# Cloud-Based Multi-Robot Path Planning in Complex and Crowded Environment with Multi-Criteria Decision Making Using Full Consistency Method

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. System Overview

#### 3.1. D* Lite Algorithm

_{0}to node s

_{к}, where s

_{0}, s

_{к}∈ S, is defined as the sum of the sequential cost of transitions between neighboring nodes (edge costs) in the set of nodes {s

_{0}, s

_{1}, …, s

_{к}

_{−1}, s

_{к}}, i.e., as (c(s

_{0}, s

_{1}) + … + c(s

_{i}

_{−1}, s

_{i}) + … + c(s

_{k}

_{−1}, s

_{k})), where c(s

_{i}

_{−1}, s

_{i}) represents the cost of moving from s

_{i}

_{−1}to s

_{i}and s

_{i}∈ Succ(s

_{i}

_{−1}), 1 ≤ i ≤ k. The least-cost path from s

_{0}to s

_{к}is denoted with c*(s

_{0}, s

_{к}). For s

_{0}= s

_{к}, we define c*(s

_{0}, s

_{к}) = 0.

_{start}to s

_{goal}whose cost is minimal, i.e., equal to c*(s

_{start}, s

_{goal}). D* Lite algorithm, as well as A*, during operation forms and maintains (updates) the value of four functions that describe cell s:

- g(s)—the minimum cost of moving from s
_{start}to s, i.e., c(s_{start}, s), found so far; - h(s) or heuristic value—estimates the minimum cost of moving from s to s
_{goal}. Using heuristic value ensures that the search tree is directed towards the most optimistic cells in terms of belonging to the optimal path from start to goal cell. This speeds up the search. - f(s) = g(s) + h(s)—estimates the minimum cost of moving from s
_{start}via s to s_{goal}; and, - parent(s) or parent pointer—points to the predecessor s’ of s from which is derived g(s), s’ is called the parent of s. The parent pointers are used to extract the path after the search terminates.

_{s’}

_{∈}

_{Succ}

_{(s)}(c(s, s’) + g(s’)) or zero if s is the goal cell. In implementation, each cell maintains a pointer to the cell from which it derives its rhs value, so the robot should follow the pointers from its current cell to pursue an optimal path to the goal.

_{1}(s), k

_{2}(s)] = [min(g(s), rhs(s)) + h(s

_{start}, s); min(g(s), rhs(s))]. The key value of the cell s is less than the key value of the cell s’, denoted key(s) ≤ key(s’), which means that s is a cell with a higher priority, if k

_{1}(s) < k

_{1}(s’) or both k

_{1}(s) = k

_{1}(s’) and k

_{2}(s) ≤ k

_{2}(s’).

_{goal}to s

_{start}). If the robot detects changes in the environment during the motion (i.e., the cost of some edge is altered), D* Lite first updates the rhs values of all of the cells directly affected by the changed edge cost. After that, priority queue OPEN is updated, i.e., the algorithm places new inconsistent cells onto the queue. Subsequently, the cells are expanded from the updated OPEN list according to the prioritization based on the assigned key value. This ensures the propagation of inconsistency. In this way, D* Lite algorithm checks the validity of the current path and corrects it if necessary. D* Lite is efficient because it processes only those cells that are directly affected by the changes. In other words, while using the previously obtained results to calculate the corrected path, D* Lite does not replan from scratch over the entire graph as A*. As a result, it can be up to two orders of magnitude more efficient than A*.

_{start}is changing). As the robot travels, it, at the same time, observes the environment. If changes in edge costs in some robot step are detected, D* Lite updates the rhs values of each cell immediately affected by the changed edge costs and places those cells that have been made inconsistent onto the OPEN queue (lines 21–23). D* Lite then propagates the effects of these rhs values changes to the rest of the cell space and checks/replans the path through recalling ComputePath() function (line 19) until it terminates again. Line 18 in real implementation means that the whole process ends when it becomes s

_{start}= s

_{goal}.

#### 3.2. Full Consistency Method (FUCOM)

_{j}, j = 1, 2, …, n and that their weight coefficients need to be determined. Subjective models for determining weights based on pairwise comparison of criteria require the decision-maker to determine the degree of influence between the criteria. In accordance with the defined settings, the next section presents the FUCOM algorithm, Figure 4 [6].

