# On the Use of a Genetic Algorithm for Determining Ho–Cook Coefficients in Continuous Path Planning of Industrial Robotic Manipulators

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Ho–Cook Path Planning

- k—the trajectory point;
- m—the total number of trajectory points, and
- B—coefficients of the interpolation polynomials.

#### 2.2. Genetic Algorithm

- Shape of the potential solutions;
- The way in which the crossover and mutation will be applied;
- The probabilities with which the evolutionary operations will occur;
- The fitness function that will evaluate the solutions;
- The number of iterations (generations) of the algorithm;
- The number of candidate solutions in the algorithm;
- The manner of the solution selection for the operations.

#### 2.2.1. Solution Construction

#### 2.2.2. Application of Evolutionary Computing Operations

#### 2.2.3. The Fitness Function

#### 2.2.4. Candidate Solution Selection

#### 2.3. Interpolation

## 3. Results and Discussion

#### 3.1. Determining the Optimal Boundary Condition

#### 3.2. Determining the GA Parameters

- Population size of 100 executed for 100 generations;
- Population size of 1000 executed for 50 generations;
- Population size of 10,000 executed for 20 generations.

#### Execution Time

#### 3.3. Result Illustration

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Simplified kinematic schematic of the ABB IRB 120 robot, with the associated D-H orthonormal coordinate systems.

**Figure 2.**An illustration of the recombination methodology used, where (

**A**,

**B**) represent two candidate solutions selected from the existing population and (

**C**) presents the resulting candidate solution after the crossover between (

**A**) and (

**B**) is performed.

**Figure 3.**An illustration of the mutation methodology used, where (

**A**) is the randomly selected candidate solution from the population and (

**B**) is the randomly modified solution.

**Figure 4.**The illustration of the fitness function, which is the sum of the distances between the first and the last elements of the segments, as indicated with red arrows.

**Figure 5.**An illustration of the roulette wheel selection process. In step (1) the fitness is calculated according to Equation (20) (lower is better). Then, in (2), the probability is calculated as the percentage of the total population fitness (in the illustration, the total sum of individual fitness is 3.0). Finally, due to the minimization problem being observed, the probability vector is inverted in (3).

**Figure 6.**The behavior of the algorithm without setting the boundary condition. The best-achieved solution is visualized on the (

**left**), and the fitness change through generations is on the (

**right**).

**Figure 7.**The behavior of the algorithm for the boundary condition set to 0.2. The best-achieved solution is visualized on the (

**left**), and the fitness change through generations is on the (

**right**).

**Figure 8.**The behavior of the algorithm for the boundary condition set to 0.1. The best-achieved solution is visualized on the (

**left**), and the fitness change through generations is on the (

**right**).

**Figure 9.**The behavior of the algorithm for the boundary condition set to 0.05. The best-achieved solution is visualized on the (

**left**), and the fitness change through generations is on the (

**right**).

**Figure 10.**The behavior of the algorithm for the boundary condition set to 0.005. The best-achieved solution is visualized on the (

**left**), and the fitness change through generations is on the (

**right**).

**Figure 11.**The achieved results for the GA with 100 candidate solutions trained for 100 generations. The best-achieved solution is visualized on the (

**left**), and the fitness change through generations is on the (

**right**).

**Figure 12.**The achieved results for the GA with 1000 candidate solutions trained for 50 generations. The best-achieved solution is visualized on the (

**left**), and the fitness change through generations is on the (

**right**).

**Figure 13.**The achieved results for the GA with 10,000 candidate solutions trained for 20 generations. The best-achieved solution is visualized on the (

**left**), and the fitness change through generations is on the (

**right**).

**Figure 15.**An illustration of the generated path. (

**a**) The initial step of the simulated trajectory. (

**b**) The sixth step of the simulated trajectory. (

**c**) The twelfth step of the simulated trajectory. (

**d**) The eighteenth step of the simulated trajectory. (

**e**) The twenty-fourth step of the simulated trajectory.

$\mathbf{\Theta}$ [rad] | d [mm] | a [mm] | $\mathit{\alpha}$ [rad] |
---|---|---|---|

${\mathrm{\Theta}}_{1}={q}_{1}$ | ${d}_{1}=290$ | ${a}_{1}=0$ | ${\alpha}_{1}=-\pi /2$ |

${\mathrm{\Theta}}_{2}={q}_{2}$ | ${d}_{2}=0$ | ${a}_{2}=270$ | ${\alpha}_{2}=0$ |

${\mathrm{\Theta}}_{3}={q}_{3}$ | ${d}_{3}=0$ | ${a}_{3}=70$ | ${\alpha}_{3}=-\pi /2$ |

${\mathrm{\Theta}}_{4}={q}_{4}$ | ${d}_{4}=302$ | ${a}_{4}=0$ | ${\alpha}_{4}=\pi /2$ |

${\mathrm{\Theta}}_{5}={q}_{5}$ | ${d}_{5}=0$ | ${a}_{5}=0$ | ${\alpha}_{5}=-\pi /2$ |

${\mathrm{\Theta}}_{6}={q}_{6}$ | ${d}_{6}=72$ | ${a}_{6}=0$ | ${\alpha}_{6}=0$ |

**Table 2.**The execution times of various configurations, averaged over ten runs, with an average time per generation and the total execution time given.

Population | Generations | Total Time [s] | Average Time per Generation [s] |
---|---|---|---|

100 | 100 | 23.4 | 0.234 |

1000 | 50 | 119.5 | 2.39 |

10,000 | 20 | 403.4 | 20.17 |

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**MDPI and ACS Style**

Grenko, T.; Baressi Šegota, S.; Anđelić, N.; Lorencin, I.; Štifanić, D.; Štifanić, J.; Glučina, M.; Franović, B.; Car, Z.
On the Use of a Genetic Algorithm for Determining Ho–Cook Coefficients in Continuous Path Planning of Industrial Robotic Manipulators. *Machines* **2023**, *11*, 167.
https://doi.org/10.3390/machines11020167

**AMA Style**

Grenko T, Baressi Šegota S, Anđelić N, Lorencin I, Štifanić D, Štifanić J, Glučina M, Franović B, Car Z.
On the Use of a Genetic Algorithm for Determining Ho–Cook Coefficients in Continuous Path Planning of Industrial Robotic Manipulators. *Machines*. 2023; 11(2):167.
https://doi.org/10.3390/machines11020167

**Chicago/Turabian Style**

Grenko, Teodor, Sandi Baressi Šegota, Nikola Anđelić, Ivan Lorencin, Daniel Štifanić, Jelena Štifanić, Matko Glučina, Borna Franović, and Zlatan Car.
2023. "On the Use of a Genetic Algorithm for Determining Ho–Cook Coefficients in Continuous Path Planning of Industrial Robotic Manipulators" *Machines* 11, no. 2: 167.
https://doi.org/10.3390/machines11020167