## 3 Experiments and Results
### Crosstalk Calibration  
The reason for setting \(\mathbf{C}_{f2t}\) to zero when designing the crosstalk matrix \(\mathbf{C}\) is that torque is minimally affected by crosstalk. Figure 5 shows the results when the wrist angle of the end-effector is rotated 360 degrees, starting from a posture with the highest torque, as in the calibration data collection postures. In this case, the CoM is estimated without inclination, and no crosstalk calibration was performed. As a result, the \(\mathbf{f}_{ext}\) exhibited significant errors due to the influence of crosstalk. In contrast, the \(\mathbf{\tau}_{ext}\) showed a maximum error of 0.1 Nm, indicating that the influence of crosstalk on torque is minimal. As shown in Table 4, the mean and variance of the external torque are very small. Therefore, our approach to modeling the crosstalk matrix is reasonable. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(\mathbf{\tau}_{\mathbf{x}}^{ext}\) & \(\mathbf{\tau}_{\mathbf{y}}^{ext}\) & \(\mathbf{\tau}_{\mathbf{z}}^{ext}\) \\ \hline \(MSE\) & 0.0075 & 0.0037 & 0.0004 \\ Mean & 0.059 & 0.062 & 0.015 \\ Var & 0.002 & 0.002 & 0.0001 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The table of the MSE (mean squared error) with respect to 0, the mean, and the variance of the external torque for each axis in the data collected by rotating only the wrist in the sampling pose, without performing crosstalk calibration. To assess the relationship with the ideal value of zero, the mean and variance were calculated based on the absolute values of the data. Figure 5: The graph of external force and torque measured when the wrist is rotated in the sampling pose posture without crosstalk calibration. Since only the gripper is attached and no external forces are applied, the values should ideally converge to zero. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Datasheet** & **Estimation** \\ \hline \(d_{x}\) [mm] & \(-8\) & 4.7 \\ \(d_{y}\) [mm] & 0.0 & \(-7.4\) \\ \(d_{z}\) [mm] & 105 1 & 104.4 \\ \hline \hline \end{tabular} 1  
These data represent the values specified in the datasheet, with the addition of the height of the mount we designed.[ENDFOOTNOTE]  
\end{table} 
Table 3: The result of CoM estimation. The CoM specified in the datasheet is also an approximation and therefore not entirely accurate. To evaluate whether the crosstalk calibration can compensate for the effects of the end tool's own weight, we conducted an experiment using the Robotiq 3F gripper, trying a pick-and-place scenario on a non-inclined base. The path of the scenario is described in Figure 6. The robot moves to predefined joint positions using its internal controller, which is based on H-infinity control [19]. When the CoM is not estimated, the data from the gripper's datasheet are used directly. During the experiment, the gripper does not grasp any object, and no external forces are applied. Ideally, the external wrench \(\mathbf{w}^{ext}=\mathbf{w}^{ft}-\mathbf{w}^{comp}\) should be zero, where \(\mathbf{w}^{comp}=\mathbf{w}^{est}_{f}+\mathbf{w}^{bias}_{f}+\mathbf{C}\! \left[\tau^{est\intercal}_{f}\quad\mathbf{0}_{1\times 3}\right]^{\mathsf{T}}\). The first experiment is conducted on a non-inclined surface to investigate the impact of crosstalk. Figure 7a shows the external force and torque graphs during the pick-and-place operation in the _No Calibration_ case, while Figure 7b shows the same data for _LSM Calibration_. As you can see in Figure 7, in postures where torque is generated due to the influence of the end tool, significant force errors are observed due to crosstalk. The analysis of the external force and torque data along each axis, as shown in Figure 7a, is summarized in Table 5.  
When crosstalk calibration was applied, the MSE reduced 55.8%, 56.2%, and 14.5% with respect to _No Calibration_ for external force. In contrast, the external torque showed almost identical results. This is because the estimated CoM closely matches the CoM specified in the datasheet, and no crosstalk compensation was applied for torque.