A behaviour based algorithm for encirclement of a dynamic target using multiple mobile robots

A BEHAVIOUR-BASED ALGORITHM FOR ENCIRCLEMENT OF A DYNAMIC TARGET USING MULTIPLE MOBILE ROBOTS LOW YEE LEONG (B.Eng.(Hons.), NUS) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2004 ACKNOWLEDGEMENTS I would like to thank my supervisor, A/P Gerard Leng Siew Bing, for his intellectual guidance and criticism, continuous support and understanding throughout my research and study. I would also like to thank my colleagues in Cooperative Systems Lab, especially Mr. Cheng Chee Kong and Mr. Ng Wee Kiat for their help and interaction. I am also thankful to staff in Dynamics Lab like Ms. Priscilla, Ms. Amy, Mr. Cheng, and Mr. Ahmad for their assistance. Last but not least, I would like to thank my family for their encouragement and support. Without the love and backing from all of you, I would not be able to finish the research. i TABLE OF CONTENTS ACKNOWLEDGEMENTS I TABLE OF CONTENTS II SUMMARY V LIST OF TABLES . VII LIST OF FIGURES VIII PROJECT DEFINITION .1 1.1 Problem Definitions and Assumptions 1.2 Definitions 1.3 1.2.1 Target .2 1.2.2 Encirclement Thesis Outline LITERATURE SURVEY 2.1 Circle Formation of Distributed Mobile Robots 2.2 Behaviour-Based Control of Multiple Robots .7 2.3 Chapter Summary DESIGN OF ROBOTIC BEHAVIOURS FOR ENCIRCLEMENT .10 3.1 3.2 Three Basic Robotic Behaviours .10 3.1.1 Obstacle-Avoidance .10 3.1.2 Target-Tracking .11 3.1.3 Target-Circumnavigation .11 Robot Controller 13 ii 3.3 Behaviour Coordination .16 3.4 Encirclement Strategy 17 3.5 Chapter Summary 17 VALIDATION OF ENCIRCLEMENT ALGORITHM VIA SIMULATION 20 4.1 Program Structure 20 4.2 Implementation of Robotic Behaviours on Simulation .22 4.2.1 Obstacle-Avoidance .22 4.2.2 Target-Tracking .22 4.2.3 Target-Circumnavigation .22 4.3 Process of Encirclement on Simulation .25 4.4 Simulation Experimental Setup .27 4.5 Analysis of Simulation Results 28 4.5.1 Definition of Non-Dimensional Performance Index 28 4.5.2 Effects of Parameters on Performance Index .29 4.6 General Law for Performance Index of Encirclement .34 4.7 Chapter Summary 36 VALIDATION OF ENCIRCLEMENT ALGORITHM VIA HARDWARE EXPERIMENTS 37 5.1 5.2 Robot Features .37 5.1.1 Obstacle-Detection Sensor .38 5.1.2 Target-Detection Sensor 40 5.1.3 Processor 41 Test of Individual Robot Behaviours .43 5.2.1 Obstacle-Avoidance .43 5.2.2 Target-Tracking .44 iii 5.2.3 Target-Circumnavigation .46 5.3 Hardware Experimental Setup .47 5.4 Comparison of Hardware and Simulation Results .49 5.5 Chapter Summary 49 CONCLUSIONS 52 6.1 Thesis Conclusions 52 6.2 Recommendations for Future Work .53 REFERENCES 54 APPENDICES .57 8.1 Simulation Results (Chapter 4) 58 8.2 Hardware Implementation Results (Chapter 5) .121 iv SUMMARY The objective of this project is to formulate an algorithm that will coordinate the movement of multiple mobile robots to encircle a dynamic target. The robots are not equipped with global coordinate system and communication system. In addition, we will study the performance of this algorithm for different number of robots used and for different speed