## 4 Calculation of Integrated Geometric Errors  
For each axis, 12 points are selected and their complete geometric errors are measured in this paper. In this way, the integrated geometric errors of near 3 million (12\({}^{6}\)) machining points are obtained. Limited to the length of the paper, as an example, this paper only performs integrated error calculation for the 36 machining points as shown in Figure 13.  
Take p\({}_{1}\) (xi, yi, xi, \(\alpha\)i, \(\beta\)i, \(\gamma\)i) = (0, 0, 0, 0, 0, 0) as an example; its complete geometric errors (\(\delta_{\rm x}\), \(\delta_{\rm y}\), \(\delta_{\rm z}\), \(\varepsilon_{\rm x}\), \(\varepsilon_{\rm y}\), \(\varepsilon_{\rm z}\)) are obtained as follows:  
X:( -1.2, 3.0, 4.2, 1.25, 0.85943, 0.61593) X:( -2.4, -4.0, -6.2, 1.0, -0.862, 0.563) Y:( -3.8, 1.0, 6.4, 0.63584, 0.56, 0.27658) B:(5.6, 0.5, -4.4, -0.267, 0.87, -0.581) Z:( -3.4, -5.6, 0.2, 0.33456, 0.42638, 0.06) C:( -4.2, -2.4, 3.2, 0.264, -0.795, 0.4)  
According to Equations (15)-(18), and (25)-(30), its actual HTMs and integrated error are obtained as follows in Equations (36)-(44):  
\[T_{X}^{actual}=\left[ \begin{array}{cccc}0.99983&-0.01064&0.01508&-0.0012\\ 0.01075&0.99992&-0.00703&0.0030\\ -0.01500&0.00724&0.99986&0.0042\\ 0&0&0&1\end{array} \right], \tag{36}\]  
\[T_{Y}^{actual}=\left[ \begin{array}{cccc}0.99997&-0.00475&0.00658&-0.0038\\ 0.00483&0.99993&-0.01107&0.0010\\ -0.00653&0.01110&0.99992&0.0064\\ 0&0&0&1\end{array} \right], \tag{37}\]  
\[T_{Z}^{actual}=\left[ \begin{array}{cccc}0.99997&0.00219&0.00743&-0.0034\\ -0.00215&0.99998&-0.00586&-0.0056\\ -0.00744&0.00584&0.99996&0.0002\\ 0&0&0&1\end{array} \right], \tag{38}\]  
\[T_{A}^{\rm actual}=\left[ \begin{array}{cccc}0.99984&-0.00982&-0.01505&-0.00 24\\ 0.00982&0.99995&0.00043&-0.0040\\ 0.01504&-0.00058&0.99989&-0.0062\\ 0&0&0&1\end{array} \right], \tag{39}\]  
Figure 13: Machining points to be calculated. \[T_{B}^{\text{actual}}=\left[ \begin{array}{cccc}0.99983&0.01007&0.01523&0.0056\\ -0.01014&0.99994&0.00451&0.0005\\ -0.01518&0.00466&0.99987&-0.0044\\ 0&0&0&1\end{array} \right], \tag{40}\]  
\[T_{C}^{\text{actual}}=\left[ \begin{array}{cccc}0.99988&-0.00705&-0.01384&-0. 0042\\ 0.00698&0.99996&-0.00470&-0.0024\\ -0.01387&0.00461&0.99989&0.0032\\ 0&0&0&1\end{array} \right], \tag{41}\]  
\[T_{t}^{actual}=\left[ \begin{array}{cccc}0.99971&-0.00836&0.02257&-0.00454\\ 0.00865&0.99988&-0.01281&-0.00264\\ -0.02245&0.01305&0.99967&0.0044\\ 0&0&0&1\end{array} \right]=\left[ \begin{array}{cccc}193.93746\\ 1.72736\\ -4.4856\\ 1\end{array} \right], \tag{42}\]  
\[T_{w}^{\text{actual}}=\left[ \begin{array}{cccc}0.99988&-0.00705&-0.01384&-0. 0042\\ 0.00698&0.99996&-0.00470&-0.0024\\ -0.01387&0.00461&0.99989&0.0032\\ 0&0&0&1\end{array} \right]\left[ \begin{array}{cccc}0\\ 0\\ 400\\ 1\end{array} \right]=\left[ \begin{array}{cccc}-5.5402\\ -1.8824\\ 399.9592\\ 1\end{array} \right],\text{ and} \tag{43}\]  
\[E=\left[ \begin{array}{cccc}200\\ 0\\ -400\\ 0\end{array} \right]-\left[ \begin{array}{cccc}193.93746\\ 1.72736\\ -4.4856\\ 1\end{array} \right]+\left[ \begin{array}{cccc}-5.5402\\ -1.8824\\ 399.9592\\ 1\end{array} \right]=\left[ \begin{array}{cccc}0.52234\\ -3.60976\\ -4.5264\\ 0\end{array} \right]. \tag{44}\]  
Similarly, geometric errors of all machining points selected are obtained as shown in Figure 14.  
In the traditional measurement method, the small geometric angle errors are not taken into consideration; they are calculated as 0. The integrated geometric error is expressed as Equation (44). The first term in Equations (44) and (45) is the expected relative distance of the workpiece from the  
Figure 14: Integrated geometric errors of machining points (3D view). tool. The second item is the actual geometric position of the workpiece, and the third item is the actual geometric position of the tool. \[E=\left[ \begin{array}{c}200\\ 0\\ -400\\ 0\end{array} \right]-\left[ \begin{array}{c}199.9948\\ -0.0051\\ 0.0002\\ 1\end{array} \right]+\left[ \begin{array}{c}-0.0042\\ -0.0024\\ 400.0032\\ 1\end{array} \right]=\left[