## 2 Coordination Space Model Based on Small Displacement Torsor  
The coordination accuracy is  
\[\
abla_{AB}=_{1}+_{2}=L_{A}-L_{B}, \tag{1}\]  
It can be seen that the coordination accuracy is the amount of the allowance on a certain dimension. In this way, the assembly coordination of a single dimension is well presented by coordination accuracy such as angle, length, etc. However, it is not suitable for complicated assembly. Therefore, the concept should be extended to pose allowance space and the space can be predicted by digital measurement data during large-scale assembly. This space is named the assembly coordination space, which is the ability of pose variation under the condition of assembly accuracy requirements, as Figure 2 shows. The parts of the assembly are divided into the reference part and the align part. The reference part is the fixed part during assembly and the align part will move to the target pose by the pose adjustment tooling. Assume that the primary measurement data of the two parts are  
\[\left\{ \begin{array}{l}\mathbf{P}^{R}=\left[\mathbf{p}_{1}^{R}\ \mathbf{p}_{2}^{R}\cdots\ \mathbf{p}_{n}^{R}\right]\\ \mathbf{P}^{A}=\left[\mathbf{p}_{1}^{A}\ \mathbf{p}_{2}^{A}\cdots\ \mathbf{p}_{n}^{A}\right] \end{array} \right., \tag{2}\]  
where \(\mathbf{P}^{R}\) and \(\mathbf{P}^{A}\) are the point sets of the reference part and align part, separately, where \(\mathbf{p}_{1}^{R}\), etc. and \(\mathbf{p}_{1}^{A}\), etc. are the points of the sets \(\mathbf{P}^{R}\) and \(\mathbf{P}^{A}\), respectively. The two parts are separated first. According to the least-squares method, the optimal assembly pose can be calculated by  
\[\left\{ \begin{array}{l}\mathbf{P}^{R}\approx\mathbf{R}\mathbf{P}^{A}+\mathbf{T}\\ e=\min\Bigl{\{}\sum_{i=1}^{n}\|\mathbf{R}\mathbf{p}_{1}^{A}+\mathbf{T}-\mathbf{P}^{R}\|^{2} \Bigr{\}}\end{array} \right., \tag{3}\]  
Figure 1: Coordination accuracy of a keyway assembly. Figure 2: The pose variations under the condition of assembly accuracy requirements. where \(R\) is the rotation matrix, \(T\) is the movement matrix, and \(e\) is the minimum residual sum of squares. The singular value decomposition method [24] is taken to calculate the parameter \(R\) and \(T\). Then, the optimal pose of the align part based on the least-squares method is  
\[\mathbf{\omega}_{0}=\left[ \begin{array}{cc}\mathbf{R}&\mathbf{T}\\ 0&1\end{array} \right], \tag{4}\]  
The assembly deviation can be predicted by the pose \(\mathbf{\omega}_{0}\) of the align part. The key assembly characteristics (KAC) [25] are the important geometric structures that have key influences on assembly quality. They are described by measurement data and some dimensions that are not necessary to be measured. \[\mathbf{K}=\{\mathbf{P},\mathbf{G}\}, \tag{5}\]  
where \(\mathbf{K}\) is the parameters of a KAC, \(\mathbf{P}\) is the measurement data, \(\mathbf{G}\) is the dimensions that are not necessary to be measured. The KACs have an irregular distribution in space during large-scale assembly. As shown in Figure 3, the wing-fuselage assembly is completed by 4 pairs of joints. There are four assembly accuracy requirements on each pair of joints: Two on coaxialities and two on clearances. The KACs are restrained by assembly accuracy requirements. The assembly accuracy is described as  
\[T_{j}^{i}=f_{j}^{i}\Big{(}\mathbf{K}_{i}^{R},\mathbf{K}_{i}^{A}\Big{)}, \tag{6}\]  
where \(\mathbf{K}_{i}^{R}\) is the parameters of the \(i\)th KAC on the reference part (fuselage), \(\mathbf{K}_{i}^{A}\) is the parameters of the \(i\)th KAC on the align part (wing), \(T_{j}^{i}\) is the \(j\)th assembly accuracy of the \(i\)th KAC, and \(f_{j}^{i}\) is the mapping from parameters to the assembly accuracy. The assembly accuracy should meet the requirements of assembly accuracy, which is formulated in Equation (7)  
\[T_{j}^{i}\in\left[T_{j}^{i-min},T_{j}^{i-max}\right], \tag{7}\]  
where \(T_{j}^{i-min}\) and \(T_{j}^{i-max}\) are the ranges of \(T_{j}^{i}\).