## 4 Pose Decoupling of Axis TF
### Mapping between the Area Coordinate System and Cartesian Coordinate System  
For axis TF with any pose, Equation (8) can be expressed by a linear matrix:  
\[\left[\begin{array}{c}L\\ M\\ N\\ P\\ Q\end{array}\right]=\left[\textbf{X}\right]\left[\begin{array}{c}\lambda_{1} \\ \lambda_{2}\\ \lambda_{3}\\ \lambda_{4}\\ \lambda_{5}\end{array}\right] \tag{13}\]Combining Equations (11)-(13), the axis of any pose in the tolerance zone can be expressed by either Plucker coordinates in the Cartesian coordinate system directly, or the linear sum of Plucker coordinates of five base lines in the area coordinate system. These two expressions are equivalent:  
\[\left[\begin{array}{c}L\\ M\\ N\\ P\\ Q\end{array}\right]=\left[\begin{array}{c}t\lambda_{4}\\ t\lambda_{5}\\ d(\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4}+\lambda_{5})\\ \frac{t}{2}d\lambda_{3}\\ -\frac{t}{2}d\lambda_{2}\end{array}\right] \tag{14}\]  
By solving the Equation (14), the \(S_{1}(x_{1},y_{1},-d/2)\) and \(S_{2}(x_{2},y_{2},d/2)\) which are coordinate values of \(S_{1}\) and \(S_{2}\) in the expression of area coordinate values (\(\lambda_{1}\), \(\lambda_{2}\), \(\lambda_{3}\), \(\lambda_{4}\), \(\lambda_{5}\)) can be obtained:  
\[\left[\begin{array}{c}x_{1}\\ y_{1}\\ x_{2}\\ y_{2}\end{array}\right]=\left[\begin{array}{c}\frac{t}{2}(\lambda_{2}- \lambda_{4})\\ \frac{t}{2}(\lambda_{3}-\lambda_{5})\\ \frac{t}{2}(\lambda_{2}+\lambda_{4})\\ \frac{t}{2}(\lambda_{3}+\lambda_{5})\end{array}\right] \tag{15}\]  
Similarly, the error transformation matrix can be expressed by area coordinate values (\(\lambda_{1}\), \(\lambda_{2}\), \(\lambda_{3}\), \(\lambda_{4}\), \(\lambda_{5}\)). It can be seen from Figure 2 that variables on four DOFs can be expressed by the area coordinate system:  
\[\left\{\begin{array}{c}u_{i}=\frac{t}{2}\lambda_{2}\\ v_{i}=\frac{t}{2}\lambda_{3}\\ \alpha_{i}=-\frac{t}{2}\lambda_{4}\\ \beta_{i}=\frac{t}{2}\lambda_{5}\end{array}\right. \tag{16}\]  
By bringing Equation (16) into Equation (2), the error transformation matrix which is expressed by the area coordinate system can be obtained.  
The analytic expression of error propagation which is expressed by area coordinate values can be gained from the tolerance analysis method based on the tracking local coordinate system [19]. Since (\(\lambda_{1}\), \(\lambda_{2}\), \(\lambda_{3}\), \(\lambda_{4}\), \(\lambda_{5}\)) range from \(-1\) to \(1\), any pose of axis TF in the tolerance zone can be decoupled by changing (\(\lambda_{1}\), \(\lambda_{2}\), \(\lambda_{3}\), \(\lambda_{4}\), \(\lambda_{5}\)).  
When calculating the limits of target feature variations, basic area coordinate values (each row) of the base line in the matrix (17) are permutated and combined, and then brought into the tolerance analysis model, so the analytic expression of the convex hull model which reflects variations in the target feature, can be found.  
\[\left[\begin{array}{cccc}\left\{\begin{array}{cccc}\lambda_{1}&\lambda_{2 }&\lambda_{3}&\lambda_{4}&\lambda_{5}\end{array}\right\}\\ -1&0&0&0&0\\ 0&-1&0&0&0\\ 0&0&-1&0&0\\ 0&0&0&-1&0\\ 0&0&0&0&-1\\ 0&0&0&0&1\\ 0&0&1&0&0\\ 0&1&0&0&0\\ 1&0&0&0&0\end{array}\right] \tag{17}\]