## 5 Experiment and Analysis
The solution of Equation (46) can be expressed by  
\[X=\left(A^{T}A\right)^{-1}A^{T}Y. \tag{47}\]  
After the measurements of the deformation and the forces and torques, Equation (47) is used to calculate the estimated values of the joint stiffness. The results of the identification were the statistical average of several calculation values, and they are listed in Tables 3 and 4. The estimated values are not big, and this is similar to the joint stiffness characteristic of the industrial robot as expected, according to Reference [9]. These two arms had the same mechanism topology, but the estimated values of the joint stiffness of R1 were not the same as those of R2. This is because the identification results were the theoretical values that can be affected by the measurement noise, calculation errors, postures of the arm, installation environment, and so on. After the identification, the estimated values of joint stiffness were used to calculate the deformation of the EE under the given external load. The pose error \(\Delta E\) between the nominal value and the calibrated value is divided into the position error \(\Delta P\) and the orientation error \(\Delta O\) in this paper:  
\[\Delta E=\Delta P+\Delta O, \tag{48}\]  
\[\Delta P=P_{cal}-P_{n}, \tag{49}\]  
\[\Delta O=O_{cal}-O_{n}, \tag{50}\]  
where \(P_{cal}\) and \(O_{cal}\) are the estimated position value and orientation values, respectively, \(P_{n}\) and \(O_{n}\) are the ideal position value and orientation values, respectively. The norm value computed by the vector components of position error \(\Delta P\) is calculated by  
\[\Delta E=\sqrt{\Delta P_{x}^{2}+\Delta P_{y}^{2}+\Delta P_{z}^{2}}, \tag{51}\]  
where \(\Delta P_{x}\) is the error along the \(x\)-axis direction, \(\Delta P_{y}\) is the error along the \(y\)-axis direction, \(\Delta P_{z}\) is the error along the \(z\)-axis direction. Similarly, the norm value computed by the vector components of orientation error \(\Delta O\) is calculated by  
\[\Delta\Omega=\sqrt{\Delta O_{x}^{2}+\Delta O_{y}^{2}+\Delta O_{z}^{2}}, \tag{52}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Joint No.** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline Estimated value & 0.06 & 0.07 & 0.45 & 0.05 & 0.10 & 0.03 & 0.21 & 9.95 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Estimated values of joint stiffness of R2 in \(\left[10^{6}\ \mathrm{Nm/rad}\right]\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Joint No.** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline Estimated value & 0.06 & 0.07 & 0.45 & 0.05 & 0.10 & 0.03 & 0.21 & 9.95 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Estimated values of joint stiffness of R1 in \(\left[10^{6}\ \mathrm{Nm/rad}\right]\). where \(\Delta O_{x}\) is the error around the \(x\)-axis direction, \(\Delta O_{y}\) is the error around the \(y\)-axis direction, \(\Delta O_{z}\) is the error around the \(z\)-axis direction. After the joint stiffness identification, a given load was added to the EE and the deformation was calculated for the simulation first. The given load was a dumbbell with a mass of 0.153 kg. Figures 4 and 5 depict the error between the nominal pose and the actual pose caused by the deformation throughout the robot workspace. The analysis result shows that the position error of R1 was in the range [0, 3.5] (in millimeters), and the orientation error of R1 was in the range [0, 0.0045] (in radians). The position error of R2 was in the range [0, 4.2] (in millimeters), and the orientation error of R2 was in the range [0, 0.0037] (in radians). The different deformation between the two arms was because of the difference in the mechanism and the measurement results. The drilling procedure is typical in manufacturing, and industrial robots can perform the drilling task as they have high flexibility and low cost. The SDASF is also a humanoid robot, which is useful in mimicking human beings during manufacturing.