# Cable-Driven Mechanism Models for Sensitive and Actuated Minimally Invasive Robotic Instruments
## 4 Multibody Design and Control of MM1 and MM2
### General Control Architectures for Each Model  
Our proposed architectures for models MM1, MM2, and MM3 (shown in Figures 15 and 16) are able to control and/or to measure the motion of part C, where forceps are attached (see Figure 3), and the interaction forces on the forceps by using only the measure of the revolution of motors (or the displacement of each spring at the end of each cable). Figures 15 and 16 show the control blocks for the sensing and actuation architectures applied to each model presented above and in our previous works [5; 12].  
Figure 14: Multi-body model MM2 implemented in the planar test bench I with big dimensions.  
Figure 13: Multi-body model MM2 in a reduced scale.  
Figure 15 shows the sensitive control architecture. The displacements of springs (\(\delta_{a}\), \(\delta_{b}\), \(\delta_{c}\), \(\delta_{b}\)) (or the displacement of the revolution of motors, obtained by \(\delta_{a}\), \(\delta_{b}\), \(\delta_{c}\), \(\delta_{b}\)) are used to calculate the applied forces and motion on the mobile part. For model MM1, studied in the last section, Equation (22) allows for the calculation of the external forces (\(F_{ec}\)). \(\Delta T_{a}\), \(\Delta T_{b}\), \(\Delta T_{c}\), \(\Delta T_{d}\) are the tensions on cables and \(k_{a}\), \(k_{b}\), \(k_{c}\), \(k_{d}\) are the equivalent stiffness coefficients of springs and cables. Calculated forces (CF) and displacements (CD) are obtained, respectively, including two proportional values: force amplification value (VF) and displacement amplification value (VD).  
If the model is used like an actuated system, inputs to the system are the revolute motions of motors (used instead of springs). A correct combination of the motion of motors permits for the movement of part C, where forceps are attached (see Figure 3). On the contrary, if the model is used like a sensitive system, the motion and forces applied on part C, where forceps are attached (see Figure 3), can be evaluated by the combination of motions and torques on motors.  
Figure 16 shows the control architecture for the actuated system. Calculated tensions on cables (\(T_{cal}\)) are used to evaluate the error between the measured one (\(T_{meta}\)). This error helps to reduce the gap between the measured and calculated external forces (\(F_{em}\) and \(F_{ec}\)). The error between the measured displacement (\(\delta_{meta}\)) and the calculated one (\(\delta_{cal}\)) allows for the reduction of the gap in the prevision of motion of the mobile part.  
Figure 16: Actuation control architecture for each model. \(\delta e\) is the error between measured (\(\delta_{meta}\)) and calculated (\(\delta_{cal}\)) displacement of the spring; \(T_{e}\) is the error between measured (\(T_{meta}\)) and calculated (\(T_{cal}\)) tensions on cables; \(k\) is the equivalent stiffness coefficient; \(F_{ec}\) is the calculated external force; \(F_{em}\) is the measured external force; \(v_{i1}\) and \(v_{i2}\) are input to have variable coefficients of stiffness. In this work, the equivalent stiffness coefficient \(k\) is constant.  
Figure 15: Sensing control architecture for each model. \(\delta_{a}\), \(\delta_{b}\), \(\delta_{c}\), \(\delta_{b}\) are displacements of springs; \(k_{a}\), \(k_{b}\), \(k_{c}\), \(k_{d}\) are equivalent stiffness coefficients, \(\Delta T_{a}\), \(\Delta T_{b}\), \(\Delta T_{c}\), \(\Delta T_{d}\) are tensions on cables; \(F_{ec}\) is the calculated external force; \(\theta\) is the angular displacement of the mobile part with respect to the fixed part; VF is the force amplification value and VD is the displacement amplification value.