An Approach for the Evaluation of Sphericity Error and Its Uncertainty

1. Introduction

Dimension and geometrical product specification and verification (GPS) is a specification of the product geometry, which covers the standard of the size, dimension, geometrical tolerancing, and surface texture of geometrical product. The important concept used in GPS standards is the concept of verification/measurement uncertainty according to the “guide to the express of uncertainty in measurement” (GUM) [1, 2]. The uncertainty management is a good system in order to explain the discrepancies between measurement results from two different parties and has led to the development of the ISO 14253-series [2–4].

The spherical surface is an important geometrical feature in industry. In heavily loaded components such as ball bearings in engines, and turbines, any defects on the surface may result in the life reduction, wear, and run-out rotation. Therefore, precise evaluation of the sphericity is an important issue. The sphericity data usually are obtained by using appropriate inspection devices such as coordinate measurement machines (CMM). Then, the data must be processed by using appropriate techniques to obtain reliable results during the evaluation of sphericity error.

Some attempts were made to develop methods for evaluating the sphericity error. The existing approaches are mainly based on the least squares sphere, minimum zone sphere, minimum circumscribed sphere, and maximum inscribed sphere, among which the technique of least squares sphere is quite mature and widely accepted [5–10]. Huang [5] developed a mathematical model based on the 3D Voronoi diagrams to get the exact minimum zone solution. This method also indicated that the minimum sphericity was determined by a small number of critical measured points that gave an efficient way to save the computation time. Fan and Lee [6] proposed an approach with minimum potential energy analogy to the minimum zone solution of spherical error, which assumed an imaginary spring was located between the concentric spheres containing all measured points. This method can also be extended to solve other minimum zone form error problems. An algorithm for the evaluation of sphericity error based on the MRS criterion was presented by wang et al. [7]. The infinitesimal analysis method was used to analyze the characteristics of the sphericity function in the neighborhood of the local optimal solution. Samuel and Shumugam [8] developed the efficient algorithms based on the computational geometry to establish Minimum Circumscribed, Maximum Inscribed and Minimum Zone Limacoids. With the help of geometry of a limacoid, the appropriate computational geometric techniques are developed. The proposed algorithms can deal with data with both uniform and nonuniform spacing. With minimum radial separation center, He et al. [9] used a geometric approximation technique to obtain the minimum sphericity error. This technique regarded the least square sphere center as the initial center of the concentric spheres, and then the center was moved gradually to reduce the radial separation till the minimum radial separation center appeared. Also the method based on the genetic algorithm [10] to evaluate the sphericity error was presented.

Meanwhile, several methods for the uncertainty have been proposed by different researchers. Yan and Menq [11] developed a method for the uncertainty analysis of coordinate estimation in which a sensitivity matrix was presented to establish a linearized relationship between the variations of the coordinate estimation and the geometric errors. Hall [12] proposed a computational technique that the uncertainty equation was automatically derived from the measurement function and need not be provided separately. Bachmann et al. [13] presented an approach based on statistical concepts, which can generate uncertainties on the verification of ISO specifications. Recently, a calculation method for the uncertainty of spatial straightness least-square verification was proposed in [14]. By this method, the coefficients of the line equation were regarded as statistical vectors, so that the result of the spatial straightness verification and the uncertainty can be obtained after the expected value and covariance matrix of the vector are determined.

Most of the current approaches mentioned above can assess sphericity error and uncertainty, respectively. But few of them considered evaluation result and the uncertainty at the same time. So the result was not in conformity with the requirement of the new generation of GPS standards. It is necessary to develop the method of uncertainty evaluation of sphericity error.

The verification uncertainty can be considered as the sum of the method and implementation uncertainties. In this paper, we only consider the effect of method uncertainty. An approach for evaluation of sphericity errors by using PSO algorithm and calculate the uncertainty of evaluation result is proposed. Therefore, the evaluation results are more accurate and consistent with requirements of new generation GPS standard.

2. Evaluation of Sphericity Errors

2.1. Mathematical Model

According to the minimum zone condition, two concentric spheres at minimum radial separation must be found so that they contain all points on the actual spherical surface. Therefore, the sphericity error evaluation under the condition of minimum zone is to search the center of sphericity, which makes the smallest separation of the maximum distance and the minimum distance between the center of sphericity and any measured point.

Suppose the center of sphericity can be represented as (a, b, c), the measured points are P _i (x _i , y _i , z _i ) (i = 1, 2, …, n). Distance from the point P _i to the center is

R i = ( x i - a ) 2 + ( y i - b ) 2 + ( z i - c ) 2 .

(1)

The sphericity error can be defined as follows:

f = R max - R min ,

(2)

where

R max = max i ( x i - a ) 2 + ( y i - b ) 2 + ( z i - c ) 2 , R min = min i ( x i - a ) 2 + ( y i - b ) 2 + ( z i - c ) 2 .

(3)

So the optimization objective function can be defined as follows:

f ( a , b , c ) = min ( R max - R min ) .

