# Transferability of Multi-Objective Neuro-Fuzzy Motion Controllers: Towards Cautious and Courageous Motion Behaviors in Rugged Terrains

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## Abstract

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## 1. Introduction

- (a)
- It is the first study that defines target meta-problems that aim to find non-specialized Neuro-Fuzzy Controllers (NFCs) that produce various motion behaviors.
- (b)
- It is the first study that applies knowledge extraction from solving multi-objective source problems to support finding non-specialized NFCs that produce various motion behaviors for environments that are different due to their motion difficulties.
- (c)
- It is the first study that raises the hypothesis that the genetic transfer of edge solutions of multi-objective source problems is preferred over that of the center solutions. Furthermore, it is the first study to substantiate the hypothesis with respect to the considered type of evolutionary neuro-fuzzy control problem.

## 2. Background

#### 2.1. Multi-Objective Optimization

**,**respectively. Then, $\mathit{a}$ is said to dominate $\mathit{b}$, denoted by $\mathit{a}\preccurlyeq \mathit{b}$, if the following condition is satisfied:

#### 2.2. Neuro-Fuzzy Systems and Control

#### 2.3. Positioning of This Research

## 3. Problem Description and Solution Approach

#### 3.1. Target Meta-Problem

#### 3.2. Source Problems and Their Solutions

#### 3.3. Knowledge Extraction and Research Hypothesis

- Define the number of transferred edge-controllers ($2{n}_{ec}$), such that ${2n}_{ec}\ll \left|NDS\right|$
- For each edge:
- Find the edge PV and store the associated solutions in a set of extracted edge controllers (EEC).
- Select the ${n}_{ec}-1$ controllers that their PVs are the closest to the edge PV and add them to EEC.

- Define the number of transferred center controllers (${n}_{cc}$), such that ${n}_{cc}\ll \left|NDS\right|$.
- Find the line connecting the ideal and the nadir points.
- Select the ${n}_{cc}$ controllers for which their PVs are the closest to the line found in 4 and store them in a set of extracted center controllers (ECC).

## 4. Numerical Study

#### 4.1. Source and Target Problems

#### 4.2. Fuzzy Controllers

#### 4.3. Comparison Methods

#### 4.4. Experimental Setup

#### 4.5. Experimental Results and Analyses

#### 4.5.1. Demonstration of Knowledge Extraction

#### 4.5.2. Results and Analysis of the Target Meta-Problems

#### 4.5.3. Detailed Demonstration of the Transferability Results

#### 4.5.4. Demonstration of the Obtained Behaviors

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 10.**Statistical results: HV versus generations. Initialization by the edge-, center- and random-controllers, in the (

**left**), (

**center**), and (

**right**) panels, respectively.

Related Study | Evolutionary Driven | Pareto-Based Source | Application |
---|---|---|---|

Aouf et al., 2018 [33] | X | X | Indoor robotics |

Juang & Bui 2019 [34] | X | X | Indoor robotics |

Sell & Coupland 2015 [35] | X | X | Classification |

Fouladvand et al., 2015 [36] | X | X | Indoor robotics |

Chou & Juang 2018 [37] | V | X | Indoor robotics |

Ferdaus et al., 2019a [38] | V | X | Aerial vehicles |

Ferdaus et al., 2019b [39] | X | X | Aerial vehicles |

Ferdaus et al., 2018 [40] | V | X | Aerial vehicles |

Current study | V | V | Offroad robotics |

Parameter | Signal Source | Target |
---|---|---|

Number of generations | $300$ | $200$ |

Population size | $100$ | $60$ |

Penalty value | $100$ | |

Weight Mutation (WM) mechanism | $\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{n}\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{a}\mathrm{l}$ | |

WM parameters (probability, distribution index) | $\left(\mathrm{0.2,15}\right)$ | |

