## 4 Conclusions  
This work presents a kinematic model to simulate the effect of two sources of uncertainty, the localization and the machine ones, on the geometrical deviation of a surface manufactured by face milling, contouring or drilling processes. The model consists of four steps. The first one evaluates the influence of the deviations in the locator positions on the workpiece reference frame. The second one estimates the deviations in the workpiece reference frame as a function of the form deviations in the workpiece datum surfaces. The third one defines the volumetric uncertainty of the machine tool. Finally, the four steps combine the three deviations. A chain of coordinate transfers from a datum reference system to another is the heart of this work, with the results being evidently right. It overcomes the literature because it considers two uncertainty sources together, such as the locator position deviations, the datum deviations and the volumetric uncertainty of the machine tool, in a simple and effective model. The proposed kinematic model was applied to a plate that is located through 3-2-1 locators, it had a flatness deviation on its datum surfaces in contact with the locators and the volumetric uncertainty of the used machine tool was known. In this way, it was possible to estimate numerically the geometrical deviations of the milled and drilled surface by saving time, energy and materials because experimental tests have been avoided. In particular, the flatness and orientation deviation from nominal were investigated for a milled surface and the orientation deviation of a drilled hole was analyzed. From the results, it is clear how the proposed model easily allows us to estimate the deviation of a milled or a drilled surface due to the combined effect of two uncertainty sources. The proposed model allows us to take into account geometrical deviations of the part datum surfaces of 0.001 mm, location deviations in the locators of \(\pm\) 0.03 mm and machine tool positional and rotational uncertainties of 0.01 mm and \(\sigma_{d}=0.01*\frac{\pi}{180}\) mm, respectively. The proposed model is kinematic and it is based on the results sought in the kinematics of robots that are validated in the literature. The idea of the model is simple enough to be evidently right, since it is based on the transfer of coordinates from a datum reference system to another. The innovative ideas are the terms of the uncertainty transfer, i.e., the locator uncertainty, the workpiece datum uncertainty and the machine tool volumetric uncertainty, that are considered together inside a simple and effective model. However, experimental work to set the parameters of each considered contribution to hole or milled plane deviation is matter for further studies. The time required by the kinematic model to simulate the two sources of uncertainty in milling or drilling was very short (a few seconds) by using a computer with an Intel core i7 950 processor running at 3.07 GHz, with 16 GB of RAM, a mechanical hard drive of 1 TB at 5400 rpm and running Windows 7 Professional. The required time is so short that it is compatible with online data processing. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Uncertainty Source** & **Conforming Fraction** & **Conforming Fraction for** \\  & **for Case 1 (\%)** & **Case 2 (\%)** \\ \hline Locator uncertainty & 99.99 & 99.99 \\ Volumetric uncertainty of machine tool & 99.48 & 99.48 \\ Datum flatness & 99.78 & 90.11 \\ Locator uncertainty + Volumetric uncertainty & 99.14 & 95.63 \\ Locator + Datum flatness & 75.89 & 83.49 \\ Volumetric uncertainty + Datum flatness & 39.15 & 46.68 \\ Locator deviation + Volumetric uncertainty + Datum flatness & 31.48 & 36.28 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results of one drilled hole. Further elements influencing the geometrical deviations of the manufactured surface, such as those connected with the process uncertainty, may be added in a subsequent development of the model. It is a subject of further research. Conceptualization, W.P.