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Kinematic Optimization for the Design of a Collaborative Robot End-Effector for Tele-Echography^{ †}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Ultrasonography

#### 2.2. Tele-USG Systems

## 3. End-Effector Design Optimization

#### 3.1. Requirements

#### Model of the Patient and Target of the ECG

#### 3.2. UR5 and Panda Kinematics

#### 3.3. End-Effector Kinematics

#### 3.4. Optimization Variables

#### 3.5. Objective Function

#### 3.6. Constraints

## 4. Implementation

Algorithm 1: Optimization algorithm |

Data: sample points ${P}_{i}$, weights ${\lambda}_{i}$ |

Result: optimal |

create the manipulator MN; |

set ${W}_{u}$, ${W}_{t}$, $\mathcal{M}=0$; |

compute ${P}_{i}$, ${\lambda}_{i}$; |

compute inverse kinematics starting points ${\mathbf{q}}_{0}$; |

set bounds in MultiStart according to Equations (13)–(15); |

## 5. Results

#### Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**The mathematical model of the patient. The heart is represented as a red asterisk, whereas target points ${E}_{i}$ are plotted in orange and the sample points ${P}_{i}$ in green.

**Figure 3.**(

**a**) The Universal Robot UR5 robot and the adopted kinematic model without and with additional joint. (

**b**) The Panda robot from Franka Emika along with the first and the last frames, ${\Sigma}_{0}$ and ${\Sigma}_{7}$, respectively.

**Figure 5.**The target metric $\tilde{\mu}$ calculated at the target points for the three arms. The color represents $\mu $ as indicated in the color bars besides the plots. The color scale is different for the three arms. Circles represent points ${E}_{i}$, whereas the star represents the heart.

**Table 1.**Denavit–Hartenberg parameters for the definition of the kinematics of the two robots. The seventh frame of the UR5 with additional joint is drawn as separate from the sixth for clarity of representation, even though their origins are superimposed. * UR5 without additional joint; ** UR5 with an additional joint; ${d}_{6}$ is a design parameter which is part of the optimization.

UR5 | ||||
---|---|---|---|---|

link | ${\mathit{a}}_{\mathit{i}}$ | ${\mathit{\alpha}}_{\mathit{i}}$ | ${\mathit{d}}_{\mathit{i}}$ | ${\mathit{\theta}}_{\mathit{i}}$ |

1 | 0 | $\pi /2$ | 0.0895 | ${\theta}_{1}$ |

2 | −0.4250 | 0 | 0 | ${\theta}_{2}$ |

3 | −0.3922 | 0 | 0 | ${\theta}_{3}$ |

4 | 0 | $\pi /2$ | 0.1091 | ${\theta}_{4}$ |

5 | 0 | $-\pi /2$ | 0.0946 | ${\theta}_{5}$ |

6 * | 0 | 0 | 0.0823 | ${\theta}_{6}$ |

6 ** | 0 | $-\pi /2$ | ${d}_{6}$ | ${\theta}_{6}$ |

7 | 0 | 0 | 0 | ${\theta}_{7}$ |

Panda | ||||

link | ${\mathit{a}}_{\mathit{i}}$ | ${\mathit{\alpha}}_{\mathit{i}}$ | ${\mathit{d}}_{\mathit{i}}$ | ${\mathit{\theta}}_{\mathit{i}}$ |

1 | 0 | 0 | 0.333 | ${\theta}_{1}$ |

2 | 0 | $-\pi /2$ | 0 | ${\theta}_{2}$ |

3 | 0 | $\pi /2$ | 0.316 | ${\theta}_{3}$ |

4 | 0.0825 | $\pi /2$ | 0 | ${\theta}_{4}$ |

5 | −0.0825 | $-\pi /2$ | 0.384 | ${\theta}_{5}$ |

6 | 0 | $\pi /2$ | 0 | ${\theta}_{6}$ |

7 | 0.088 | $\pi /2$ | 0.107 | ${\theta}_{7}$ |

**Table 2.**Optimization results for the three manipulators. UR5+1 stands for Ur5 with additional joint.

Robot | ${\mathit{x}}_{\mathit{S}}$ [m] | ${\mathit{y}}_{\mathit{S}}$ [m] | ${\mathit{z}}_{\mathit{S}}$ [m] | ${\mathit{x}}_{\mathit{G}}$ [m] | ${\mathit{y}}_{\mathit{G}}$ [m] | ${\mathit{z}}_{\mathit{G}}$ [m] | $\mathit{\beta}$ [deg] | $\mathcal{M}$ [${10}^{-4}$] | $\mathit{\u03f5}$ |
---|---|---|---|---|---|---|---|---|---|

UR5 | 0.55 | −0.23 | 0.11 | 0.017 | −0.019 | 0.161 | −65.9 | 5.56 | 8.54 |

UR5 + 1 | 0.73 | −0.33 | 0.20 | 0.037 | 0.005 | 0.123 | - | 1.97 | 3.23 |

Panda | 0.39 | −0.17 | 0.31 | 0.023 | 0.071 | 0.146 | - | 5.28 | 14.1 |

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**MDPI and ACS Style**

Filippeschi, A.; Griffa, P.; Avizzano, C.A.
Kinematic Optimization for the Design of a Collaborative Robot End-Effector for Tele-Echography. *Robotics* **2021**, *10*, 8.
https://doi.org/10.3390/robotics10010008

**AMA Style**

Filippeschi A, Griffa P, Avizzano CA.
Kinematic Optimization for the Design of a Collaborative Robot End-Effector for Tele-Echography. *Robotics*. 2021; 10(1):8.
https://doi.org/10.3390/robotics10010008

**Chicago/Turabian Style**

Filippeschi, Alessandro, Pietro Griffa, and Carlo Alberto Avizzano.
2021. "Kinematic Optimization for the Design of a Collaborative Robot End-Effector for Tele-Echography" *Robotics* 10, no. 1: 8.
https://doi.org/10.3390/robotics10010008