A Fuzzy Analytic Hierarchy Process and Cooperative Game Theory Combined Multiple Mobile Robot Navigation Algorithm
Abstract
:1. Introduction
2. Problem Description
3. Fuzzy-Based Analytic Hierarchy Process
- Step 1: Model the problem as a hierarchy: the decision goal, the alternatives as solution candidates, and the objectives to evaluate the candidates.
- Step 2: Establish priorities among the considered objectives: define the relative importance of the objectives by comparing them in pairs using a nine-point scale.
- Step 3: Synthesize the user’s priorities to yield a set of overall priorities for the hierarchy.
- Step 4: Check the consistency of the decision making.
- Step 5: Evaluate the candidates considering the weighted importance matrix.
- Step 1: Definition of the relative importance among objectives.
- Step 2: Consistency check of the relative important matrix.
- Step 3: Fuzzification of the relative importance matrix.
- Step 4: Calculation of fuzzy synthetic extent.
- Step 5: Calculation of weight vectors of FRM.
4. Application of FAHP to Multi-Robot Collision-Free Navigation
4.1. FAHP Algorithm-Based Mobile Robot Navigation
4.2. FAHP and Cooperative Game Theory Combined Algorithm-Based Mobile Robot Navigation
5. Simulation and Results
5.1. Simulation I (FAHP-Based Single Mobile Robot Navigation)
5.2. Simulation II (FAHP-CGT-Based Multi-Robot Navigation under No Obstacle Condition)
5.3. Simulation III (FAHP-CGT Based Multi-Robot Navigation under a Warehouse Environment)
6. Conclusions
Author Contributions
Conflicts of Interest
References
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Intensity of Importance | Definition |
---|---|
(1, 1, 1 + d) | Equal importance of objectives |
(3 − d, 3, 3 + d) | Moderate importance of one objective relative to another |
(5 − d, 5, 5 + d) | Strong importance of one objective relative to another |
(7 − d, 7, 7 + d) | Very strong importance of one objective relative to another |
(9 − d, 9, 9) | Extreme importance of one objective relative to another |
(x − d, x, x + d), x = 2, 4, 6, 8 | Intermediate values between two adjacent judgements |
Number of Objectives | RC |
---|---|
3 | 0.58 |
4 | 0.90 |
5 | 1.12 |
6 | 1.24 |
7 | 1.32 |
8 | 1.41 |
9 | 1.45 |
Notation | Definition | Multi-Robot Application |
---|---|---|
n | Number of players | Number of robots |
Strategy space of player i | All the candidates for robot i | |
Strategy of player i | Selection of robot i among candidates | |
Strategy profile of n players | Each robot’s solution | |
Strategy profile of n − 1 players | A set without the selection of robot i | |
U | Domain of all possible outcomes | A set of possible benefit for robots |
Payoffs given to players under strategy s | Benefit of robots | |
Payoff to player i under strategy s | Benefit of robot i |
Two Robots | Four Robots | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
R | R | Average | ROA | SDA | R | R | R | R | Average | ROA | SDA | |
Initial position | (70, 140) | (690, 140) | (160, 160) | (600, 160) | (600, 600) | (160, 600) | ||||||
Target position | (690, 140) | (70, 140) | (600, 600) | (160, 600) | (160, 160) | (600, 160) | ||||||
Travel distance (pixel) | 630.09 | 630.42 | 630.26 | 644.00 | 641.00 | 641.42 | 639.23 | 639.07 | 642.82 | 640.63 | 661.00 | 649.00 |
Increased distance(%) | 1.6 | 1.7 | 1.7 | 3.9 | 3.4 | 3.5 | 3.1 | 3.1 | 3.7 | 3.3 | 6.6 | 4.7 |
Eight Robots | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
R | R | R | R | R | R | R | R | Average | ROA | SDA | |
Initial position | (380, 611) | (160, 520) | (69, 300) | (160, 80) | (380, −11) | (600, 80) | (691, 300) | (600, 520) | |||
Target position | (380, −11) | (600, 80) | (691, 300) | (600, 520) | (380, 611) | (160, 520) | (69, 300) | (160, 80) | |||
Travel distance (pixel) | 651.26 | 652.90 | 657.42 | 656.89 | 654.59 | 658.05 | 657.24 | 655.07 | 655.43 | 687.00 | 652.00 |
Increased distance (%) | 5.0 | 5.3 | 6.0 | 5.9 | 5.6 | 6.1 | 6.0 | 5.7 | 5.7 | 10.8 | 5.2 |
12 Robots | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
R | R | R | R | R | R | R | R | R | R | R | R | Average | ROA | SDA | |
Travel distance (pixel) | 674.15 | 679.24 | 679.44 | 681.68 | 678.37 | 676.24 | 675.31 | 680.48 | 679.76 | 678.30 | 679.99 | 675.87 | 678.04 | 713.00 | 675.00 |
Increased distance (%) | 8.73 | 9.55 | 9.59 | 9.95 | 9.42 | 9.07 | 8.92 | 9.75 | 9.64 | 9.40 | 9.68 | 9.01 | 9.36 | 15.00 | 8.87 |
R | R | R | R | R | R | R | R | R | |
---|---|---|---|---|---|---|---|---|---|
Initial position | (70, 42) | (22, 35) | (45, 60) | (45, 42) | (100, 51) | (70, 24.5) | (70, 21.5) | (83, 40) | (78, 12) |
Target position | (22, 15) | (100,33) | (65, 42) | (65, 60) | (20, 51) | (100, 35) | (100, 15) | (87, 10) | (79, 50) |
Scenario 1 | Scenario 2 | Scenario 3 | |||||||
---|---|---|---|---|---|---|---|---|---|
R | R | R | R | R | R | R | R | R | |
Original travel distance (m) | 58.03 | 78.00 | 28.79 | 28.66 | 80.04 | 32.62 | 31.05 | 29.53 | 42.06 |
Cooperative travel distance (m) | 58.95 | 78.54 | 28.94 | 29.18 | 82.16 | 34.28 | 31.35 | 29.63 | 42.71 |
Increased distance (%) | 1.6 | 0.7 | 0.5 | 1.8 | 2.6 | 5.1 | 1.0 | 0.3 | 1.6 |
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Kim, C.; Won, J.-S. A Fuzzy Analytic Hierarchy Process and Cooperative Game Theory Combined Multiple Mobile Robot Navigation Algorithm. Sensors 2020, 20, 2827. https://doi.org/10.3390/s20102827
Kim C, Won J-S. A Fuzzy Analytic Hierarchy Process and Cooperative Game Theory Combined Multiple Mobile Robot Navigation Algorithm. Sensors. 2020; 20(10):2827. https://doi.org/10.3390/s20102827
Chicago/Turabian StyleKim, Changwon, and Jong-Seob Won. 2020. "A Fuzzy Analytic Hierarchy Process and Cooperative Game Theory Combined Multiple Mobile Robot Navigation Algorithm" Sensors 20, no. 10: 2827. https://doi.org/10.3390/s20102827
APA StyleKim, C., & Won, J.-S. (2020). A Fuzzy Analytic Hierarchy Process and Cooperative Game Theory Combined Multiple Mobile Robot Navigation Algorithm. Sensors, 20(10), 2827. https://doi.org/10.3390/s20102827