# Cable-Driven Mechanism Models for Sensitive and Actuated Minimally Invasive Robotic Instruments
## 2 Modeling of Sensitive Cable-Driven Mechanisms
### General Algorithms and Formulations  
Figure 7 shows the general scheme used to determine three cable-driven mechanism models presented in [5] (MM1 and MM2) and in [12] (MM3).  
Figure 6: Section of the Planar Sensitive Cable-Driven Mechanism Model with 1 degree of freedom and two cables, presented in [12] (MM3).  
Figure 7: General scheme used to determine cable-driven mechanism models presented in [5] (MM1 and MM2) and [12] (MM3).  
\(O_{A}-X_{A}Y_{A}Z_{A}\) is the reference system of mobile part A, and \(O_{B}-X_{B}Y_{B}Z_{B}\) is the reference system of fixed part B, each one applied in its center of mass. \(N_{Bi}\) and \(N_{Ai}\) are anchor points of the cables, respectively, connected to part B and part A. \(\mathbf{p}_{Bi}\) and \(\mathbf{p}_{Ai}\) are position vectors. The vector of the cables is \(\mathbf{C}_{i}\), and \(\mathbf{r}_{i}\) is the unit vector of the cables. \(\mathbf{p}_{G}\) is the position vector of mobile platform A with respect to fixed platform B. \(\mathbf{p}_{P}\) and \(\mathbf{p}_{PG}\) are position vectors of the general point P with respect to the reference systems of the fixed (B) and mobile (A) parts.  
The correlation among the length of cables (\(\mathbf{C}_{i}\)), the position vectors of anchor points (\(\mathbf{p}_{Bi}\) and \(\mathbf{p}_{Ai}\)), and the position vector of mobile part A with respect to fixed part B (\(\mathbf{p}_{G}\)) is shown by the following equations for eight cables, including the rotational matrix \(S\) to transform the vectors calculated in mobile platform A with respect to fixed platform B:  
\[\mathbf{C}_{i}=+\mathbf{p}_{Bi}-\mathbf{p}_{G}-\mathbf{Sp}_{Ai};i=1,...,8; \tag{1}\]  
\(\sum_{i=1}^{8}\mathbf{T}_{i}=\sum_{i=1}^{8}T_{i}\mathbf{r}_{i}\) are tensions on the cables applied to mobile platform A in \(N_{Ai}\); \(\sum_{i=1}^{8}\left(\mathbf{Sp}_{Ai}\wedge\mathbf{T}_{i}\right)\) are torques produced by tensions on the cables; \(m\mathbf{g}_{G}\) is gravity; \(\mathbf{F}_{P}\) and \(\mathbf{M}_{P}\) are, respectively, the external forces and torques applied to point P shown in Figure 7:  
\[\begin{bmatrix}\mathbf{r}_{1}&\ldots&\mathbf{r}_{8}\\ \mathbf{Sp}_{A1}\wedge\mathbf{r}_{1}&\ldots&\mathbf{Sp}_{A8}\wedge\mathbf{r}_ {8}\end{bmatrix}\begin{bmatrix}T_{1}\\ \vdots\\ T_{8}\end{bmatrix}=-\begin{bmatrix}m\mathbf{g}_{G}\\ \left(\mathbf{p}_{G}\wedge m\mathbf{g}_{G}\right)\end{bmatrix}-\begin{bmatrix} \mathbf{F}_{P}\\ \left(\mathbf{p}_{P}\wedge\mathbf{F}_{P}\right)+\mathbf{M}_{P}\end{bmatrix} \tag{2}\]  
\[\mathbf{QT}=-\mathbf{U}_{G}-\mathbf{U}_{P} \tag{3}\]  
where \(\mathbf{Q}\) is the force transformation matrix [34]; \(\mathbf{T}\) is the vector of the cables' tension; \(\mathbf{U}_{G}\) is the gravity vector [15]. The vector of external forces and torques, applied to point P (\(\mathbf{U}_{P}\)), is calculated by the analysis of variation of the tension of cables \(\Delta\mathbf{T}=\mathbf{T}_{1}-\mathbf{T}_{0}\); \(\mathbf{T}_{0}\) is the vector of the tension on the cables used as an offset in the starting balancing configuration, and \(\mathbf{T}_{1}\) is the vector of the tension on the cables in the new balancing configuration:  
\[\mathbf{U}_{P}=\begin{bmatrix}F_{PX},F_{PY},F_{PZ},M_{PX},M_{PY},M_{PZ}\end{bmatrix} ^{T}=\mathbf{U}_{P0}-\mathbf{Q}\Delta\mathbf{T} \tag{4}\]  
\(\mathbf{U}_{P0}\) is the vector of external forces and torques in the starting balancing configuration. In our models, we used \(\mathbf{U}_{P0}=0\), and Equation (4) can be rewritten with the following Equation (5). All details of the used formulations and models can be found in [5; 12].  
\[\mathbf{U}_{P}=\begin{bmatrix}F_{PX},F_{PY},F_{PZ},M_{PX},M_{PY},M_{PZ} \end{bmatrix}^{T}=-\mathbf{Q}\Delta\mathbf{T} \tag{5}\]