Weight Crossover (WC) mechanism | $\mathrm{S}\mathrm{B}\mathrm{X}$ | |

WC parameters (probability, distribution index) | $\left(\mathrm{0.8,20}\right)$ | |

Simulation time limit ${T}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ | $20\text{}\left[\mathrm{s}\right]$ | |

Cart mass $m$ | $1\text{}\left[\mathrm{k}\mathrm{g}\right]$ | |

Force saturation value ${U}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ | $50\text{}\left[\mathrm{N}\right]$ | |

Starting point ${x}_{0}$ | $-10\text{}\left[\mathrm{m}\right]$ | |

Starting Velocity $\dot{x}\left(0\right)$ | $0\text{}[\mathrm{m}/\mathrm{s}]$ | |

Goal point ${x}_{g}$ | $0\text{}\left[\mathrm{m}\right]$ | |

Speed limit ${V}_{\mathrm{max}\text{}}$ | $5\text{}[\mathrm{m}/\mathrm{s}]$ |

Problem | Indicator | Random | Center | Edge |
---|---|---|---|---|

$MO{P}_{1}$ Target set 1 | ToT | 33 (±20) | 12 (±7.5) | 3 (±2) |

AP | 0.88124 (±0.0601) | 0.91951 (±0.0314) | 0.93812 (±0.0107) | |

Best-F1 | 0.00829 (±0.0079) | 0.00230 (±0.0018) | 0.00355 (±0.0011) | |

Best-F2 | 0.01216 (±0.0098) | 0.00258 (±0.0012) | 0.00127 (±0.0001) | |

Best-F3 | 0.00546 (±0.0022) | 0.00493 (±0.0020) | 0.00375 (±0.0010) | |

$MO{P}_{2}$ Target set 1 | ToT | 24 (±197) | 2 (±2) | 2 (±149) |

AP | 0.90159 (±0.944) | 0.95214 (±0.094) | 0.96308 (±0.730) | |

Best-F1 | 0.03682 (±1.071) | 0.02305 (±0.023) | 0.01784 (±0.816) | |

Best-F2 | 0.01970 (±1.094) | 0.00620 (±0.037) | 0.00387 (±0.825) | |

Best-F3 | 0.02136 (±1.087) | 0.01029 (±0.018) | 0.00849 (±0.820) | |

$MO{P}_{3}$ Target set 1 | ToT | 43 (±150) | 8 (±148) | 5 (±149) |

AP | 0.75407 (±0.589) | 0.79441 (±0.640) | 0.85067 (±0.643) | |

Best-F1 | 0.00830 (±0.063) | 0.00425 (±0.183) | 0.00596 (±0.085) | |

Best-F2 | 0.01917 (±0.067) | 0.00570 (±0.210) | 0.00183 (±0.096) | |

Best-F3 | 0.00492 (±0.032) | 0.01093 (±0.466) | 0.00799 (±0.048) |

Problem | Indicator | Random | Center | Edge |
---|---|---|---|---|

$MO{P}_{1}$ Target set 2 | ToT | 24 (±197) | 2 (±2) | 2 (±149) |

AP | 0.21958 (±0.387) | 0.67058 (±0.236) | 0.70545 (±0.116) | |

Best-F1 | 0.00568 (±0.007) | 0.00258 (±0.003) | 0.00081 (±0.001) | |

Best-F2 | 0.00480 (±0.003) | 0.00135 (±0.001) | 0.00151 (±0.001) | |

Best-F3 | 0.00369 (±0.005) | 0.00220 (±0.002) | 0.00082 (±0.001) | |

$MO{P}_{2}$ Target set 2 | ToT | 4 (±163) | 1 (±1) | 2 (±150) |

AP | 0.95268 (±0.780) | 0.96454 (±0.077) | 0.97688 (±0.770) | |

Best-F1 | 0.00818 (±0.836) | 0.00625 (±0.020) | 0.00398 (±0.831) | |

