# Study of the Error Caused by Camera Movement for the Stereo-Vision System
## 3 Camera Movement Errors
### Identical Rotations  
The identical rotations of two cameras in a stereo-vision system imply that the global world coordinate system experiences a rigid body rotation, as shown in Figure 3. The 3D coordinate error \(\Delta\mathbf{X}_{w}\) can be expressed as  
\[\left[\begin{array}{c}\Delta X_{w}\\ \Delta Y_{w}\\ \Delta Z_{w}\end{array}\right]=\left[\begin{array}{c}X^{\prime}_{w}\\ Y^{\prime}_{w}\\ Z^{\prime}_{w}\end{array}\right]-\left[\begin{array}{c}X_{w}\\ Y_{w}\\ Z_{w}\end{array}\right]=\Delta\mathbf{R}_{w}\left[\begin{array}{c}X_{w}\\ Y_{w}\\ Z_{w}\end{array}\right]-\left[\begin{array}{c}X_{w}\\ Y_{w}\\ Z_{w}\end{array}\right] \tag{8}\]  
Figure 2: The relationship between a pair of camera coordinate systems prior and post identical translations in a stereo-vision arrangement.  
where \(\Delta R_{w}\) denotes the transformation from the initial world coordinate system to the final world coordinate system owing to the identical rotations of both cameras.  
The rotation matrix \(\Delta R_{w}\) is a \(3\times 3\)-unit orthogonal matrix, and depends only on three independent variables. Therefore, \(\Delta R_{w}\) can be represented as  
\[\Delta R_{w}=\left[\begin{array}{ccc}\cos\widetilde{\gamma}&-\sin\widetilde{ \gamma}&0\\ \sin\widetilde{\gamma}&\cos\widetilde{\gamma}&0\\ 0&0&1\end{array}\right]\left[\begin{array}{ccc}1&0&0\\ 0&\cos\widetilde{\alpha}&-\sin\widetilde{\alpha}\\ 0&\sin\widetilde{\alpha}&\cos\widetilde{\alpha}\end{array}\right]\left[ \begin{array}{ccc}\cos\widetilde{\beta}&0&-\sin\widetilde{\beta}\\ 0&1&0\\ \sin\widetilde{\beta}&0&\cos\widetilde{\beta}\end{array}\right] \tag{9}\]  
where \(\left(\widetilde{\alpha},\widetilde{\beta},\ \widetilde{\gamma}\right)\) are defined as the rotation angles that the initial world coordinate system is transformed to the orientation that coincides with the final world coordinate system.  
The unavoidable camera movements are often slight, and then the rotation angles \(\left(\widetilde{\alpha},\widetilde{\beta},\ \widetilde{\gamma}\right)\) can be set to a small value. By taking the small angle approximation, i.e., \(\sin\widetilde{\alpha}\approx\widetilde{\alpha},\ \cos\widetilde{\alpha}\approx 1,\ \sin \widetilde{\beta}\approx\widetilde{\beta},\ \cos\widetilde{\beta}\approx 1,\ \sin \widetilde{\gamma}\approx\widetilde{\gamma},\ \cos\widetilde{\gamma}\approx 1\), we have  
\[\left\{\begin{array}{c}\Delta X_{w}\approx\widetilde{\beta}\widetilde{ \alpha}\widetilde{\gamma}X_{w}-\widetilde{\gamma}Y_{w}+\left(-\widetilde{ \beta}+\widetilde{\alpha}\widetilde{\gamma}\right)Z_{w}\\ \Delta Y_{w}\approx\left(\widetilde{\gamma}-\widetilde{\beta}\widetilde{ \alpha}\right)X_{w}+\left(-\widetilde{\beta}\widetilde{\gamma}-\widetilde{ \alpha}\right)Z_{w}\\ \Delta Z_{w}\approx\widetilde{\beta}X_{w}+\widetilde{\alpha}Y_{w}\end{array}\right. \tag{10}\]  
From Equation (10), the 3D coordinate error \(\Delta\mathbf{X}_{w}\) has a linear relationship with the identical rotation angle of the two cameras.  
According to Equation (8), the distance between the two measured points after camera movement can be described as  
\[L^{\prime}_{w}=\|\mathbf{X}^{\prime}_{w1}-\mathbf{X}^{\prime}_{w2}\|=\| \Delta\mathbf{R}_{w}(\mathbf{X}_{w1}-\mathbf{X}_{w2})\| \tag{11}\]  
Since \(\|\ \Delta\mathbf{R}_{w}\ \|=1\), we have \(L_{w}=L^{\prime}_{w}\). It indicates that the distance between any two points is unaffected by the identical rotations of two cameras.