# Optimization Method of Assembly Tolerance Types Based on Degree of Freedom
## 6 Case Study
The assembly constraint relationship of the assembly and the assembly feature surface of each part are marked before automatic reasoning, as shown in Figure 8b. Select part P4 as the representative to carry out tolerance project reasoning, and obtain the markable shape tolerance of the assembly feature cylindrical surfaces P4S1 and P4S2 of workpiece P4. With the central element of feature plane P4S2 as the reference datum, the optional directional position tolerance of the target assembly feature plane P4S1 can be obtained, as shown in Table 3.
* Step 5: Determine the DOF vector \(V_{4(j)\_k}\) of the tolerance zone for all recommended tolerance types of part P4 based on the global coordinate system. Then obtain the DOF vector for all recommended tolerance types based on Table 1 (Note the order of exchanging the DOF and DOI in the DOF vector of the tolerance zone when the coordinate system in the tolerance band in the table is oriented differently from the overall coordinate system. ), as shown in Table 4.
* Step 6: Calculate the comparison freedom vectors \(V^{\prime}_{4(j)\_k}\). Use Equation (11) to perform a Boolean operation on the DOF vector \(V_{4(j)\_k}\) of the tolerance zones and the CPDF vector \(V_{0}\) for different tolerance types; the operation results are shown in Table 3.
* Step 7: Screening optimization of tolerance types. Based on the decision rule of the optimization, the comparison freedom vector \(V^{\prime}_{4(j)\_k}\) of different tolerance types is compared with the CPDF vector \(V_{0}=\)(0 0 0 1 1 0) to obtain the optimization of tolerance types. Through comparison, it can be seen that in the form tolerance class of reference datum feature plane P4S2, the comparison freedom vector \(V^{\prime}_{4(j)\_k}\) of straightness and roundness in the \(X\)-\(Z\) plane is not equal to the CPDF vector \(V_{0}\), straightness, or  
Figure 8: Form and position tolerance specification of target part P4. cylindricity in any direction, and the other two form tolerances meeting the geometric functional requirements of the assembly are obtained through screening optimization. The non-reference feature plane P4S1 is optimized to obtain three directional position tolerances, including total run-out, position, and coaxiality, and two form tolerances, including straightness and cylindricity in any direction. It can be seen from the above optimization results of tolerance types that the number of tolerance types is optimized, which not only reduces the recommended number of tolerance items, but also avoids marking unreasonable tolerance types. * Step 8: Label the detailed tolerance information of the part. From the optimization results, it can be seen that when the detailed tolerance specifications for part P4 are finally completed, further screening of tolerance types and adding tolerance symbols, such as material conditions and tolerance domain feature symbols [39; 40], are needed. According to the fit characteristics and kinematic characteristics of workpiece P4, the assembly feature surface P4S2 is the first assembly datum, and the cylindricity shape tolerance is selected [41]. The feature surface P4S1 is selected as cylindricity shape tolerance and coaxiality tolerance [42], and the final labeling of the tolerances of part P4 is shown in Figure 8. Refer to the determination steps of part P4 for detailed tolerance specifications of other assembly parts to obtain the marked tolerance as shown in Figure 9.  
* AFS represents assembly feature surface. \end{table}
Table 4: Pre-selected form tolerances and associated degrees of freedom for mating feature features.