# Identification of Inertial Parameters for Position and Force Control of Surgical Assistance Robots

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Kinematic Model of Surgical Assistance Robot

#### 2.1. Design Specifications

_{3}), a sensor has been arranged to know the forces performed at the extreme of the manipulator. A universal joint has been interposed between the sensor and the uterine manipulator, which initially increases the number of DOF to 5. However, 2 DOFs have been restricted by setting a support point at O

_{G}that limits the lineal movement along Y

_{0}and Z

_{0}, turning them into twists.

#### 2.2. Denavit–Hartenberg Parameters

_{1}, q

_{2}and q

_{3}), which represent the three linear movements along the cartesian axes, and two passive universal joints (q

_{4}and q

_{5}). The parameters q

_{1}, q

_{2}and q

_{3}are the active variables and represent the three linear movement of prismatic joints, while q

_{4}and q

_{5}are the passive variables and represent the generalized coordinates of the universal joint. The manipulator is considered like a straight stick—parameter L of the Table 1 indicates distance between of center O

_{4}and O

_{5}at X

_{5}axis, parameter d it the distance between center O

_{4}and O

_{5}Z

_{5}axis.

_{4}and q

_{5}) is solved by equating the unit director vector of the points O

_{4}and O

_{5}, with the unit director vector between points O

_{4}and O

_{G}. According to Equation (1), using this equation, Equations (2) and (3) are obtained with the value of the passive variables of the SAR.

_{5}:

## 3. Identification of Inertial Parameters

_{4}as A, O

_{G}as B, O

_{COM}as C and O

_{5}as D, as is shown in Figure 3.

## 4. Calculation of External Force at End of the Manipulator

_{B1}and a second force perpendicular to the vector $\overrightarrow{{r}_{AB}}$ and belonging to the anterior plane (Figure 4), that is called F

_{B2}.

## 5. Methodology: Implementation in the Actual Robot

#### 5.1. Surgery Assistance Robot

^{®}800 Mhz microprocessor, 256 Mb RAM and 256 Mb Flash. In addition, the embedded PC has Ethernet, EtherCAT, USB ports, digital and analog inputs and outputs.

#### 5.2. Calculation of Inertial Parameters

_{0}and Z

_{0}axes disappear. Measuring the value of the force on the Y

_{0}axis at the force sensor on the Equation (19), the Equation (25) is obtained.

#### 5.3. Calculation of External Forces

## 6. Results and Discussion

_{4}, in the direction given by X

_{5}. For the identification process, point O

_{4}has been arranged as close as possible to point O

_{G}, without any difference in length in Y

_{0}axis between both points, and a movement has been made in the negative direction X

_{5}, until reaching almost the manipulator limit, leaving a margin of 5 cm (Figure 6).

_{5}, and it has been checked if the method could solve the correctly applied force (Table 3), with an acceptable error (Figure 8).

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Prototype of the surgical assistance robot (SAR) showing the cartesian robot’s three active coordinates (where q

_{1}is the axis Z, q

_{2}is the axis Y and q

_{3}is the axis X), the force sensor, the support element that is restrict movement at the XY plane and a uterine manipulator.

j | α_{1j} | a_{1j} | d_{1j} | θ_{1j} |
---|---|---|---|---|

1 | 0 | q_{1} | 0 | −π/2 |

2 | −π/2 | q_{2} | 0 | −π/2 |

3 | 0 | q_{3} | 0 | 0 |

4 | q_{4} | 0 | 0 | π/2 |

5 | q_{5} | d | L | 0 |

Real | Calculated | Error | ||||
---|---|---|---|---|---|---|

Name | Mass (kg) | COM (m) | Mass (kg) | COM (m) | Mass (kg) | COM (m) |

Manipulator 1 | 1.611 | 0.115 | 1.538 | 0.1129 | −4.53% | −1.83% |

Manipulator 2 | 2.593 | 0.103 | 2.5602 | 0.1055 | −1.26% | 2.43% |

Manipulator 3 | 3.198 | 0.125 | 3.2208 | 0.1303 | 0.71% | 4.24% |

Manipulator 4 | 4.159 | 0.1456 | 4.206 | 0.1398 | 1.13% | −3.98% |

Mean | −0.99% | 0.21% | ||||

Standard deviation | 2.58% | 3.78% |

Mass Applicated (kg) | Mass Calculated (kg) | Error |
---|---|---|

0.51 | 0.5061 | −0.765% |

1.03 | 1.013 | −1.650% |

2.04 | 2.030 | 0.490% |

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**MDPI and ACS Style**

Zamora-Ortiz, P.; Carral-Alvaro, J.; Valera, Á.; Pulloquinga, J.L.; Escarabajal, R.J.; Mata, V.
Identification of Inertial Parameters for Position and Force Control of Surgical Assistance Robots. *Mathematics* **2021**, *9*, 773.
https://doi.org/10.3390/math9070773

**AMA Style**

Zamora-Ortiz P, Carral-Alvaro J, Valera Á, Pulloquinga JL, Escarabajal RJ, Mata V.
Identification of Inertial Parameters for Position and Force Control of Surgical Assistance Robots. *Mathematics*. 2021; 9(7):773.
https://doi.org/10.3390/math9070773

**Chicago/Turabian Style**

Zamora-Ortiz, Pau, Javier Carral-Alvaro, Ángel Valera, José L. Pulloquinga, Rafael J. Escarabajal, and Vicente Mata.
2021. "Identification of Inertial Parameters for Position and Force Control of Surgical Assistance Robots" *Mathematics* 9, no. 7: 773.
https://doi.org/10.3390/math9070773