## 2 Materials and Methods
### Single-Cable-Pulley Transmission  
Figure 2 is a schematic diagram of the transmission of an infinitesimal cable segment of a single cable-pulley system with a given length \(dl=Rd\theta\) at an angular position \(\theta\). The tangential forces on both sides of the infinitesimal cable segment are \(T\) and \(T+dT\), the shear forces are \(Q\) and \(Q+dQ\), and the bending moments around the rotation center \(O\) are \(M\) and \(M+dM\), respectively. The infinitesimal cable segment is subjected to the normal force \(dN\) and friction \(dF_{u}\) varying with the angular position \(\theta\). If the inertia force of the infinitesimal cable segment is neglected, the force balance and the moment balance equation around the center of rotation can be obtained as follows:  
\[\left\{ \begin{array}{l}(T+dT)\cos\frac{d\theta}{2}-T\cos(\frac{d\theta}{2}) Q\sin\frac{d\theta}{2}+(Q+dQ)\sin\frac{d\theta}{2}-dF_{u}=0\\ (Q+dQ)\cos\frac{d\theta}{2}-Q\cos\frac{d\theta}{2}-(T+dT)\sin\frac{d\theta}{2 }-T\sin(\frac{d\theta}{2})+dN=0\\ (M+dM)-M-(Q+dQ)\frac{Rd\theta}{2}-Q\frac{Rd\theta}{2}+rdF_{u}=0\end{array} \right., \tag{1}\]  
where \(r,r_{1}\) and \(R\) are, respectively, the cable radius, the pulley radius and the bending radius of the cable, \(T_{1}\) and \(T_{2}\) are the incoming tension and outcoming tension, and \(\theta_{slip}\) and \(\theta_{wrap}\) are the slip angle and wrap angle, respectively. Figure 1: The cable-driven transmission structure of surgical instruments. Since \(d\theta\) is infinitely small, \(\cos(d\theta/2)\to 1\) and \(\sin(d\theta/2)\to d\theta/2\), Equation (1) can be simplified as:  
\[\left\{ \begin{array}{l}dT+Qd\theta-dF_{u}=0\\ dQ-Td\theta+dN=0\\ dM-QRd\theta+rdF_{u}=0\end{array} \right., \tag{2}\]  
Since the movement of the cable on the pulley is a sliding one between the rigid bodies, Amontons' law of friction can be used, and then the following can be calculated as:  
\[dF_{u}=\mu dN, \tag{3}\]  
where \(\mu\) is the friction coefficient. As the bending stiffness is considered, the cable is modeled as a linear elastic material, so the bending constitutive equation [33] of the cable is as follows:  
\[M=\frac{EI}{R}=\frac{E}{R}\frac{\pi r^{A}}{4},\ dM=0, \tag{4}\]  
where \(E\) and \(I\) are the tensile modulus and cross-sectional area of the cable, and \(M\) is the bending moment of the cable. Substituting Equations (3) and (4) into Equation (2), the following expression is obtained:  
\[dT+(Q-\mu T)d\theta+\mu dQ=0, \tag{5}\]  
\[Q=\frac{r}{r_{1}}\frac{dT}{d\theta}=\frac{1}{\rho}\frac{dT}{d\theta}, \tag{6}\]  
where \(\rho=r_{1}/r\) is the radius ratio of the pulley to the cable. Substituting Equation (6) into Equation (5) can yield:  
\[\mu\frac{d^{2}T}{d\theta^{2}}+(1+\rho)\frac{dT}{d\theta}-\rho\mu T=0, \tag{7}\]  
Equation (7) is a second-order linear ordinary differential equation whose general solution can be expressed as:  
\[T(\theta)=C_{1}e^{\theta\theta}+C_{2}e^{b\theta},\ b=\frac{-1-\rho-\sqrt{(1+ \rho)^{2}+4\mu^{2}\rho}}{2\mu}<0\]  
To determine the constant \(C_{1}\) and \(C_{2}\), the boundary conditions of Equation (8) need to be known. According to [34], the boundary condition of Equation (8) is:  
\[\left\{ \begin{array}{l}\theta\to 0\\ Q(0)=T_{1}\sin\vartheta=0\\ T(0)=T_{1}\cos\vartheta=T_{1}\end{array} \right., \tag{9}\]  
Figure 2: Free-body diagram of a wrapped cable segment. where \(Q(0)\) and \(T(0)\) are the incoming shear force and the incoming tension force in the contact region, respectively, and \(\vartheta\) is the inclined angle of the outcoming cable, which is zero in the contact region. Combining Equations (6) and (9), the following can be formulated as:  
\[\left\{ \begin{array}{l}T(0)=T_{1}\\ \frac{dT}{d\vartheta}|_{\theta=0}=\rho Q(0)=0\end{array} \right., \tag{10}\]  
Combining Equations (8) and (10) can yield:  
\[C_{1}=\frac{bT_{1}}{b-a},\;C_{2}=\frac{aT_{1}}{a-b}, \tag{11}\]  
Let \(\lambda(\rho,\mu,\theta)=\left(ae^{b\theta}-be^{a\theta}\right)/(a-b)\), and by substituting Equation (11) into Equation (8), the following can be obtained as:  
\[