Assessing the uncertainty of the means for determining the positioning accuracy of the machine

Positioning deviations such as systematic and random errors lend a substantial influence to finish machining. In this article is carried a theoretical research of the uncertainty when determining the repeatability of positioning with standard methodology, regulated in ISO 230-2. It is carried out by simulating different levels of random deviation for the position reached and the number of points on the controlled axis. The laws of distribution are established on standard assessment of positioning repeatability and the relative share of uncertainty regarding the actual (set) random positional deviation.


Introduction
Positional errors of the machine have substantial influence on finish machining.During turning their influence, especially in diametrical dimension on axis X, increases due to the following reasons: -Accuracy of the diametrical dimensions is higher; -Positional errors are doubled.Systematic positional error during contour turning is hard to detect and compensate by controlling the finished surface.The rational approach is to determine positional deviations on both axis and introduce them into the CNC system, and to remove their influence by correcting the tool trajectory.Dominant in finish turning becomes the random positional error.

Exposition
Information basis, in relation to positional errors, provides the following tasks: 1. Defining the level of machine accuracy, by contrasting certain parameters connected to them; 2. Diagnosing the current equipment to establish its condition; 3. Compensating systematic errors; 4. Retrieving data for random positional errors, required in order to predict the accuracy during design of operations.
Parameters summarizing the entire axis are used for the first two tasks.Test required to determine them are regulated by the standard ISO 230-2 [1].In it is described the methodology for determining positional repeatability.The main points in it are: -On machines with 2000mm axial travel, at least 5 positions per meter on the axis must be measured, but no less than 5 positions; -In any position  = 1 …  measures are carried out and mean value of positional deviations   is being calculated: (1) -For each position a standard uncertainty of positional deviation is calculated   , acquired as a result of series of  ( = 5) unidirectional approaches to position   by equations: (2) -Unidirectional positional repeatability   at position   by utilizing the coefficient of stock  = 2 is defined by dependencies:   ↑= 4  ↑ (3)   ↓= 4  ↓ -Unidirectional positional repeatability along the axis is: According to standard the values  ↑ and  ↓, which are shown as second parameter of positional accuracy along the given axis, summarizing uncertainty, that define accuracy of compensation to positional deviation [2].It accounts the influence of: random factors related to positional accuracy of translation; the use of measuring device; temperature stability of the environment.Beside the applied in the standard quasi-static measurement a rapid measuring in dynamic conditions techniques are applied [4,5,6].
Standard deviations, defined by machines positional accuracy are important parameter affecting the: -accuracy of compensation of characteristic systematic errors, connected to machines geometrical accuracy [3]; -automatic measurement part surfaces [7]; -compensation of systematic technological errors [8]; -formation of resulting random deviation of dimensions on finishing processes [9,10].
From technological point of view, a question arises whether the uncertainty according to the standard value  = 4  (  -the maximum sample mean squared value of those calculated along the axis), corresponds to the field of random error, influencing the resulting size.The latter is accepted as   = 6  (  -mean squared value of random positional error), the field of error is calculated as in probability = 99,73% , such as is accepted for the other technological errors.The two values  and   have the same physical composition, as the way of defining the influences their values.
The parameter  is summarizing characteristic of random error of positioning for the specified machine axis, while her influence on machining accuracy is defined in most cases by its local values.Defining the positional repeatability  is connected with accounting of the following two reasons causing change of the mean square dissipation   in the tested positions:  Positional dissipation of the support in given position along the axis is not the same because of the variable influence of the tightness and quality of the fractioning surfaces in the guides and the screw, and also the accuracy of the feedback.During exploitation because of the wear, values of those factors worsen additionally to varying degrees in different sections along the axis.
 The sample approach in defining dissipation in the different controlled points leads to methodical error -dissipation of calculated mean square values (  ).Limiting the repeated measurements to 5 in the controlled point is not connected only with lowering the complexity of the test being performed, but with excluding the influence of heat deformations during longer test.
