# Robot Coverage Path Planning under Uncertainty Using Knowledge Inference and Hedge Algebras

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## Abstract

**:**

## 1. Introduction

- How can a high-level probabilistic representation (a model) of the environment be created;
- How can understanding and reasoning about the environment be achieved to enable CPP with completion of the required task(s) while a robot is in motion.

## 2. Related Research

## 3. Problem Formulation

#### 3.1. Awareness in CPP with Decision-Support

- $\left(G\right)$ is a collection of birth elements of linguistic variable;
- $\left(H\right)$ is a set of hedges; (and)
- $\left(\le \right)$ is semantic relation on $\left(T\left(X\right)\right)$.

- Each element is either positive or negative for any part in $\left(HA\right)$ including itself;
- The two elements $\left(u\right)$ and $\left(v\right)$ are independent. That is: $\left(u\ne H\left(v\right)\right)$ $AND$ $\left(v\ne H\left(u\right)\right)$ are comparable with $\left(\forall x\in H\left(u\right)\right)$ $AND$ $\left(x\in H\left(v\right)\right)$. $IF$ $\left(u\right)$ $AND$ $\left(v\right)$ are not comparable, $THEN$ $\left(\forall x\in H\left(u\right)\right)$ $AND$ $\left(\forall y\in H\left(v\right)\right)$ are $NOT$ comparable;
- $IF$$\left(x\ne {h}_{x}\right)$$THEN$$\left(x\in H\left({h}_{x}\right)\right)$$AND$$IF$$\left(h\ne k\right)$$AND$$\left({h}_{x}\le {k}_{x}\right)$$THEN$$\left({h}^{\prime}{h}_{x}\le {k}^{\prime}{k}_{x}\right)$ with $\left(\forall {h}^{\prime},{k}^{\prime},h,k,\in H\right)$;
- $IF$$\left(u\in H\left(v\right)\right)$$AND$$\left(u\le v\right)$$OR$$\left(u\ge v\right)$$THEN$$\left(u\le {h}_{v}\right)$$OR$$\left(u\ge {h}_{v}\right)$ where $\left(\forall h\in H\right)$;

#### Fuzzy Linguistic Representation

- $\left(T\left(X\right)\right)$: is a set of linguistic values for (dom($TRUTH$);
- $\left(G\right)$: is a set of primitive words - birth elements (true, false);
- $\left(H\right)$: is a set of linguistic hedges (very, more, little);
- $\left(\le \right)$: is the semantic relation on “words" (a fuzzy concept). The semantic relations are the ordered relations derived from the natural language meaning, i.e., $\left(false\le true\right)$, $\left(moretrue\le verytrue\right)$, $\left(veryfalse\le moretrue\right)$, $\left(possibletrue\le true\right)$, $\left(false\le possiblefalse\right)$, ….

- $\left(x={h}_{n},{h}_{n-1},\dots {h}_{1}g,G\in G\right)$
- $\left(H\left(x\right)\right)$ is set of elements is resulting from $\left(x\right)$
- Considering $\left(V\in {H}^{+}\left(V-very\right)\right)$, $\left(L\in {H}^{-}\left(L-little\right)\right)$, $\left(g\in G\right)$ is positive $IF$ $\left(g\le Vg\right)$ and is negative $IF$ $\left(g\ge Vg\right)$ $\left(OR\right)$ $\left(g\in G\right)$ is positive $IF$ $\left(g\ge Lg\right)$ and is negative $IF$ $\left(g\le Lg\right)$
- $IF$$\left(G\right)$ has exactly two fuzzy primitive elements $\left({g}^{+}\right)$ and $\left({g}^{-}\right)$, then $\left({g}^{+}\right)$ is called a positive birth element and $\left({g}^{-}\right)$ is called a negative birth element and $\left({g}^{-}<{g}^{+}\right)$

#### 3.2. Quantitative Semantic Mapping

## 4. The Methodology

- Using the optimal path traverse the complete environment and visit all nodes without repeating or overlapping paths;
- Identify if the nodes (cells) are: (a) clear; (b) occupied by an obstacle (static or moving); or (c) are bounded by walls;
- Avoid all static and moving obstacles;
- Find the optimal CPP and traverse operating environment with multiple decision-making.

