# Assembly Tolerance Design Based on Skin Model Shapes Considering Processing Feature Degradation
## 4 Case Study
### Stochastic Process Training and Parameter Calculation  
Twenty outsourced parts were randomly chosen. Parallel lines in two directions were drawn on assembly surfaces with a uniform interval of 0.01 mm. Sampling points are defined as the intersection points of these lines. We used a non-contact coordinate measuring machine with an accuracy of at least 0.001 mm to measure the actual location of sampling points. The measured coordinates amounted to 20-point cloud sets \(\{T_{0},T_{1},\cdots,T_{20}\}\). Take surface F as an example: the measured point clouds sets are \(\{T_{F0},T_{F1},\ \cdots,T_{F20}\}\). Two of the three coordinates are determined by sampling points. The third coordinate is called the feature parameter. A first-order gradient object function of maximum likelihood estimation with weight decay is shown in Equation (17):  
\[g(w)=\sum_{i=1}^{N}\sum_{t=1}^{T}f_{k}(x_{t},y_{t},t)-\sum_{i=1}^{N}\sum_{y} \left[p(y|x,t)\times\sum_{t=1}^{T}f_{k}(x_{t},y_{t},t)\right]-\frac{\sum w^{2} }{2\delta^{2}} \tag{17}\]  
Set the initial parameter weights as \(w^{(0)}=\{0.1,0.1,\cdots,0.1\}\) and the initial positive definite matrix as \(M^{(0)}=diag(0.1,0.2,\cdots,2.0)\). The coefficient of the correct term is set to 0.5. The maximum number of iterations is 1000. The feature weights of \(P(y|x,t)\) corresponding to processing features are calculated according to Section 2.1. Some of the feature weights of surface F are shown in Table 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Target Accuracy Term** & **Explanation** & **Symbol** & **Design Requirement** \\ \hline \(FR_{F}\) & Runout of surface F & \(e_{f}\) & 0.020 mm \\ \hline
**Accuracy Terms of the Shaft Part** & **Explanation** & **Symbol** & **Nominal Value** \\ \hline \(FL_{F}\) & Flatness of surface F & \(\varepsilon_{1}\) & 0.010 mm \\ \(TR_{C1}\) & Total runout of the distal & \(\varepsilon_{2}\) & 0.020 mm \\ \(TR_{C2}\) & Total runout of the proximal end surface & \(\varepsilon_{3}\) & 0.010 mm \\ \(FL_{G}\) & Flatness of shoulder & \(\varepsilon_{4}\) & 0.010 mm \\  & surface G & & \\  & Verticality between the & & \\ \(VT_{CF}\) & distal end surface and & \(\varepsilon_{5}\) & 0.010 mm \\  & axis C & & \\ \hline
**Accuracy Terms of the Hole Part** & **Explanation** & **Symbol** & **Nominal Value** \\ \hline \(FL_{E}\) & Flatness of the shoulder & \(e_{1}\) & 0.010 mm \\ \(TR_{D}\) & Total runout of axis D & \(e_{2}\) & 0.012 mm \\  & Verticality between axis & \(e_{3}\) & 0.005 mm \\  & D and surface A & \(e_{3}\) & 0.005 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of the target accuracy term of the assembly, the nominal value of the accuracy terms of the shaft part, and the to-be-designed accuracy terms of the hole part. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Feature Terms** & **Feature Weight Value** & **Feature Terms** & **Feature Weight Value** \\ \hline \(w_{1}\) & 0.036 & \(w_{8}\) & 0.124 \\ \(w_{2}\) & 0.498 & \(w_{9}\) & 0.965 \\ \(w_{3}\) & 0.015 & \(w_{10}\) & 2.326 \\ \(w_{4}\) & 0.968 & \(w_{11}\) & 0.216 \\ \(w_{5}\) & 1.526 & \(w_{12}\) & 2.979 \\ \(wThe prediction of degraded machined surface F is obtained using the methods in Section 2.3. Consider a particular sampling point \((i,j)\), with the assumption that its feature parameter of the location on the predictive machined surface is \(y^{*}_{i,j}\) and its feature parameter of the location on the real surface is measured as \(y^{(1)}_{i,j}\). The geometric average of the overall location residuals \(E_{r}\) is set as an estimation of the precision of the predictive machined surface model, as shown in Equation (18):  
\[E_{r}=\frac{\sum\limits_{N=1}^{20}\sqrt{\sum\limits_{i,j}\left(y^{(N)}_{i,j}- \overline{y^{(N)}_{i,j}}\right)^{2}}}{\sum\limits_{N=1}^{20}\sqrt{\sum \limits_{i,j}\left(y^{*}_{i,j}-y^{(N)}_{i,j}\right)^{2}}} \tag{18}\]  
The value of \(E_{r}\) decreases when the difference between the predictive location and actual location of all the sampling points grows larger.