## 2 Method
### Calibration Model  
The typical crosstalk value can be interpreted based on changes in the structure of the FT sensor due to deformation. These changes are estimated by analyzing the variation in strain gauges or capacitance and are then converted into force or torque [16]. For strain gauges, the variation in the sensors installed on the structure is calculated through an internal matrix and converted into a force and torque that the user can understand. On the other hand, in the case of capacitance-type sensors, raw data are generated by utilizing changes in the distance between electrodes on the installed substrate (parallel plates with normal displacement), the area changes due to shear force (parallel plates with shear displacement), and the angle and distance changes between electrodes orthogonal to the direction of the shear force (orthogonal plates with shear displacement). Similar to the strain gauge type, the data are processed through internal calculations to provide force and torque values to the user [17]. We model the wrench measured by the FT sensor as follows. Let the superscript \(ft\) denote the value measured by the FT sensor, \(abs\) the theoretical applied value, \(bias\) the bias present in the sensor, and \(est\) our estimated value. Then, the wrench read by the sensor is defined as shown in Equation (1). \[\mathbf{w}_{f}^{ft}=\mathbf{w}_{f}^{abs}+\mathbf{w}_{f}^{bias}+\mathbf{C} \begin{bmatrix}\mathbf{r}_{f}^{est}\\ \mathbf{0}_{3\times 1}\end{bmatrix}  \tag{1}\]  
\(\mathbf{w}^{abs}\) represents the ideal force applied by the end tool attached to the sensor, where force is expressed as \(m\mathbf{g}\) and torque as \(\mathbf{r}\times m\mathbf{g}\). \(\mathbf{w}^{bias}\) is the initial default value of each data point when the sensor is first activated, which varies with each boot. The \(C\) matrix is the crosstalk matrix, which maps the influence of torque on force. Our objective is to determine all the terms on the right side of Equation (1) to compensate for the wrench caused by the end tool attached to the sensor, thereby enabling the measurement of the pure external force. Since the \(\mathbf{w}^{abs}\) caused by the end tool is due to gravity, it is necessary to calculate the body gravity in the \(\{f\}\) coordinate. If the base frame coincides with the global coordinate system \(\{g\}\), this calculation can be simplified using the robot's forward kinematics. However, if the base is inclined, it is essential to know the angle to accurately determine the direction of gravity which is described in frame \(\{b\}\). \[\mathbf{g}_{b}=\mathbf{R}_{0}\mathbf{g}_{g}=g \begin{bmatrix}\sin\beta\\ -\sin\alpha\cdot\cos\beta\\ -\cos\alpha\cdot\cos\beta\end{bmatrix}  \tag{2}\]  
Defining the orientation of the \(\{b\}\) frame relative to the global \(\{g\}\) frame by the roll, pitch, and yaw angles \(\alpha\), \(\beta\), and \(\gamma\), respectively, the gravity acting in the \(\{b\}\) frame can be formulated as in Equation (2). It is important to note that since the normal vectors of the \(\{g\}\) and \(\{b\}\) frames can be aligned in opposite directions through two rotations, as in a pan-tilt configuration, it is sufficient to determine only \(\alpha\) and \(\beta\) to obtain \(\mathbf{g}_{g}\) without fully calculating \(\mathbf{R}_{0}\). To estimate the force \(\mathbf{f}_{f}^{abs}\) physically exerted by the end tool on the sensor, \(\mathbf{f}_{f}^{est}\) is defined as shown in Equation (3). \[\mathbf{f}_{f}^{est}=m\mathbf{R}_{s}\mathbf{R}_{f}\mathbf{R}_{0}\mathbf{g}_{ g}=m\mathbf{R}_{f}\mathbf{g}_{b} \tag{3}\]  
The body gravity of frame \(\{b\}\) obtained from Equation (2) is transformed to the sensor frame \(\{f\}\) using forward kinematics, based on data received from the controller. The transformation matrix between the end-effector \(\{e\}\) frame and the force sensor \(f\) framerepresents any rotational misalignment due to sensor assembly error; however, this effect is minimal relative to other factors and can be reasonably approximated as an identity matrix.