# Cable-Driven Mechanism Models for Sensitive and Actuated Minimally Invasive Robotic Instruments
## 5 Results and Discussion  
Figure 17 shows the results of a simulation if a measured force (MF) of 1 N is applied to the sphere of multi-body model MM1, shown in Figure 12. On the top left of Figure 17, forces are calculated (CF) modifying the damping coefficient \(c\) from 0.2 Nms/deg to 3 Nms/deg and using stiffness coefficient \(k=0.0033\) Nmm/deg for the four revolute joints. In the multi-body environment the cable is assumed inextensible, and it is not included in the calculation of the equivalent stiffness coefficient. On the top right of Figure 17, forces are calculated (CF) with a fixed damping coefficient \(c=1\) Nms/deg and modifying the stiffness coefficient \(k\) from 0.0033 Nmm/deg to 330 Nmm/deg for the four revolute joints. In the center (down), the simulation of model MM1 is presented with six different measured force inputs: 1 N, 2 N, 3 N, 4 N, 5 N, 6 N. Forces are calculated (CF) with damping coefficient \(c=1\) Nms/deg and stiffness coefficient \(k=0.0033\) Nmm/deg and using the following coefficients for displacement \(VD=0.074\) and force \(VF=0.137\). Results presented in Figure 17 underline that the formulation conceived for MM1 requests high damping to converge, and high forces cannot be managed correctly. These results confirm the analysis performed in the last section and shown in Figure 9, in which the equilibrium of model MM1 was noted as not stable for its geometrical characteristics with respect to model MM2. Figure 17: Simulation of model MM1 shown in Figure 12 with 1 N of measured force (MF). Left (top): calculated force (CF) with damping coefficient \(c\) variable from 0.2 Nms/deg to 3 Nms/deg and fixed stiffness coefficient \(k=0.0033\) Nmm/deg for the four revolute joints. Right (top): calculated force (CF) with fixed damping coefficient \(c=1\) Nms/deg and variable stiffness coefficient \(k\) from 0.0033 Nmm/deg to 330 Nmm/deg for the four revolute joints. Center (down): simulation of model MM1 with six different measured force inputs: 1 N, 2 N, 3 N, 4 N, 5 N, 6 N. Forces are calculated (CF) with damping coefficient \(c=1\) Nms/deg and stiffness coefficient \(k=0.0033\) Nmm/deg. \(VD=0.074\); \(VF=0.137\). Figure 18 shows the virtual simulation of model MM2 (shown in Figure 13) with 1 N of measured force (MF). In the top of the figure, forces are calculated (CF) with a variable damping coefficient \(c\) from 0.2 Nms/deg to 3 Nms/deg and a fixed stiffness coefficient \(k=0.0033\) Nmm/deg for the four revolute joints. In the multi-body environment the cable is assumed inextensible, and it is not included in the calculation of the equivalent stiffness coefficient. In the lower part of the figure, forces are calculated (CF) with fixed damping coefficient \(c=1\) Nms/deg and variable stiffness coefficient \(k\) from 0.0033 Nmm/deg to 330 Nmm/deg for the four revolute joints. The following coefficients for displacement and force are used: \(VD=0.5\), \(VF=0.83\). On the contrary to model MM1, model MM2 is stable and more efficient for use as the next generation of MIRIs. This result was also noted and underlined in [5]. This is also confirmed in Figure 19, in which six different measured force inputs are applied to MM2. It can be seen that force and displacement errors are very low, underlining the very good quality of model MM2. The quality of model MM2 is also validated with the multi-body model with bigger dimensions, shown in Figure 14, for test bench I, shown in Figure 10 and presented in [5]. Figure 20 confirms the very good quality of the formulation presented in [5]. The five external loads applied to the mobile part in Figure 21 are the same applied in the experimental tests for test bench I presented in [5]. Figure 18: Simulation of model MM2 shown in Figure 13 with 1 N of measured force (MF). Top: calculated force (CF) with damping coefficient \(c\) variable from 0.2 Nms/deg to 3 Nms/deg and fixed stiffness coefficient \(k=0.0033\) Nmm/deg for the four revolute joints. Left (top): measured (MF) and calculated (CF) forces; right (top): force error between MF and CF.