#### 4.1. The Coverage Path Planning Problem

- Hedge_DSS_Robot: the objective optimisation function to maximise the operational efficiency of the robot in CPP and enable multiple robot decision-making objectives;
- $\left({w}_{j}\right)$: the weight which is representative of $\left({S}_{i}\right)$ and $\left({w}_{j}\right)$. The weight is a value of the linguistic variable that can recognised as the value in range: important, very important, more important, little important, very little important, possibly important, ...;
- Based on quantitative semantic mapping of $\left(HA\right)$ linguistic values for $\left({w}_{j}\right)$ fall in the range $\left[0,\phantom{\rule{0.277778em}{0ex}}1\right]$ and are used in multiple decision-making objectives for the Robot tasks $\left({S}_{i}\right)$ where $\left(\left({S}_{i}\cap {S}_{j}\right)=\left(\theta ,\forall \phantom{\rule{0.277778em}{0ex}}i,j\in \{1,2,\cdots r\}\right)\right)$;
- $\left({Q}_{j}\left(X\right)\right)$: the objective function for the multiple decision-making objective. $\left({Q}_{j}\left(X\right)\right)$ recognises the linguistic value of the linguistic variable used in the quantitative semantics mapping of $\left(HA\right)$ and transfers the linguistic value in the range $\left[0,\phantom{\rule{0.277778em}{0ex}}1\right]$;
- The decision variable $\left({X}_{ij}\right)$ is binary and defines the tasks for the multiple decision- making objectives;
- Calculate the objective function value for $\left({Q}_{j}\left(X\right)\right)$.

#### 4.2. The Proposed Coverage Path Planning Algorithm

**Step 1**: Identify the objectives the for the moving robot in the operating environment;

**Step 2**: The applied STC algorithm for Robot travel in a graph representation:

**Initialization**: Call STC2 $\left(Null,S\right)$ where $\left(S\right)$ is the starting cell. While spanning tree construction, the robot subdivides every cell it encounters into four identical sub-cells

**Procedure STC2 $\left(w,x\right)$**:

**While**x has a new free or partially occupied neighbouring cell $\left(Y\right)$ where $\left(x\ne \theta \right)$:

**while**loop.

**Step 3**: Apply reasoning techniques using rules in the knowledge-based

**$\left({w}_{i}\right)$**of $Rule\phantom{\rule{0.277778em}{0ex}}i$ with certain factor weighting $\left(c\right)$. In typical rules we can consider reasoning techniques combined with events and as shown in Equation (4):

**Step 4**: Process rules with reasoning forward chaining

**Step 5**: Find the appropriate rules applied in Knowledge Base (KB)

## 5. Evaluation

#### 5.1. The Robot Case Study

- $\left(Q\left(1\right)\right)$: Traverse the operating environment visiting all the nodes in the operating environment;
- $\left(Q\left(2\right)\right)$: Identify if the nodes (cells) are: (a) clear, (b) occupied by an obstacle(s), or (c) bounded by wall(s);
- $\left(Q\left(3\right)\right)$: Complete its traverse over the operating environment without repeating or overlapping paths;
- $\left(Q\left(4\right)\right)$: Avoid all static and dynamic (independently moving) obstacles;
- $\left(Q\left(5\right)\right)$: Apply CPP to find the “optimal path” to traverse operating environment with multiple decision-making;

- ${S}_{1}$ Simple cleaning function;
- ${S}_{2}$ Cleaning and picking up rubbish;
- ${S}_{3}$ Cleaning while avoiding objects;
- ${S}_{4}$ Intelligent cleaning with decision-support in selecting tasks;
- ${S}_{5}$ Heavy cleaning.

## 6. Experimental Testing and Comparative Analysis

#### 6.1. The Experimental Results

#### 6.2. A Comparative Analysis for Moving Obstacles

## 7. Discussion

#### 7.1. The Concept of “Self”

#### 7.2. Machine Cognition

#### 7.3. Future Directions for Research

- Extending the proposed rule-based linguistic approach using semantics with kansei engineering in combination with hedge algebras forms an interesting research direction;
- A further potentially profitable direction for research (in computing terms) lies in the use of semiotics [20] and SC [60] to recognise the type and nature of obstacles or other robots operating in the environment. Semiotics employs both linguistics and images to create a representative model, their combined use in context-aware intelligent robotic systems is a potentially profitable direction for robotics research;
- There are potential use-cases where multiple mobile robots may operate collaboratively using for example “forward chaining” [62,63,64]; in such a use-case awareness of their environment and other robots operating in the same environment is required. For example, in a large search area multiple robots may be deployed to investigate an environment where efficient search requires both CPP for each robot while avoiding duplication in the search activity.

