# A New Cable-Driven Model for Under-Actuated Force–Torque Sensitive Mechanisms

^{*}

## Abstract

**:**

## 1. Introduction

- Their convenient disassembly/reassembly;
- Their light-weight structures;
- Their high payload-to-weight ratios.

## 2. Problem Statement and Proposed Solution

#### 2.1. Da Vinci Robotic Tools (dVRTs)

- Local part: located near the surgeon and controlled directly by the hands of the surgeon;
- Remote part: located near the patient and tele-operated by the local part.

#### 2.2. Problem Statement: Force Feedback in Robotic Surgery

#### 2.3. Proposed Solution: Cable-Driven Force–Torque Sensing Mechanism

- Hybrid System: the mechanism can be used as a sensor and/or actuator;
- Modelling: the model of the mechanism is constituted by an external mobile part connected with cables to an internal fixed structure;
- Application: micro- and macro- applications can be performed with the proposed mechanism;
- Under-actuation: the mechanism is composed of a cable-driven, under-actuated system.

#### 2.3.1. Hybrid System: Sensor and/or Actuator

#### 2.3.2. Modelling: An Internal Cable-Driven Fixed Structure

#### 2.3.3. Application: Micro/Macro Cable-Driven Applications

#### 2.3.4. Under-Actuation: Cable-Driven System

## 3. Analysis of the Mechanism’s Configurations

#### 3.1. Details of the Kinematic Model

#### 3.2. Configuration I

- External forces and torques in condition $j=1$: ${\mathbf{W}}_{P1}={[{F}_{PX},{F}_{PY},{F}_{PZ},{M}_{PX},{M}_{PY},{M}_{PZ}]}^{T}$;
- Gravity: ${\mathbf{W}}_{G}={[0,0,-m{g}_{G},0,0,0]}^{T}$;
- Tension of cables in condition $j=1$: ${\mathbf{T}}_{1}={[{T}_{11},{T}_{21},{T}_{31},{T}_{41},{T}_{51},{T}_{61},{T}_{71},{T}_{81}]}^{T}$;
- Offset in condition $j=0$: ${\mathbf{T}}_{0}={[{T}_{10},{T}_{20},{T}_{30},{T}_{40},{T}_{50},{T}_{60},{T}_{70},{T}_{80}]}^{T}$.

#### 3.3. Configuration II

## 4. Configurations I and II: Comparison and Simulation

- The two matrices ${\mathbf{S}}_{I0}$ and ${\mathbf{S}}_{II0}$ have the same rank, equal to 5;
- $\parallel {S}_{I0}{\parallel}_{2}<{\parallel {S}_{II0}\parallel}_{2}$.

## 5. Analytical Formulation of Configuration II

#### 5.1. Force Transformation Matrix

#### 5.2. Vector of the Cable’s Tension

## 6. Model Validation and Physical Implementation with Two Test Bench Prototypes

#### 6.1. Model Validation

#### 6.2. Planar Test Bench Prototype: TBI

#### 6.3. Spatial Test Bench Prototype: TBII

#### 6.4. Springs and Cables

#### 6.5. Implementation

- 1
- Attachment of the spring to the cable;
- 2
- Attachment of the spring to the fixed part;
- 3
- Centring of part A and calibration using screws in the fixed part;
- 4
- Measuring the spring’s length in the condition $j=0$;
- 5
- Applying weights to the point of attachment for the weights;
- 6
- Measuring the spring’s length in the condition $j=1$.

## 7. Comparison between Measured and Calculated Forces: Results and Discussion

#### 7.1. Results

#### 7.2. Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CDPR | cable-driven parallel robots |

dVRT | da Vinci Robotic Tool |

MIRS | Minimally Invasive Robotic Surgery |

dVRA | da Vinci Robotic Arm |

MIRT | Minimally Invasive Robotic Tool |

dVRK | da Vinci Robotic Kit |

E-E | End-Effector |

TBI | Test Bench I |

TBII | Test Bench II |

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**Figure 1.**Remote part of the da Vinci Robotic Kit (dVRK) constituted by da Vinci Robotic Tools (dVRTs) and da Vinci Robotic Arms (dVRAs). Courtesy of the Altair Robotic Lab, University of Verona.

**Figure 2.**General scheme of the working principle of the invention: trigonometric view (on the left) and section (on the right). Element A can be stationary or in motion with respect to the fixed element B. The C zone can be constituted by different types of elements or connections between A and B: membranes, cables, tendons, magnets, etc.

**Figure 3.**Two configurations used for the cable-driven mechanism: configuration I (on the left) and configuration II (on the right). Part A is the mobile part and part B is the fixed part.

**Figure 4.**Example of implemented configurations on the tube of the surgical instrument. Part A is the mobile part and part B is the fixed part.

**Figure 5.**Sketch to determine the kinematic model of the cable-driven mechanism. A is the local reference system and B is the absolute reference system.

**Figure 6.**Configurations I (on the left) and II (on the right): spatial and planar representations. ${\mathbf{W}}_{P1}={[{F}_{PX},{F}_{PY},{F}_{PZ},{M}_{PX},{M}_{PY},{M}_{PZ}]}^{T}$ are the external applied vectors of forces and torques; ${\mathbf{T}}_{0}={[{T}_{10},{T}_{20},{T}_{30},{T}_{40},{T}_{50},{T}_{60},{T}_{70},{T}_{80}]}^{T}$ are the vectors of the tension of the cables in condition $j=0$.

