planes

in the orthonormal base of the reference system.

Plane that contains the point p of coordinate s

(8)

=

min fk fl–( )2 fk fp fl≤ ≤ p∀ 1 … n, ,=

min fk fl–( )2[ ] 2 min fp f*–( )2p

2

∑⋅= p k l,=

f*fk fl+

2-----------=

min fp f*–( )2p

2

∑ f*fk fl+

2-----------= fk fp fl≤ ≤ p∀ 1 … n, ,=

min fk fl–( )2[ ] fk fp fl≤ ≤ p∀ 1 … n, ,=

m∃ s, 1 … n, ,[ ]∈ fpfm fs+

2-------------–⎝ ⎠

⎛ ⎞2

p

2

∑

fk fp fl≤ ≤

fk f*– fp f*– fl f*–≤ ≤ f*fk fl+

2-----------=

n fk f*–( )2 fp f*–( )p

n

∑ n fl f*–( )2≤ ≤

n

4--- fk fl–( )2 fp f*–( )2

p

n

∑n

4--- fk fl–( )2≤ ≤

4

n--- fp f*–( )2

p

n

∑ fk fl–( )2=

fk fl–( )2

fp f*–( )2p

n

∑

min fp f*–( )2[ ]p

n

∑ f1

f*–( )2 … fn f*–( )2+[ ]= f*

fpp

n

∑

n---------=

xp0

yp0

zp0, , axp

0byp

0czp

0+ + fp=

min fpfm fs+

2-------------–⎝ ⎠

⎛ ⎞2

p

n

∑

min a xp0 xm

0xs0

+

2---------------–⎝ ⎠

⎛ ⎞ b yp0 ym

0ys0

+

2---------------–⎝ ⎠

⎛ ⎞ c zp0 zm

0zs0

+

2--------------–⎝ ⎠

⎛ ⎞+ +

2

p

n

∑

Fig. 1 The possible configurations of 4 points on the envelop planes,

departing from one point on each envelop plane Pk, Pl


subject to

The minimization problem (8) can be expressed as a constrained

singular linear problem (9), where the constraint G allows discarding the

trivial solution a = b = c = 0 from the singular linear problem. Then, we

formulate the minimization problem by Laplace multipliers (10).

; ; (9)

; subject to

(10)

where ; ;

As a result, the minimization problem is transformed into an

eigenvector problem of the matrix . This matrix admits the

decomposition, also used in least-squares matrix formulation, known as

the normal form, (11). Note that any real mxn matrix M can be

decomposed uniquely as M = UHV T, where V is nxn orthogonal and its

columns are eigenvectors of MTM, because (MTM = VHUT.UHVT =

VH2VT) , where H = diag(σ1, σ2, …, σn) ordered so that σ1 ≥ σ2 ≥…

≥ σn. In consequence, if S is a singular value of M, its square is an

eigenvalue of MTM.

The existence of the singular value decomposition of a rectangular

matrix allows to state that the squares of the singular values of D are

the eigenvalues of DTD = . Multiplying in (10) each row by a, b

and c respectively, we obtain (12). The value of 2λ is the value of the

function we want to minimize (8). We look for the minimum solution

of the singular linear problem, so we take the minimum of those three

eigenvalues. Its corresponding eigenvector represents the solution B,

the normal vector to the envelop planes. In order to get the solution to

(10) the direct inversion of is not necessary, using the singular

value decomposition of by (11). This avoids a possible bad

conditioning of . For speedy calculation with increasing size

problems, we used in our algorithm directly the SVD or the 3×3 matrix

DTD to get the eigenvalues.

(11)

vm,s the eigenvecto r associated with

Solution / m,s i,j

(12)

We have finally reduced the optimization problem (5) to

eigenvectors problems. It requires the solution of n(n-1)/2 linear

problems retaining the solution that gives the minimum distance

between the envelop planes of the data set.

The first part of the algorithm has allowed identifying one point on

each envelop plane as a first necessary condition. The second part

completes the construction, looking for the other two points that

defines the solution out of the four configurations of minimum zone

already analyzed. This second task is accomplished by evaluating the

(n-2)(n-3)/2 possible pairs from the rest (n-2) points. The distances

between the envelop planes are dA … dD. Next, we do simply to

calculate with the four configurations of Fig. 1 by vectorial calculus

(13), retaining the configuration of minimum zone.

; ; ; (13)

;

It is evident that the four conceptual configurations are not compatible

between them, but they just enumerate all possible solutions based on

adding two points i, j to the initial two points k, l identified in the first

step. Before using a pair candidate for minimum in (13), the configuration

is checked for compliance of the envelop condition -first part of (5)-.

