# A Robot Arm Design Optimization Method by Using a Kinematic Redundancy Resolution Technique

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## Abstract

**:**

## 1. Introduction

## 2. A Brief Review of Redundancy Resolution Techniques

## 3. Description of the New Mechanical Design Optimization Method for Robot Manipulators

## 4. Optimization Methodology Described for the Passive Arm of the NeuRoboScope System

#### 4.1. Mechanism of the Robotic Manipulator and the Description of the Case Scenario

#### 4.2. Design Optimization Constraints

- The surgeon can insert the endoscope from either nostril.
- The endoscope and the active robot arm should not interfere with the surgeon’s hands, and they should not block the surgeon’s view of the monitor, see Figure 4.
- The passive arm should locate the active arm inside the surgery workspace by approaching from behind the patient’s head.
- The passive arm should be fixed to the surgery table.
- Physical dimensions of the links should not be large, and they should not be heavy, but they should be rigid enough to compose an inertial frame for the active arm when their brakes at the joints of the passive arm are activated.
- There should be no actuators on the joints of the passive robot arm.
- When the passive arm’s brakes are released, the surgeon should be able to move the endoscope freely while the endoscope is still attached to the active robot arm.

#### 4.3. Optimization through Mechanical Redundancy

- The surgeon should have minimal effort when he/she intends to push the active robot arm in or away from the surgery zone.
- The parallelogram loop in the passive robot arm is utilized with no modifications since it is designed with counter-spring for gravity compensation. This linkage is responsible for providing vertical motion of the base of the active arm.
- The optimization is related only to the ease of manipulation only on the horizontal plane.
- The fixing point position on the y-axis (with respect to reference-frame in Figure 5) is selected as a possible design parameter, which is related to another design parameter that is the first link’s length.
- MP position is fixed at the coordinate (−20, −30) cm (which can be considered as an average position of workspace required by the surgeon) relative to the reference frame in Figure 5.
- The effective link length of the first link should be limited depending on its manufacturability, final weight and allowed compliant displacements due to loads.
- The linear density of the first link is taken as follows: mass/length = 1 kg/m.
- The third joint variable ${\theta}_{3}$ in Figure 3 is fixed at −30° which is the condition when the endoscope is located just above the patient’s nostrils.

## 5. Implementation of the New Optimization Strategy

#### 5.1. Manipulability Ellipsoid and Singular Value Decomposition (SVD)

#### 5.2. The Modified Condition Number

#### 5.3. Generalized Inertia Matrix

## 6. Simulation Tests and Results

#### 6.1. Simulation Test with the Modified Condition Number Performance Index

#### 6.2. Simulation Test with the Modified Condition Number Performance Index and Generalized Inertia Matrix

## 7. Validation of the Proposed Optimum Design Approach

**Find design variables**: ${Y}_{0}$, ${\theta}_{1}$, ${\theta}_{2}$, ${a}_{1}$

**Subjected to**: $-1\le {Y}_{0}\le 0,\text{}0{a}_{1}0.5$

## 8. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Nelder–Mead algorithm:**

**Simulated Annealing algorithm:**

- (1)
- Introduce an initial guess ${z}_{i}$.
- (2)
- Generate next point, ${z}_{i+1}$, in the neighboring point of ${z}_{i}$.
- (3)
- The main goal of step 2 is to obtain smaller radius of the neighborhood for each iteration.
- (4)
- If $g\left({z}_{best}\right)=g\left({z}_{i+n}\right)$, ${z}_{i+n}$ replaces ${z}_{best}$ and $z$.
- (5)
- Boltzmann’s probability distribution function is used to measure the distance between these two points

**Differential Evolution algorithm:**

- (1)
- Introduce a population of $h$ points.
- (2)
- Produce randomly generated population points.
- (3)
- Use the real scaling factor $rsF$ and select Cross-Probability value in the interval [0, 1].
- (4)
- Compare the difference between the two most recently generated points.

**Random Search algorithm:**

- (1)
- Enter the start parameter.
- (2)
- Create working group point ${V}_{k+1}$.
- (3)
- Update ${X}_{k+1}$, ${Q}_{k+1}$ for ${V}_{k+1}$
- (4)
- Compare the difference between the two most recently generated points.

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**Figure 5.**Coordinate system fixed on the surgery table (where the unit vector along the x-axis is ${\overrightarrow{u}}_{1}^{\left(0\right)}$ and the unit vector along the y-axis is ${\overrightarrow{u}}_{2}^{\left(0\right)}$ ).

**Figure 6.**Change of the robot arm structure and the manipulability ellipse (printed in blue color) during the optimization routine by using the modified condition number.

**Figure 7.**Variation of the manipulability index and singular values during the optimization routine by using the modified condition number.

**Figure 8.**The optimization procedure with the modified condition number: (

**a**) Variation of the generalized inertia matrix, (

**b**) variation of components of the objective function, (

**c**) variation in the values of design parameters.

**Figure 9.**Variation of the robot arm structure and the manipulability ellipse (printed in blue color) during the optimization routine by using the modified condition number and the generalized inertia matrix.

**Figure 10.**Variation of the manipulability index and singular values during the optimization process by using the modified condition number and the inertia matrix.

**Figure 11.**The optimization procedure with modified condition number and inertia matrix: (

**a**) Variation of the generalized inertia matrix, (

**b**) variation of components of the objective function, (

**c**) variation in the values of design parameters.

**Figure 12.**Pareto set and the initiation-termination points of the optimization procedure for test B for minimizing the modified condition number, ${C}_{n}$, and the determinant of the generalized inertia matrix, $ImN$.