Best-F2 | 0.01925 (±0.828) | 0.01686 (±0.014) | 0.01099 (±0.831) | |

Best-F3 | 0.00772 (±0.836) | 0.00648 (±0.0236) | 0.00475 (±0.830) | |

$MO{P}_{3}$ Target set 2 | ToT | 129 (±122) | 15 (±9) | 13 (±9) |

AP | 0.61719 (±0.178) | 0.73587 (±0.080) | 0.73441 (±0.057) | |

Best-F1 | 0.00525 (±0.003) | 0.00293 (±0.002) | 0.00093 (±0.001) | |

Best-F2 | 0.00571 (±0.003) | 0.00374 (±0.002) | 0.00462 (±0.001) | |

Best-F3 | 0.00445 (±0.004) | 0.00246 (±0.001) | 0.00145 (±0.001) |

Indicator | Edge Better Than Center | Edge Better Than Random | Center Better Than Random |
---|---|---|---|

ToT (+/−/≈) | 3/1/2 | 5/0/1 | 6/0/0 |

AP (+/−/≈) | 5/0/1 | 5/0/1 | 4/1/1 |

Best-F1 (+/−/≈) | 4/2/0 | 6/0/0 | 5/0/1 |

Best-F2 (+/−/≈) | 3/1/2 | 5/0/1 | 5/0/1 |

Best-F3 (+/−/≈) | 5/0/1 | 5/0/1 | 5/0/1 |

Total (+/−/≈) | 20/4/6 | 26/0/4 | 25/1/4 |

Indicator | Center vs. Edge | Random vs. Edge | Random vs. Center |
---|---|---|---|

ToT | 2 × 10^{−11} | 1 × 10^{−11} | 5 × 10^{−9} |

AP | 8 × 10^{−5} | 3 × 10^{−10} | 4 × 10^{−6} |

Best-F1 | 7 × 10^{−4} | 7 × 10^{−9} | 4 × 10^{−9} |

Best-F2 | 1 × 10^{−7} | 1 × 10^{−11} | 9 × 10^{−11} |

Best-F3 | 7 × 10^{−5} | 2 × 10^{−4} | 4 × 10^{−1} |

Target Set | Scenario | Edge Initialization | |
---|---|---|---|

$\mathit{M}\mathit{O}{\mathit{P}}_{2}$ | $\mathit{M}\mathit{O}{\mathit{P}}_{3}$ | ||

1 | 1 | 8.140 (±0.66) | 15.73 (±4.70) |

2 | 18.70 (±1.09) | 19.97 (±2.40) | |

3 | 9.650 (±0.80) | 16.70 (±4.28) | |

2 | 1 | 17.45 (±0.52) | 19.02 (±0.32) |

2 | 15.35 (±0.35) | 17.30 (±0.60) | |

3 | 19.95 (±0.06) | 19.99 (±0.02) | |

(+/−/≈) | (6/0/0) |

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**MDPI and ACS Style**

Salih, A.; Gabbay, J.; Moshaiov, A.
Transferability of Multi-Objective Neuro-Fuzzy Motion Controllers: Towards Cautious and Courageous Motion Behaviors in Rugged Terrains. *Mathematics* **2024**, *12*, 992.
https://doi.org/10.3390/math12070992

**AMA Style**

Salih A, Gabbay J, Moshaiov A.
Transferability of Multi-Objective Neuro-Fuzzy Motion Controllers: Towards Cautious and Courageous Motion Behaviors in Rugged Terrains. *Mathematics*. 2024; 12(7):992.
https://doi.org/10.3390/math12070992

**Chicago/Turabian Style**

Salih, Adham, Joseph Gabbay, and Amiram Moshaiov.
2024. "Transferability of Multi-Objective Neuro-Fuzzy Motion Controllers: Towards Cautious and Courageous Motion Behaviors in Rugged Terrains" *Mathematics* 12, no. 7: 992.
https://doi.org/10.3390/math12070992