(4)

Solveing sphericity error can be transformed to search the value of (a, b, c) in whichminimize the objective function.

2.2. PSO Algorithm

PSO is a population-based stochastic optimization technique developed by Kennedy and Eberhart [15] in 1995. It is similar to evolutionary computation techniques in that a population of potential solutions to the optimal problem under consideration is used to probe the searching space. Each solution is also assigned a randomized velocity, and the potential solutions, called particles, correspond to individuals. Figure 1 shows the flowchart of PSO.

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Figure 1:

Implementation flowchart of PSO.

PSO is initialized with a group of random particles and then searches for optima by updating generations. In every iteration, each particle is updated by “pbest” and “gbest” values. Suppose that the search space is d-dimensional, then the ith particle of the population can be represented by d-dimensional vector X _i = (x_i1, x_i2, x_i3, …, x _id ). The velocity of this particle can be represented by another d-dimensional vector V _i = (v_i1, v_i2, v_i3, …, v _id ). The best previously visited position of the ith particle is denoted as P _i = (p_i1, p_i2, p_i3, …, p _id ). The best position of the all particles is denoted as P _g = (p_g1, p_g2, p_g3, …, p _gd ). After finding the two best values, the particle updates its velocity and positions according to the following equations:

v i d k + 1 = w × v i d + c 1 × r 1 × ( P i d - x i d ) + c 2 × r 2 × ( P g d - x i d k ) , x i d k + 1 = x i d k + v i d k + 1 ,

(5)

where w is the inertia weight; d is the dimension of search space; c₁ and c₂ are learning factors; k denotes the iteration number; and r₁ and r₂ are random numbers uniformly distributed in the range.

3. Uncertainty Evaluation

In the above paragraph, the evaluation of sphericity error based on PSO is presented. In the following paragraph, the uncertainty of sphericity error will be calculated.

Suppose the two peak values for sphericity error are (x₁, y₁, z₁) and (x₂, y₂, z₂), then the sphericity error t can be written as

t = ( x 1 - a ) 2 + ( y 1 - b ) 2 + ( z 1 - c ) 2 - ( x 2 - a ) 2 + ( y 2 - b ) 2 + ( z 2 - c ) 2 .

(6)

Based on the uncertainty theory of GPS, in order to calculate the uncertainty of the sphericity error verification, it must determine the propagation coefficients of each of the elements and correlation coefficients. Obviously, it is considered that a, b, and c are correlated in all the elements.

The calculating formula for the uncertainty of sphericity error verification can be written as

t c 2 = ( ∂ t ∂ x 1 u x 1 ) 2 + ( ∂ t ∂ x 2 u x 2 ) 2 + ( ∂ t ∂ y 1 u y 1 ) 2 + ( ∂ t ∂ y 2 u y 2 ) 2 + ( ∂ t ∂ z 1 u z 1 ) 2 + ( ∂ t ∂ z 2 u z 2 ) 2 + ( ∂ t ∂ a u a ) 2 + ( ∂ t ∂ b u b ) 2 + ( ∂ t ∂ c u c ) 2 + 2 ∂ t ∂ a ∂ t ∂ b ρ a b u a u b + 2 ∂ t ∂ b ∂ t ∂ c ρ b c u b u c + 2 ∂ t ∂ a ∂ t ∂ c ρ a c u a u c ,

(7)

where t _c is uncertainty of sphericity error, the differential coefficients are propagation coefficients, ρ is correlation coefficient, and u is the uncertainty of each element.

The propagation coefficient of every element is deduced as follows:

∂ t ∂ a = a - x 1 ( x 1 - a ) 2 + ( y 1 - b ) 2 + ( z 1 - c ) 2 - a - x 2 ( x 2 - a ) 2 + ( y 2 - b ) 2 + ( z 2 - c ) 2 , ∂ t ∂ b = b - y 1 ( x 1 - a ) 2 + ( y 1 - b ) 2 + ( z 1 - c ) 2 - b - y 2 ( x 2 - a ) 2 + ( y 2 - b ) 2 + ( z 2 - c ) 2 , ∂ t ∂ c = c - z 1 ( x 1 - a ) 2 + ( y 1 - b ) 2 + ( z 1 - c ) 2 - c - z 2 ( x 2 - a ) 2 + ( y 2 - b ) 2 + ( z 2 - c ) 2 , ∂ t ∂ x 1 = x 1 - a ( x 1 - a ) 2 + ( y 1 - b ) 2 + ( z 1 - c ) 2 , ∂ t ∂ x 2 = - x 2 - a ( x 2 - a ) 2 + ( y 2 - b ) 2 + ( z 2 - c ) 2 , ∂ t ∂ y 1 = y 1 - b ( x 1 - a ) 2 + ( y 1 - b ) 2 + ( z 1 - c ) 2 , ∂ t ∂ y 2 = - y 2 - b ( x 2 - a ) 2 + ( y 2 - b ) 2 + ( z 2 - c ) 2 , ∂ t ∂ z 1 = z 1 - c ( x 1 - a ) 2 + ( y 1 - b ) 2 + ( z 1 - c ) 2 , ∂ t ∂ z 2 = - z 2 - c ( x 2 - a ) 2 + ( y 2 - b ) 2 + ( z 2 - c ) 2 .

