# Impact of a Robot Manipulation on the Dimensional Measurements in an SPC-Based Robot Cell
## 2 Process Capability--Capability Indices  
To quantify process capability, a statistical analysis of the process itself, considering the process mean \(\mu\) and the standard deviation \(\sigma\), has to be taken into account. Normally, \(\mu\) and \(\sigma\) are unknown and approximated by \(\bar{x}\) and \(s\), respectively. For a statistical evaluation, several capability indices are used in manufacturing, such as precision-to-tolerance (PTR) [27; 28; 29; 30; 31; 32; 33; 34], number of distinct categories (NDC) [28; 34], repeatability and reproducibility (R&R) [31; 34; 35], and discrimination ratio (DC) [34; 35]. The indices can be roughly divided into two subgroups for:  
1. comparing the tolerance of the part's specifications with the measurement system's variability \(\sigma_{m}\), and
2. matching the process variability \(\sigma_{p}\) to the measurement system's variability. The PTR is the most commonly used criterion concerning the first group because of its simplicity. It represents the percentage of tolerance consumed by the variability of the measurement. The general form is shown in (1), where \(k\) is a constant and corresponds to the limiting number of standard deviations. Normally, k equals 6. \[PTR=\frac{k\,\sigma_{m}}{TOL}. \tag{1}\]  
Figure 1: The primary division of robot grasping, manipulation, and measurements. The NDC, the percentage of repeatability and reproducibility (%GR&R), and DC are the main criteria based on process and measurement variability. The NDC criterion is often reported as the signal-to-noise ratio (SNR). The first group of capability indices should be taken into account during part inspection and measurement-system selection. In the case of process control, measurement-system selection has to be made using the second group's criteria. In many cases, when the data from parts are used to determine the process variability, measurement-system selection must consider both groups of criteria simultaneously. In the literature, a lower PTR value than the threshold value (commonly set between 5% [36; 37] and 10% [31]) indicates an acceptable measurement method. The NDC has the opposite behavior to that of the PTR. A larger value indicates a more acceptable measurement method, and the threshold value is commonly set to 5 [1; 38]. It must be pointed out that there is no indication of how these thresholds were established. Referencing the MSA study of Reference [23; 24], a type-1 study corresponds to the capability of the measurement equipment (the measurement process) and type-2 and -3 studies correspond to a repeatability and reproducibility study (with and without the operator's influence). The measurement equipment's capability is commonly calculated as the C\({}_{p}\) factor, which is called the precision index and is calculated as:  
\[ \begin{split} C_{p}&=\frac{\textit{allowable process spread}}{\textit{actual process spread}}\\ &=\frac{\textit{TOL}}{6\;\sigma_{p}}\\ &=\frac{\textit{USL}-\textit{LSL}}{6\;\sigma_{p}}\\ \end{split} . \tag{2}\]  
USL and LSL are the upper and lower specification limits. Due to its simplicity, C\({}_{p}\) cannot provide an assessment of the process centering. To measure the degree of process centering relative to the manufacturing tolerance, the accuracy index C\({}_{a}\) is defined as:  
\[ \begin{split} C_{a}=1-\frac{|\mu-m|}{d},\end{split}  \tag{3}\]  
where \(\mu\) is the process mean, \(d\;=\;\frac{\textit{USL}-\textit{LSL}}{2}\) is the half specification width, and \(m\;=\;\frac{\textit{USL}+\textit{LSL}}{2}\) is the midpoint between the upper and lower specification limits. As can be seen from (3), any deviation from the centered point results in a reduction of the C\({}_{a}\) value. Combining the C\({}_{p}\) and C\({}_{a}\) factors results in a definition of the C\({}_{pk}\) factor as:  
\[ \begin{split} C_{pk}=min\left\{\frac{\textit{USL}-\mu}{3\;\sigma },\frac{\mu-\textit{LSL}}{3\;\sigma}\right\}=\frac{d-|\mu-m|}{3\;\sigma}.\end{split}