## 3 Assemblability Analysis Based on Coordination Space Model  
Assemblability refers to the geometric consistency of the matching geometric structures of the two assembling parts. It can be judged whether the assembly can directly be carried out by assemblability prediction. The assemblability is good if the coordination space is greater than 0, which means at least one pose conforms to Equation (15). Otherwise, the assemblability is bad. Therefore, the assemblability analysis flow is shown in Figure 5.  
Firstly, the KACs are measured by a laser tracker or other digital measurement devices. Then, the coordination space model is constructed based on the assembly accuracy requirements. The volume of the coordination space is solved to judge whether it is assemblable. It will be assemblable when \(\varnothing_{CS}\) is greater than 0. The assembly can be executed by calculating the optimal pose and aligning the parts. It will be uncoordinated when \(\varnothing_{CS}\) is 0. Then, the assembly deviation should be analyzed and compensated to make it assemblable. The solution process of the coordination space is based on the Monte Carlo method:  
1. Calculate the optimal pose based on the least-squares method. 2. According to the dimensions and assembly accuracy requirements, a maximum pose space is assumed, as Equation (16) shows. All poses out of the space are not valid for any assembly accuracy requirements. \[\omega_{d}:(-x_{d},-y_{d},-z_{d},-\alpha_{d},-\beta_{d},-\gamma_{d})\to(x_{d},y_{d},z_{d},\alpha_{d},\beta_{d},\gamma_{d}),\] (16)
3. Generate a random SDT uniformly for \(n_{T}\) times and check the SDTs by Equation (15). 4. If \(n_{j}\) of \(n_{T}\) SDTs are valid, the coordination space is \[\varnothing_{CS}=\frac{n_{j}}{n_{T}}64x_{d}y_{d}z_{d}\alpha_{d}\beta_{d})^{ \prime}_{d},\] (17)  
In the case of incoordination, the coordination space should be further analyzed. According to Equation (15), the coordination space is the intersection of KAC's constraint equations. All constraints are divided by KACs. Equation (15) will be translated to  
Figure 5: Assemblability analysis process. \[\left\{\omega_{\Delta}\right| \begin{array}{l}\forall f_{j}^{1}\big{(}\mathbf{K}_{1 }^{R},\mathbf{K}_{1}^{A}\boldsymbol{\omega}_{\Delta}^{H}\big{)}\in\left[T_{j}^ {1-min},T_{j}^{1-max}\right],j\in[1,j_{1}],\\ \forall f_{j}^{2}\big{(}\mathbf{K}_{2}^{R},\mathbf{K}_{2}^{A}\boldsymbol{\omega }_{\Delta}^{H}\big{)}\in\left[T_{j}^{2-min},T_{j}^{2-max}\right],j\in[1,j_{2} ],\\ \ldots\\ \forall f_{j}^{n}\big{(}\mathbf{K}_{n}^{R},\boldsymbol{\omega}_{\Delta}^{H} \big{)}\in\left[T_{j}^{n-min},T_{j}^{n-max}\right],j\in[1,j_{n}]\end{array}  \right\}, \tag{18}\]  
Let  
\[\varnothing_{i}^{KAC}=\left\{\omega_{\Delta}^{i-KAC}\right|\ \forall f_{j}^{i}\big{(} \mathbf{K}_{i}^{R},\mathbf{K}_{i}^{A}\boldsymbol{\omega}_{\Delta}^{H}\big{)} \in\left[T_{j}^{i-min},T_{j}^{i-max}\right],j\in[1,j_{i}]\ \ \ \ \right\}, \tag{19}\]  
where \(\varnothing_{\Delta}^{KAC}\) is the \(KAC\) coordination space formed by the assembly accuracy requirements of a \(KAC\), and \(\omega_{\Delta}^{i-KAC}\) is an SDT in the \(KAC\) coordination space. The \(\varnothing_{CS}\) would be  
\[\varnothing_{CS}=\bigcap_{i=1}^{n}\varnothing_{i}^{KAC}, \tag{20}\]  
The relationship between the KAC coordination space and the assembly coordination space is shown in Figure 5(a). Figure 5(b) shows the status of the \(KAC\) coordination space when the assembly is uncoordinated. Each area of the same color represents a \(KAC\) coordination space. The divided zone is named the coordination zone. The accuracy compensation method is needed to improve the assemblability. According to Figure 5(b), set the union of \(KAC\) coordination space as a union coordination space. It is formulated as  
\[\varnothing_{UCS}=\bigcup_{i=1}^{n}\varnothing_{i}^{KAC}, \tag{21}\]  
where \(\varnothing_{UCS}\) is the union coordination space. In the union coordination space, all poses are valid for some KACs but not valid for all.