## 4 Simulation  
Influence of each deviation on the eccentricity is investigated by using the above mathematical model. The commands separately given to each axis during simulation are shown in Table 2.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Cylindrical**} & \multicolumn{4}{c}{**B Axis**} & \multicolumn{4}{c}{**C Axis**} \\ \cline{2-7}  & **Radial** & **Tangential** & **Axial** & **Radial** & **Tangential** & **Axial** \\ \hline Command X & \(\left(L_{B}+R_{B}\right)\sin\varphi\) & \( \begin{array}{c}R_{\text{RZ}}\sin\varphi\\ \tan^{-1}L_{B}/R_{B}\end{array} \right)^{a}\) & \(R_{B}\sin\varphi\) & \(\left(R_{\text{C}}-L_{B}\right)\cos\theta\) & \( \begin{array}{c}R_{\text{CT}}\cos\theta\\ \tan^{-1}L_{B}/R_{\text{C}}\end{array} \)b & \(R_{\text{C}}\cos\theta\) \\ \hline Command Y & \(0\) & \(0\) & \(L_{B}\) & \(\left(R_{\text{C}}-L_{B}\right)\sin\theta\) & \( \begin{array}{c}R_{\text{CT}}\sin\theta\\ \tan^{-1}L_{B}/R_{\text{C}}\end{array} \)b & \(R_{\text{C}}\sin\theta\) \\ \hline Command Z & \(Z_{\text{C}}+\left(L_{B}+R_{B}\right)\cos\varphi\) & \(Z_{\text{C}}+R_{B}\cos\varphi\) & \(Z_{\text{C}}+R_{B}\) & \(Z_{\text{C}}+R_{B}\) & \(Z_{\text{C}}+R_{B}\) & \(Z_{\text{C}}+L_{B}+R_{B}\) \\ \hline Command B & \(\varphi\) & \(\varphi\) & \(\varphi\) & \(0\) & \(0\) & \(0\) \\ Command C & \(0\) & \(0\) & \(0\) & \(\theta\) & \(\theta\) \\ \hline
**Cartesian** & X-direction & Y-direction & Z-direction & X-direction & Y-direction & Z-direction \\ \hline Command X & \(L_{B}+R_{B}\sin\varphi\) & \(R_{B}\sin\varphi\) & \(R_{B}\sin\varphi\) & \(L_{B}+R_{\text{C}}\cos\theta\) & \(R_{\text{C}}\cos\theta\) & \(R_{\text{C}}\cos\theta\) \\ Command Y & \(0\) & \(L_{B}\) & \(0\) & \(R_{\text{C}}\sin\theta\) & \(The simulation of the simultaneous three-axis control movements is conducted at the condition of \(L_{B}\) = 100 mm, \(Z_{\text{C}}\) = 100 mm, \(R_{\text{B}}\) = 200 mm, \(R_{\text{C}}\) = 50 mm. \(\pm\)0.005 degrees and \(\pm\)20 \(\upmu\)m are given as angular deviations and positional deviations, respectively. Then, simulation results are obtained as shown in Table 3. The dotted circle represents theoretical trajectory when there is no geometric deviation and the red or blue one represents changed trajectory affecting by geometric deviations. The figures show that if only one of the thirteen geometric deviations exist, the red or blue circular trajectory will appear and reflect the effect of the given deviation on the eccentricity. The blank part shows that there is no influence of the geometric deviation on trajectory. For example, for B axis radial direction measurement, when a value of +20 \(\upmu\)m is given to \(\delta x_{\text{BT}}\) while other twelve geometric deviations are all zero, eccentricity of trajectory occurs in -X axis direction. On the contrary, when a value of \(-\)20 \(\upmu\)m is given to \(\delta x_{\text{BT}}\), eccentricity of trajectory occurs in +X axis direction. The eccentricities occur by the following three reasons. * The position of trajectory center is changed; for example, the effect of \(\delta x_{\text{BT}}\) on eccentricity in case of the B axis radial measurement. * The size of trajectory radius is changed; for example, the effect of \(\delta z_{\text{BT}}\) on eccentricity in case of the B axis radial measurement. * The shape of trajectory is changed; for example, the effect of \(\beta_{\text{YZ}}\) on eccentricity in case of the B axis radial measurement.