The two envelop planes are defined by the four points (i, j, k, l).

3. Specifics for Planar Straightness

The proposed algorithm in the former section is directly applicable

to planar straightness, by adapting the eigenvector problem and the

vectorial calculation to a bidimensional problem.

The constraint of (5) is formulated by (14)

Distance form the origin to the straight line in the plane XY

that contains the point p of coordinates

(14)

Where the vector (a, b) is normal to the line and normalized

a2

b2

c2

1–+ + 0=

xp xp0

xp0

p

n

∑–= yp yp0

yp0

p

n

∑–= zp zp0

zp0

p

n

∑–=

∇Am s, B⋅ 0= Ba

b

c

= G: a2

b2

c2

1–+ + 0=

∇A m s,

xp x–( )2

p

n

∑ xp x–( ) yp y–( )⋅p

n

∑ xp x–( ) zp z–( )⋅p

n

∑

xp x–( ) yp y–( )⋅p

n

∑ yp y–( )2

p

n

∑ yp y–( ) zp z–( )⋅p

n

∑

xp x–( ) zp z–( )⋅p

n

∑ yp y–( ) zp z–( )⋅p

n

∑ zp z–( )2

p

n

∑

=

xxm xs+

2--------------= y

ym ys+

2--------------= z

zm zs+

2--------------=

∇A B⋅ λ∇G– B⋅ 0;=

subject to

G: a2

b2

c2

1–+ + 0 ∇G⇒2 0 0

0 2 0

0 0 2

= =

⎭⎪⎪⎪⎬⎪⎪⎪⎫

∇A B⋅⇒ 2λB=

∇A

∇A

∇A

∇A

∇A

∇A B⋅ DT

D B⋅ ⋅ 2λ;= =

Dm s,

x1

xm xs+

2--------------– y

1

ym ys+

2--------------– z

1

zm zs+

2--------------–

… … …

xpxm xs+

2--------------– yp

ym ys+

2--------------– zp

zm zs+

2--------------–

=

min eigenvalue Dm s,

TDm s,( )[ ]

k l,( ) i j,( )= min maxvm s,

vm s,

----------- PiPj⋅⎝ ⎠⎛ ⎞

a xpxm xs+

2--------------–⎝ ⎠

⎛ ⎞p

n

∑ b ypym ys+

2--------------–⎝ ⎠

⎛ ⎞p

n

∑ c zpzm zs+

2--------------–⎝ ⎠

⎛ ⎞p

n

∑+ +

2

= 2λ a2

b2

c2

+ +( ) 2λ=

nij

Pi Pj

Pi Pj

------------= dA

nki nkj×nki nkj×--------------------= dB

nli nlj×nli nlj×-------------------=

dC

nki nlj×nki nlj×--------------------= dD

nkj nli×nkj nli×--------------------=

xp0

yp0,

axp0

byp0

+xp0

1/a--------

yp0

1/b--------+ fp= =


subject to

The corresponding eigenvalue problem is (15)

; (15)

vm,s eigenvector associated with min[eigenvalue(Dm,sT Dm,s)]

Solution / i,j,m,s = 1,…,n m,s i,j

Note that in the case of planar straightness the solution vector (a,b)

is the normal to the envelop lines and it is defined by the eigenvector

corresponding to the smaller of the two eigenvalues of DTm,n Dm,n Where

(m,n) is the pair, out of the n(n-1)/2 possible, that gives the minimum

zone of the envelop lines defined by k,l, that is the first statement of (5).

The second part of the algorithm requires the examination of only

n configurations result of adding one more point to the pair. We can

transpose the reasoning about the configurations for flatness, to state:

- At least one point of the data set must lie in each envelop line.

- The envelop lines contain at minimum 1 or 2 points of the data

set under the configuration 1-2 or 2-1.

The vectorial calculations in the planar straightness case are given

by (16) that correspond to the geometrical configurations of Fig 2.

; (16)

;

Solution defined by 3 points [i, k, l] / min[min(dA, dB)], i

A blocks diagram of the algorithm for flatness is sketched in Fig. 3.

The planar straightness algorithm can be derived immediately from it.

In this work, both algorithms have been coded in Matlab for an easy

programming.

4. Algorithm Testing

The proposed algorithm, Vectorial Minimum Zone (VMZ), is tested

in Table 1 with datasets from the literature and the reported results of

selected methods are compared with those obtained by the VMZ

method.

We note VMZ reaches or surpasses the precision of all the methods

in the comparison: Those based on meta-heuristics or least-squares, but

also the geometry computational methods based on convex hull.

