## 2 Coordination Space Model Based on Small Displacement Torsor
Substitute Equation (6) into Equation (7):  
\[f_{j}^{i}\Big{(}\mathbf{K}_{i}^{R},\mathbf{K}_{i}^{A}\Big{)}\in\left[T_{j}^{i-min},T_{ j}^{i-max}\right], \tag{8}\]  
For the m assembly accuracy requirements on n KACs of the assembly, the constraint equations can be expressed as  
\[\forall f_{j}^{i}\Big{(}\mathbf{K}_{i}^{R},\mathbf{K}_{i}^{A}\Big{)}\in\left[T_{j}^{i -min},T_{j}^{i-max}\right],i\in[1,n],j\in[1,j_{i}], \tag{9}\]  
Figure 3: Wing-fuselage assembly. where \(j_{i}\) is the assembly accuracy requirement number of the \(i\)th KAC, \(m=\sum_{i=1}^{n}j_{i}\). When all KACs satisfy their assembly accuracy requirements, the pose is a valid pose to be aligned. As shown in Figure 4, the valid pose may not be the only one that satisfies all assembly accuracy requirements. Therefore, the adjacent poses of the primary pose shown in Figure 4a can be analyzed. A small displacement torsor (SDT) [26] represents a tiny rigid body's pose variation. It is described as  
\[\mathbf{\omega}_{\Delta}=(x_{\Delta},y_{\Delta},z_{\Delta},\alpha_{\Delta},\beta_ {\Delta},\gamma_{\Delta}), \tag{10}\]  
The homogeneous transformation matrix of an SDT is  
\[\mathbf{\omega}_{\Delta}^{H}=\left[ \begin{array}{cccc}C_{\gamma\Delta}C\beta_{ \Delta}&-S\gamma_{\Delta}C\beta_{\Delta}&S\beta_{\Delta}&x_{\Delta}\\ S_{\gamma\Delta}C\alpha_{\Delta}+C\gamma_{\Delta}S\beta_{\Delta}s\alpha_{ \Delta}&C_{\gamma\Delta}C\alpha_{\Delta}-S\gamma_{\Delta}S\beta_{\Delta}s\alpha _{\Delta}&-C\beta_{\Delta}s\alpha_{\Delta}&y_{\Delta}\\ S_{\gamma\Delta}s\alpha_{\Delta}-C\gamma_{\Delta}S\beta_{\Delta}c\alpha_{ \Delta}&C\gamma_{\Delta}S\alpha_{\Delta}+S\gamma_{\Delta}S\beta_{\Delta}c \alpha_{\Delta}&C\beta_{\Delta}c\alpha_{\Delta}&z_{\Delta}\\ 0&0&0&1\end{array} \right]\approx\left[ \begin{array}{cccc}1&-\gamma_{\Delta}& \beta_{\Delta}&x_{\Delta}\\ \gamma_{\Delta}&1&-\alpha_{\Delta}&y_{\Delta}\\ -\beta_{\Delta}&\alpha_{\Delta}&1&z_{\Delta}\\ 0&0&0&1\end{array} \right], \tag{11}\]  
where \(S\) is \(\sin\), \(C\) is \(\cos\), \(\lim\limits_{\Delta_{\Delta}\to 0}C\alpha_{\Delta}=1\), \(\lim\limits_{\Delta_{\Delta}\to 0}S\alpha_{\Delta}=\alpha_{\Delta}\), and \(\lim\limits_{\Delta_{\Delta}\beta_{\Delta}\to 0}S\alpha_{\Delta}S\beta_{ \Delta}=0\). The change in point \(\mathbf{p}=(x,y,z)\) after a slight change in the rigid body's pose is  
\[\mathbf{p}_{\alpha}^{\prime}=\left[x\ y\ z\ 1\right]\left[ \begin{array}{cccc}1&- \gamma_{\Delta}&\beta_{\Delta}&x_{\Delta}\\ \gamma_{\Delta}&1&-\alpha_{\Delta}&y_{\Delta}\\ -\beta_{\Delta}&\alpha_{\Delta}&1&z_{\Delta}\\ 0&0&0&1\end{array} \right]=\left[ \begin{array}{cccc}x+x_{\Delta}+\beta_{ \Delta}z-\gamma_{\Delta}z\\ y+y_{\Delta}-\alpha_{\Delta}z+\gamma_{\Delta}x\\ z+z_{\Delta}+\alpha_{\Delta}y-\beta_{\Delta}x\\ 1\end{array} \right]^{T}, \tag{12}\]  
Then, the assembly accuracy would be  
\[T_{j}^{i-\omega_{\Delta}}=f_{j}^{i}\big{(}K_{i}^{R},K_{i}^{A}\mathbf{\omega}_{ \Delta}^{H}\big{)}=f_{j}^{i}\big{(}P_{i}^{R},\mathbf{G}_{i}^{R},\mathbf{P}_{i}^{A}\mathbf{ \omega}_{\Delta}^{H},\mathbf{G}_{i}^{A}\big{)}, \tag{13}\]  
On this pose, if the assembly accuracy requirements are still satisfied as  
\[\forall f_{j}^{i}\big{(}K_{i}^{R},K_{i}^{A}\mathbf{\omega}_{\Delta}^{H}\big{)} \in\left[T_{j}^{i-min},T_{j}^{i-max}\right],i\in\left[1,n\right],j\in\left[1, j_{i}\right], \tag{14}\]  
the pose is still a valid pose. The coordination space model can, hence, be expressed as  
\[\varnothing_{\text{CS}}=\Big{\{}\mathbf{\omega}_{\Delta}\Big{|}\forall f_{j}^{i} \big{(}K_{i}^{R},K_{i}^{A}\mathbf{\omega}_{\Delta}^{H}\big{)}\in\left[T_{j}^{i-min}, T_{j}^{i-max}\right],i\in\left[1,n\right],j\in\left[1,j_{i}\right]\Big{\}}, \tag{15}\]  
where \(\varnothing_{\text{CS}}\) is the coordination space, which is the whole pose variation space under the condition of assembly accuracy requirements. Figure 4: Valid poses of wing-fuselage assembly.