ratio (target speed / robot speed). Part of the results of this project has been presented in the 2004 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) held in Japan. In order to realize the algorithm, we have formulated three different reactive behaviours for all the robots. The first behaviour is obstacle-avoidance, which makes sure that a robot will not collide with obstacle. The second behaviour is targettracking, which guides a robot towards the target. The third behaviour is targetcircumnavigation, which leads a robot to move around the target. By adopting subsumption-based coordination, a robot will execute one of these behaviours at any one time according to the priority of the behaviour. Obstacle-avoidance has the highest priority followed by target-circumnavigation and target-tracking. We have also designed a simple neural controller to execute all these three behaviours. We have implemented our algorithm on an object-oriented simulation C++ program. Multiple simulations were performed to find out how the time taken changed for different number of robots used and for different speed ratio. A general law governing the performance and speed ratio of our algorithm was deduced from the simulation v results. The performance is quantified by a non-dimensional index. We can use this general law to predict the performance of the encirclement experiment as long as we know the speed ratio regardless of the size of operational area, the speed of robots or the speed of target. We have also validated our algorithm by implementing it on physical robots that we built. These robots have been used to perform encirclement experiments to validate the feasibility of the simulation program. The results obtained from hardware experiments agree with the simulation. Thus, we can use the general law deduced from simulation to extrapolate the performance of hardware experiments. vi LIST OF TABLES Table 3.1: Different sets of weights for different robot behaviours…………….16 Table 4.1: Relationship between speed ratio, target speed, and robot speed……28 vii LIST OF FIGURES Figure 1.1: Definition of encirclement. Highlighted sensors detect the robot within the preset distance or the radius of encirclement…………………… .3 Figure 2.1: Reuleaux’s triangle……………………………………………………6 Figure 2.2: Motor schema architecture……………………………………………8 Figure 2.3: Example of subsumption architecture……………………………… .9 Figure 3.1: Basic behaviour 1: Obstacle-avoidance…………………………… .11 Figure 3.2: Basic behaviour 2: Target-tracking………………………………… 12 Figure 3.3: Basic behaviour 3: Target-circumnavigation……………………… .12 Figure 3.4: Network structure of the robot controller…………………………….14 Figure 3.5: Relationship between weights numbering and robot’s directions……15 Figure 3.6: Subsumption-based coordination of behaviours…………………… 16 Figure 3.7: Switching between target-tracking behaviour and targetcircumnavigation behaviour………………………………………….18 Figure 3.8: Obstacle-avoidance behaviour makes robots distribute more evenly around the target…………………………………………………… .19 Figure 4.1: Class structure of simulation program……………………………… 21 Figure 4.2: Implementation