An Approach for the Unified Evaluation of Orientation Tolerances

An Approach for the Unified Evaluation of Orientation Tolerances

G. Rajamohan, Ph. D

Dept. of Manufacturing Engineering National Institute of Foundry and Forge Technology

Hatia, Ranchi – 834 003 INDIA

Abstract – Orientation tolerances are specified to control the parallel, perpendicular and other angular relationships between two adjacent features of manufactured parts. One feature acts as datum feature and the other as measured or controlled feature.

Verification of orientation tolerances require the establishment of

two parallel planes at required parallel, perpendicular or angular A

1. A

   30o 45

   Datum A

   0.020

   30o

   orientation with respect to the datum feature and encompassing

   all the data points of measured feature with minimum spacing. In this paper, a new approach for unified evaluation of orientation tolerances of straight line features is proposed first followed by its implementation using the random walk, simplex search and normal least squares methods. The relative performances of these methods have been studied using simulated data.

   Keywords – Angularity; Datum Feature; Measured Feature; Normal Least Squares Method; Orientation Tolerances; Parallelism; Perpendicularity; Random Walk; Simplex Search. B

   12

   60 25

   (a) Angularity specification (b) Interpretation

   			0.020	B

   0.020

   Datum B

   1. INTRODUCTION

      The orientation tolerances are geometric tolerances that are specified with one or more datums. They are used to control the parallel (parallelism), perpendicular (perpendicularity) and

      30o 45

      12

      60 25

      90o

      other angular relationships (angularity) between two adjacent features in manufactured parts. Parallelism can be defined as the condition of a surface, median plane or axis being parallel to a datum plane or axis. Perpendicularity can be defined as the condition of a surface, median plane or axis being at right angle

      (c) Perpendicularity specification (d) Interpretation

      		0.020	C

      		0.020	C

      Datum C

      to a datum plane or axis. Angularity refers to the condition of a C surface, median plane or axis being at some specified angle to a

      datum plane or axis. In all cases, the tolerance zone is defined by two parallel planes established at 0° for parallelism, 90° for perpendicularity and specified angle for angularity with respect to datum plane or axis. The elements of measured or controlled

      25 45

      60 25

      0.020

      12 12

      feature must lie between these planes [1]. Fig. 1(a-b), Fig. 1(c-

      d) and Fig. 1(e-f) respectively show the example specifications of angularity, perpendicularity and parallelism tolerances and their interpretations.

      Verification of orientation tolerances on measured feature requires the establishment of an ideal datum feature, based on the measurement data of datum feature, using suitable methods. The standards recommend minimum zone evaluation but do not suggest any specific method for finding the minimum zone. Despite not guaranteeing the minimum zone solution, the least squares method (LSM) is commonly used for this purpose due to its sound mathematical basis [2-4]. Numerous algorithms for finding the minimum zone solutions have been reported. Such algorithms are based on some optimization, soft computing and geometry-based computational techniques. A review of some of these works is presented here.

      (e) Parallelism specification (f) Interpretation

      Fig. 1. Examples of orientation tolerances

      Computation of minimum zone form errors using numerical methods such as the Monte-Carlo method, discrete Chebyshev approximation, min-max approximation, simplex search, spiral search, median technique, etc. [4-11], enclosing polygon based methods such as the convex hull method, Eigen-polyhedral method, etc. [10, 12-14] and other methods such as control line rotation scheme [2], nonlinear optimization approach [3-4] and linearizing nonlinear problems using combined coordinate and scaling transformations [15] have been attempted. Use of soft computing tools, such as Genetic Algorithms (GAs) have been shown to be robust in form tolerance evaluation, e.g. circularity evaluation [16]. Computational geometric techniques were also developed for dealing with datum related features [17]. Least squares method based evaluation of the geometric tolerances in relationships, viz. parallelism, run-out and concentricity, has been reported [18]. Extension of straightness evaluation using

      convex hull based approach to perpendicularity evaluation has been reported [19]. An improved Particle Swarm Optimization (PSO) algorithm has been used to evaluate the perpendicularity error between two planar lines by formulating it as a linear optimization problem [20]. The authors have used maximum absolute distance as straightness error on datum feature, which violates the standards. A random walk based algorithm for the

      angle for angularity (Fig. 2), 90° for perpendicularity (Fig. 3) and 0° for parallelism (Fig. 4).

      Y Measured feature

      t

      perpendicularity evaluation has also been reported [21].

      There is still a need for developing effective algorithms for the evaluation of orientation tolerances. Their evaluation also lacks a unified approach. The present work attempts to address both these issues. A unified approach is proposed first for the planar straight line features. This approach is a generalization of an earlier work meant for perpendicularity evaluation [21].

