# Simulation Model for Correction and Modeling of Probe Head Errors in Five-Axis Coordinate Systems
## 1 Introduction
A probe head error model utilizing an artificial neural network was developed at the Laboratory of Coordinate Metrology (LCM) of the Cracow University of Technology as part of a Virtual CMM based on assumptions of the matrix method [16]. The model utilizes the Probing Error Function (PEF) described in [13; 16]. Probing Error (PE) depends on a number of factors, among them: deformation of the measuring ball during contact, deflection of the stylus under the force of the measurement, probe pretravel, the form error of the measuring ball, non-linearity and differentiation of the transducer characteristics in different directions of deflection, _i.e._, the directions of probe head movements \((u,v,w)\), reaction to uneven load of the probe head, _etc._ The PE can be defined as a sum of a number of contributing factors. The value of PE changes significantly for different probing directions, so the PEF may be obtained in relation to the probe deflection angle (\(\alpha\)). Values of PE can be determined through measurements of calibrated material standards with known form deviations. The value of the PE at a certain measured point, measured at a specified angle, may be expressed as a radial error obtained during point measurement [16]. The model is divided into two modules: one responsible for systematic error simulation and the second for random error modeling. The ring gauge of a 25 mm diameter has to be measured 32 times, in 64 evenly distributed points, to gather data for model preparation. It was assumed that the 64 measured points are sufficient to model the probe head functioning for any probing direction. On the basis of obtained results, the mean value of the PEF should be attributed to each measured point. These values are treated as a systematic part of the probing system error. The obtained data is used as a learning data set for the constructed artificial neural network, which links the probe head deflection angle and systematic error. The network consists of three layers, and utilizes back propagation and different activation functions (Figure 1). The module was prepared both for inner dimensions--using the gauge ring as a reference--and outer dimensions. In the case of the latter, the entire procedure should be repeated, except that instead of a ring, the spherical standard of a 25 mm diameter should be measured at the reference sphere equator. The second module is responsible for a simulation of probe head random errors. The chi-squared test should be used to check the distribution of results obtained during the gauge ring measurement. Tests performed previously indicated that the resulting distributions are mostly similar to a normal distribution. In this case, the random error of the probe head connected with a certain probing direction could be modeled using the standard deviation obtained for the considered point. In order to interpolate the values of standard deviation between measured points, the artificial neural network with identical construction to the one in the previous case was used. Finally, a normal distribution generator based on the Monte Carlo method may be used in order to simulate the random error. More details regarding this model may be found in [16]. Another probe head error model developed at LCM is based on the Monte Carlo method. The main idea of the model is the same as in the previously discussed model, namely to link probe head errors with the probing direction; however, this model was adapted to describe the probe spatial approach to the measured surface. In such a situation at least two angles defining tip deflection should be used to sufficiently describe the probe head behavior.