of (a) obstacle-avoidance and (b) target-tracking on simulation. Robot’s direction is indicated by the white arrow on the robot body…………………………………………………………….23 Figure 4.3: Implementation of target-circumnavigation on simulation. Robot’s direction is indicated by the white arrow on the robot body…………24 Figure 4.4: Selected images of the encirclement simulation: (a) the initial distribution, (b) – (e) intermediate steps, and (f) completion of encirclement………………………………………………………….26 Figure 4.5: Graph of the non-dimensional performance index versus speed ratio………………………………………………………………… 30 viii Figure 4.6: Graph of non-dimensional performance index versus number of robots…………………………………………………………………31 Figure 4.7: Graph of success rate versus speed ratio…………………………… 32 Figure 4.8: Maximum speed ratio for encirclement. R can be considered as the radius of encirclement……………………………………………… 33 Figure 4.9: Graph of non-dimensional performance index vs smaller speed ratio………………………………………………………………… .34 Figure 4.10: Graph of log(non-dimensional performance index) vs log(speed ratio)………………………………………………………………….35 Figure 5.1: Devantech SRF08 Ultrasonic Ranger……………………………… .38 Figure 5.2: Field of view of SRF08 Ultrasonic Ranger. (From the website of the manufacturer, http://www.acroname.com/ [16])…………………… 39 Figure 5.3: Calibration graph of sonar sensor couple in Devantech SRF08 Ultrasonic Ranger…………………………………………………….39 Figure 5.4: Calibration graph of light sensor coupled in Devantech SRF08 Ultrasonic Ranger…………………………………………………….40 Figure 5.5: Microcontroller used: Acroname BrainStem GP 1.0……………… .41 Figure 5.6: A photograph of the robot. It becomes a target when a light bulb is mounted on it…………………………………………………………42 Figure 5.7: Robots displaying obstacle-avoidance behaviour……………………44 Figure 5.8: Robots displaying target-tracking behaviour……………………… .45 Figure 5.9: Robot displaying target-circumnavigation behaviour……………… 46 Figure 5.10: Snapshots of hardware experiment………………………………… .48 Figure 5.11: Comparison of hardware and simulation results. (Speed Ratio: 0.2) .50 Figure 5.12: Comparison of hardware and simulation results. (Speed Ratio: 0.3) .50 Figure 5.13: Comparison of hardware and simulation results. (Speed Ratio: 0.4) .51 ix Robots, Speed Ratio: 0.5 Number of Categories Max P.I. Min P.I. Category Length Category 10 10 2.318841 0.834637 0.15 Lower Upper Mean Value Frequency Bound Bound 0.82 0.97 0.895 0.97 1.12 1.045 1.12 1.27 1.195 1.27 1.42 1.345 1.42 1.57 1.495 1.57 1.72 1.645 1.72 1.87 1.795 1.87 2.02 1.945 2.02 2.17 2.095 2.17 2.32 2.245 Total 50 Frequency Distribution of P.I. (9 Robots, Speed Ratio: 0.5) (Mean: 1.59, Standard Deviation: 0.35) 25 20 Frequency No. TimeSteps P.I. 465 1.146953 368 1.449275 305 1.748634 417 1.278977 387 1.378122 266 2.005013 399 1.336675 349 1.528176 383 1.392515 10 453 1.177336 11 287 1.858304 12 361 1.477378 13 639 0.834637 14 357 1.493931 15 435 1.226054 16 331 1.611279 17 281 1.897983 18 286 1.864802 19 288 1.851852 20 465 1.146953 21 443 1.203913 22 397 1.343409 23 277 1.925391 24 281 1.897983 25 360 1.481481 26 370 1.441441 27 230 2.318841 28 300 1.777778 