(8)

Obviously, the key to solve the uncertainty issue is how to obtain the correlation coefficients of a, b, and c. Bachmann et al. [13] proposed a method to get the correlation coefficients. In the method, the coefficients were regarded as a statistical vector v. The covariance matrix of vector can be calculated after the measured points are evaluated many times using the same sampling strategy. Therefore, the calculating formula for the uncertainty of sphericity error verification can be written as

t c 2 = u x 1 2 ( x 1 - a ) 2 + u y 1 2 ( y 1 - b ) 2 + u z 1 2 ( z 1 - c ) 2 p 2 + u x 2 2 ( x 2 - a ) 2 + u y 2 2 ( y 2 - b ) 2 + u z 2 2 ( z 2 - c ) 2 q 2 + ( a - x 1 p - a - x 2 q ) 2 u a 2 + ( b - y 1 p - b - y 2 q ) 2 u b 2 + ( c - z 1 p - c - z 2 q ) 2 u c 2 + 2 ρ a b u a u b ( a - x 1 p - a - x 2 q ) × ( b - y 1 p - b - y 2 q ) + 2 ρ b c u b u c ( b - y 1 p - b - y 2 q ) × ( c - z 1 p - c - z 2 q ) + 2 ρ a c u a u c ( a - x 1 p - a - x 2 q ) × ( c - z 1 p - c - z 2 q ) ,

(9)

where

p = ( x 1 - a ) 2 + ( y 1 - b ) 2 + ( z 1 - c ) 2 , q = ( x 2 - a ) 2 + ( y 2 - b ) 2 + ( z 2 - c ) 2 .

(10)

Therefore, the uncertainty of sphericity error verification can be obtained.

4. Uncertainty Application

According to the new generation GPS standard, the measured results consist of the tolerance of workpiece and its uncertainty. As shown in Figure 2, according to GPS standard, only when measured value of a workpiece falls in conformance zone, the workpiece can be regarded as “accepted”; also, only when measured value falls in nonconformance zone, it is safe to say the workpiece “rejected.” But in the case when evaluation value is in uncertainty zone, the judgment of “accepted” or “rejected” for this workpiece will depend on the consultant result between supplier and consumer. So the unnecessary failed products can be avoided without the expense of losing product function, and finally, it will save the manufacturing cost.

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Figure 2:

The uncertainty of sphericity error verification.

5. Case Study

According to the above paragraph the example in [6] is quoted to evaluate the sphericity error and calculate the uncertainty. The data points are generated randomly so that they are all located between two concentric spherical surfaces of radii equal to 0.995 mm and 1.005 mm, respectively. So we can regard the sphericity error as 0.01 mm. Using the same sampling strategy we get three group data.

Based on PSO algorithm, the parameters can be set as follows: (1) the number of particles s = 30. (2) Dimension of particles d = 3. (3) Inertia weight ω = 0.3. (4) V_max = 5. (5) The stop condition: if the loop number reaches the preset max loop number or the value of the optimal sufficiency function has no obvious variation in continuous 50 steps, the iteration circle will be stopped.

As shown in Table 1, the precision of PSO is superior to that given by the Least Square Method (LSM) and equal to those given by GA. Though it is poor to that of MZS, PSO is not only effective but also simple and easy to implement.

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Table 1:

Result comparison of different methods (unit: mm).

	a	b	c	f
LSM [6, 10]				0.008486
MZS [6]				0.00766
GA [10]	0.000993	0.000024	0.000058	0.007928
PSO	0.001053	0.000022	0.000037	0.0079

According to (9), the uncertainty of sphericity error verification is 0.0020 mm. As we know, the error is 0.01 mm. Therefore, the uncertainty zone is [0.0080, 0.0120]. Using proposed method, the error is 0.0079 mm. According to the decision rules in GPS, if the measured value is inside the “conformance zone,” the result is accepted. If the result is inside the “uncertainty,” it must negotiate between the supplier and the customer whether the result is rejected or not.

6. Conclusions

Uncertainty management is a powerful system to communicate magnitudes of “risk” on decision-making based on verification results. Uncertainty will be helpful to promote the efficient distribution of resources between specification and verification. In this paper, particle swarm optimization algorithm has been proposed to evaluate the sphericity error, and then the uncertainty of sphericity error was calculated according to new generation GPS standard. The results of presented method and other methods are compared. It can be seen that the precision of PSO is better than that of LSM and equal to those given by GA. Furthermore, the PSO is easily implemented and has a rapid convergent speed. Also the presented method solves the problem of uncertainty and consistent with requirements of new generation GPS standard. It can improve the integrity and veracity of verification. In the future, we will work on how to use this method to evaluate other form errors, especially to calculate the uncertainty of form errors based on uncertainty theory.