In particular, it can be mentioned the computational precision

inconsistencies found in a recent improved convex hull method,

min fpfm fs+

2-------------–⎝ ⎠

⎛ ⎞2

p

n

∑ min a xp0 xm

0xs0

+

2---------------–⎝ ⎠

⎛ ⎞ b yp0 ym

0ys0

+

2---------------–⎝ ⎠

⎛ ⎞+

2

p

n

∑=

a2

b2

1–+ 0=

∇A B⋅ 2λB= DT

D B⋅ ⋅ 2λB=

Dm s,

x1

xm xs+

2--------------– y

1

ym ys+

2--------------–

… …

xpxm xs+

2--------------– yp

ym ys+

2--------------–

=

k l,( ) i j,( )= min maxvm s,

vm s,

----------- PiPj⋅⎝ ⎠⎛ ⎞ ∀

nij

Pi Pj

Pi Pj

------------ nxij nyij,( )= = n⊥ij nyij– nxij,( )=

dA n⊥li Pk Pl⋅= dB n⊥ki Pk Pl⋅=

i∀ 1 … n, ,[ ] k l,{ }–∈

Fig. 2 Minimum zone defined by 3 points on the envelop lines, in 2

configurations

Fig. 3 Flow chart of the VMZ algorithm for flatness


ECHEM. For instance, one of the results marked with (*) in Table 1:

They reported minimum zone is 0.0261282, but the reported points

that define the envelop planes are (3,19)-(4,11), in agreement with the

VMZ solution.

Based on these four points coordinates the minimum zone can be

calculated as 0.0236128872, by (17).

; ; (17)

In the Table 2 we apply the VMZ algorithm for planar straightness

and we reach the better solutions found to the different problems,

giving more meaningful decimal places.

Reported by Zhu,23 the faster algorithm that guarantees the minimum

zone of flatness was presented by Houle and Toussaint34 with complexity

O(n2) in the worst case, while Traband35 is O(n3) in the worst case

p3

p19

,( ) 1.7277 2.4538 13.6542–, ,[ ] 0.1132 0.3538– 13.6468–, ,[ ],( )=

p4

p11

,( ) 3.1273 2.4538 13.6531–, ,[ ] 0.1095– 0.5439 13.6451–, ,[ ],( )=

x3 19, p

3p19

–= x4 11, p

4p11

–= x3 4, p

3p11

–=

minimum zonex3 19, x

4 11,×x3 19, x

4 11,×----------------------------- x

3 4,⋅ 0.0026128872= =

Table 1 Trials of VMZ for flatness and comparison


Lee.41 Other methods have been reported23 with lower complexity than

O(n2), but only approximate algorithms.

The VMZ algorithm for flatness guarantees the MZ solution based

on its construction. We transform the original minimax formulation into

n(n-1) eigenvector problems, inspecting all of them to yield the better

possible MZ based on the demonstrated equivalence to the problem (5).

It is built-in two loops, Fig 3. The first loop requires the resolution of

n(n-1)/2 eigenvector problems. It catches the two pivot points in the

envelop planes. The second loop requires the inspection of the minimum

zone value after adding 2 additional points through vectorial calculus,

so (n-2)(n-3)/2 pairs inspection. This second loop inspects all the

possible solutions, retaining the best one. The complete algorithm

conceptually brings approximately to the order O(n2). After preliminary

testing for speed of our Matlab code, the computing time increases by

O(n2.2) for flatness and O(n2) for planar straightness, function of the

sample size n. We note that speed improvement would be a next step

by programming optimization or low-level programming language.

5. About the Estimation of Flatness and Straightness

Uncertainty

Flatness or planar straightness is the subject of measurement.

Nevertheless, we determine the position of the points with a coordinates

measuring machine CMM and the flatness value under the minimum

zone criteria is the output of an algorithm, so it is an indirect measure

of flatness. The standard way of specifying a measurement includes the

better expected value and its uncertainty. When a direct measurement

is made, for instance the coordinates of a point, the mean value from

the sample of m measures is taken as the best expected value. The

estimation of the variance of this mean is the uncertainty, directly from

the distribution, when known, or estimated from the experimental

standard deviation of the sample.

Under the GUM, the ISO Guide of Uncertainty Measurement, the

expected value of flatness could be approached by the mean flatness of

m samples. The uncertainty can be calculated by the uncertainty

propagation law by GUM or the alternative Monte Carlo Method

(MCM). The last one propagates the uncertainty distribution of the

points through the model -plane or straight line equation- to estimate the

uncertainty of flatness or planar straightness. Noteworthy, both GUM

and MCM requires a function or model of the mensurand that links its

uncertainty factors. Nevertheless, uncertainty comes directly from the

distribution of the mensurand in the case of a direct measure. Both, direct

and indirect measurement are conciliated under the same framework by

considering that the calculated flatness through the algorithm is the

mensurand and the measured points uncertainty is a contribution to

flatness or straightness uncertainty. Other factors contributing to

uncertainty will come from the model or the sampling procedure.

