# Assembly Tolerance Design Based on Skin Model Shapes Considering Processing Feature Degradation
## 3 Constrained 3D Assembly Simulation and Tolerance Synthesis
### Static and Dynamic Tolerance Synthesis  
In this paper, tolerance synthesis assigns tolerance indices \(X\), minimizes manufacturing cost, maximizes assembling probability, and meets the assembly functional requirement \(Y\) at the same time. The \(X\) tolerance index comprises the given indices \(X_{g}\) (mainly in outsourced parts) and to-be-designed indices \(X_{d}\) (mainly in self-made parts). \(X_{g}\) and \(X_{d}\) are independent from the perspective of manufacturing, but they become relevant because of the assembly functional requirement \(Y\). Additional objectives including the manufacturing cost and assembling probability turn this issue into an optimization problem.  
To increase the robustness of the design scheme, feature degradation during the machining process is considered when generating observable parts. Predictions of part surfaces at any time during the designed life are made using the methods in Section 2. Generally, surfaces in their late design age with seriously deteriorated geometric features are used in tolerance synthesis. The surfaces of predictable parts are simulated by a multivariate Gaussian process (MGP) based on given tolerance indices. Repetitive experiments and variated models are applied in the design process due to the stochastic characteristic of the MGP. Numerical experiments of assembly simulation provide assembly tolerance indices, which are usually categorized into static tolerance indices (such as verticality and parallelism) and dynamic tolerance indices (such as circular runout and end face runout). Static tolerance indices are certain indices based on measurements and the analysis of assembled Skin Model Shapes. These only vary when the tolerance models are varied (in Figure 4a). However, dynamic tolerance indices are obtained through monitoring a series of static tolerance indices by analyzing the possible motion and translational/rotational samplings. Measuring positions are sampled for experiments through a Monte Carlo process. As shown in Figure 4b, the upper end face parallelism of a sliding table on a high-precision guide is a typical dynamic tolerance index, which is calculated  
Figure 3: Main steps in the computation of assembling condition when considering assembly force constraints.  
through a series of assembly simulations after sampling measuring positions in its translational motion. Similarly, common runout errors of cylindrical parts (such as axle parts) can also be obtained by sampling measuring orientations in the rotational motion of those parts. A small sampling interval and large sampling quantity lead to a precise estimation.  
As shown in Figure 4, assembly tolerance indices are estimated through data analysis in each measuring position/orientation. The distribution of estimated assembly tolerance indices to the to-be-designed tolerance indices is summarized in the form of tables or figures. Then, an integration of assembly functional requirements and manufacturing cost estimation is introduced to outline the objectives. Due to the stochastic characteristic of tolerance models, not all the examples with the same assigned tolerance scheme can meet the assembly functional requirements. Generally, if the ratio of conformation reaches a certain threshold (such as 95%), the tolerance scheme is treated as reliable. On that basis, the manufacturing costs of all the reliable tolerance schemes are computed and compared based on industrial statistics and empirical formulation. That with the lowest costs is selected as the designed tolerance scheme.  
Figure 4: Synthesis of static and dynamic assembly tolerance indices based on assembly simulation.