# 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
\[ \begin{array}{c}PEF(A_{\mathrm{s}},\ B_{\mathrm{s}},\ \alpha_{\mathrm{s}})\ =\ \left((A_{\mathrm{s+1}}-A_{\mathrm{s}})/(A_{\mathrm{s+1}}-A_{\mathrm{s-1}}) \times((B_{\mathrm{s+1}}-B_{\mathrm{s}})/(B_{\mathrm{s+1}}-B_{\mathrm{s-1}}) \times P2+\right.\\ \left.(B_{\mathrm{s}}-B_{\mathrm{s-1}})/(B_{\mathrm{s+1}}-B_{\mathrm{s-1}}) \times P4)\right)\ +\ \left((A_{\mathrm{s}}-A_{\mathrm{s-1}})/(A_{\mathrm{s+1}}-A_{\mathrm{s-1}}) \times((B_{\mathrm{s+1}}-B_{\mathrm{s}})/(B_{\mathrm{s+1}}-B_{\mathrm{s-1}}) \times P1+\right.\\ \left.(B_{\mathrm{s}}-B_{\mathrm{s-1}})/(B_{\mathrm{s+1}}-B_{\mathrm{s-1}}) \times P3)\right)\end{array}  \tag{6}\]  
where:  
\[ \begin{array}{c}\mathrm{P1}=\left(((\alpha_{\mathrm{s+1}}-\alpha_{\mathrm{ s}})/(\alpha_{\mathrm{s+1}}-\alpha_{\mathrm{s-1}}))\times\mathrm{PEF}(A_{ \mathrm{s+1}},\ B_{\mathrm{s-1}},\ \alpha_{\mathrm{s-1}})\right)+\left(((\alpha_{\mathrm{s}}-\alpha_{\mathrm{s-1}}) /(\alpha_{\mathrm{s+1}}-\alpha_{\mathrm{s-1}}))\times\mathrm{PEF}(A_{\mathrm{ s+1}},\ B_{\mathrm{s-1}},\ \alpha_{\mathrm{s+1}})\right)\\ \mathrm{P2}=\left(((\alpha_{\mathrm{s+1}}-\alpha_{\mathrm{s}})/(\alpha_{\mathrm{ s+1}}-\alpha_{\mathrm{s-1}}))\times\mathrm{PEF}(A_{\mathrm{s-1}},\ B_{\mathrm{s-1}},\ \alpha_{\mathrm{s-1}})+\left(((\alpha_{\mathrm{s}}-\alpha_{\mathrm{s-1}})/(\alpha_{ \mathrm{s+1}}-\alpha_{\mathrm{s-1}}))\times\mathrm{PEF}(A_{\mathrm{s-1}},\ B_{\mathrm{s-1}},\ \alpha_{\mathrm{s+1}})\right)\\ \mathrm{P3}=\left(((\alpha_{\mathrm{s+1}}-\alpha_{\mathrm{s}})/(\alpha_{\mathrm{ s+1}}-\alpha_{\mathrm{s-1}}))\times\mathrm{PEF}(A_{\mathrm{s+1}},\ B_{\mathrm{s+1}},\ \alpha_{\mathrm{s-1}})\right)+\left(((\alpha_{\mathrm{s}}-\alpha_{\mathrm{s-1}})/( \alpha_{\mathrm{s+1}}-\alpha_{\mathrm{s-1}}))\times\mathrm{PEF}(A_{\mathrm{s+1}}, \ B_{\mathrm{s+1}},\ \alpha_{\mathrm{s+1}})\right)\\ \mathrm{P4}=\left(((\alpha_{\mathrm{s+1}}-\alpha_{\mathrm{s}})/(\alpha_{\mathrm{ s+1}}-\alpha_{\mathrm{s-1}}))\times\mathrm{PEF}(A_{\mathrm{s-1}},\ B_{\mathrm{s+1}},\ \alpha_{\mathrm{s-1}})\right)+\left(((\alpha_{\mathrm{s}}-\alpha_{\mathrm{s-1}})/( \alpha_{\mathrm{s+1}}-\alpha_{\mathrm{s-1}}))\times\mathrm{PEF}(A_{\mathrm{ s-1}},\ B_{\mathrm{s+1}},\ \alpha_{\mathrm{s+1}})\right)\end{array} \]  
The presented model may serve two primary purposes. The first involves its use as part of virtual model of a five-axis coordinate system. In this application, it should be connected with one of the modules responsible for the simulation of the CMM's kinematic system errors (presented in [10; 12; 13; 16]). For all of the points considered in a simulated measurement, the kinematic errors should first be simulated, and then the probe head errors should be simulated using the presented model. A single simulation of the probe head error for the considered point is done using the methodology presented above. For proper functioning of the virtual model, the simulation should be repeated \(n\) times, where \(n\) is the number of Monte Carlo trials set by the user or manufacturer of the virtual CMM software. The second possible usage of the developed model is to utilize it for simulation and correction of probe head errors during the measurements performed in five-axis mode. A detailed description of this process is presented in Section 2.4.