## 3 Deviation Analysis of Weakly Rigid Parts
### Part Deviation  
In terms of the spatial coordinate system, part deviations can be decomposed into rigid translations and rigid rotations along the x, y and z directions and, for weakly rigid parts, flexible bending deformations in specific directions. 1 Typical structural dimensional planes:  
For weakly rigid thin-walled parts of the wall plate type shown in Figure 1, there are deviations along the x and y directions and rotations in the \(\sigma\) plane. 2 Atypical structural dimensional planes:  
Figure 2: (**a**) Fixture layout diagram in plane xoy. (**b**) Fixture layout diagram in plan yoz. To reduce the redundancy of the state space model, the wear of the locating block and the deviation of the part from rigid rotation in the \(\tau\) and \(\varphi\) planes are neglected. The deviations of the part in the atypical structural dimension plane are projected in the direction of its long axis, i.e., to the \(\varphi\) plane, as shown in Figure 1.  
At station \(k\), for a point \(A\) on the part, the part deviation vector is noted as:  
\[\Delta X_{A} =[\Delta x_{A}(k),\Delta y_{A}(k),\Delta z_{A}(k),\Delta\alpha(k), \Delta\beta_{A}(k)]^{T} \tag{1}\]  
In Equation (1), \(\{\Delta x_{A}(k),\Delta y_{A}(k),\Delta z_{A}(k)\}\) are the offsets of point \(A\) on the part along the x, y and z directions, \(\{\Delta\alpha\}\) is the rigid rotation deviation of the typical structural dimension plane of the part on the part, and \(\{\Delta\beta_{A}\}\) is the flexible rotation deviation of point A on the part in the projection plane. For any two points \(A\) and \(B\) on part 1, their deviations in the \(\sigma\) plane are shown in Figure 3. \(A\) and \(B\) are the theoretical positions of the two points, \(A^{\prime}\) and \(B^{\prime}\) are the actual positions of the two points and the equidistant parallel line \(A^{\prime}B^{\prime\prime}\) of \(AB\) is made with point \(A^{\prime}\) as the base point. According to the vector operation theorem, the relative deviation (line \(B^{\prime}A^{\prime}\)) between point \(A^{\prime}\) and point \(B^{\prime}\) has:  
\[\overline{B^{\prime}A^{\prime}} = \overline{B^{\prime}B}-\overline{A^{\prime}A}\] \[\overline{B^{\prime\prime}B} = \overline{A^{\prime}A}\] \[\overline{B^{\prime}A^{\prime}} = \overline{B^{\prime}B}-\overline{B^{\prime\prime}B}=\overline{B^{ \prime}B^{\prime\prime}} \tag{2}\]  
At this time, the relative deviation between \(A^{\prime}\) and \(B^{\prime}\) (\(B^{\prime}A^{\prime}\)) is transformed into the relative deviation between \(B^{\prime}\) and \(B^{\prime\prime}\) (\(B^{\prime}B^{\prime\prime}\)). Assuming that the deviations of point \(A\) and point \(B\) in the \(\sigma\) plane are \(\Delta x_{A},\Delta y_{A},\Delta\alpha\) and \(\Delta x_{B},\Delta y_{B},\Delta\alpha\), respectively, then the deviations of point \(B\) from point \(A\) in the \(\sigma\) plane are related as:  
\[ \begin{bmatrix}\Delta x_{B}\\ \Delta y_{B}\end{bmatrix}  = \begin{bmatrix}\Delta x_{A}\\ \Delta y_{A}\end{bmatrix} +\begin{bmatrix}\left\lVert H_{B,A^{\sigma}}\right \rVert_{2}\Delta\alpha\text{\em{\em{\em{\em{\em{\em{\em{\em{\em{\em{\em{\em{ \em{\em{\em{\em{\em{\em{\em{\em{\em{\em{\, \,}}}}}}}{}{}{}{}{}{}{}{}{}{ {{{{{{{  
uneven distribution of support forces and other factors, it can be considered that the flexible deformation of the part in the \(\varphi\) plane can be regarded as a smooth arc distributed along the y axis. Suppose the deviations of point \(A\) and point \(C\) in the \(\varphi\) plane are \(\Delta y_{A},\Delta z_{A},\Delta\beta\) and \(\Delta y_{C},\Delta z_{C},\Delta\beta\), respectively, then the deviation of point \(C\) from point \(A\) in the \(\varphi\) plane is related to  
\[ \begin{array}{c}\left[\Delta z_{C}\right]=\left[\Delta z_{A}\right]+\left[ \left\|H_{C,A}^{\varphi}\right\|_{2}\Delta\beta\text{\em cose}_{\varphi} \right]\\ \left[\Delta z_{C}\right]=\left[1\quad\left\|H_{C,A}^{\varphi}\right\|_{2} \text{\em cose}_{\varphi}\right]\cdot\left[\begin{array}{c}\Delta z_{C}\\ \Delta\beta\end{array}\right]\end{array}