29 325 1.641026 30 312 1.709402 31 305 1.748634 32 326 1.635992 33 264 2.020202 34 267 1.997503 35 416 1.282051 36 275 1.939394 37 585 0.911681 38 271 1.96802 39 385 1.385281 40 293 1.82025 41 451 1.182557 42 275 1.939394 43 518 1.029601 44 388 1.37457 45 294 1.814059 46 248 2.150538 47 391 1.364024 48 230 2.318841 49 371 1.437556 50 297 1.795735 Mean 1.59122794 Std Dev 0.35148032 15 Data Mean 10 0 0.5 1.5 2.5 P.I. Note: P.I. is the non-dimensional performance index. TimeSteps is the number of time steps the robots spent to encircle the target. Category Length = (Max P.I. – Min P.I.) / Number of Categories 107 Robots, Speed Ratio: 0.6 Number of Categories Max P.I. Min P.I. Category Length Category 10 10 1.911589 0.737667 0.12 Lower Upper Mean Value Frequency Bound Bound 0.72 0.84 0.78 0.84 0.96 0.9 0.96 1.08 1.02 1.08 1.20 1.14 1.20 1.32 1.26 1.32 1.44 1.38 1.44 1.56 1.5 1.56 1.68 1.62 11 1.68 1.80 1.74 1.80 1.92 1.86 Total 50 Frequency Distribution of P.I. (9 Robots, Speed Ratio: 0.6) (Mean: 1.44, Standard Deviation: 0.27) 25 20 Frequency No. TimeSteps P.I. 676 0.788955 335 1.59204 449 1.187825 380 1.403509 444 1.201201 318 1.677149 356 1.498127 476 1.120448 492 1.084011 10 328 1.626016 11 460 1.15942 12 350 1.52381 13 279 1.911589 14 342 1.559454 15 394 1.353638 16 290 1.83908 17 378 1.410935 18 361 1.477378 19 338 1.577909 20 317 1.68244 21 499 1.068804 22 336 1.587302 23 328 1.626016 24 347 1.536984 25 375 1.422222 26 341 1.564027 27 341 1.564027 28 338 1.577909 29 389 1.371037 30 355 1.502347 31 377 1.414677 32 723 0.737667 33 361 1.477378 34 322 1.656315 35 509 1.047806 36 499 1.068804 37 441 1.209373 38 319 1.671891 39 342 1.559454 40 296 1.801802 41 489 1.090661 42 314 1.698514 43 369 1.445348 44 481 1.108801 45 403 1.323408 46 302 1.766004 47 289 1.845444 48 442 1.206637 49 291 1.832761 50 409 1.303993 Mean 1.43524694 Std Dev 0.27173549 15 Data 10 Mean 0 0.5 1.5 P.I. Note: P.I. is the non-dimensional performance index. TimeSteps is the number of time steps the robots spent to encircle the target. Category Length = (Max P.I. – Min P.I.) / Number of Categories 108 Robots, Speed Ratio: 0.7 Number of Categories Max P.I. Min P.I. Category Length Category 10 10 1.661475 0.665834 0.11 Lower Upper Mean Value Frequency Bound Bound 0.61 0.72 0.665 0.72 0.83 0.775 0.83 0.94 0.885 0.94 1.05 0.995 1.05 1.16 1.105 10 1.16 1.27 1.215 1.27 1.38 1.325 1.38 1.49 1.435 11 1.49 1.60 1.545 1.60 1.71 1.655 Total 50 Frequency Distribution of P.I. (9 Robots, Speed Ratio: 0.7) (Mean: 1.27, Standard Deviation: 0.24) 25 20 Frequency No. TimeSteps P.I. 510 1.045752 400 1.333333 474 1.125176 506 1.054018 361 1.477378 412 1.294498 465 1.146953 443 1.203913 431 1.237432 10 725 0.735632 11 801 0.665834 12 565 0.943953 13 510 1.045752 14 717 0.74384 15 372 1.433692 16 333 1.601602 17 369 1.445348 18 382 1.396161 19 438 1.217656 20 407 1.310401 21 321 1.661475 22 384 1.388889 23 368 1.449275 24 550 0.969697 25 492 1.084011 26 460 1.15942 27 334 1.596806 28 378 1.410935 29 410 1.300813 30 414 1.288245 31 339 1.573255 32 369 1.445348 33 474 1.125176 34 502 1.062417 35 643 0.829445 36 491 1.086219 37 406 