# Simulation Model for Correction and Modeling of Probe Head Errors in Five-Axis Coordinate Systems
## 2 Developed Simulative Model and Steps to Its Implementation
### Description of Simulative Model  
The developed simulative model is based on the concept of PEF described in Section 1. In this case, the number of input quantities that has to be specified in order to obtain the probe error (PE) is equal to three. The PEF may be written as presented in Equation (5):  
\[\text{PEF}\,=\,(A,\,B,\,\alpha,\,\,\text{PE}) \tag{5}\]  
where:  
\(A\)--rotation angle along the horizontal axis of the probe head,  
\(B\)--rotation angle along the vertical axis of the probe head,  
\(\alpha\)--angle in which the touch-trigger module is working,  
PE--probe error given as a result of PEF usage for considered angles. The \(A\) and \(B\) angles were presented in Figure 2. They have their zero positions when the probe is oriented vertically. The \(\alpha\) angle is dependent on the \(A\) and \(B\) angles. It is defined on the plane perpendicular to the probe when it is oriented using \(A\) and \(B\) angular positions, and has its zero indication in the direction in which the probe rotates along the \(A\) axis in the positive direction. The angle increases in the counterclockwise direction. Depending on those three input parameters, the model provides the output value, which is the simulated value of the PE. The simulation is based on the Monte Carlo method and trilinear  
Figure 2: Articulated probe head used in five-axis coordinate systems with revolution axes marked. \(A\): rotation angle along the horizontal axis of the probe head; \(B\): rotation angle along the vertical axis of the probe head. interpolation adapted for usage in polar systems. It uses the values of PE identified experimentally for selected \(A\), \(B\) and \(\alpha\) angles (experiments aimed at determining these values are presented in Section 2.2). The points determined using these angles for the chosen stylus used during measurements may be treated as nodes of the reference point grid. Let \(A_{\mathrm{s}}\), \(B_{\mathrm{s}}\) and \(\alpha_{\mathrm{s}}\) denote the values of angles for which the simulation of errors has to be performed. \(A_{\mathrm{s-1}}\), \(B_{\mathrm{s-1}}\) and \(\alpha_{\mathrm{s-1}}\) are the values of angles for the nearest node with angles lower than \(A_{\mathrm{s}}\), \(B_{\mathrm{s}}\) and \(\alpha_{\mathrm{s}}\), respectively, and \(A_{\mathrm{s+1}}\), \(B_{\mathrm{s+1}}\) and \(\alpha_{\mathrm{s+1}}\) are the values of angles for the nearest node with corresponding angles higher than \(A_{\mathrm{s}}\), \(B_{\mathrm{s}}\) and \(\alpha_{\mathrm{s}}\). The values of PEs in nodes surrounding the point defined using \(A_{\mathrm{s}}\), \(B_{\mathrm{s}}\) and \(\alpha_{\mathrm{s}}\) have to be used. In order to simulate the PE value for the point defined using \(A_{\mathrm{s}}\), \(B_{\mathrm{s}}\) and \(\alpha_{\mathrm{s}}\), the values of PEs in nodes surrounding this point have to be simulated using the Monte Carlo method. For all of the simulations presented here, the Monte Carlo method uses the scaled and shifted t-distributions with parameters (\(\overline{x}\), \(s\), \(v\)), where \(\overline{x}\) denotes the mean radial PE, \(s\) is the standard deviation associated with \(\overline{x}\) and \(v\) is the number of degrees of freedom. The parameters of these distributions are determined using the experiment presented in Section 2.2. Hence, the simulation should be performed for nodes (\(A_{\mathrm{s-1}}\), \(B_{\mathrm{s-1}}\), \(\alpha_{\mathrm{s-1}}\)), (\(A_{\mathrm{s-1}}\), \(B_{\mathrm{s-1}}\), \(\alpha_{\mathrm{s+1}}\)), (\(A_{\mathrm{s-1}}\), \(B_{\mathrm{s+1}}\), \(\alpha_{\mathrm{s-1}}\)), (\(A_{\mathrm{s-1}}\), \(B_{\mathrm{s+1}}\), \(\alpha_{\mathrm{s+1}}\)), (\(A_{\mathrm{s+1}}\), \(B_{\mathrm{s-1}}\), \(\alpha_{\mathrm{s+1}}\)), (\(A_{\mathrm{s+1}}\), \(B_{\mathrm{s-1}}\), \(\alpha_{\mathrm{s+1}}\)), (\(A_{\mathrm{s+1}}\), \(B_{\mathrm{s-1}}\), \(\alpha_{\mathrm{s+1}}\)) and (\(A_{\mathrm{s+1}}\), \(B_{\mathrm{s+1}}\), \(\alpha_{\mathrm{s+1}}\)). Then, a trilinear interpolation according to Formula (6) should be performed in order to obtain the PE value for the simulated point.