# Potential Energy Distribution of Redundant Cable-Driven Robot Applied to Compliant Grippers: Method and Computational Analysis

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## Abstract

**:**

## 1. Introduction

#### 1.1. Contribution

#### 1.2. Mechanism Description

#### 1.3. Article Scheme

## 2. Methodology

#### 2.1. Potential Energy of Cables

_{i}. The energy stored in each spring is defined as

_{i}is the linear deformation of the i-cable. This linear deformation is proportional to the mechanical tension of the i-cable, as seen in the following equation:

#### 2.2. Energy Boundaries

#### 2.3. Selection of Single Cable Energy

**λ**in order to set the corresponding desired potential energy values in each cable. For each variable

**λ**whose value is established, the volume of the total potential energy feasible region, Ω, is reduced because, by setting the value of a nullspace variable, the dimension corresponding to that variable collapses into a point.

#### 2.4. Energy Analysis of the Reconfigurable End-Effector

_{1}, has to exert

_{A}have to be preserved. Those energy boundaries are defined as

## 3. Theoretical Results

#### 3.1. Energy Distribution of the Rigid End-Effector

^{3}N/m. The boundary values for the energy are obtained with Equations (17) and (18), and they are 1.25 and 320 J. The regions of feasible energy for each cable $\left({\mathsf{\Omega}}_{\mathrm{i}}\right)$ are obtained by using Equations (19) and (20). Those regions and their boundaries ${\mathsf{\Lambda}}_{\mathrm{i}}{}_{min}$ and ${\mathsf{\Lambda}}_{\mathrm{i}}{}_{max}$ are shown in Figure 6.

#### 3.2. Energy Distribution of the Reconfigurable End-Effector

## 4. Simulated Results

#### 4.1. Rigid Solid End-Effector

#### 4.2. Reconfigurable End-Effector

## 5. Discussion

## 6. Conclusions

## 7. Patents

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## Appendix B

**M**is

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**Figure 1.**Scheme of the proposed end-effector with a grasping tool based on a reconfigurable end-effector. One cable provides the energy for the tension distribution of the main body and for the displacement of the grasping tool. The cable is coiled around a winch that is able to deploy the desired amount and to impose the desired tension.

**Figure 2.**Scheme of the proposed methodology for obtaining the aperture and force exerted by the gripper of the reconfigurable end-effector by considering the boundaries imposed by the rigid end-effector model.

**Figure 3.**Weight force acting on the gripper. It is projected in parallel to the cable, imposing the tension ${\tau}_{JB},$ and perpendicular to the cable, acting on the main body of the end effector.

**Figure 4.**Cable-driven parallel robot scheme and the points for cable connections between the inertial frame and the end effector.

**Figure 5.**Surface corresponding to the potential energy of each cable as a function of the nullspace variables.

**Figure 6.**Red region $\left({\mathsf{\Omega}}_{\mathrm{i}}\right)$: feasible region in the nullspace that provides a cable tension inside the allowable value. Black line $\left({\mathsf{\Lambda}}_{\mathrm{i}}{}_{\mathrm{min}}\right)$: region where one or more cables achieve the minimum allowable tension value. Red line $\left({\mathsf{\Lambda}}_{\mathrm{i}}{}_{\mathrm{max}}\right)$: region where one or more cables achieve the maximum allowable tension value.

**Figure 7.**Surface section that represents the potential energy of all cables. The red polygon defines the region of the nullspace that provides feasible tension values in all cables.

**Figure 8.**Red region $\left(\mathsf{\Omega}\right)$: region of the nullspace that provides a feasible tension in all cables. Black region $\left({\mathsf{\Omega}}^{*}\right)$: region of the nullspace that provides feasible efforts in all the rigid links of a parallel robot with the same configuration of the cable robot.

**Figure 9.**The blue 3D curve represents the potential energy stored in all cables that can be achieved by maintaining the condition of setting a deformation in the cable 8 of 0.2 m. The projection of this curve over the horizontal plane provides feasible nullspace variables. Coordinates are [${\lambda}_{\mathbf{1}}$, ${\lambda}_{\mathbf{2}}$, U].

**Figure 10.**The black 3D curve represents the potential energy stored in cable 1 that can be achieved by maintaining the condition of setting a deformation in cable 8 of 0.2 m. The projection of this curve over the horizontal plane provides feasible nullspace variables. Coordinates are [${\lambda}_{\mathbf{1}}$, ${\lambda}_{\mathbf{2}}$, U

_{1}].

**Figure 11.**Surface section that represents the potential energy of all the cables when the two grippers are attached in cables 1 and 8. The green polygon defines the region of the nullspace that provides feasible tension values in all cables. It is different form the red polygon from Figure 7 because in this case the weight of the two grippers is being considered.

**Figure 12.**The blue 3D curve represents the potential energy stored in all cables that can be achieved by maintaining the condition of setting a deformation in cable 8 of 0.447 m and two grippers hanging from cables 1 and 8.

**Figure 13.**The black 3D curve represents the potential energy stored in cable 1

_{A}by maintaining the condition of setting an elongation in cable 8 of 0.2 m and two grippers hanging from cables 1 and 8.

**Figure 14.**Feasible region of the energy imposed by actuator 1 for setting the position of the gripper attached to its compliant actuator. The range of feasible energy imposed by the actuator is related to the tension range available in the springs. Four different spring stiffness were analyzed for cable section ${1}_{A}$. Boundaries were applied for the case of ${K}_{A}=1000\text{}\mathrm{N}/\mathrm{m}$.

