## 3 Experiments and Results
### Inclination Calibration  
To investigate the impact of the inclined base, we conduct the second experiment with three calibration steps: _No Calibration_, _LSM Calibration_, and _Full Calibration_. The parameters and initial values for non-linear optimization are set as follows: the learning rate for CoM is 0.00005, the learning rate for alpha and beta is 0.0001, and the learning rate for crosstalk is 0.001. For the ground inclination, since it is measured in radians and the base inclination is generally small, it is advantageous to optimize with a small learning rate. Similarly, for CoM, as its value is in meters and relatively small, a small learning rate is also appropriate. For initial values, the CoM can converge easily by directly using the values provided in the datasheet, as they are close to the optimal point. For ground inclination, even if the inclination is unknown, setting it to 0 allows convergence due to the small learning rate. For crosstalk, it is recommended to set the initial value to 0, as sensor manufacturers usually perform internal calibration. As will be shown in the experimental results discussed later, the inclination significantly impacts accurate force-torque decoupling. Therefore, setting the tolerance to \(1\times 10^{-6}\) to ensure sufficient convergence is advantageous for obtaining precise results. The results of the experiment are shown in Figure 8. The base is inclined by approximately 5 degrees as we set up in Figure 4. Comparing Figure 7(a),b, one can see that while the external force error in Figure 7(b) has decreased due to crosstalk compensation, the torque error remains due to the base's incline, indicating that the estimated CoM and crosstalk are insufficient to reduce the torque error. In fact, the torque error is the same as in Figure 7(a), where _No Calibration_ is applied. Comparing Figure 7(b),c, one can observe that both force and torque have converged close to zero in Figure 7(c). This result demonstrates effective compensation for the errors in force and torque caused by the inclined base, and compared to Figure 7(a), where _No Calibration_ is applied, it shows that the influence of the end tool has been effectively compensated for. Additionally, as mentioned in the CoM results, our  
Figure 7: Two different data comparing the external force and torque when (**a**) only _No Calibration_ is applied and when (**b**) _LSM Calibration_ is applied on a non-inclined base. Since there is no incline, _Full Calibration_ does not affect the result. calibration method compensates using the data provided by the sensor, enabling effective calibration from the sensor's perspective even if there are discrepancies with the datasheet. The analyzed experimental data are presented in the table. Figure 8a,b show a trend similar to that in Figure 7. Although crosstalk compensation reduced the force error, the presence of the base incline prevented convergence close to zero. Comparing the data in Figure 8b,c, one can see that estimating the incline angle allowed the error to converge close to zero. The _Full Calibration_ reduced MSE with respect to _No Calibration_ by 85.8%, 5.4%, and 56.8% for external force and 27.3% and 65.3% for external torque as you can see in Table 6. Additionally, the final results in Table 5, obtained from experiments on a non-inclined base, are nearly identical to the results obtained on an inclined base. Therefore, our calibration method achieves the same external force estimation accuracy on an inclined base as on an even floor. From Equations (3) and (4), it can be observed that the predicted force and torque are dependent on the ground inclination and the CoM. In the case of the crosstalk matrix used in Equation (8) with LSM or in Equation (9) in the non-linear form, it can be interpreted as a constant value multiplied by the error of the predicted force and torque. Therefore, in the absence of inclination, multiplying the torque by the crosstalk matrix can reduce the error.