Flatness or straightness assessment will be expressed by the calculated

value through the proposed algorithm and the estimation of its

uncertainty. Not only the total number of data points or the size of each

sample will influence the uncertainty. The form under the minimum

zone criteria is determined by extreme values in the tails of the

distribution of points, so it is difficult to collect them in a small sample.

Obviously, the spatial strategy of sampling on the specimen surface

will condition the results. Diverse techniques of surface sampling has

been studied and proposed for an effective process, trying to catch the

form boundaries of the surface with small samples. An up-to-date brief

revision of the methods and last advances can be found in Rosa48, along

references to the different methods. Our work does not deal with this

aspect of an efficient sampling strategy for flatness or straightness

evaluation. So forth, we consider that a proper strategy of sampling has

been adopted to make the sample useful, representative and valuable

for flatness or straightness evaluation. Otherwise, we will probably

have a bias in the flatness value, in despite of the accuracy of the

minimum zone algorithm and the estimated uncertainty. Complementary

to the better flatness figure, its uncertainty estimation establishes an

interval where the flatness value is probably present.

We can express the uncertainty in a privileged reference system,

oriented by the mean direction of the m normal vectors to the envelop

planes solution of m samples (18). We develop in the Appendix C the

expression of uncertainty in this particular reference system. The input

variables are the position vector coordinates of two points in the envelop

planes and the components of the normal vector to the envelop plane.

The output variable is the minimum zone gap f.

(18)

This expression of uncertainty contains clearly separated the influence

factors: The point coordinates uncertainty estimation in the direction of

the normal to the surface z’ from the CMM, and the model and sampling

influence. We call this last contribution the parallel component, in a

resemblance of the expression (18) to the vector module from the

decomposition in two orthogonal directions. Note that the normal to the

surface is part of the solution. It also contains the influence of the number

u2

2uz'1

2x'1

x'2

–( )2uA'

2y'1

y'2

–( )2uB'

22 x'

1x'2

–( ) y'1

y'2

–( )uA'B'

2+ + +=

=u⊥

2CMM( ) u//

2MODEL mn,( )+

Table 2 Trials of VMZ for planar straightness and comparison


of the samples used to evaluate the covariances. We can determine the

variances and covariances of the normal vector from a set m samples49

using the same sampling method.

The uncertainty of the CMM is the uncertainty of the measured

points, but the model incorporates additional uncertainty to the

mensurand. In practical terms, we would like to estimate whether

extensive sampling could be worthy to proper uncertainty estimation,

in addition to the CMM contribution.

As an example of application, we take the 3 datasets used in a recent

work. Its origin is an earlier work46 by Huang, who reports an uncertainty

estimation of 0.005. The same data sets have been used by Wen et al.44

presenting the minimum zone uncertainty following the ISO standards.

They apply a GA (their results are included in Table 1) and estimate the

uncertainty based on those 3 sets of 25 points under the same sampling

conditions of the same surface, using the uncertainty propagation law

by GUM and MCM. The minimum squares criterion of Batchman49 is

followed to calculate the variance and covariance. Finally, they report

under GUM a mean flatness of 0.0187 and expanded uncertainty of

0.0041 coverage factor 2. When using MCM the reported result is a

mean flatness of 0.0187 and estimated uncertainty 0.0042 coverage

factor 2. The standard uncertainty of the CMM is 0.0015, equal on the

three axis of the machine.

We use the propagation of uncertainties by (18) to evaluate the former

example of 3 samples (19). Note that after changing the reference system

we evaluate the sensitivity coefficients in the mean solution of the input

variable vectors estimated from the m samples. We approach f evaluated

in the points of the real solution (x’1 and x’2) nearest to the mean

minimum zone solution.

For low or moderate low number samples, the GUM50 points out,

paragraph 4.3.2, that the bias of the variance estimated from the

experimental standard deviation could be important for a small number

of samples.

Evaluation of flatness based on m=3 datasets of n=25 points each one.

Meand flatness =0.01867538

Two points on the envelop planes of the first dataset solution (f=

0.0189)

x1=(34.9974 54.9996 3.9987); x2=(34,9993 104,9982 4,0201)

The normal vectors (a, b, c) to the envelop planes for the 3 samples

are in Table 1.

