# Cable-Driven Mechanism Models for Sensitive and Actuated Minimally Invasive Robotic Instruments
## 2 Modeling of Sensitive Cable-Driven Mechanisms
### Modeling of the Mechanism Model MM1  
The general mechanical behavior of the three mechanism models (MM1, MM2, and MM3) is described with the formulations presented above. The mechanism models MM1 and MM2 (shown in Figures 4 and 5) have been presented in [5]. Formulations for MM2 have been included and validated with experimental tests. Formulations for MM3 (shown in Figure 6) have also been included and validated with virtual simulation and experimental tests, as shown in [12]. On the contrary, model MM1 was not analyzed in detail in [5], and formulations for MM1 will be included in the following equations. A comparison between the two models MM1 and MM2 underlines the convenience for using one model instead of the other one. Figure 8 shows the sketch of model MM1 in a starting position (dashed lines in Figure 8: points \(C_{0}\), \(A_{0}\), \(B_{0}\), \(D_{0}\)) and in motion (continuous lines in Figure 8: points \(C_{1}\), \(A_{1}\), \(B_{1}\), \(D_{1}\)). The mobile part is connected to the fixed part by using four cables, four springs, and four pulleys. Each cable is connected with one spring, interacting with only one pulley. The starting balancing configuration of the mobile part (dashed lines in Figure 8: points \(C_{0}\), \(A_{0}\), \(B_{0}\), \(D_{0}\)) can be perturbed by external resultant forces (\(F_{c}\)) applied in point P, generating the motion of the mobile part. The new balancing configuration (continuous  
lines in Figure 8: points \(C_{1}\), \(A_{1}\), \(B_{1}\), \(D_{1}\)) and the external applied forces can be found, respectively, with the geometrical analysis of the whole system, and with the analysis of the equilibrium among \(F_{c}\) and the tension on the cables (\(\Delta T_{a}\), \(\Delta T_{b}\), \(\Delta T_{c}\), \(\Delta T_{d}\)), obtained by the following equations:  
\[\Delta T_{a}=T_{a11}-T_{a10};\Delta T_{b}=T_{b11}-T_{b10};\Delta T_{c}=T_{a21} -T_{a20};\Delta T_{d}=T_{b21}-T_{b20}; \tag{6}\]  
The following parameters (shown in Figure 8 and in Equation (7)) will be used in the next few equations:  
* \(T_{10}\), \(Tb_{10}\), \(T_{20}\), \(Tb_{20}\) are tensions on four cables in the starting position of the mobile part;
* \(T_{11}\), \(Tb_{11}\), \(T_{21}\), \(Tb_{21}\) are tensions on four cables in the new configuration of the mobile part;
* \(a_{10}\), \(b_{10}\), \(a_{20}\), \(b_{20}\) are lengths of the cables in the starting configuration from points \(A_{0}\), \(B_{0}\), \(C_{0}\), \(D_{0}\) to each tangent of the pulleys centered in \(O_{a}\), \(O_{b}\), \(O_{c}\), \(O_{d}\);
* \(a_{11}\), \(b_{11}\), \(a_{21}\), \(b_{21}\) are lengths of the cables in the new configuration, calculated from the points \(A_{1}\), \(B_{1}\), \(C_{1}\), \(D_{1}\), respectively, to each tangent of the pulleys centered in \(O_{a}\), \(O_{b}\), \(O_{c}\), \(O_{d}\);
* \(R\) is the radius of the pulleys;
* \(L_{1}\), \(L_{2}\), \(L_{3}\), \(L_{4}\), \(L_{5}\), \(L_{6}\) are the dimensions of the whole system shown in Figure 8;
* \(\alpha_{a}\), \(\beta_{a}\), \(\sigma_{a}\), \(\alpha_{b}\), \(\beta_{b}\), \(\varphi_{b}\), \(\alpha_{c}\), \(\beta_{c}\), \(\sigma_{c}\), \(\alpha_{d}\), \(\beta_{d}\), \(\sigma_{d}\) are the angles shown in Figure 8;
* \(k_{a}\), \(k_{b}\), \(k_{c}\), \(k_{d}\) are the equivalent stiffness coefficients of the springs and cables in the series;  
Figure 8: Mechanism model MM1: planar scheme with four cables, four pulleys, and four springs. Mobile part in the starting position (dashed lines: points \(C_{0}\), \(A_{0}\), \(B_{0}\), \(D_{0}\)) and new balancing configuration (continuous lines: points \(C_{1}\), \(A_{1}\), \(B_{1}\), \(D_{1}\)). * \(\delta a\), \(\delta b\), \(\delta c\), \(\delta d\) are displacements of the springs from the starting position to the new balancing configuration;
* \(\delta a\), \(\delta b\), \(\delta c\), \(\delta d\) are displacements of springs in the starting configuration for preloading;
* \(m_{A}\) is the mass of the mobile part;
* \(g\) is gravity.