# Optimization Method of Assembly Tolerance Types Based on Degree of Freedom
## 4 CPDF for Geometric Functional Tolerancing of Assemblies
### CPDF Acquisition
#### 4.1.3 Analysis of the Impact of Contact Change Displacement on the Geometric Function
\begin{pmatrix}0&l&n&1\end{pmatrix} \), \(-k\leq n\leq k\), \(d_{G}=r_{o}= \begin{pmatrix}0&b&k&1\end{pmatrix} \). Set the position vector of the measured point \(P\) after the position is changed as \(r_{G,m}\). From Equations (5) and (9), \(r_{G,m}\) can be obtained:  
\[r_{G,m}= \begin{pmatrix}X_{P}\\ Y_{P}\\ Z_{P}\\ 1\end{pmatrix} = \begin{pmatrix}1&0&0&0\\ 0&\cos\alpha_{2-1}&-\sin\alpha_{2-1}&b\\ 0&\sin\alpha_{2-1}&\cos\alpha_{2-1}&k\\ 0&0&0&1\end{pmatrix}\begin{pmatrix}1\\ n\\ 1\end{pmatrix} = \begin{pmatrix}0\\ l\cos\alpha_{2-1}-n\sin\alpha_{2-1}+b\\ l\sin\alpha_{2-1}+n\cos\alpha_{2-1}+k\\ 1\end{pmatrix}  \tag{11}\]  
When part\({}_{1}\) undergoes a rotational displacement around the X-axis at the assembly feature surface F\({}_{11}\), the displacement change vector \(\Delta r_{m}\) at point P within the ideal surface is obtained from Equations (10) and (11):  
\[\Delta r_{m}= \begin{pmatrix}\Delta X_{P}\\ \Delta Y_{P}\\ Z\Delta_{P}\end{pmatrix} = \begin{pmatrix}0\\ l(\cos\alpha_{2-1}-1)-n\sin\alpha_{2-1}\\ l\sin\alpha_{2-1}+n\cos\alpha_{2-1}-n\end{pmatrix}  \tag{12}\]Similarly, according to Formulas (5) and (6), calculate \(\Delta r_{l}\) when Part\({}_{1}\) rotates around the Y-axis and \(\Delta r_{n}\) when Part\({}_{1}\) rotates around the Z-axis on the assembly feature surface F\({}_{11}\), and the results are as follows:  
\[\Delta r_{l}= \begin{pmatrix}\Delta X_{p}\\ \Delta Y_{p}\\ \Delta Z_{p}\end{pmatrix} = \begin{pmatrix}-n\sin\beta_{2-1}\\ 0\\ n\sin\beta_{2-1}-n\end{pmatrix} ,\Delta r_{n}= \begin{pmatrix}\Delta X_{P}\\ \Delta Y_{p}\\ \Delta Z_{P}\end{pmatrix} = \begin{pmatrix}-l\sin\gamma_{2-1}\\ l\cos\gamma_{2-1}-l\\ 0\end{pmatrix} \]  
It can be seen from the calculation results that in the global coordinate, when the ideal assembly connection has a rotation shift change, the influence on the measurement of the perpendicularity error is as follows: When the assembly feature surface F\({}_{11}\) rotates around the \(X\)-axis, it can be seen from the change vector \(\Delta r_{m}\) that the displacement variation of the measured point \(P\) in the \(X\)-axis measurement direction is 0. When rotating around the \(Y\)-axis, it can be seen from the change vector \(\Delta r_{l}\) that the displacement variation of the measured point P in the measurement direction of the \(X\)-axis is \(n\sin\beta_{2-1}\). Based on the perpendicularity measurement method, the ideal feature plane P\({}_{2}\) to be measured is always parallel to the measurement plane P\({}_{3}\). The measurement plane P\({}_{3}\) rotates around the \(Z\)-axis synchronously with the ideal feature plane P\({}_{2}\) to be measured, and the distance between the measurement point and the placement point of the measurement instrument does not change, then the relative variation of the measurement point and the placement point of the measurement instrument is constant\({}^{\prime\prime}\) 0\({}^{\prime\prime}\), so when the measurement value of the perpendicularity error rotates around the \(Z\)-axis, the \(X\)-axial variation of the measurement value under the overall coordinate is also constant at \({}^{\prime\prime}\)0\({}^{\prime\prime}\). In summary, when the measurement reference element is a plane and the measured element is also a plane, the rotational displacement that affects the perpendicularity error is only a rotational variation around the \(Y\)-axis, and the direction of this displacement variation is the CPDF of the assembly shown in Figure 2, and the displacement variations in the other five directions do not affect the assembly geometrical function, and together they form the CPDF vector \(V_{0}\), denoted as \(V_{0}=(0\;0\;0\;0\;1\;0)\).