# Study of the Error Caused by Camera Movement for the Stereo-Vision System
## 4 Experiments and Results
### Identical Rotation Experiment  
The experimental setup for identical rotation of the two cameras is the same as that shown in Figure 6. First, a pair of images of the checkerboard are acquired without camera rotation, as shown in Figure 7b. Then both cameras are rotated together from 0\({}^{\circ}\) to 10\({}^{\circ}\) in an angular step of 1\({}^{\circ}\) around the \(X\)-axis and \(Y\)-axis, respectively. In each rotation pose, a pair of checkerboard images are taken by the left and right cameras, simultaneously. Figure 7b  
Figure 7: (**a**) The stereo image pairs captured in the initial state and the state of identical translations (44 mm) along \(X\)-axis; (**b**) The stereo image pairs captured in the initial state and the state of identical rotations (10\({}^{\circ}\)) around \(Y\)-axis; (**c**) The stereo image pairs captured in the initial state and the state of relative translation (44 mm) along \(X\)-axis; (**d**) The stereo image pairs captured in the initial state and the state of relative rotation (21\({}^{\circ}\)) around \(X\)-axis.  
also shows the stereo image pair captured in the state of identical rotations of 10\({}^{\circ}\) around the \(Y\)-axis.  
Figure 9a shows the mean value of the 3D coordinate errors of all the corner points in each rotation pose, while the two cameras are rotated around the \(X\)-axis. The 3D coordinate error in \(Y_{w}\) direction is significantly proportional to the rotation angle, and the error coefficient is \(-22.1\) mm/degree. The error in the \(Z_{w}\) direction experiences minuscule change, whereas the error in the \(X_{w}\) direction is always zero. This result can be explained based on the analysis given in Section 3.2. The rotation of both cameras around the \(X\)-axis implies that \(\widehat{\beta}=0,~{}\widetilde{\gamma}=0\). Using Equation (10), we can obtain that \(~{}\Delta X_{w}\approx 0\), \(~{}\Delta Y_{w}\approx-\widetilde{\pi}Z_{w0},\Delta Z_{w}\approx\widetilde{ \pi}Y_{w0}\). Initially, the mean \(Z_{w0}\) of all the corner points is 1174 mm. Therefore, the mean error of all corner points in the \(Y_{w}\) direction can be approximated as \(~{}\Delta\widetilde{Y}_{w}\approx 20.49\widetilde{\alpha}\). Considering the small deviation between the \(Z\)-axis of the rotational stage and the \(Z_{w}\) direction of the world coordinate system, the present experimental results succeed in verifying the theoretical model. When the two cameras are rotated around the \(Y\)-axis, similar trends have been observed in Figure 9b. We can conclude that the identical rotation of the two cameras in a stereo-vision system produces errors in coordinates only in the rotating plane.  
Figure 9c,d shows the measured distance between the adjacent corner points compared to the real distance of 25 mm. The maximum difference is less than 0.08 mm. Therefore, the effect of the identical rotation on the distance measurement can be negligible. This result agrees with the theoretical prediction from Section 3.2 that the angle of identical rotation should have produced zero distance error.  
Figure 8: The 3D coordinates and distance measurement errors introduced by identical translation: (**a,b**) The average 3D coordinate errors of all corner points at each translational position when the two cameras are translated along the \(X\)-axis and \(Z\)-axis, respectively; (**c,d**) The comparison between the measured and real distance between adjacent corner points.