1.1 Least Squares Method2
1.2 Minimum Zone Method3
6.3 Closing error9-10
9.1 Appendix A-Procedure11-13
9.2 Appendix B-Certificates of calibration14-15
The aim of this experiment was to examine three methods for determining the straightness and flatness of a horizontal granite surface. The first method was manual and the other two (Least Squares method and Minimum Zone method) were analysed by the computer, after a set of data was inputted. The main equipment used in the experiment was the granite surface, the ''Talyvel'' electronic level, the PC and the software (SURFSURE). After obtaining a number of results and analyzing them it was deduced that the computer analysis was more accurate than the manual calibration. It also produced post-processed data that could be printed and analyzed further.
Calibration is a method by which the straightness and flatness of a surface can be determined (within specified tolerances) and hence taken into account. There are many applications where the calibration of straightness and flatness of surfaces is crucial (e.g. high precision surfaces). Possible examples could be the surface bed of a milling machine and lathe-bed guide ways. Straightness is defined as ''A condition in which an element of a surface or an axis is a straight line''. Flatness is said to exist if the following conditions are satisfied: 1) ''all generators (lines) must be straight'' and/or 2) ''all generators (lines) must lie in the same plane''.
Fig. 2.1 A surface, all of whose generators (lines) parallel to the sides are straight, but which is not flat
There are several ways with which the straightness and flatness of a surface can be determined. Some examples are: straight edge method, wedge method, with sharp-stylus (roughness) instruments, with roundness and cylindricity instruments, with co-ordinate measuring machines (CMM), auto-collimator method and with electronic levels, e.g. ''Talyvel'' electronic level with stride base.
Fig. 2.2 Historical surface metrology measurement spectrum
Fig. 2.3 Current surface metrology measurement spectrum
In order to determine the flatness of a surface several methods can be employed to analyze the obtained results. Two of these methods are the ''Least Squares'' and ''Minimum Zone''.
1.1 Least Squares Method
The most frequently used method, but has no general optimal properties to recommend it. For problems where the parameter dependence is linear, the ''Least Squares'' (LS) method produces unbiased estimators of minimum variance. The method of least squares requires that a straight line be fitted to a set of data points such that the sum of the squares of the vertical deviations from the points to the line is minimized, if the regression is on y.
Fig. 2.4 Linear regression on y-axis
At observational points x1, ……..., xN we measure experimental values of y1, ………, yN. The true functional form is defined by L parameters, fi = fi (θ1, ……, θL).
To find the parameter estimates, θ1, ……, θL, minimise X2 = Σwi (yi-fi)2, where wi is the weight that expresses the accuracy of yi.
wi varies according to the accuracy of yi. For example, if yi is constant then wi = 1. If the accuracy of yi is given by σi the wi = 1/σ2. If yi represents a Poisson distributed random number then wi = 1/fi (or sometimes wi = 1/yi) and so on.
1.2 Minimum Zone Method
The ''Minimum Zone'' (MZ) is the minimum distance between two offset features such that all data points are included in the zone. The minimum zone is the zone contained between two lines of minimal...