## 4 Assemblability Optimization Based on the Union Coordination Space  
The accuracy compensation process is time- and effort- consuming [22] when the assemblability is poor. For example, it needs programming, clamping, tool setting, machining, loosen clamping, and other steps when finishing a KAC with cutting. Therefore, reducing the number of KACs to  
Figure 6: The relationship between the part and feature PFS. be processed is an effective means to improve the assembly efficiency in many cases. The optimal pose is usually obtained under the condition of optimal assembly accuracy. If each unqualified KAC is compensated one by one under the optimal pose, more work may be needed and the assembly quality might not be good, due to the unknown assembly quality after accuracy compensation. If the assembly quality is bad after compensation, there are no alternative compensation schemes based on the least-squares method. Therefore, the assemblability optimization method is proposed to solve the incoordination problem. The key to optimize the assemblability is whether there is one or more coordination zones that can satisfy assembly accuracy requirements with fewer KACs to be compensated and a better or approximate volume of coordination space. The coordination zone index shows the valid KACs in the certain coordination zone. The total number of all KACs is \(n_{KAC}\). The incoordination zone index shows the number of uncoordinated KACs in the coordination zone. Their relationship is  
\[n_{IZI}=n_{KAC}-n_{CZI}, \tag{22}\]  
where \(n_{IZI}\) is the incoordination zone index and \(n_{CZI}\) is the coordination zone index. If the accuracy of uncoordinated KACs is compensated well in the coordination zone, this coordination zone will change to the assembly coordination space, as Figure 7 shows. In this way, each coordination zone can be analyzed to check whether it is good to be compensated or not. Two indicators of the coordination zone should be analyzed, one is the incoordination zone index, and the other is the volume of the coordination zone. The Monte Carlo method of Section 3 is improved to judge the state of each coordination zone one by one, and the optimal assemblability optimization schemes of the coordination zone are selected for recording. The solution process based on the Monte Carlo method is as follows:  
1. Solve the optimal pose of the align part;
2. Set a pose space as the pose boundary as shown by the square box of Figure 8;
3. Generate a random SDT in the pose space;
4. According to Equation (19), judge which KAC equations are satisfied (coordination zone index) and which are not (incoordination zone index);
5. Cluster the analysis results of each SDT. The SDTs in the same coordination zone are clustered together;
6. Put the clustered results into the data structure of Equation (23). The KAC number to be compensated is the incoordination zone index. Select the scheme with a better KAC number and space volume of the coordination zone. \[\left\{\Gamma\middle|\Gamma_{i}=\left(n_{IZI},V_{CZ},b_{f},s_{u}\right),i< \Gamma_{num}\right\},\] (23)  
Figure 7: The coordination zone variation process by accuracy compensation. where \(\Gamma_{i}\) is the \(i\)th scheme, \(n_{IZI}\) is the incoordination zone index, \(V_{\text{CZ}}\) is the space amount of the coordination zone, \(b_{f}\) is the information of uncoordinated KACs, \(s_{\omega}\) is the SDT set, and \(\Gamma_{num}\) is the max number of the schemes. 7. Calculate the center SDT of the SDTs in the selected scheme. The assembly deviation of target features under the SDT is analyzed and the accuracy compensation is carried out. \[\mathbf{\omega}_{\Delta}^{c}=\frac{1}{n_{s}}\sum_{i=1}^{n_{s}}\mathbf{\omega}_{\Delta i},\] (24) where \(\mathbf{\omega}_{\Delta}^{c}\) is the center SDT, \(n_{s}\) is the SDT number of \(s_{\omega}\), and \(\mathbf{\omega}_{\Delta i}\) is an SDT of \(s_{\omega}\). All assembly accuracies on \(\mathbf{\omega}_{\Delta}^{c}\) are calculated. Then, the deviations on the excessive KACs will be compensated.