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Considering the characteristics of spatial straightness error, this paper puts forward a kind of evaluation method of spatial straightness error using Geometric Approximation Searching Algorithm (GASA). According to the minimum condition principle of form error evaluation, the mathematic model and optimization objective of the GASA are given. The algorithm avoids the optimization and linearization, and can be fulfilled in three steps. First construct two parallel quadrates based on the preset two reference points of the spatial line respectively; second construct centerlines by connecting one quadrate each vertices to another quadrate each vertices; after that, calculate the distances between measured points and the constructed centerlines. The minimum zone straightness error is obtained by repeating comparing and reconstructing quadrates. The principle and steps of the algorithm to evaluate spatial straightness error is described in detail, and the mathematical formula and program flowchart are given also. Results show that this algorithm can evaluate spatial straightness error more effectively and exactly.

Spatial straightness is one of important geometric elements. Spatial straightness error is defined as “the minimum cylindrical diameter subsuming measured line” in ISO/TS 12780-1 [

LSM for spatial straightness error first uses linear least square fitting to obtain a line as centerline, then calculates the distances between measured points and the centerline, at last constructs a cylinder including all measured points by taking the maximum distance as radius, and the maximum distance is spatial straightness error by least square method. Owing to its ease of computation and uniqueness of solution, it is now widely applied in engineering. It can be easily found that LSM does not exactly coincide with spatial straightness error definition and brings some defectiveness in error evaluation such as spatial straightness error by LSM is usually not the minimal [

MZM is used to determine the minimum cylinder in diameter to enclose the measured spatial line. The evaluation of spatial straightness error can be carried out based on the minimum cylinder diameter which is an accurate and applicable method conforming to the definition of form error. However, how to find the minimum zone enclosing all measured points of the spatial line and verify if the minimum zone is the smallest zone is still a problem received the widespread attention and urgent to be solved. Unfortunately there is no final conclusion in the academia in recent years.

Because of the complexity of data process in MZM, many approximate methods with relative higher accuracy were proposed. Yue and Wu [

We know from the references that the assessment methods of spatial straightness error are very important in metrology. The construction of the cylinder containing and enclosing all measured points of the line is a very complex geometric problem and can be formulated as a nonlinear optimization. In the process of the nonlinear optimization, the selection and application of algorithms are of great importance. Meanwhile, the convergence rate, precision of result and reliability of the algorithm directly affect the evaluating precision. In other words, the simplified algorithms might not bring about accurate evaluation; the computation process of the optimization algorithms might be very complex and may be not understood by the non-professional.

According to the definition of spatial straightness error, an innovative and simple evaluation method, named as Geometric Approximation Searching Algorithm (GASA) is presented, in which the spatial straightness error can be gained by repeatedly calculating spatial distance and simple judgment without any optimization and linearization processing.

The essence of spatial straightness error evaluation methods is to resolve the parameters of the centerline of cylindrical surface containing all real measuring points [1-10]. It is obvious that the ideal centerline must be closed to the least square centerline, or closed to the connecting line between the starting measured point and end measured point of measured straight line. If take the two endpoints the of the least square centerline as reference points, or take the two endpoints the of the measured line as reference points, the straightness error of the measured line can easily to get and expressed as, evidently , it is not an outcome we look forward to. Now, take the two endpoints as reference points, a square is allocated respectively (the length of side is f, f is estimated value according to the machining accuracy of the measured straight line, or f is the error of the LSM), and then, the 16 lines can be arranged by connecting each vertices of one square to another square (

Assuming that the measuring points are, the LSM spatial straightness error is f (the LSM method has been discoursed in many papers, and its computational process is not given in this paper).

It can be seen that from Refs [1-10] and actual measurements, the ideal center line for evaluating spatial straightness error must be closed the least square center line and the line between the initial measured point and end measured point. Then the two endpoints of measured line or the least square center line can be chosen as the initial reference points for simplicity, the two initial reference points are expressed by and respectively.

