## 4 Kinematic Analysis of the Rope Drive Mechanism  
### Forward Kinematic Analysis of the Mechanism  
The motion of a rigid body in three-dimensional space \(R^{3}\) can be defined as \(g(p)=Rp+t\), where \(t\in R^{3}\), \(R\in SO(3)\). The entire mapping of the motion forms a six-dimensional Lie group [22; 23; 24; 25], which is called a special Euclidean group, is indicated as \(SE(3)\)  
\[SE(3)=(R,t), \tag{1}\]  
where \(R\) represents the \(3\times 3\) attitude rotation matrix, \(t\) is the position vector. According to [PERSON]'s theorem, there is always a rotation matrix corresponding to each rotational motion of a rigid body. Set the unit vectors \(\omega\) representing the rotation axis and \(\theta\) representing the rotation angle, then \(R\) can be determined as a function containing \(\omega\) and \(\theta\):  
\[R=e^{\theta\omega} \tag{2}\]  
where \(\hat{\omega}\) is the antisymmetric matrix. According to the mapping relationship of the index, it can be concluded that:  
\[ \begin{split} e^{\theta\hat{\omega}}=E+\hat{\omega}sin\theta+ \hat{\omega}^{2}(1-cos\theta),\\ \hat{\xi}=\begin{pmatrix}\hat{\omega}&v\\ 0&0\end{pmatrix}_{4\times 4}\in R^{4\times 4}\\ \hat{\xi}=\begin{pmatrix}\hat{\omega}&v\\ 0&0\end{pmatrix}_{4\times 4}^{\vee}=\begin{pmatrix}\omega\\ v\end{pmatrix}_{6\times 1}\end{split}  \tag{3}\]  
where \(\hat{\xi}\) is the motion screw, \(\hat{\xi}\) is the screw coordinates of \(\hat{\xi}\). Based on [PERSON] theorem [26; 27; 28; 29], any rigid body motion can be achieved through spiral motion, which is a composite motion containing the rotation around a certain axis and the movement along that axis. Therefore, a rigid body motion can be expressed as the exponential product of screw:  
\[g=e^{\theta\hat{\xi}}= \begin{bmatrix}e^{\theta\hat{\omega}}&\big{(}E-e^{ \theta\hat{\omega}}\big{)}(\omega\times v)+\theta\omega\omega^{T}v\\ 0&1\end{bmatrix}  \tag{4}\]  
Therefore, the product of exponentials (POE) formula for the forward kinematics of serial robots is:  
\[ \begin{smallmatrix}S\\ T\end{smallmatrix} g(\theta)=e^{\theta_{1}\hat{\xi}_{1}}e^{\theta_{2}\hat{\xi }_{2}}\ldots e^{\theta_{n}\hat{\xi}_{n}}T\hat{\xi}(0) \tag{5}\]  
According to the initial configuration and size of the rope drive shown in Figure 6, the unit vector of each joint axis direction can be derived as:  
\[ \begin{split}\omega_{1}=\omega_{3}=\begin{bmatrix}0&0&1 \end{bmatrix}^{T},\;\omega_{2}=\omega_{4}=\begin{bmatrix}0&1&0\end{bmatrix}^ {T}\\ r_{1}=r_{2}=\begin{bmatrix}L_{1}&0&0\end{bmatrix}^{T},\;r_{3}=r_{4}=\begin{bmatrix} L_{1}+L_{2}&0&0\end{bmatrix}^{T}\end{split}  \tag{6}\]In the initial state, the pose matrix of the coordinate system \(T\) for the rope drive's end effectors, which is relative to the base coordinate system \(S\), can be represented as:  
\[\lx@sectionsign_{T}g(0)= \begin{bmatrix}1&0&0&L\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}  \tag{7}\]  
where \(L=L_{1}+L_{2}+L_{3}\). The forward kinematic solution of the rope drive mechanism is:  
\[\lx@sectionsign_{T}g(\theta)=e^{\theta_{1}\lx@sectionsign_{1}}e^{\theta_{2} \lx@sectionsign_{2}}e^{\theta_{3}\lx@sectionsign_{3}}e^{\theta_{4}\lx@sectionsign_{4 }}\lx@sectionsign_{T}g(0) \tag{8}\]  
According to designed dimensions of the rope drive, we obtain:  
\[L_{1}=0.6\text{ mm};L_{2}=L_{3}=1.2\text{ mm}\;;\;L_{4}=1\text{ mm}\]  
The rope drive operation is driven by the rope, which can only be stretched but not compressed. Therefore, the structural design of the conduit should meet the controllable requirements of the conduit to ensure that the tension of the rope cannot exceed the rotation centerline of the joint, and, thereby, the system can rotate flexibly under the action of rope tightening and loosening, as shown in Figure 7.  
Figure 6: (a) Three-dimensional diagram of the rope drive mechanism; (b) schematic diagram of the rope drive mechanism. Figure 7: The limit position of the joint angle.