## 4. Multi-Criteria Decision Making Model and Procedure

- C
_{1}—the convenience of the terrain configuration for robots motion, a 0–10 scaled grading scheme (0 = ‘favorable terrain’ to 10 = ‘extremely unfavorable terrain’), - C
_{2}—the risk related to the loss of communication with the cloud, a 0–10 scaled grading scheme (0 = ‘low risk’ to 10 = ‘high risk’), - C
_{3}—the risk related to the human-robot interactions (slows down the robots motion), a 0–10 scaled grading scheme (0 = ‘low risk’ to 10 = ‘high risk’), and - C
_{4}—the robot safety related to conditions dependent on specific mission (0 = ‘safe conditions’ to 10 = ‘extremely unsafe conditions’).

_{ij}as the weight of factor j defined by expert i (i = 1,2, …, I; j = 1,2, …, J) and C

^{m}

_{ij}as the score of jth factor for cell m provided by the ith expert (i = 1,2, …, I; j = 1,2, …, J; m = 1,2, …, M), we find C

^{m}, the ‘path planning index’ of the mth cell. Here, equation (1) is used to aggregate the score of jth factor given by I experts for cell m, as well as its total score for all J factors:

^{m}≤ 2.5. Yellow marks a moderate risk level and it represents cells with 2.5 ≤ C

^{m}≤ 5. Orange indicates a high risk and it represents cells with 5.0 ≤ C

^{m}≤ 7.5 and red symbolises severe risk and represents cells with 7.5 ≤ C

^{m}≤ 10. Each risk level implies its unique cost of transition through the cell. This approach provides risk-sensitive planning.

## 5. Simulation and Results

- start1 (x, y) = (1, 21), goal1 (x, y) = (73, 100)
- start2 (x, y) = (20, 1), goal2 (x, y) = (100, 91)
- start3 (x, y) = (35, 15), goal3 (x, y) = (100, 100).

_{3}≥ C

_{2}≥ C

_{1}> C

_{4}.

_{3}criterion. The comparison was based on the scale $\left[1,9\right]$. Thus, the priorities of the criteria (${\varpi}_{{C}_{j(k)}}$) for all of the criteria ranked in Step 1 were obtained (Table 1).

_{3}> C

_{2}> C

_{1}≥ C

_{4}.

_{3}criterion. The comparison was based on the scale $\left[1,9\right]$. Thus, the priorities of the criteria (${\varpi}_{{C}_{j(k)}}$) for all of the criteria ranked in Step 1 were obtained (Table 2).

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 5.**Scenario cost map (left) and path planning results (right): (

**a**) D* Lite without multi-criteria decision making (MCDM); (

**b**) D* Lite with MCDM1; and, (

**c**) D* Lite with MCDM2.

Criteria | C_{3} | C_{2} | C_{1} | C_{4} |
---|---|---|---|---|

${\varpi}_{{C}_{j(k)}}$ | 1 | 1 | 1 | 5 |

Criteria | C_{3} | C_{2} | C_{1} | C_{4} |
---|---|---|---|---|

${\varpi}_{{C}_{j(k)}}$ | 1 | 4 | 7 | 7 |

D* Lite without MCDM | D* Lite with MCDM1 | D* Lite with MCDM2 | |
---|---|---|---|

Risky Actions | 887 | 798 (−10.1%) | 719 (−18.9%) |

Distance | 407.16 | 425.32 (+4.4%) | 435.35 (+6.9%) |

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**MDPI and ACS Style**

Zagradjanin, N.; Pamucar, D.; Jovanovic, K.
Cloud-Based Multi-Robot Path Planning in Complex and Crowded Environment with Multi-Criteria Decision Making Using Full Consistency Method. *Symmetry* **2019**, *11*, 1241.
https://doi.org/10.3390/sym11101241

**AMA Style**

Zagradjanin N, Pamucar D, Jovanovic K.
Cloud-Based Multi-Robot Path Planning in Complex and Crowded Environment with Multi-Criteria Decision Making Using Full Consistency Method. *Symmetry*. 2019; 11(10):1241.
https://doi.org/10.3390/sym11101241

**Chicago/Turabian Style**

Zagradjanin, Novak, Dragan Pamucar, and Kosta Jovanovic.
2019. "Cloud-Based Multi-Robot Path Planning in Complex and Crowded Environment with Multi-Criteria Decision Making Using Full Consistency Method" *Symmetry* 11, no. 10: 1241.
https://doi.org/10.3390/sym11101241