In order to account the change of dissipation of mean square values (  ) for the reasons discussed, in the standard methodology is accepted that the summarizing parameter  to be calculated with   (Maximum mean quadratic deviation in all positions).The lack of information for what is the influence of the methodology (sample approach) on uncertainty in defining positional deviations imposes its investigation.It is carried out with computer simulation of the acquired results by the methodology, regulated in the standard.Due to small number of factors a non-planar factorial experiment is applied.
It researches the influence of: -The number of measured points (positions) along the axis length (i=5 and i=10); -Dissipation of real random factors, set with standard deviations  = 0,5,0,75  1.
The number of controlled (tested) points is based on the requirements of the standard -that there are no less than 5 for each 1000mm of the support movement.In the research they are tailored with axis X, which on most of the machines have the movement of less than 500mm.Attentions is pointed to axis X, because: -Accuracy of diametrical dimensions is higher; -The random positional error   is doubled in diametrical surfaces, while along axis Z its influence on the distance between two flat surfaces is √2  .
The simulation experiment is realized with the software MS EXCEL.For imitation of random positioning deviations a random numbers generator is used, distributed by normal law ( = 0,5, 0,75  1).For the accurate defining of distribution law of R and its parameters  experiments ( = 1 … ) are carried out.At  = 5, a  = 120 repeated tests are conducted, at  = 10 they are  = 60.The goal is with both research the product  *  =  (results are obtained from the same number of random numbers).In accordance with the standard for each of the controlled position  a  = 5 random numbers are generated and their arithmetic mean   ̅ and standard deviation   are calculated in each of the tested positions -dependencies 1 and 2.
, is defined from the acquired   and the unidirectional repeatability of positioning   = 4 , (dependence 4) is calculated for each sequential number of the experiment .Arithmetic mean value  ̅ = ∑    =1 and squared mean deviation   of the repeatability   (table 1) is calculated.The parameter  characterizes the influence of random factors on positioning.The difference  ̅ −   reflects compliance between mean random error, defined by the methodology accepted in the standard and set (actual) during the experiment   .In figure 1 the difference is shown with the parameter   ̅ , and in table 1the discrepancy is given in percentages as relative error ∆  ̅ .The parameters    and    show the limit deviations of the field   = 4  in relations to the value of   .In table 1 with parameter  is given the experimentally determined value of positional repeatability as mean value  ̅ and her random dispersion   .With parameters (   ) and (   ) are evaluated the possible values of field  which set higher and respectively lower values to the actual ( set with simulation) value of random positional dissipation   = 6, where  + is the number of values of  >   , and  − −  <   .Table 1.Experimentally defined positional repeatability, its random dissipation and discrepancy between mean random error, defined using methodology from the standard and the set error.From results obtained from the study the following regularities are formulated: 1.The mean value of positional repeatability  ̅ determined with testing using standard methodology is moved in relations to the actual (set) dissipation   .The magnitude of this movement ∆  ̅ substantially reduces with increase of the number  of tested positions, 2. Relative difference ∆    and ∆    between repeatability , defined with testing according to standard methodology is in the region of 20 ÷ 45% from the actual   , also the deviations are not symmetrical at minimal number of control positions ( = 5).This is also confirmed by the probabilistic indicators (   ) and (   ).With higher probability are the values of  <   .The law of the field distribution  is also an important influence on these parameters.The form of distribution of   is checked with four theoretical distribution laws.The degree of compliance is given in figure 2,a to figure 2,f for the different experimental conditions.Due to their form, resembling the normal distribution, their influence is insignificant.

Conclusions
Despite the relatively high uncertainty of the standard methodology   =   = 4  in the boundaries of 40-60% of the actual positional deviation, it can be used as summarizing characteristic of the degree of accuracy in relation to the random factors.About solving the other two taskscompensation of systematic errors and receiving data about random positional errors necessary for forecasting the accuracy of technological design, a further research and analysis are required regarding the permissible uncertainty of methodologies used [11].It must be tailored with the following specific requirements connected with these tasks: it`s values must be connected with the accuracy required of the implemented technological transitions; to be specific for each position of the examined axis rather than generalizing; in predicting the random positional deviations its influence over general uncertainty of the general deviation of the received size must be accounted for.

Figure 1 .
Figure 1.Compliance between mean random error, defined with the methodology, accepted in the standard and the set error (actual)

Figure 2 .
Figure 2. The laws of distribution of positional repeatability check