1.313629 38 439 1.214882 39 423 1.260835 40 355 1.502347 41 338 1.577909 42 362 1.473297 43 366 1.457195 44 388 1.37457 45 461 1.156905 46 336 1.587302 47 331 1.611279 48 365 1.461187 49 470 1.134752 50 394 1.353638 Mean 1.26738954 Std Dev 0.24199726 15 Data Mean 10 0 0.5 1.5 P.I. Note: P.I. is the non-dimensional performance index. TimeSteps is the number of time steps the robots spent to encircle the target. Category Length = (Max P.I. – Min P.I.) / Number of Categories 109 Robots, Speed Ratio: 0.8 Number of Categories Max P.I. Min P.I. Category Length Category 10 10 1.616162 0.572861 0.11 Lower Upper Mean Value Frequency Bound Bound 0.54 0.65 0.595 0.65 0.76 0.705 0.76 0.87 0.815 0.87 0.98 0.925 10 0.98 1.09 1.035 12 1.09 1.20 1.145 1.20 1.31 1.255 1.31 1.42 1.365 1.42 1.53 1.475 1.53 1.64 1.585 Total 50 Frequency Distribution of P.I. (9 Robots, Speed Ratio: 0.8) (Mean: 1.08, Standard Deviation: 0.23) 25 20 Frequency No. TimeSteps P.I. 603 0.884467 428 1.246106 475 1.122807 394 1.353638 600 0.888889 446 1.195815 420 1.269841 840 0.634921 545 0.978593 10 414 1.288245 11 694 0.768492 12 472 1.129944 13 514 1.037613 14 421 1.266825 15 343 1.554908 16 547 0.975015 17 555 0.960961 18 482 1.106501 19 334 1.596806 20 383 1.392515 21 504 1.058201 22 450 1.185185 23 582 0.91638 24 898 0.593912 25 470 1.134752 26 625 0.853333 27 552 0.966184 28 420 1.269841 29 433 1.231717 30 508 1.049869 31 527 1.012018 32 853 0.625244 33 490 1.088435 34 406 1.313629 35 546 0.976801 36 330 1.616162 37 459 1.161946 38 539 0.989487 39 495 1.077441 40 511 1.043705 41 583 0.914808 42 509 1.047806 43 441 1.209373 44 931 0.572861 45 465 1.146953 46 492 1.084011 47 387 1.378122 48 506 1.054018 49 524 1.017812 50 558 0.955795 Mean 1.08397406 Std Dev 0.23166644 15 Data Mean 10 0 0.5 1.5 P.I. Note: P.I. is the non-dimensional performance index. TimeSteps is the number of time steps the robots spent to encircle the target. Category Length = (Max P.I. – Min P.I.) / Number of Categories 110 Robots, Speed Ratio: 0.9 Number of Categories Max P.I. Min P.I. Category Length Category 10 10 1.300813 0.530152 0.08 Lower Upper Mean Value Frequency Bound Bound 0.52 0.60 0.56 0.60 0.68 0.64 0.68 0.76 0.72 0.76 0.84 0.8 0.84 0.92 0.88 0.92 1.00 0.96 1.00 1.08 1.04 1.08 1.16 1.12 1.16 1.24 1.2 1.24 1.32 1.28 Total 50 Frequency Distribution of P.I. (9 Robots, Speed Ratio: 0.9) (Mean: 0.92, Standard Deviation: 0.18) 25 20 Frequency No. TimeSteps P.I. 1006 0.530152 528 1.010101 631 0.845219 654 0.815494 629 0.847907 485 1.099656 734 0.726612 543 0.982198 517 1.031593 10 675 0.790123 11 558 0.955795 12 903 0.590624 13 564 0.945626 14 526 1.013942 15 561 0.950683 16 410 1.300813 17 680 0.784314 18 458 1.164483 19 604 0.883002 20 493 1.081812 21 478 1.11576 22 426 1.251956 23 858 0.621601 24 479 1.113431 25 533 1.000625 26 522 1.021711 27 465 1.146953 28 619 0.861605 29 730 0.730594 30 516 1.033592 31 550 0.969697 32 477 1.118099 33 563 0.947306 34 500 1.066667 35 580 0.91954 36 528 1.010101 37 821 0.649614 38 482 1.106501 39 576 0.925926 40 663 0.804424 41 420 1.269841 42 771 0.691742 43 766 0.696258 44 591 0.902425 45 678 0.786627 