## 3 Selecting Optimal Measurement Poses Based on Observability Index  
### Introduction to Observability Indices  
Observability indices are used to select optimal measurement poses. Using these poses for calibration can improve the problem of measurement noise reducing calibration accuracy. When the measurement noise level is maintained, selecting different measurement poses for robot calibration will result in different calibration accuracies [22]. Theoretically, it is possible to use multiple sets of poses spread throughout the workspace of the robot for calibration, and then retain the set with the highest accuracy after calibration. However, in order to spread throughout the entire workspace, multiple sets of poses have to be selected, as well as perform many calibrations, which is time-consuming, labor-intensive, and generally difficult to operate in practice. Therefore, to efficiently select suitable measurement poses for kinematic calibration, five observability indices, \(O_{1}\)-\(O_{5}\), have been proposed [18; 19; 20; 21; 22; 23]. For several groups of measurement poses, the group with a larger observability index is more likely to achieve higher calibration accuracy. In addition, the calibration based on the observability index can achieve high calibration accuracy and only requires a limited number of measurement poses. To calculate observability indices, the singular value decomposition (SVD) of the identification Jacobian matrix \(J_{\mathrm{B}}\) is required:  
\[J_{B}=\mathbf{U}\Sigma\mathbf{V}^{T} \tag{30}\]  
Then, substituting Equation (30) into Equation (27) yields:  
\[dq_{B}=\mathbf{U}\Sigma\mathbf{V}^{T}d\mathbf{C} \tag{31}\]  
where:  
\[\Sigma= \begin{bmatrix}\mathbf{S}\\ \mathbf{O}\end{bmatrix}  \tag{32}\]  
where \(\mathbf{S}=\text{diag}\) (\(\sigma_{1}\), \(\sigma_{2}\),..., \(\sigma_{\mathrm{L}}\)) is a diagonal matrix composed of singular values of \(J_{\mathrm{B}}\). \(\sigma_{1}\) and \(\sigma_{\mathrm{L}}\) are the maximum and minimum singular values, respectively. \(L\) denotes the number of kinematic parameters. \(\mathbf{O}\) is a 0 matrix; the size of \(\mathbf{O}\) is \((6n-42)\times 42\). According to the geometric significance of SVD, as shown in Figure 3, if the kinematic parameter error vector \(\mathrm{d}C\) is on the surface of a hypersphere with a unit radius, then the pose error vector \(\mathrm{d}q_{\mathrm{B}}\) is on the surface of an L-dimensional hyperellipsoid. The semi-axes length of the hyperellipsoid is \(\sigma_{1}\), \(\sigma_{2}\),..., \(\sigma_{L}\), the singular values of the identification Jacobian matrix. The principles of observability indices can be explained geometrically. The properties of SVD also reveal the following relationship between the norms of \(\mathrm{d}\mathbf{q}_{\mathrm{B}}\) and \(\mathrm{d}\mathbf{C}\):  
\[\sigma_{L}\leq\frac{|d\mathbf{q}_{\mathrm{B}}|}{|d\mathbf{C}|}\leq\sigma_{1} \tag{33}\]  
The measurement noise is determined by the measurement equipment and pose resolving method, so the level of the measurement noise remains unchanged during the calibration process. In addition, the kinematic parameter error vector \(\mathrm{d}\mathbf{C}\) is constant. Therefore, the larger the value of \(|\mathrm{d}\mathbf{q}_{\mathrm{B}}|\), the smaller the proportion of measurement noise in \(\mathrm{d}\mathbf{q}_{\mathrm{B}}\). The smaller the proportion, the smaller the negative impact of measurement noise, and the higher the accuracy of kinematic calibration. The first observability index \(O_{1}\) is calculated as follows:  
\[O_{1}=\frac{\sqrt[4]{\sigma_{1}\sigma_{2}\cdots\sigma_{L}}}{\sqrt{n}} \tag{34}\]  
\(O_{1}\) is the geometric mean of singular values, which is proportional to the volume of the hyperellipsoid in Figure 3. Maximizing \(O_{1}\) requires the volume of the hyperellipsoid to be as large as possible, which will increase the pose error \(\mathrm{d}\mathbf{q}_{\mathrm{B}}\) in general [18; 19].