      Bmax

      O

      Bmin

      Ideal datum feature

      s

      X

      The proposed approach is implemented using random walk, simplex search and normal least squares methods so as to find an effective algorithm. The effectiveness of these algorithms is tested using simulated data. The results obtained are presented and discussed.

      Remainder of the paper is organized as follows. Section II pronounces the proposed unified approach. Section III briefs about the algorithms used for fitting the ideal datum features. Section IV presents the evaluation approach used and Section V presents and discusses the results obtained using different algorithms. Section VI states the conclusions and future scope.

   2. THE PROPOSED UNIFIED APPROACH

      As mentioned earlier, an ideal datum feature (a straight line in present case [21]) has to be established first for evaluation of orientation tolerances. Let this feature be represented as in (1), with usual notations.

      Datum feature

      Fig. 2. Angularity analysis ( = Specified angle)

      Y

      Bmax

      t

      Measured feature

      Ideal datum feature

      Datum feature

      s

      O

      O

      y ax b

      (1)

      Bmin X

      Distances dj (j = 1, 2 m; m is the number of measurement points) between measured points Pj (xj, yj) of datum feature to the ideal datum feature may be calculated using (2).

      y j ax j b

      Y

      Bmax

      Fig. 3. Perpendicularity analysis ( = 90°)

      Measured feature

      t

      d j

      (1 a2 )

      ; j 1,2,…,m

      (2)

      Distance dj is taken as positive when the measured point is above the ideal datum feature and negative when i is below. Straightness error (s) of datum feature can be expressed as:

      Bmin

      Ideal datum

      s feature

      j

      j

      s d

      max

      In (3), d j

      • d

      j

      j

      min

      j

      j

      and d

      min

      max

      (3) O

      represent the minimum and maximum

      X

      Datum feature

      values of dj respectively. Estimation of a and b for computing the straightness error (s) of datum feature can be stated as an unconstrained minimization problem, satisfying the minimum zone condition, as follows [20]:

      Fig. 4. Parallelism analysis ( = 0°)

      Lines are drawn through all the points PJ (xJ, yJ) (J = 1, 2

      M; M is the number of points) of the measured feature. Let the equations of such lines be described using (6).

      Minimize f (a, b) = s (4)

      Y AX B

      (6)

      If (a*, b*) is the optimal solution for (4), equation of ideal datum feature can be written as:

      In (6), A denotes the slopes and B denotes the y-intercepts

      of those lines. If Bmax and Bmin are the maximum (i.e. largest Y-

      y a*x b*

      (5)

      intercept) and minimum (i.e. smallest Y-intercept) values of B

      respectively, is the angle between ideal datum feature and the

      Depending on the orientation tolerance to be evaluated, two parallel lines are established at specified angular orientation, with respect to ideal datum feature, which will be the specified

      x-axis and is the specified angular orientation of measured or controlled features in relation to the datum feature, orientation tolerance (t) can be expressed as:

      t (Bmax Bmin)cos( )

      (7)

      Start

      The use of Bmax and Bmin in (7) ensures that all the points of measured feature lies within the tolerance zone. The tolerance value (t) computed using (7) is therefore the minimum value, which implies that orientation tolerances obtained using this equation follows the minimum zone evaluation.

   3. ALGORITHMS FOR FITTING THE IDEAL DATUM FEATURE

1. Random Walk Method

   In random walk, a sequence of improved approximations to the minimum value are generated based on the preceding approximation. If Xi is the approximation to minimum value obtained in (i-1)th iteration, improved approximation in the ith iteration is obtainable from (8).

   Establish the ideal datum feature using the selected algorithm

   Establish the ideal datum feature using the selected algorithm

   Draw lines through all points on the measured feature at specified angular orientation to the ideal datum feature

   Draw lines through all points on the measured feature at specified angular orientation to the ideal datum feature

   Compute the orientation tolerance

   Compute the orientation tolerance

   Stop

   Fig. 5. Flowchart for evaluation of orientation tolerances (modified from [21])

   Xi 1 Xi ui

   (8)

   V. RESULTS AND DISCUSSION

   where, is a prescribed scalar step length and ui is a unit random vector generated in ith stage. The detailed procedure of random walk method may be found in [22].

2. Simplex Search Method

   Simplex is a geometric figure formed by (n + 1) points in an n-dimensional space. The simplex is gradually moved towards the optimal point by comparing the objective function values at (n + 1) vertices of the general simplex. Three operations, viz. reflection, contraction and expansion, are performed to achieve the desired movement of the simplex. More details of simplex search method and its algorithm may be found in [22].

3. Normal Least Squares Method

The objective of normal LSM is to obtain the best fit line (feature), as given in (1), that minimizes the sum of squares of normal distances (EN), between the measurement points Pj (xj, yj) and the best fit line (feature), as given in (9) [23].