## 8. Concluding Observations

## Author Contributions

## Funding

## Conflicts of Interest

## References

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${\mathit{Q}}_{1}$ | ${\mathit{Q}}_{2}$ | ${\mathit{Q}}_{3}$ | ${\mathit{Q}}_{4}$ | |
---|---|---|---|---|

${S}_{1}$ | very high | very low | little very low | low |

${S}_{2}$ | low | very low | high | little low |

${S}_{3}$ | very low | little very low | little high | low |

${S}_{4}$ | little high | little little high | little low | little low |

${S}_{5}$ | high | very high | little very high | very high |

${\mathit{W}}_{1}$ | ${\mathit{W}}_{2}$ | ${\mathit{W}}_{3}$ | ${\mathit{W}}_{4}$ | |
---|---|---|---|---|

${S}_{1}$ | imp | imp | unimp | very very imp |

${S}_{2}$ | unimp | unimp | imp | unimp |

${S}_{3}$ | unimp | unimp | unimp | unimp |

${S}_{4}$ | little imp | little imp | very unimp | little imp |

${S}_{5}$ | little imp | little imp | very imp | very very imp |

${\mathit{Q}}_{1}$ | ${\mathit{Q}}_{2}$ | ${\mathit{Q}}_{3}$ | ${\mathit{Q}}_{4}$ | |
---|---|---|---|---|

${S}_{1}$ | 0.875 | 0.125 | 0.1875 | 0.25 |

${S}_{2}$ | 0.25 | 0.125 | 0.75 | 0.375 |

${S}_{3}$ | 0.125 | 0.1875 | 0.625 | 0.25 |

${S}_{4}$ | 0.625 | 0.6875 | 0.375 | 0.375 |

${S}_{5}$ | 0.75 | 0.1375 | 0.1125 | 0.875 |

μ (very very unimportant) = 0.10985 | μ (very unimportant) = 0.169 |

$\mu $ (little very unimportant) = 0.20085 | $\mu $ (unimportant) = 0.26 |

$\mu $ (little little unimportant) = 0.29185 | $\mu $ (little unimportant) = 0.309 |

$\mu $ (very little unimportant) = 0.34085 | $\mu $ (very little important) = 0.488725 |

$\mu $ (little important) = 0.5365 | $\mu $ (little little important) = 0.562225 |

$\mu $ (important) = 0.61 | $\mu $ (little very important) = 0.698725 |

$\mu $ (very important) = 0.7465 | $\mu $ (very very important) = 0.835225 |

${\mathit{W}}_{1}$ | ${\mathit{W}}_{2}$ | ${\mathit{W}}_{3}$ | ${\mathit{W}}_{4}$ | |
---|---|---|---|---|

${S}_{1}$ | 0.61 | 0.61 | 0.26 | 0.835225 |

${S}_{2}$ | 0.26 | 0.26 | 0.61 | 0.26 |

${S}_{3}$ | 0.26 | 0.26 | 0.26 | 0.26 |

${S}_{4}$ | 0.5365 | 0.5365 | 0.169 | 0.5365 |

${S}_{5}$ | 0.5365 | 0.5365 | 0.7465 | 0.15225 |

Methods | Obstacles | Multiple Decision Making Objectives | |||
---|---|---|---|---|---|

Regular (%) | Irregular (%) | Multiple Regular (%) | Multiple Irregular (%) | Average (%) | |

BFS | 4.00 | 3.10 | 36.50 | 32.50 | 50 |

ISS | 7.00 | 20.50 | 19.50 | 26.10 | 53 |

UAPP | 5.00 | 5.40 | 8.85 | 14.40 | 67 |

CPP | 0.00 | 2.20 | 2.00 | 7.30 | 96 |

Methods | Duration (s)/Obstacles | ||||
---|---|---|---|---|---|

Regular | Irregular | Multiple Regular | Multiple Irregular | Average | |

BFS | 134 | 154 | 150 | 144 | 140 |

ISS | 115 | 135 | 130 | 125 | 120 |

UAPP | 95 | 115 | 95 | 110 | 100 |

CPP | 66 | 78 | 79 | 74 | 82 |

Methods | Repetition Rate (%)/Obstacles | Repetition Rate (%)/Multiple Decision Making | |||
---|---|---|---|---|---|

Regular (%) | Irregular (%) | Multiple Regular (%) | Multiple Irregular (%) | Average (%) | |

BFS | 14 | 29 | 38.5 | 38 | 40 |

ISS | 16 | 25 | 29.5 | 32 | 35 |

UAPP | 8 | 12 | 15 | 25 | 29 |

CPP | 3 | 4 | 3 | 11.3 | 13.2 |

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Van Pham, H.; Moore, P. Robot Coverage Path Planning under Uncertainty Using Knowledge Inference and Hedge Algebras. *Machines* **2018**, *6*, 46.
https://doi.org/10.3390/machines6040046

**AMA Style**

Van Pham H, Moore P. Robot Coverage Path Planning under Uncertainty Using Knowledge Inference and Hedge Algebras. *Machines*. 2018; 6(4):46.
https://doi.org/10.3390/machines6040046

**Chicago/Turabian Style**

Van Pham, Hai, and Philip Moore. 2018. "Robot Coverage Path Planning under Uncertainty Using Knowledge Inference and Hedge Algebras" *Machines* 6, no. 4: 46.
https://doi.org/10.3390/machines6040046