The mean vector is

(19)

From the Appendix A we obtain

The normal vectors of the m=3 samples and envelop planes points

in the new reference system

From which we obtain for m=3

; ;

; ;

For

and

In many practical cases and related with the cost of sampling, the

number of samples will be low. In our example, we use just 3 samples

to estimate de variances of the vectors. The bias of the experimental

standard deviation was studied by Craig.51 It has been revisited by

Huang,52 showing from simulation a significant lower bias of the

uncertainty in the Craig model than the ordinary estimation based on a

t-Student population, in special for small samples. In accordance with

this simulation study, the calculation of the variances and covariances

of the model parameters would admit an alternative calculation to the

standard GUM of lower bias, adopting the Craig model (20), where Γ( )

stands for Gamma function.

; s is the experimental standard deviation

; ; (20)

In Table 3, we present the results of uncertainty based on the example

datasets, calculating variances with the Craig model and under three

different sampling setups (N=mxn). Uncertainty could be reduced to a

half sampling up to 75 points instead 30 points. A sampling setup N=70

with m=3 would allow an estimation of the uncertainty that can be

improved by additional sampling no more than 17%.

Noteworthy, the lower limit of the uncertainty is established by the

fix contribution from the CMM in the normal direction to the surface

u(CMM). The 3 dataset of n=10 has been generated retaining randomly

f

n 3.6656.105– 4.2588.10

9– −1( ) A B 1–, ,( )= =

CT

C1–

4.9030.101–

– 8.7155.101–

– 3.6656.105–

8.7155.101–

– 4.9030.101–

6.5158.105–

7.4761.105–

– 0 1–

= =

n'1

8.7847.106– 5.9084.10

6– 1( )=

n'2

5.3396.106– −1.1861.10

6– 1–( )=

n'3

3.4440.106– −4.7223.10

6– 1–( )=

x'1

65.0946 −3.5355 −3.9938–( )=

x'2

108.6719 20.9774 −4.0120–( )=

uA'

2

A'i2

i

m

∑

m m 1–( )-------------------- 1.9591.10

11–= = uB'

2

B'i2

i

m

∑

m m 1–( )-------------------- 7.9666.10

12–= =

uA'B'

2

A'iB'ii

m

∑

m m 1–( )-------------------- 1,0953.10

11–= =

x'i x'2

–( )2 1898.98= y'1

y'2

–( )2 600.88=

x'1

x'2

–( ) y'1

y'2

–( )⋅ 1068.21–=

uz'1 0.0015=

u2

2uz'1

2x'1

x'2

–( )2uA'

2y'1

y'2

–( )2uB'

22 x'

1x'2

–( ) y'1

y'2

–( )uA'B'

2+ + +=

=u2

CMM( ) u2

model m n, ,( )+ 4.50.106–

3.0289.108–

+=

u 0.00213= Uk=2 0.00426=

u2 m 1–

2-----------

Γm 1–

2-----------⎝ ⎠⎛ ⎞

Γm

2----⎝ ⎠⎛ ⎞

--------------------

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

2

s2

=

uA'

2 m 1–

2-----------

Γm 1–

2-----------⎝ ⎠⎛ ⎞

Γm

2----⎝ ⎠⎛ ⎞

--------------------

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

2

A'2

i

m

∑

m 1–------------= uB'

2

Γm 1–

2-----------⎝ ⎠⎛ ⎞

Γm

2----⎝ ⎠⎛ ⎞

--------------------

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

2

B'2

i

m

∑

2------------=

uA'B'

2

Γm 1–

2-----------⎝ ⎠⎛ ⎞

Γm

2----⎝ ⎠⎛ ⎞

--------------------

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

2

A'B'i

m

∑

2---------------=


10 points from each sample of n=25.

Reducing the number of n points has an important effect in increasing

the uncertainty contribution of the model. In other terms, for a low n

each dataset is a poor representation of the real surface, so the variation

of the normal vectors of the envelop planes of each flatness solution

generates high uncertainty. In other cases, the high relative contribution

of the model could be due to the relative low contribution of a high

quality CMM machine, so the total uncertainty could be highly sensitive

to the sampling setup. The main influence on uncertainty of the sample

size n over the spatial sampling strategy has been formerly pointed out

in other studies.7 The ratio between the parallel component (model and

sampling contribution) and the normal component of the uncertainty is

a measure of the variability of the normal vectors found in the m

samples.

The uncertainty expression (18) makes explicit that the minimum

uncertainty is finally established in its lower bound by the uncertainty

associated to the points that define the envelop planes. The asymptotic

solution with increasing sample size (n→∞) will lead to the progressive

reduction of the variance parallel to the surface that is associated to the

model and sampling. In fact, all our analysis of flatness departs from

the assumption that the surface or its normal direction exists (‘true

value’) univocally defined by the outstanding points of the surface after

proper measurement and data processing. The limit situation would be

(21).

