## 4 Deviations in the Fixture Positioning and the Resulting Part Deviations
### Part Deviations Due to Fixture Deviations  
1 Typical structural dimensional planes:  
Figure 6: Diagram of the deviation of two positioning points \(P_{1}\) and \(P_{2}\) on the \(\varphi\) plane.  
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At station \(k\), it follows from Equation (3) that the deviation of any point \(A\) on part 1 from the positioning point \(P_{1}\) is as follows:  
\[\begin{bmatrix}\Delta x_{A}\\ \Delta y_{A}\end{bmatrix}=\begin{bmatrix}1&0&\lVert H_{A,P_{1}}{}^{\sigma} \rVert_{2}\,cose_{\sigma}\\ 0&1&-\lVert H_{A,P_{1}}{}^{\sigma}\rVert_{2}\,sine_{\sigma}\end{bmatrix}. \begin{bmatrix}\Delta x_{P_{1}}\\ \Delta y_{P_{1}}\\ \Delta\alpha\end{bmatrix}\] (16)  
\(\copyright\) Atypical structural dimensional planes:  
As can be seen from the previous section, the flexible deformation of the part in the \(\varphi\) plane can be regarded as an arc, and the flexible bending deformation angle \(\Delta\beta\) varies at different coordinate points. As shown in Figure 7, any point \(A\) on the part, \(A^{\prime}\) is the actual position of the point, and \(P_{1}{}^{\prime}\) and \(P_{2}{}^{\prime}\) are taken as the base points to make equal parallel lines \(P_{1}{}^{\prime}P_{2}{}^{\prime\prime}\) and \(P_{2}{}^{\prime}P_{2}{}^{\prime\prime\prime}\) of line \(P_{1}P_{2}\), respectively, and \(\angle P_{2}{}^{\prime}P_{1}{}^{\prime}P_{2}{}^{\prime\prime}\) is noted as \(\Delta\beta^{\prime}\), \(\angle A^{\prime}P_{1}{}^{\prime}P_{2}{}^{\prime\prime\prime}\) is \(\Delta\beta\). When the point \(A^{\prime}\) is located at position \(A_{a}{}^{\prime}\) in Figure 7a, the relationship between the corresponding \(\Delta\beta\) and \(\Delta\beta^{\prime}\) is as follows:  
\[\Delta\beta=\Delta\beta^{\prime}-\frac{180\cdot\lVert H_{A,P_{2}}{}^{\prime} \rVert_{2}}{2\pi\cdot\rho_{A,P_{1},P_{2}}},\ \Delta\beta^{\prime}>0\] (17)  
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When the point \(A^{\prime}\) is located at position \(A_{b}{}^{\prime}\) in Figure 7a, the relationship between the corresponding \(\Delta\beta\) and \(\Delta\beta^{\prime}\) is as follows:  
\[\Delta\beta=\Delta\beta^{\prime}-\frac{180\cdot\left\|H_{A,P_{2}}{}^{\varphi} \right\|_{2}}{2\pi\cdot\rho_{A,P_{1},P_{2}}},\ \Delta\beta^{\prime}>0 \tag{18}\]  
When the point \(A^{\prime}\) is located at position \(A_{c}{}^{\prime}\) in Figure 7c, the relationship between the corresponding \(\Delta\beta\) and \(\Delta\beta^{\prime}\) is as follows:  
\[\Delta\beta=\Delta\beta^{\prime}+\frac{180\cdot\left\|H_{A,P_{2}}{}^{\varphi} \right\|_{2}}{2\pi\cdot\rho_{A,P_{1},P_{2}}},\ \Delta\beta^{\prime}>0 \tag{19}\]  
When the point \(A^{\prime}\) is located at position \(A_{d}{}^{\prime}\) in Figure 7d, the relationship between the corresponding \(\Delta\beta\) and \(\Delta\beta^{\prime}\) is as follows:  
\[\Delta\beta=\Delta\beta^{\prime}+\frac{180\cdot\left\|H_{A,P_{2}}{}^{\varphi} \right\|_{2}}{2\pi\cdot\rho_{A,P_{1},P_{2}}},\ \Delta\beta^{\prime}>0 \tag{20}\]  
From Equation (4), the deviation of any point \(A\) on the part from the positioning point \(P_{1}\) at station \(k\) is related as follows:  
\[\left[\Delta z_{A}\right]=\left[1\quad\left\|H_{A,P_{1}}{}^{\varphi}\right\|_ {2}sine_{\varphi}\right]\cdot\left[\begin{matrix}\Delta zP_{1}\\ \Delta\beta\end{matrix}\right] \tag{21}\]  
In summary, the deviation of any point \(A\) on the part is related to the deviation of the positioning point \(P_{1}\) as follows:  
\[\Delta X_{A}=E_{A,P_{1}}\cdot F_{A,P_{1}}\cdot\Delta X_{P_{1}} \tag{22}\]  
\[E_{A,P_{1}}=\begin{bmatrix}1&0&0&\left\|H_{A,P_{1}}{}^{\varphi}\right\|_{2} cos\epsilon_{\sigma}&0\\ 0&1&0&-\left\|H_{A,P_{1}}{}^{\varphi}\right\|_{2}sine_{\varphi}&0\\ 0&0&1&0&\left\|H_{A,P_{1}}{}^{\varphi}\right\|_{2}cos\epsilon_{\varphi}\\ 0&0&0&1&0\\ 0&0&0&0&1\end{bmatrix} \tag{23}\]  
\[F_{A,P_{1}}=\begin{bmatrix}1&0&0&0&0\\ 0&1&0&0&0\\ 0&0&1&0&0\\ 0&0&0&1&0\\ 0&0&0&1&0\\ 0&0&0&0&1\pm sin\left(\frac{180\cdot\left\|H_{A,P_{1}}{}^{\varphi}\right\|_{2} }{2\pi\cdot\rho_{A,P_{1},P_{2}}}\right)\cdot\frac{1}{\Delta\beta^{\prime}} \end{bmatrix} \tag{24}\]