## 3 Kinematics Modeling of Self-Insulating Mechanical Arm Joints
### Forward Kinematics  
The homogeneous coordinates of \(P_{1}\) and \(P_{2}\) in the fixed platform's global coordinate system \(OXYZ\) are denoted as \({}^{O}\mathbf{p}_{1}\) and \({}^{O}\mathbf{p}_{2}\), while \(Q_{1}\) and \(Q_{2}\)'s coordinates in the moving platform's local frame \(oxyz\) are expressed as \({}^{o}\mathbf{q}_{1}\) and \({}^{o}\mathbf{q}_{2}\), respectively:  
\[ \begin{split}{}^{O}\mathbf{p}_{1}&=(0,-p,-d_{1},1)^{T},{}^{O}\mathbf{p}_{2}=(0,p,-d_{1},1)^{T}\\ {}^{o}\mathbf{q}_{1}&=(-q,0,0,1)^{T},{}^{o}\mathbf{q}_{2}=(q,0,0,1)^{T}\end{split}  \tag{1}\]  
The upper platform's motion is described by three angles: \(\theta_{m}\) (the angle between \(o^{\prime}O\) and \(OX\)-axis, representing bending direction), \(\theta_{b}\) (the angle between \(OXY\) plane and upper platform, representing bending magnitude), and \(\theta_{r}\) (the counterclockwise rotation angle about the \(z\)-axis, representing yaw angle in the local frame). Here, \(o^{\prime}\) represents the projection of point o onto the \(OXY\) plane. Using [PERSON] rotation formula, the rotation matrix \(\mathbf{R}\) from local to global coordinates is obtained, and the transformation matrix \(\mathbf{R}_{h}\) is defined as follows:  
\[ \begin{split}\mathbf{R}=\hat{e}^{\hat{\mathbf{q}}_{b}}=\mathbf{I}+\hat{\mathbf{i }}\sin\theta_{b}+\hat{\mathbf{i}}^{2}(1-\cos\theta_{b})\\ \mathbf{R}_{h}=\left[\begin{array}{cc}\mathbf{R}&\mathrm{O}\\ \mathrm{O}&1\end{array}\right]\end{split}  \tag{2}\]  
The specific elements in the equation are as follows:  
\[ \begin{split}\hat{\mathbf{i}}=[-\sin\theta_{m},\cos\theta_{m},0]^{T} \end{split}  \tag{3}\]  
\[ \begin{split}\hat{\mathbf{i}}=\left[\begin{array}{cccc}0&0& \cos\theta_{m}\\ 0&0&\sin\theta_{m}\\ -\cos\theta_{m}&-\sin\theta_{m}&0\end{array}\right]\end{split}  \tag{4}\]  
The rotation matrix \(M_{rot\_oz}\) about the \(z\)-axis is given by:  
\[ \begin{split}\mathbf{M}_{rot\_oz}=\left[\begin{matrix}\cos\theta_{r }&-\sin\theta_{r}&0&0\\ \sin\theta_{r}&\cos\theta_{r}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{matrix}\right]\end{split}  \tag{5}\]  
The translation matrix \(\mathbf{M}_{trans}\) along the \(z\)-axis is given by:  
\[ \begin{split}\mathbf{M}_{trans}=\left[\begin{matrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&d_{2}\\ 0&0&0&1\end{matrix}\right]\end{split}  \tag{6}\]  
Finally, the transformation matrix \({}^{O}\mathbf{T}_{o}\) from the local coordinate system to the global coordinate system is  
\[{}^{O}\mathbf{T}_{o}=\mathbf{R}_{h}\mathbf{M}_{trans}\mathbf{M}_{rot\_oz}=\left[ \begin{matrix}R _{{}_{O}_{1}\sigma_{1}}&P_{{}_{O}_{1}\sigma_{1}}\\ 0&1\end{matrix} \right] \tag{7}\]Similarly, a transformation matrix from \(\mathbf{o}_{1}\) to \(\mathbf{C}\) is established. Since there is only a translational relationship rather than a rotational relationship between them, the transformation matrix is  
\[{}^{o_{1}}\mathbf{T}_{\text{C}}=M_{trans2}= \begin{bmatrix}I&P_{o_{1}C}\\ 0&1\end{bmatrix}  \tag{8}\]  
Similarly, the transformation matrix from \(\mathbf{C}\) to \(\mathbf{O}_{2}\) is derived. Given the presence of both a rotational relationship (with a rotation angle of \(\alpha\)) and a translational relationship between them, the transformation matrix is therefore:  
\[{}^{C}\mathbf{T}_{O_{2}}=M_{trans3}M_{rot.ox}= \begin{bmatrix}R_{o_{1}C}&P_{o_{1}C} \\ 0&1\end{bmatrix}  \tag{9}\]  
Similarly, the transformation matrix from \(\mathbf{O}_{2}\) to \(\mathbf{o}_{2}\) is derived. Given that the upper and lower joint components are identical, the entire derivation process and matrix structure are exactly the same as those of the \(\mathbf{O}\) to \(\mathbf{o}\) matrix. Therefore, the result is  
\[{}^{O_{2}}\mathbf{T}_{o_{2}}=\mathbf{R}_{h}\mathbf{M}_{trans}\mathbf{M}_{rot.oz}= \begin{bmatrix} R_{O_{2}o_{2}}&P_{O_{2}o_{2}}\\ 0&1\end{bmatrix}