(21)

The minimum zone value is the minimum distance between the

envelop planes and it is measured by its normal direction, so the gap

uncertainty will be reduced to the normal component when the envelop

planes are finally determined (n→∞). That is, we will have no parallel

component of the uncertainty, when a sufficient sampling is provided.

This result could correspond to the following intuitive heuristic

geometrical interpretation:

The definition of uncertainty is the dispersion of the mensurand as

a stochastic variable. The solution to the flatness problem gives the

minimum zone envelop planes, defined by their common normal vector,

where at least four points lie in the envelop planes. The distance between

two points of both planes is the same measured in the direction of the

normal to the plane, so only one point on each envelop plane is necessary

to determine the minimum zone value in that direction. The distance f

to each envelop plane will be the projection of the position vector of

the point on the normal direction to the plane (part of the solution), so

a stochastic variable. In an average isotropic behavior of the CMM, the

uncertainty is the same regardless the direction, but in other particular

cases the variance in the normal direction to the envelop planes will be

the relevant one. The variance of the difference fmax-fmin of two stochastic

variables independent equally distributed with variance so is the variance

of the convolution. In fact, this produces the same rule of uncertainty

propagation than GUM with sensitivity coefficients 1, for the very simple

model of the difference of two input variables that are independent, (22).

(22)

Where

This plain reasoning includes implicitly the fact that all the points

between the two envelop planes do not affect the uncertainty of the

solution for samples of sufficient size: When sampling with increasing

sample points to n→∞, the definitive or ‘real’ envelop planes should be

finally identified at some point during sampling and the probability of

finding a point outside the minimum zone tends to 0 afterwards. The

variability of the normal vectors to the surface becomes progressively

reduced, because once established the envelop planes, the next points

added inside the minimum zone to the sample will not modify the normal

vector, so the variances and covariance uA’, uB’ , uA’B’ will tend to 0, as

n→∞.

Applying it directly to the former example, with coverage factor k=2,

we calculate (23).

(23)

This simple calculation can be compared with the MCM reported

output of 0.0042 from a million shot simulation. The MCM is expected

to get the parent distribution of the output after propagating the point

distributions through the model. The standard recommendation of a

million shot run is an operative approximation of n→∞. Our

decomposition (18) has as a first and fix term the asymptotic

uncertainty, in fact the irreducible uncertainty of the width of the set of

points by the CMM uncertainty in the normal direction to the surface.

We retake the case samples and their results to remark a feature on

the standard MCM. We could point out the high quality of the 3 samples

of 25 points with mean flatness of 0.0187, the same than the result of

the simulated parent distribution through MCM. Conversely, we could

interpret it in a reverse manner. The possible bias of the flatness present

at the 3 samples remains in the MCM, because we simulate imposing

on every point of the sample a N(0, σ0) through each of the three axis.

Only the outstanding points on the envelop planes are required to

defining the minimum zone. The mean position of the points will be the

same than the initial sample, so flatness as the expected value or mean

of the distribution from a million shots is the same than the calculated

from the 75 original points. Note that the uncertainty calculated from

a simulated million shot on the distribution and from only 75 points are

both similar. The difference of using a larger sample across the whole

surface could represent the possibility of catching the more outstanding

picks and valleys. This will offer a lower biased value to minimum zone.

Meanwhile, the minimum zone value could be underestimated for small

samples even with a good value of the uncertainty interval. Increasing

the sample size with points between the envelop planes would not

improve the expected value of flatness and it would only reduce the

uncertainty to the lower limit of the CMM in the normal direction to

the surface. This decomposition makes explicit how the minimum zone

criteria of flatness is appropriated for relative low uncertainty CMM’s,

so the irreducible contribution of the CMM to the uncertainty interval

is small and it does not ruin the operative use of low flatness values

from very flat surfaces.

u2

2uz'1

2u⊥

2CMM( )= =

VAR MZ( ) VAR min max fi fj–( )[ ] VAR min fmax fmin–( )[ ]= =

=VAR f*max f*min–( )

=VAR f*max( ) VAR f*min( )+ 2σ0

2=

VAR fi( ) σ0

2= i∀ 1 … n, ,=

Uk=2 k VAR MZ( ) k 2 σ0

⋅ ⋅ 2 2 0.0015⋅ ⋅ 0.00424= = = =

Table 3 Uncertainty contributions by sampling size


As an overall assessment of the former considerations, the expression

of flatness will be influenced by the spatial strategy in the expected

value in relation to the chances of catching the outstanding picks and

valleys on the surface. Nevertheless, the number of sampled points is

a main influence factor of its uncertainty, with the lower limit of the

CMM uncertainty in the normal direction to the surface. Since the

assessment of the flatness or straightness tolerance is composed by the

best expected value and its uncertainty interval, both the sample size and

the spatial sampling path are important.

