## 4 Conclusions  
In this paper, we introduce a calibration method for accurately obtaining external forces by modeling an FT sensor to determine essential parameters, including bias, crosstalk, CoM, and inclination. Although discrepancies can exist between the FT sensor's measured data and actual physical quantities, our goal is to model the sensor to minimize the error between the FT sensor data and the predicted values to obtain accurate external forces. Tasks requiring interaction with the environment through a robot necessitate the attachment of an end tool. To isolate contact forces with the external environment, it is essential to compensate for the wrench due to the end tool's weight, which we achieve through calibration. Notably, adding incline angle calibration, found to be the most critical factor, improves compensation performance. In Section 3, we evaluated whether calibration was appropriately applied in response to changes in the environment. Our calibration results \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline  & & & & & & & & & & & \\  & \multicolumn{3}{c}{**Pose 1**} & \multicolumn{3}{c}{**Pose 2**} & \multicolumn{3}{c}{**Pose 4**} & \multicolumn{3}{c}{**Pose 5**} & \\  & \(\alpha:288.^{\circ}\) & \(\beta:-171.4^{\circ}\) & \(\alpha:140.6^{\circ}\) & \(\beta:-154.2^{\circ}\) & \(\beta:-170.7^{\circ}\) & & & & & \\  & \(\gamma:27.7^{\circ}\) & \(\gamma:199.7^{\circ}\) & \(\gamma:109.2^{\circ}\) & \(\gamma:46.1^{\circ}\) & \(\gamma:179.74^{\circ}\) & & & & & \\ \hline _No Calib._ & 2.58 & 2.26 & 1.14 & 1.08 & 1.25 & 0.511 & 1.66 & 0.674 & 0.496 \\ _LSM_ & 2.26 & 2.28 & 1.64 & 2.10 & 2.07 & 0.058 & 2.07 & 0.214 & 0.067 \\ _Cully Callib._ & 2.22 & 2.28 & 1.70 & 1.97 & 2.08 & 0.044 & 2.05 & 0.182 & 0.053 \\ \hline \end{tabular}
\end{table} 
Table 8: This table summarizes the mass estimation results [kg] across five different orientations for each of the three calibration scenarios. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline  & & & & & & & & & & \\ \cline{3-10}  & & & & & & & & & & \\ \cline{3-10}  & & & & & & & & & & \\ \cline{3-10}  & & & & & & & & & & \\ \hline \multirow{4}{*}{MSE} & \multirow{4}{*}{w/o mass} & \multirow{4}{*}{\(\mathbf{LSM}\)} & \multirow{4}{*}{\(\mathbf{LSM}\)} & \multirow{4}{*}{\(\mathbf{LSM}\)} & \multirow{4}{*}{\(\mathbf{LSM}\)} & \multirow{4}{*}{\(\mathbf{LSM}\)} & \multirow{4}{*}{\(\mathbf{LSM}\)} & \multirow{4}{*}{\(\mathbf{LSM}\)} & \multirow{4}{*}{\(\mathbf{LSM}\)} & \multirow{4}{*}{\(\mathbf{LSM}\)} \\  & & & & & & & & & & \\ \cline{3-10}  & & & & & & & & & & \\ \cline{3-10}  & & & & & & & & & & \\ \hline \end{tabular}
\end{table} 
Table 9: This table summarizes the calibration results obtained using Robotiq and Barrett grippers under various base angles. It includes the MSE calculated relative to an external wrench of 0 when no object was grasped and the MSE calculated relative to a 2 kg reference when a 2 kg object was grasped. showed that on a non-inclined base, crosstalk and CoM adjustments reduced the MSE by 55.8%, 56.2%, and 14.5% for external force with respect to _No Calibration_. On an inclined base, the MSE decreased by 85.8%, 5.4%, and 56.8% for external force and 27.3% and 65.3% for external torque with respect to _No Calibration_. These results indicate that working on an inclined base can significantly impair accurate external force estimation using FT sensors, underscoring the necessity of our calibration method, including incline compensation, to ensure consistent FT sensor performance. We expect that the proposed calibration method will be valuable in applications like mobile manipulators, where environments change frequently and interaction with the environment is essential.