46 509 1.047806 47 554 0.962696 48 814 0.655201 49 677 0.787789 50 816 0.653595 Mean 0.92439664 Std Dev 0.18422027 15 Data Mean 10 0 0.5 1.5 P.I. Note: P.I. is the non-dimensional performance index. TimeSteps is the number of time steps the robots spent to encircle the target. Category Length = (Max P.I. – Min P.I.) / Number of Categories 111 10 Robots, Speed Ratio: 0.1 Number of Categories Max P.I. Min P.I. Category Length Category 10 10 9.52381 3.333333 0.63 Lower Upper Mean Value Frequency Bound Bound 3.28 3.91 3.60 3.91 4.54 4.225 4.54 5.17 4.855 5.17 5.80 5.485 5.80 6.43 6.115 6.43 7.06 6.745 7.06 7.69 7.375 7.69 8.32 8.005 8.32 8.95 8.635 8.95 9.58 9.265 Total 50 Frequency Distribution of P.I. (10 Robots, Speed Ratio: 0.1) (Mean: 6.31, Standard Deviation: 1.77) 25 20 Frequency No. TimeSteps P.I. 56 9.52381 71 7.511737 64 8.333333 110 4.848485 109 4.892966 62 8.602151 72 7.407407 96 5.555556 119 4.481793 10 140 3.809524 11 64 8.333333 12 62 8.602151 13 118 4.519774 14 87 6.130268 15 66 8.080808 16 67 7.960199 17 59 9.039548 18 79 6.751055 19 66 8.080808 20 57 9.356725 21 103 5.177994 22 156 3.418803 23 80 6.666667 24 149 3.579418 25 56 9.52381 26 99 5.387205 27 111 4.804805 28 112 4.761905 29 83 6.425703 30 100 5.333333 31 68 7.843137 32 111 4.804805 33 66 8.080808 34 60 8.888889 35 81 6.584362 36 117 4.558405 37 107 4.984424 38 86 6.20155 39 65 8.205128 40 160 3.333333 41 93 5.734767 42 94 5.673759 43 80 6.666667 44 116 4.597701 45 123 4.336043 46 89 5.992509 47 89 5.992509 48 82 6.504065 49 126 4.232804 50 98 5.442177 Mean 6.31117832 Std Dev 1.77245754 15 Data Mean 10 0 10 P.I. Note: P.I. is the non-dimensional performance index. TimeSteps is the number of time steps the robots spent to encircle the target. Category Length = (Max P.I. – Min P.I.) / Number of Categories 112 10 Robots, Speed Ratio: 0.2 Number of Categories Max P.I. Min P.I. Category Length Category 10 10 5.614035 1.012018 0.47 Lower Upper Mean Value Frequency Bound Bound 0.96 1.43 1.195 1.43 1.90 1.665 1.90 2.37 2.135 2.37 2.84 2.605 2.84 3.31 3.075 12 3.31 3.78 3.545 10 3.78 4.25 4.015 4.25 4.72 4.485 4.72 5.19 4.955 5.19 5.66 5.425 Total 50 Frequency Distribution of P.I. (10 Robots, Speed Ratio: 0.2) (Mean: 3.56, Standard Deviation: 0.96) 25 20 Frequency No. TimeSteps P.I. 241 2.213001 127 4.199475 136 3.921569 241 2.213001 163 3.271984 167 3.193613 169 3.155819 357 1.493931 177 3.013183 10 148 3.603604 11 98 5.442177 12 169 3.155819 13 95 5.614035 14 152 3.508772 15 133 4.010025 16 118 4.519774 17 159 3.354298 18 527 1.012018 19 151 3.532009 20 127 4.199475 21 199 2.680067 22 157 3.397028 23 110 4.848485 24 172 3.100775 25 138 3.864734 26 297 1.795735 27 111 4.804805 28 162 3.292181 29 164 3.252033 30 168 3.174603 31 125 4.266667 32 142 3.755869 33 146 3.652968 34 147 3.628118 35 115 4.637681 36 188 2.836879 37 118 4.519774 38 103 5.177994 39 184 2.898551 40 141 3.782506 41 205 2.601626 42 170 3.137255 43 162 3.292181 44 161 3.312629 45 140 3.809524 46 130 4.102564 47 158 3.375527 48 98 5.442177 49 194 2.749141 50 125 4.266667 Mean 3.56168652 Std Dev 0.96296439 15 Data Mean 10 0 P.I. Note: P.I. is the