## 2 Method
### Calibration Model
Since the torque arises from the CoM displacement of the end tool and its weight in the \(\{f\}\) frame, it can be expressed as shown in Equation (4) using Equation (2). \[ \begin{split}\tau_{f}^{est}&=\mathbf{r}_{f}\times \mathbf{f}_{f}^{est}\\ &=\begin{bmatrix}\left(r_{31}d_{y}-r_{21}d_{z}\right)\sin\beta- \left(r_{33}d_{y}-r_{23}d_{z}\right)\cos\alpha\cos\beta-\left(r_{32}d_{y}-r_{2 2}d_{z}\right)\cos\beta\sin\alpha\\ \left(r_{33}d_{x}-r_{13}d_{z}\right)\cos\alpha\cos\beta-\left(r_{31}d_{x}-r_{1 1}d_{z}\right)\sin\beta+\left(r_{32}d_{x}-r_{12}d_{z}\right)\cos\beta\sin \alpha\\ \left(r_{21}d_{x}-r_{11}d_{y}\right)\sin\beta-\cos\alpha\left(r_{23}d_{x}-r_{1 3}d_{y}\right)\cos\beta-\left(r_{22}d_{x}-r_{12}d_{y}\right)\cos\beta\sin \alpha\\ \end{bmatrix}\end{split}  \tag{4}\]  
The \(\mathbf{r}_{f}\) vector represents the vector of CoM on the end tool in the \(\{f\}\) coordinate system. Expanding the forward kinematics from the \(\{b\}\) coordinate system to the \(\{e\}\) coordinate system, if each element of \(\mathbf{R}_{f}\) is defined as \(r_{i,f}\), \(\tau_{f}^{est}\) can be expressed in terms of the known variables \(r_{11}\) through \(r_{33}\) and the unknown variables \(d_{x}\), \(d_{y}\), \(d_{z}\), \(\alpha\), and \(\beta\). By using the sensor values obtained from the four postures shown in Figure 2, the bias values can be determined through Equation (5). \[ \begin{split}\mathbf{w}_{1}^{ft}&=\begin{bmatrix} \mathbf{f}_{f}^{bs}\\ \tau_{f}^{abs}\end{bmatrix}_{1}+\mathbf{w}^{bias}+\mathbf{C}\begin{bmatrix} \tau_{f}^{est}\\ \mathbf{0}_{3\times 1}\end{bmatrix}_{1},\quad\text{where}\quad\mathbf{f}_{f,1}^{bs }=\begin{bmatrix}+f_{x}\\ +f_{y}\\ +f_{z}\end{bmatrix},\quad\tau_{f,1}^{abs}=\begin{bmatrix}-d_{z}f_{y}+d_{y}f_{z }\\ +d_{z}f_{x}-d_{x}f_{z}\\ -d_{y}f_{x}+d_{x}f_{y}\end{bmatrix}\\ \mathbf{w}_{2}^{ft}&=\begin{bmatrix}\mathbf{f}_{f}^{abs}\\ \tau_{f}^{abs}\end{bmatrix}_{2}+\mathbf{w}^{bias}+\mathbf{C}\begin{bmatrix} \tau_{f}^{est}\\ \mathbf{0}_{3\times 1}\end{bmatrix}_{2},\quad\text{where}\quad\mathbf{f}_{f,2}^{ abs}=\begin{bmatrix}-f_{x}\\ -f_{y}\\ +f_{z}\end{bmatrix},\quad\tau_{f,2}^{abs}=\begin{bmatrix}+d_{z}f_{y}+d_{y}f_{z }\\ -d_{z}f_{x}-d_{x}f_{z}\\ +d_{y}f_{x}-d_{x}f_{y}\end{bmatrix}\\ \mathbf{w}_{3}^{ft}&=\begin{bmatrix}\mathbf{f}_{f}^{abs}\\ \tau_{f}^{abs}\end{bmatrix}_{3}+\mathbf{w}^{bias}+\mathbf{C}\begin{bmatrix} \tau_{f}^{est}\\ \mathbf{0}_{3\times 1}\end{bmatrix}_{3},\quad\text{where}\quad\mathbf{f}_{f,3}^{ abs}=\begin{bmatrix}-f_{x}\\ +f_{y}\\ -f_{z}\end{bmatrix},\quad\tau_{f,3}^{abs}=\begin{bmatrix}-d_{z}f_{y}-d_{y}f_{z }\\ -d_{z}f_{x}+d_{x}f_{z}\\ +d_{y}f_{x}+d_{x}f_{y}\end{bmatrix}\\ \end{split}  \tag{5}\]  
When the same force is applied in the \(\{g\}\) frame, the forces and torques from the four postures in Figure 2 are composed of elements with identical magnitudes but opposite signs. Additionally, since the crosstalk effect we modeled is a linear combination of \(\tau_{f}^{est}\), it is dependent on the sign of \(\tau_{f}^{est}\). If two postures could be identified where each of the three axes has an opposite sign, it would be possible to cancel out all terms except \(\mathbf{w}^{bias}\) using only those two postures. However, as such postures do not exist, we achieve the same effect through a combination of four postures. Consequently, by summing the sensor data from all four postures, we isolate the term composed solely of \(\mathbf{w}^{bias}\). Figure 2: The four initialization postures for bias estimation. When the same force is applied in the \(\{g\}\) frame, the forces represented in the \(\{f\}\) frame for each of the four postures are shown. The crosstalk matrix \(\mathbf{C}\) is defined as shown in Equation (6). This matrix represents the effect that a linear combination of torque values exerts on the force term. As mentioned above, forces and torques along the same axis do not affect each other in a capacitance-type FT sensor.