**Figure 15.**Feasible region of the energy imposed by actuator 1 for setting the position of the gripper attached to its compliant actuator without closing the gripper (black line) and with the gripper closed around a rigid solid (blue line). Four different springs were analyzed for cable section ${1}_{\mathrm{B}}$. The contact was produced at 30 J, or ${\mathrm{L}}_{21}=0.16\text{}\mathrm{m}$ (green line), and consequently, the gripper was determined to be closed. Boundaries were applied for the case of ${\mathrm{K}}_{\mathrm{B}}=1000\text{}\mathrm{N}/\mathrm{m}$.

**Figure 16.**Feasible region of the energy imposed by actuator 1 for exerting force on the gripper attached to its compliant actuator. Four different springs were analyzed for cable section ${1}_{B}$. The boundaries were applied for the case of ${K}_{B}=1000\text{}\mathrm{N}/\mathrm{m}$, where the range of feasible force is ${F}_{1BA}\in \left(0,\text{}72.8\right)\text{}\mathrm{N}$.

**Figure 17.**Dynamic multibody model tested in MSC ADAMS. Table 4 defines the robot dimensions and configuration. Cables are actuated by pulling from their ends situated in the basement. Those cables have to pass around two pulleys, one situated in the pillar base and the other in the top of the pillar.

**Figure 18.**Dynamic model of one of the grippers. By pulling “Cable j”, the spring ${K}_{A}$ elongates and the gripper begins to close, moving towards the end-effector. This gripper moves along a linear guide, and the secondary cable moves inside the end-effector. The pulley is designed to align this cable in a vertical way.

**Figure 20.**Elongation of the spring situated between cable 8 and the end-effector. The value oscillates between 0.451 and 0.448 m after the stabilization of the measure.

**Figure 21.**Forces acting on the end-effector. (

**a**) Force in the x-axis from 215 to 199 N. (

**b**) Force in the y-axis from 5.75 to −12 N. (

**c**) Force in the Z axis from −22.8 to −13.3 N.

**Figure 22.**Comparison between the theoretical and simulated gripper aperture (measuring the value of ${L}_{21}$) and the energy of actuator 1. At 30 J or 0.16 m, the gripper closes, increasing the error between the theoretical and simulated results.

**Figure 23.**Comparison between the theoretical and simulated actuator displacement compared with the actuator energy when the gripper is already closed.

**Figure 24.**Comparison between the theoretical and simulated displacement of the cable in the actuator when the gripper is open or closed by considering the two different expressions.

Inertial Frame | End-Effector | Cable Name |
---|---|---|

P1 | U4, D2 | Cable1, Cable2 |

P2 | U1, D3 | Cable3, Cable4 |

P3 | U2, D4 | Cable5, Cable6 |

P4 | U3, D1 | Cable7, Cable8 |

Nullspace Position | Tension Distribution (N) |
---|---|

(−120.1, 84.39) | [191.6, 83.1, 50.0, 199.7, 531.4, 422.3, 280.3, 447.2] |

(−12.39, 101.5) | [137.5, 72.9, 97.4, 196.2, 484.0, 434.23, 345.23, 447.2] |

(161.6, 129.1) | [50.0, 56.5, 174.0, 190.5, 407.5, 453.5, 450.0, 447.2] |

Nullspace Position | Total Potential Energy (J) | Cable 1 Potential Energy (J) | End-Effector Wrench (N|Nm) |
---|---|---|---|

(−120.1, 84.39) | 412.67 | 18.36 | [200, 0, −1470|−500, 0, 0] |

(−12.39, 101.5) | 407.1 | 9.45 | [200, 0, −1470|−500, 0, 0] |

(161.6, 129.1) | 423.23 | 1.25 | [200, 0, −1470|−500, 0, 0] |

Parameter | Value |
---|---|

Frame dimensions (X, Y, Z) [m] | [10, 5, 6] |

End-effector mass [kg] | 150 |

Gripper mass [kg] | 10.2 |

End-effector center position (X, Y, Z, R, P, Y) [m, rad] | [7, 4, 1.5, 0, 0, $\frac{\pi}{6}$] |

External wrench (F_{X}, F_{Y}, F_{Z}, M_{X}, M_{Y}, M_{Z}) [N, Nm] | [200, 0, −1470, −500, 0, 0] |

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**MDPI and ACS Style**

Rodriguez-Barroso, A.; Saltaren, R.; A. Portilla, G.; S. Cely, J.; Yakrangi, O.
Potential Energy Distribution of Redundant Cable-Driven Robot Applied to Compliant Grippers: Method and Computational Analysis. *Sensors* **2019**, *19*, 3403.
https://doi.org/10.3390/s19153403

**AMA Style**

Rodriguez-Barroso A, Saltaren R, A. Portilla G, S. Cely J, Yakrangi O.
Potential Energy Distribution of Redundant Cable-Driven Robot Applied to Compliant Grippers: Method and Computational Analysis. *Sensors*. 2019; 19(15):3403.
https://doi.org/10.3390/s19153403

**Chicago/Turabian Style**

Rodriguez-Barroso, Alejandro, Roque Saltaren, Gerardo A. Portilla, Juan S. Cely, and Oz Yakrangi.
2019. "Potential Energy Distribution of Redundant Cable-Driven Robot Applied to Compliant Grippers: Method and Computational Analysis" *Sensors* 19, no. 15: 3403.
https://doi.org/10.3390/s19153403