Orientation tolerances refer to the geometric tolerances that are specified with datums to control the geometric deviations in manufactured parts. The verification of orientation tolerances, viz. angularity, perpendicularity and parallelism, is considered in the present work due to their importance in machine tools and coordinate measuring machines. A new unified approach for the evaluation of orientation tolerances in planar straight line features is presented in this paper. This approach has been implemented using random walk, simplex search and normal least squares algorithms using C++ language under the Visual Studio 2015 environment and run on a Microsoft Windows 10 powered PC equipped with 4 GB RAM. Performance of these algorithms were studied using the simulated datasets shown in Appendices A to C. The results of evaluation of angularity, perpendicularity and parallelism tolerances are shown in Table I, II and III respectively. As the desired criteria, evaluation algorithms are expected to yield smaller straightness values on the datum feature and preferably smaller values of orientation tolerances on the measured feature. The preferably adjective is

m

m

EN

(9)

due to the fact that these geometrical tolerances are estimated

2

2

j 1

Substituting for y = axj + b and we get

cos

1

(1 a)2

in (9),

in sequence and not simultaneously. Precisely, smaller values of straightness error does not necessarily mean smaller values of orientation tolerances.

Table I reveals that the most commonly used normal least squares method overestimates the straightness (7.009 µm for A

EN

EN

( y ax b)2

and 8.944 µm for B) and angularity (16.120 µm for A and

m

j 1

j j

(1 a2 )

(10)

16.323 µm for B) errors. Random walk based algorithm yields smallest values of straightness (5.502 µm for A and 5.287 µm for B) and angularity (14.311 µm for A and 15.346 µm for B)

The details on normal least squares method and the method

of solving the coefficients a and b can be found in [23].

IV. APPROACH TO COMPUTATION OF ORIENTATION TOLERANCES

Flowchart for the approach used to evaluate the orientation tolerances is shown in Fig. 5. The random walk, simplex search and normal least squares method based algorithms are used only for obtaining the ideal datum feature. After this, lines are drawn through all points of measured or controller feature at specified angular orientation with reference to the ideal datum feature. If Bmax is the y-intercept of topmost line and Bmin is the y-intercept of bottommost line, the orientation tolerance (t) can be computed by substituting these values in (7).

errors and simplex search comes next. Table II reveals similar trends in perpendicularity evaluation. The results of parallelism error evaluation, shown in Table III, reveals that random walk algorithm outperforms the other algorithms, i.e. yields lowest straightness and parallelism errors. The simplex search shows a slightly poor performance in parallelism error in comparison to normal LSM, however, the straightness error is still less than that of normal LSM. Thus, the random walk based algorithm is found to perform consistently in evaluating the orientation tolerances better than the other two algorithms. As a general statement, it may be said that normal LSM overestimates the geometrical tolerances always despite its sound mathematical basis and wide application in measuring instruments. Random walk based algorithm also follows the standards.

TABLE I. RESULTS OF ANGULARITY EVALUATION

TABLE II. RESULTS OF PERPENDICULARITY EVALUATION

TABLE III. RESULTS OF PARALLELISM EVALUATION

VI. CONCLUSIONS AND FUTURE SCOPE Geometrical deviations in manufactured parts are caused by

systematic and random errors that occur during manufacturing. The geometrical tolerances are used to specify and control such deviations. Their evaluation requires effective algorithms that follow the standards. The evaluation of orientation tolerances, viz. angularity, perpendicularity and parallelism in straight line features has been considered in the present work. An approach for unified evaluation orientation tolerances has been proposed

APPENDIX B: PERPENDICULARITY DATASET

APPENDIX C: PARALLELISM DATASET

Dataset E Dataset F

S# S#

and implemented using different numerical algorithms, viz. random walk, simplex search and normal least squares method. The performance of these algorithms has been evaluated using simulated data. Random walk algorithm has been found to be most effective among the three algorithms. The simplex search algorithm has also been found to yield better results. Both these algorithms outperform the most commonly used normal least squares based algorithm. The extension of proposed approach to three-dimensional features may form the future work.

APPENDIX A: ANGULARITY DATASET

Datum Measured

xj yj xj yj

1 0.00 2.5158 0.00 47.5341 1

Datum Measured

xj yj xj yj

0.00 2.5170 0.00 77.5280

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BIOGRAPHY

G. Rajamohan received his B. E. (Mechanical Engg.) from College of Engineering, Anna University, Chennai, M. E. (Manufacturing Tech.) from Regional Engineering College, Tiruchirappalli and Ph. D. (Mechanical Engg.) from Indian Institute of Technology Madras. He is associated with the

National Institute of Foundry & Forge Technology, Ranchi since 1998. His research interests include metrology and computer aided inspection, machining, image processing and applications of computers in design and manufacturing.