non-dimensional performance index. TimeSteps is the number of time steps the robots spent to encircle the target. Category Length = (Max P.I. – Min P.I.) / Number of Categories 113 10 Robots, Speed Ratio: 0.3 Number of Categories Max P.I. Min P.I. Category Length Category 10 10 4.071247 1.228879 0.29 Lower Upper Mean Value Frequency Bound Bound 1.20 1.49 1.35 1.49 1.78 1.635 1.78 2.07 1.925 2.07 2.36 2.215 2.36 2.65 2.505 11 2.65 2.94 2.795 2.94 3.23 3.085 3.23 3.52 3.375 3.52 3.81 3.665 3.81 4.10 3.955 Total 50 Frequency Distribution of P.I. (10 Robots, Speed Ratio: 0.3) (Mean: 2.62, Standard Deviation: 0.67) 25 20 Frequency No. TimeSteps P.I. 168 3.174603 164 3.252033 179 2.979516 380 1.403509 313 1.70394 152 3.508772 434 1.228879 190 2.807018 195 2.735043 10 140 3.809524 11 220 2.424242 12 219 2.435312 13 258 2.067183 14 185 2.882883 15 195 2.735043 16 153 3.485839 17 193 2.763385 18 201 2.6534 19 180 2.962963 20 224 2.380952 21 274 1.946472 22 178 2.996255 23 230 2.318841 24 370 1.441441 25 173 3.082852 26 223 2.391629 27 200 2.666667 28 164 3.252033 29 402 1.3267 30 222 2.402402 31 142 3.755869 32 253 2.108037 33 290 1.83908 34 204 2.614379 35 198 2.693603 36 236 2.259887 37 336 1.587302 38 204 2.614379 39 204 2.614379 40 131 4.071247 41 206 2.588997 42 159 3.354298 43 152 3.508772 44 218 2.446483 45 167 3.193613 46 262 2.035623 47 284 1.877934 48 156 3.418803 49 185 2.882883 50 211 2.527646 Mean 2.6242509 Std Dev 0.66974169 15 Data 10 Mean 0 0.5 1.5 2.5 3.5 4.5 P.I. Note: P.I. is the non-dimensional performance index. TimeSteps is the number of time steps the robots spent to encircle the target. Category Length = (Max P.I. – Min P.I.) / Number of Categories 114 10 Robots, Speed Ratio: 0.4 Number of Categories Max P.I. Min P.I. Category Length Category 10 10 3.155819 0.554401 0.27 Lower Upper Mean Value Frequency Bound Bound 0.50 0.77 0.635 0.77 1.04 0.905 1.04 1.31 1.175 1.31 1.58 1.445 1.58 1.85 1.715 1.85 2.12 1.985 14 2.12 2.39 2.255 2.39 2.66 2.525 2.66 2.93 2.795 2.93 3.20 3.065 Total 50 Frequency Distribution of P.I. (10 Robots, Speed Ratio: 0.4) (Mean: 1.99, Standard Deviation: 0.55) 25 20 Frequency No. TimeSteps P.I. 353 1.510859 318 1.677149 275 1.939394 242 2.203857 357 1.493931 255 2.091503 265 2.012579 215 2.48062 234 2.279202 10 278 1.918465 11 243 2.194787 12 235 2.269504 13 288 1.851852 14 218 2.446483 15 962 0.554401 16 227 2.349486 17 219 2.435312 18 196 2.721088 19 237 2.250352 20 300 1.777778 21 296 1.801802 22 169 3.155819 23 273 1.953602 24 527 1.012018 25 175 3.047619 26 263 2.027883 27 208 2.564103 28 354 1.506591 29 300 1.777778 30 425 1.254902 31 311 1.714898 32 264 2.020202 33 283 1.88457 34 329 1.621074 35 191 2.792321 36 600 0.888889 37 325 1.641026 38 211 2.527646 39 420 1.269841 40 236 2.259887 41 264 2.020202 42 558 0.955795 43 352 1.515152 44 184 2.898551 45 268 1.99005 46 207 2.57649 47 283 1.88457 48 268 1.99005 49 224 2.380952 50 277 1.925391 Mean 1.98636552 Std Dev 0.54737961 15 Data Mean 10 0 0.5 1.5 2.5 3.5 P.I. Note: P.I. is the non-dimensional performance index. TimeSteps is the number of time steps the robots spent to encircle the