## 5 Experiment and Analysis  
In order to verify the effectiveness of the stiffness identification algorithm and the deformation compensation algorithm presented above, a real experiment has been conducted on an industrial robot. The stiffness parameters were identified during the experiment first, and then the estimated parameters were used to calculate the deformation caused by the given load for the simulation. After that, the deformation compensation algorithm was applied to the real drilling operation which was performed by an industrial robot. Figure 3 shows the experimental setup. The experiment contained a dual arm industrial robot SDA5F, Leica Geosystems Absolute Tracker (AT960), and tracker machine control probe (T-MAC). T-MAC stands for tracker machine control probe, which was used to measure the pose of the rigid body. The DH parameters of robot SDA5F were used to formulate the forward kinematic model based on dual quaternion algebra, and then the stiffness modeling was established. Each arm of SDA5F was a serial chain manipulator, and these two arms were called R1 and R2, respectively. R1 was the left arm of SDA5F, R2 was the right arm of SDA5F. The R2 had the same joint configuration as R1, and they were symmetric about the base axis. Each arm had 7 revolute joints, and the robot had 15 revolute joints in total. The DH parameters of R1 and R2 are shown in Tables 1 and 2, respectively. Figure 3a shows that a mass block was applied to the robot EE. The Leica Geosystems Absolute Tracker (AT960) was fixed outside the workspace of SDA5F. Figure 3b shows the installation of the TMAC. The force sensor was mounted on the EE of the robot. The T-MAC was mounted on the force sensor, and a spherically mounted reflector (SMR) was mounted near the mounting interface of T-MAC. The measurement uncertainty of the measuring machine was \(\pm 15\mu\text{m}+6\mu\text{m}/\text{m}\). From Equation (35), the joint stiffness can be identified by the given forces and torques measured at the EE. Furthermore, by giving joint stiffness parameters, external forces and torques, it is convenient to calculate the deflection or deformation \(\Delta X\) in return. First, the force sensor measured the forces and torques caused by the mass at the EE when the arm was at different poses. Then, the measurements \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline
**Joint No.** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline \(a_{i-1}\)(rad) & 0 & \(-\pi/2\) & \(\pi/2\) & \(-\pi/2\) & \(\pi/2\) & \(-\pi/2\) & \(\pi/2\) & \(-\pi/2\) \\ \(a_{i-1}\)(mm) & 0 & 150 & 0 & 0 & 85 & 60 & 0 & 0 \\ \(\theta_{i}\)(rad) & 0 & \(\pi/2\) & 0 & \(\pi\) & \(\pi/2\) & 0 & 0 & 0 \\ \(\mathbf{d}_{i}\)(mm) & 0 & 243 & 0 & 270 & 0 & \(-270\) & 0 & \(-148\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Denavit–Hartenberg (DH) parameters of R1. \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline
**Joint No.** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline \(a_{i-1}\)(rad) & 0 & \(-\pi/2\) & \(-\pi/2\) & \(\pi/2\) & \(-\pi/2\) & \(\pi/2\) & \(-\pi/2\) & \(\pi/2\) \\ \(a_{i-1}\)(mm) & 0 & 150 & 0 & 0 & 85 & 60 & 0 & 0 \\ \(\theta_{i}\)(rad) & 0 & \(\pi/2\) & 0 & \(-\pi\) & \(\pi/2\) & 0 & 0 & 0 \\ \(\mathbf{d}_{i}\)(mm) & 0 & \(-243\) & 0 & \(-270\) & 0 & 270 & 0 & \(-148\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: DH parameters of R2. Figure 3: Experimental setup. (**a**) SDA5F robot and AT960; (**b**) The installation of T-MAC and SMR. were taken into the transformation in order to obtain vertical elements that were used to identify the joint stiffness by Equation (35). The procedure usually needs to measure several poses of the EE and then the following equation can be obtained:  
\[Y=AX, \tag{46}\]  
where \(A=\left[A_{1}\ \cdots\ A_{m}\right]^{T}\), and \(Y=\left[y_{1}\ \cdots\ y_{m}\right]^{T}\), m is the number of measurements.