Using the points and as reference points, a square is set by the f as side length in initial measured section and end measured section respectively (as shown in

And then, the 16 supposed ideal centerlines can be constructed by connecting vertex of the one square to another square, the lines can be expressed in the form of (x − a)/P = (y − b)/Q = z, based on geometric principle, the direction cosines of the lines are defined in Equation (9).

The spatial distance from measured points to the constructed ideal centerlines can be calculated by Equation (10).

There are 16 presumptive ideal centerlines lines, therefore the 16 maximum distances R_{max} can be gained. Take one of the 16 maximum distances as radius and the corresponding assumption ideal centerlines as axis, the cylindrical surface containing all measuring points can be constituted. According to definition of the spatial straightness error, we can know, the diameter of the cylindrical surface is spatial straightness error. There are 16 presumptive ideal centerlines lines and the 16 maximum distances, therefore there are 16 cylindrical surface containing all measuring points and 16 diameters. Among the 16 diameter, the minimum diameter is expressed by, and then:

The spatial distance from measured point to the line of two reference points and can be calculated by Equation (12).

where, is the direction cosines of the line linking the pointand,

,.

Take maximum distances as radius and the line of two reference points as axis; one cylindrical surface containing all measuring points can be constituted. According to definition of the spatial straightness error, we can know, the diameter of the cylindrical surface is spatial straightness error and expressed, and then

Comparing and, if, the reference points change to the endpoints of the assumption ideal centerline corresponding with the, that is, the reference point change to the pre-set square vertex (expressed asandrespecttively) correspond with the (for example, the assumption ideal centerline correspond with theis line,so the reference pointandchange to the pointand), the new square are re-established by using f as the side length, repeat steps 3.2 - 3.5.

If, the reference points remain unchanged and the new square are re-establish by using the 0.618f (i.e. f = 0.618f) as the new side length, repeat steps 3.2 - 3.5.

When the square side length f is less than a given value (normally, J < 0.0001 mm), it could be considered that the searched assumption ideal centerline is getting close to the actual centerline of the cylinders which the minimum radius and contain all the data points, the search terminates.

And then, the minimum between and is spatial straightness error of the MZM and expressed. That is

The flowchart of the GASA is shown in

The measurement data from reference [

The LSM spatial straightness is 0.031408 mm [

In order to validate the correctness of the proposed algorithm, the initial conditions are given as follows:

a) The endpoints of the least-square centerline as initial reference points and the straightness error of the least square method as the side length of the initial square.

Table1. Measurement data.

b) The endpoints of the measured line as initial reference points and the straightness error the least square method as the side length of the initial square.

c) The endpoints of the measured line as initial reference points and the side length of the initial square is assumed.

The initial condition is shown in

The straightness errors and cycle index of three initial conditions by the proposed searching algorithm are carried out.

It is seen from Tables 3 and 4 that, the straightness errors obtained by the proposed evaluation algorithm is the nearly same in despite of the initial reference point and the side length of the initial square are different, only a few nanometer difference. Therefore, the initial reference does not affect straightness error evaluation resultgained by this algorithm. For the sake of calculation, the initial reference points usually choose the endpoints of the measured line and the side length of the initial square is assumed by the manufacturing precision of parts.

It was also seen by comparing Tables 3 and 4 that the results are of no significant difference with different termination conditions such as f < 0.001 mm and f < 0.0001 mm. Generally, mm can satisfy straightness error evaluating requirement.

The straightness errors obtained by the proposed evaluation algorithm are in accordance with the results in

reference [

The GASA for spatial straightness error presented in this paper is a new spatial straightness evaluation algorithm. It is not necessary to require evenly distributed sampling interval and assume small error or deviation in this algorithm, the evaluating accuracy of the GASA depends on the pre-set termination condition, the smaller the values of the termination condition is, the more precise the evaluation is. Generally, take the endpoints of the measured line as initial reference points and the side length of the initial square is assumed by the manufacturing precision of parts and the termination condition is mm can satisfy spatial straightness error evaluating requirement. The algorithm is simple, intuitive and easy to program, and it has the commonality and better practicability.

The authors gratefully acknowledge the National Key Technology R&D Program of China (2012ZX04004011) and Innovation Scientists and Technicians Troop Construction Projects of Henan Province for financial support of this research work.