The reduction of the uncertainty with the sample size was a priori

expected, but the explicit decomposition (18) allows estimating how far

from the minimum uncertainty we are. This effect separation helps to

evaluate whether is worthy or not increasing the number of samples m

or the points per sample n for uncertainty estimation, because the

contribution in the normal direction to the surface -width of the dataset-

is its lower bound. This analysis is directly applicable -mutatis mutandis-

to planar straightness with a simplification of the equations, result of

reducing one dimension in the problem.

6. Concluding Remarks

We have introduced two accurate and fast algorithms for flatness and

planar straightness calculation from data point coordinates and proposed

a practical improvement for their uncertainty estimation. We demonstrate

the transformation from a hard minimax problem into minisum problems,

and finally to eigenvector problems. Its accuracy has been satisfactory

tested, with the same good results or outperforming many well-known

methods.

We revise their uncertainty estimation and in particular the separation

of contributions through a vectorial approach. In order to complement

or monitor an effective sampling strategy, we suggest an integrated

analysis of the uncertainty of the measured points (CMM), the number

of samples m and the size of the samples n. We apply it to the real and

practical situation of a low number of samples, looking for a low biased

estimate of its uncertainty. We finally remark that the proposed flatness

and planar straightness assessment rely on the algebra of vectorial

calculus that is a natural technique in dealing with the coordinates of

the points, as well as a familiar tool in engineering for practitioners and

researchers.

ACKNOWLEDGEMENT

This work has received financial support by the Spanish Ministry of

Economy and Competitiveness (Directorate General of Scientific and

Technical Research), under project DPI2011-27135 and the Industrial

Engineering School-UNED through the project REF2012-ICF03.

APPENDIX A

In the Euclidean space, noting by d(k, l) the distance between the

points k and l, in general

;

when the 3 points are colinear.

Noting by fp the distance from the origin O to each parallel plane

that passes through xp, with unitary normal n(a, b, c) of orientation n

from the origin to the plane,

By the definition of the distance to the family of parallel planes in

the direction to its common normal n

when

For a given pair of points k, l, and a direction n, it exists a point p*

which plane is at a distance f*, that makes

We prove , so

, ⇔

It can be decoupled in two statements

, ⇔ (a)

, (b)

Prove of (a)

, ⇔

Hipotesis fp is a C2 function R3→R, but only takes values in the

points of the dataset, that is a compact set of points (close and bounded).

Therefore the necessary condition of relative extreme or critic point of

this differentiable function is

;

⇒

f* is a relative extreme

and

⇒

At the relative extreme f*, d2fp is definite positive, and d3∆=0; that

is a sufficient condition of relative minimum, thus f*=(fk+fl) / 2 is a

relative minimum of a continuous function in a compact so by the

Weierstrass theorem the absolute min. of ∆.

The second statement (b). Given k, l and n. The minimum distance

between two parallel planes of normal n, that lie in k, l, is n(xk-xl)=|fk-fl|d k l,( ) d k p,( ) d l p,( )+≤ d k l,( ) d k p,( ) d l p,( )+=

d O x,( ) n x xp–( ) a x xp–( ) b y yp–( ) c z zp–( )+ + fp= = =

fk fl–( ) fk fm–( ) fm fl–( )+= fl fm fk≤ ≤

min fk fl–( )2 2min fk f*–( )2 fl f*–( )2+[ ]=

f*fk fl+

2-----------=


2

∑⋅= p k l,= f*fk fl+

2-----------=

min fp f*–( )2p

2

∑ p k l,= f*fk fl+

2-----------=


2

∑⋅= p k l,=

min fp f*–( )2p

2

∑ p k l,= f*fk fl+

2-----------=

min fp f*–( )2p

2

∑ min ∆( ) d∆⇒ 0= =∂∂fp------- fp f*–( )2

p

2

∑ dfp 0=

∂∂fp------- fp f*–( )2

p

2

∑ 0 2 fp f*–( )p=k l,∑–⇒ 0 f*⇒

fk fl+

2-----------= = =

dfp∂fp

∂x-------dx

∂fp

∂y-------dy

∂fp

∂z-------dz+ + a dx b dy c dz+ +[ ]= =

∂2fp

∂x2

---------∂2fp

∂y2

---------∂2fp

∂y2

---------∂2fp∂y∂x----------- …

∂2fp∂y∂z----------- 0= = = = = = d

2fp 0=

d2∆ ∂2

dfp2

--------- fp f*–( )2p

2

∑ dfp 2dfp= =


Thus

APPENDIX B

APPENDIX C

In the reference OXYZ of ortonormal base {e1, e2, e3}

Equation of the envelop planes of the sample i, Aix+Biy+Ci=z; Ci a

constant, i=1, …, m;

Where

minimum zone value or min. distance between two points of the

envelop planes (x1, y1, z1) and (x2, y2, z2).