**Figure 7.**Comparison between the four h values (${h}_{I0}$, ${h}_{I1}$, ${h}_{II0}$, ${h}_{II1}$) of the two matrices ${\mathbf{S}}_{I0}$ and ${\mathbf{S}}_{II0}$.

**Figure 8.**Trigonometric view (left) and planar XZ representation (right) of the architecture of the mechanism in $j=0$ condition.

**Figure 9.**Sketch of the trigonometric view in translation and rotation of the local reference system (centred in ${O}_{A}$) with respect to the absolute reference system (centred in ${O}_{B}$). Displacement and rotation of ${O}_{A}-{X}_{A}{Y}_{A}{Z}_{A}$ are in the direction of the applied force F.

**Figure 10.**Sketch in the XY plane of the translation and rotation of the reference system centred in ${O}_{A}$ with respect to the reference system centred in ${O}_{B}$. Displacement and rotation of ${O}_{A}-{X}_{A}{Y}_{A}{Z}_{A}$ are in the direction of the applied force F.

**Figure 11.**Planar sketch of the behaviour of the sensitive mechanism in the two conditions ($j=0$ and $j=1$) (left) and example of the calculation of the cable’s tension in cables 1 and 2 (right). Each cable is considered inextensible, and the motion of the condition $j=1$ is possible thanks to the displacement of the springs: ${k}_{1}$, ${k}_{2}$, ${k}_{3}$, ${k}_{4}$.

**Figure 12.**Scheme of the planar test bench prototype (TBI) to study the behaviour of the sensitive mechanism.

**Figure 13.**Planar test bench prototype (TBI) to study the behaviour of the sensitive mechanism. A is the mobile part and B is the fixed part.

**Figure 14.**Spatial test bench prototype (TBII) to study the behaviour of the sensing mechanism: section with cables inside the test bench; realized prototype (with a 3D printer) of parts A (mobile part) and B (fixed part).

**Figure 15.**Experiments with the TBII for the behavioural analysis of the sensitive mechanism: external applied force $\mathbf{F}$ and eight cables’ tensions: ${\mathbf{T}}_{1jm}$, ${\mathbf{T}}_{2jm}$, ${\mathbf{T}}_{3jm}$, ${\mathbf{T}}_{4jm}$, ${\mathbf{T}}_{5jm}$, ${\mathbf{T}}_{6jm}$, ${\mathbf{T}}_{7jm}$, ${\mathbf{T}}_{8jm}$. A is the mobile part and B is the fixed part.

**Figure 16.**TBI: Comparison between measured and calculated external forces on the point P. The error is obtained by the absolute difference between measured and calculated values of the force.

**Figure 17.**TBII: Comparison between measured and calculated external forces on the point P using two different values of friction: $\mu =0.25$ and $\mu =0.3$ [34]. The error is obtained by the absolute difference between measured and calculated values of the force.

**Table 1.**Dimensions of the test bench prototypes. Measures (${p}_{1}$, ${p}_{2}$, ${p}_{3}$, ${p}_{4}$, ${p}_{6}$, ${p}_{10}$) are shown in Figure 8.

${\mathit{p}}_{1}$ (mm) | ${\mathit{p}}_{2}$ (mm) | ${\mathit{p}}_{3}$ (mm) | ${\mathit{p}}_{4}$ (mm) | ${\mathit{p}}_{6}$ (mm) | ${\mathit{p}}_{10}$ (mm) | $\mathit{\mu}$ | |
---|---|---|---|---|---|---|---|

TBI | 23 | 85 | 100 | 43 | 7 | 82 | 0 |

TBII | 16 | 8.75 | 13.5 | 11.8 | 12 | 2.5 | 0.3; 0.25 [34] |

**Table 2.**Dimensions of the type of spring used in the experiment: ${D}_{e}$ is the external diameter; d is the diameter of the wire; ${L}_{m}$ is the initial length of the spring; n is the number of wraps; k is the stiffness coefficient.

${\mathit{D}}_{\mathit{e}}$ (mm) | d (mm) | ${\mathit{L}}_{\mathit{m}}$ (mm) | $\mathit{n}$ | k (N/mm) | |
---|---|---|---|---|---|

Spring | 9.4 | 1 | 24.8 | 25 | 0.6786 |

1 (N) | 2 (N) | 3 (N) | 4 (N) | 5 (N) | 6 (N) | 7 (N) | |
---|---|---|---|---|---|---|---|

TBI | 6.46 | 7.48 | 8.49 | 9.85 | 10.94 | ||

TBII | 3.25 | 4.26 | 5.27 | 6.46 | 7.48 | 8.49 | 9.57 |

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**MDPI and ACS Style**

Muscolo, G.G.; Fiorini, P.
A New Cable-Driven Model for Under-Actuated Force–Torque Sensitive Mechanisms. *Machines* **2023**, *11*, 617.
https://doi.org/10.3390/machines11060617

**AMA Style**

Muscolo GG, Fiorini P.
A New Cable-Driven Model for Under-Actuated Force–Torque Sensitive Mechanisms. *Machines*. 2023; 11(6):617.
https://doi.org/10.3390/machines11060617

**Chicago/Turabian Style**

Muscolo, Giovanni Gerardo, and Paolo Fiorini.
2023. "A New Cable-Driven Model for Under-Actuated Force–Torque Sensitive Mechanisms" *Machines* 11, no. 6: 617.
https://doi.org/10.3390/machines11060617