## 3 Modeling for Typical Task Uncertainty Evaluation
### Uncertainty Model for Location and Orientation Errors Measurement Tasks  
The location and orientation errors show the positional relationship between two or more geometric elements, i.e. the relationship between the measured elements and the reference elements. It is defined as the variation of the measured actual elements to the ideal elements with certain direction or location, able to be divided into orientation error and location error [33]. The form and position errors are both variations between the actual and ideal elements, but the methods are different for evaluation. Form measurement does not require reference elements. By sampling on the surface of the workpiece to be measured, enough sampling points of the measured elements can be obtained. Computer software can be used to fit lines, planes, circles or cylinders, and calculate the maximum variation from sampling points to fitting elements as the value of the current form error. However, the location and orientation errors are the positional relationship between two elements. It is necessary to measure the two elements, work out one fitting element as a benchmark and calculate the maximum variation from the other element to the benchmark as the measurement value of the current position error. In the principle of measurement, it is basically the same as the dimensional measurement, so the location and orientation error measurement can be concluded as miniature dimensional measurements. While evaluating the uncertainty of the CMM location and orientation tolerances measurement task, \(E_{\text{L, MPE}}\)=\(A\)+\(B\)-\(L\) may be used to represent the influence of its indication error. However, the location and orientation errors are still greatly different from dimensional measurement due to the impacts of the instrument offset and linearity; different types of position errors are differently affected by the offset and linearity of the measuring instrument. Here follows the analysis and discussion of the uncertainty components for position errors caused by the indication errors of the measuring instrument. CMM parallelism measurement task: First measure the reference plane, fit out its plane formula, then sample a characteristic point on the measured plane, calculate the longest distance \(l_{\text{max}}\) and the shortest distance \(l_{\text{min}}\) from the sampling point to the reference fitting plane, and the maximum distance difference is the parallelism tolerance \(t\) between the two planes:  
\[t=l_{\text{max}}-l_{\text{min}} \tag{11}\]  
Where, the influence of the indication errors is both \(\text{MPE}_{E}\) for the longest distance \(l_{\text{max}}\) and the shortest distance \(l_{\text{min}}\), then  
\[t=\left(l_{\text{max}}\pm E_{\text{L, MPE}}\right)-\left(l_{\text{min}}\pm E_{ \text{L, MPE}}\right) \tag{12}\]  
Fully consider the influence of \(E_{\text{L, MPE}}\) and the error range of the overestimated parallelism tolerance \(t\) to get:  
\[t=\left(l_{\text{max}}-l_{\text{min}}\right)\pm 2E_{\text{L, MPE}} \tag{13}\]If it is given that the composite distribution of two identical uniform distributions is a triangular distribution, the uncertainty component \(u_{\text{EW}_{1}}\) of the CMM parallelism measurement task caused by the indication error is as follows according to type B evaluation method [32]:  
\[u_{\text{EW}_{1}}=\frac{2E_{\text{L, MPE}}}{\sqrt{6}} \tag{14}\]  
The tolerance zone of angularity and perpendicularity is the range defined by two parallel planes whose spacing is equal to the tolerance value \(t\). Both indicate the degree to which the measured element maintains a certain angle relative to the reference element. The difference is that the two parallel planes of angularity incline to the reference at a given theoretical angle (excluding 0\({}^{\circ}\), 90\({}^{\circ}\) and 180\({}^{\circ}\)) and such two parallel planes are vertical (or parallel) to the reference.