The uncertainty expression retaining terms of the leading order

u2=fA2uA

2+fB2uB

2+fAfBuAB2+fx1

2ux12+fx2

2ux22+fy1

2uy12+fy2

2uy22+

fz12uz1

2+fz22uz2

2

The change to the reference system of base is

given D:

Where is a unitary vector by the normal direction to the mean

plane

; ;

min fp f*–( )2p

2

∑ fpfk fl+

2-----------–⎝ ⎠

⎛ ⎞2

p=k l,∑ fk

fk fl+

2-----------–⎝ ⎠

⎛ ⎞2

flfk fl+

2-----------–⎝ ⎠

⎛ ⎞2

+= =

=fl fk–

2-----------⎝ ⎠⎛ ⎞

2 fk fl–

2-----------⎝ ⎠⎛ ⎞

2

+1

2--- fk fl–( )2 1

2---min fk fl–( )2[ ]= =

fz1

z2

–( ) A x1

x2

–( )– B y1

y2

–( )–

1 A2

B2

+ +

-----------------------------------------------------------------------=

fx fx1 fx2–A–

1 A2

B2

+ +

----------------------------= = =

fy fy fy2–B–

1 A2

B2

+ +

----------------------------= = =

fz fz1 fz2–1

1 A2

B2

+ +

----------------------------= = =

fAx1

– x2

+

1 A2

B2

+ +

----------------------------A z

1z2

–( ) A x1

x2

–( )– B y1

y2

–( )–[ ]

1 A2

B2

+ +( )3/2

--------------------------------------------------------------------------------–=

fBy1

– y2

+

1 A2

B2

+ +

----------------------------B z

1z2

–( ) A x1

x2

–( )– B y1

y2

–( )–[ ]

1 A2

B2

+ +( )3/2

--------------------------------------------------------------------------------–=

O'Y'Z' e'1

e'2

e'3

, ,{ }

e'3

A

1 A2

B2

+ +( )--------------------------------

B

1 A2

B2

+ +( )--------------------------------

1–

1 A2

B2

+ +( )--------------------------------, ,

⎝ ⎠⎜ ⎟⎛ ⎞ e'

1

e'2

e'3

C

e1

e2

e3

⋅=

C

A–

1 A2

B2

+ +( ) A2

B2

+( )

------------------------------------------------------B–

1 A2

B2

+ +( ) A2

B2

+( )

------------------------------------------------------A2

B2

+( )–

1 A2

B2

+ +( ) A2

B2

+( )

------------------------------------------------------

B–

A2

B2

+( )

------------------------A

A2

B2

+( )

------------------------ 0

A

1 A2

B2

+ +( )

--------------------------------B

1 A2

B2

+ +( )

--------------------------------1–

1 A2

B2

+ +( )

--------------------------------

=


C is an orthogonal matrix in an orthonormal base so C-1=CT

And the change of coordinates or a vector components is given by

;

Where

;

In the reference of orthonormal base

Equation of the envelop planes of the sample i: ;

a constant, i=1, …, m;

The mean plane solution will be with

The partial derivatives evaluated at the mean plane

; ;

;

Thus

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x' y' z'[ ] x y z[ ] C1–⋅ x y z[ ] C

T⋅= =

A

Aii

m

∑

m----------= B

Bii

m

∑

m----------=

OX'Y'Z' e'1

e'2

e'3

, ,{ }

A'ix' B'iy' C'i+ + z'=

C'i

z' C'= A' B' 0= =

fx' fx'1 fx'2– 0= = = fy' fy'1 fy'2– 0= = = fz' fz'1 fz'2– 1= = =

fA' x'1

– x'2

+= fB' y'1

– y'2

+=

u2

fA'2uA'

2fB'

2uB'

2fA'fB'uA'B'

2fx'1

2ux'1

2fx'2

2ux'2

2+ + + +=

+fy'12uy'1

2fy'2

2uy'2

2fz'1

2uz'1

2fz'2

2uz'2

2+ + +

=2uz'1

2fA'

2uA'

2fB'

2uB'

22fA'fB'uA'B'

2+ + +

u2

2uz'1

2x'1

x'2

–( )2uA'

2y'1

y'2

–( )2uB'

22 x'

1x'2

–( ) y'1

y'2

–( )uA'B'

2+ + +=


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