Iterative solution of least-squares problems applied to flatness and grid measurements

This paper describes an iterative method which can be applied to geometrical problems in whicha large number of measurements are available out of which a large number of parameters are to becalculated. The method is based on the fact that a minimum sum of squared differences between anassumed value of a parameter and a number of independent measurements, is achieved when theparameter has the average value. This principle is applied to the problem of flatness measurementwith a gradient measuring device such as an electronic level (typically 180 measurements, 100parameters for a 10 x 10 grid) and to the measurement of a grid using distance measurements alongthe axes and along the diagonals(e.g. 342 measurements, 200 parameters for a 10 x 10 grid). The type A uncertainty of the result is estimated from simulated measurements in which simulated random measurements are superimposed on exact measurements derived from the known solution.This enables the derivation of uncertainties in coordinates relative to a defined orientation; e.g. theminimum zone plane in flatness measurements. It is shown that in both problems an uncertainty inthe results can be achieved which is about equal to the uncertainty of the single measurements.Methods are proposed to combine these uncertainties with type B uncertainties such as an uncertaintyin the temperature gradient in flatness measurements or the temperature uncertainty in grid measurements.