**Figure 13.**Flowchart for the implementation of the new design optimization designated for robot manipulators.

$\mathit{k}$ | ${\mathit{d}}_{\mathit{k}}$ | ${\mathit{\theta}}_{\mathit{k}}$ | ${\mathit{a}}_{\mathit{k}}$ | ${\mathit{\alpha}}_{\mathit{k}}$ |
---|---|---|---|---|

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

2 | ${d}_{2}$ | ${\theta}_{2}$ | ${a}_{2}$ | $-\pi /2$ |

3 | 0 | ${\theta}_{3}$ | ${a}_{3}$ | 0 |

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

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

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

7 | ${d}_{7}$ | ${\theta}_{7}$ | 0 | $\pi /2$ |

8 | 0 | ${\delta}_{8}$ | 0 | $-\pi /2$ |

Options | DE | NM | RS | SA |
---|---|---|---|---|

Crossover fractions | 0.5 | - | - | - |

Random Seed | 1 | 5/10 | 0 | 2 |

Scaling factor | 0.6 | - | - | - |

Tolerance | 0.001 | 0.001 | 0.001 | 0.001 |

Contact ratio | - | 0.5 | - | - |

Expand ratio | - | 2.0 | - | - |

Reflect ratio | - | 1.0 | - | - |

Shrink ratio | - | 0.5 | - | - |

Level iterations | - | - | - | 50 |

Perturbation scale | - | - | - | 1.0 |

Penalty Function | - | - | Automatic | - |

Search Points | - | - | 2 | - |

Method | - | - | Interior Point | - |

Optimization Algorithm | Objective Function | ${\mathit{Y}}_{0}\text{}\left(\mathbf{cm}\right)$ | ${\mathit{\theta}}_{1}\text{}\left(\mathbf{rad}\right)$ | ${\mathit{\theta}}_{2}\text{}\left(\mathbf{rad}\right)$ | ${\mathit{a}}_{1}\text{}\left(\mathbf{cm}\right)$ |
---|---|---|---|---|---|

SA | 2.21008 × 10^{−14} | −11.56 | −1.611 | −2.356 | 38.48 |

DE | 1.21496 × 10^{−13} | −11.56 | 3.079 | 2.356 | 38.48 |

NM1 | 8.39033 × 10^{−15} | −48.44 | 1.611 | 2.356 | 38.48 |

NM2 | 2.21008 × 10^{−14} | −11.56 | −1.611 | −2.356 | 38.48 |

RS | 2.21008 × 10^{−14} | −11.56 | −1.611 | −2.356 | 38.48 |

ORR | 5.03038 × 10^{−9} | −11.57 | −1.611 | −2.356 | 38.47 |

Optimization Algorithm | Objective Function | ${\mathit{Y}}_{0}\text{}\left(\mathbf{cm}\right)$ | ${\mathit{\theta}}_{1}\text{}\left(\mathbf{rad}\right)$ | ${\mathit{\theta}}_{2}\text{}\left(\mathbf{rad}\right)$ | ${\mathit{a}}_{1}\text{}\left(\mathbf{cm}\right)$ |
---|---|---|---|---|---|

SA | 0.0363314 | −30.00 | −1.998 | −2.408 | 28.50 |

DE | 0.0363314 | −30.00 | 1.998 | 2.408 | 28.50 |

NM1 | 0.0363314 | −29.997 | −1.998 | −2.408 | 28.49 |

NM2 | 0.0363314 | −29.994 | 1.998 | 2.408 | 28.50 |

RS | 0.0363314 | −30.00 | 1.998 | 2.408 | 28.50 |

ORR | 0.036349 | −26.57 | −1.835 | −2.398 | 28.55 |

**Table 5.**The optimization results for Scenario 2 obtained via ORR algorithm for different simulation durations.

Method—Duration | Objective Function | ${\mathit{Y}}_{0}\text{}\left(\mathbf{cm}\right)$ | ${\mathit{\theta}}_{1}\text{}\left(\mathbf{rad}\right)$ | ${\mathit{\theta}}_{2}\text{}\left(\mathbf{rad}\right)$ | ${\mathit{a}}_{1}\text{}\left(\mathbf{cm}\right)$ |
---|---|---|---|---|---|

ORR—200 s | 0.0363496 | −26.57 | −1.835 | −2.398 | 28.55 |

ORR—300 s | 0.0363404 | −27.22 | −1.864 | −2.402 | 28.54 |

ORR—400 s | 0.0363369 | −27.63 | −1.883 | −2.404 | 28.53 |

ORR—500 s | 0.0363351 | −27.92 | −1.897 | −2.405 | 28.52 |

ORR—800 s | 0.0363329 | −28.48 | −1.923 | −2.406 | 28.51 |

ORR—1600 s | 0.0363317 | −29.22 | −1.959 | −2.408 | 28.50 |

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**MDPI and ACS Style**

Maaroof, O.W.; Dede, M.İ.C.; Aydin, L.
A Robot Arm Design Optimization Method by Using a Kinematic Redundancy Resolution Technique. *Robotics* **2022**, *11*, 1.
https://doi.org/10.3390/robotics11010001

**AMA Style**

Maaroof OW, Dede MİC, Aydin L.
A Robot Arm Design Optimization Method by Using a Kinematic Redundancy Resolution Technique. *Robotics*. 2022; 11(1):1.
https://doi.org/10.3390/robotics11010001

**Chicago/Turabian Style**

Maaroof, Omar W., Mehmet İsmet Can Dede, and Levent Aydin.
2022. "A Robot Arm Design Optimization Method by Using a Kinematic Redundancy Resolution Technique" *Robotics* 11, no. 1: 1.
https://doi.org/10.3390/robotics11010001