target. Category Length = (Max P.I. – Min P.I.) / Number of Categories 115 10 Robots, Speed Ratio: 0.5 Number of Categories Max P.I. Min P.I. Category Length Category 10 10 2.391629 0.932401 0.15 Lower Upper Mean Value Frequency Bound Bound 0.91 1.06 0.985 1.06 1.21 1.135 1.21 1.36 1.285 10 1.36 1.51 1.435 1.51 1.66 1.585 1.66 1.81 1.735 1.81 1.96 1.885 1.96 2.11 2.035 2.11 2.26 2.185 2.26 2.41 2.335 Total 50 Frequency Distribution of P.I. (10 Robots, Speed Ratio: 0.5) (Mean: 1.61, Standard Deviation: 0.38) 25 20 Frequency No. TimeSteps P.I. 572 0.932401 411 1.297648 231 2.308802 396 1.346801 294 1.814059 366 1.457195 247 2.159244 335 1.59204 344 1.550388 10 321 1.661475 11 302 1.766004 12 254 2.099738 13 422 1.263823 14 467 1.142041 15 336 1.587302 16 277 1.925391 17 259 2.059202 18 263 2.027883 19 418 1.275917 20 355 1.502347 21 332 1.606426 22 288 1.851852 23 311 1.714898 24 555 0.960961 25 353 1.510859 26 418 1.275917 27 243 2.194787 28 415 1.285141 29 262 2.035623 30 258 2.067183 31 389 1.371037 32 257 2.075227 33 406 1.313629 34 441 1.209373 35 503 1.060305 36 247 2.159244 37 301 1.771872 38 432 1.234568 39 387 1.378122 40 318 1.677149 41 314 1.698514 42 246 2.168022 43 357 1.493931 44 357 1.493931 45 223 2.391629 46 316 1.687764 47 536 0.995025 48 407 1.310401 49 360 1.481481 50 402 1.3267 Mean 1.61142544 Std Dev 0.37847421 15 Data Mean 10 0 0.5 1.5 2.5 P.I. Note: P.I. is the non-dimensional performance index. TimeSteps is the number of time steps the robots spent to encircle the target. Category Length = (Max P.I. – Min P.I.) / Number of Categories 116 10 Robots, Speed Ratio: 0.6 Number of Categories Max P.I. Min P.I. Category Length Category 10 10 2.141901 0.614439 0.16 Lower Upper Mean Value Frequency Bound Bound 0.58 0.74 0.66 0.74 0.90 0.82 0.90 1.06 0.98 1.06 1.22 1.14 1.22 1.38 1.3 1.38 1.54 1.46 13 1.54 1.70 1.62 1.70 1.86 1.78 1.86 2.02 1.94 2.02 2.18 2.1 Total 50 Frequency Distribution of P.I. (10 Robots, Speed Ratio: 0.6) (Mean: 1.32, Standard Deviation: 0.30) 25 20 Frequency No. TimeSteps P.I. 268 1.99005 578 0.922722 294 1.814059 464 1.149425 351 1.519468 377 1.414677 249 2.141901 401 1.330008 347 1.536984 10 382 1.396161 11 436 1.223242 12 431 1.237432 13 442 1.206637 14 506 1.054018 15 334 1.596806 16 370 1.441441 17 348 1.532567 18 609 0.875753 19 336 1.587302 20 518 1.029601 21 395 1.350211 22 767 0.69535 23 370 1.441441 24 369 1.445348 25 402 1.3267 26 345 1.545894 27 581 0.917958 28 454 1.174743 29 321 1.661475 30 539 0.989487 31 412 1.294498 32 398 1.340034 33 370 1.441441 34 380 1.403509 35 349 1.528176 36 468 1.139601 37 523 1.019758 38 414 1.288245 39 359 1.485608 40 511 1.043705 41 343 1.554908 42 312 1.709402 43 449 1.187825 44 456 1.169591 45 571 0.934034 46 525 1.015873 47 342 1.559454 48 868 0.614439 49 345 1.545894 50 380 1.403509 Mean 1.3245673 Std Dev 0.30378509 15 Data Mean 10 0 0.5 1.5 2.5 P.I. Note: P.I. is the non-dimensional performance index. TimeSteps is the number of time steps the robots spent to encircle the target. Category Length = (Max P.I. – Min P.I.) / Number of Categories 117 10 Robots, Speed Ratio: 0.7 Number of Categories Max P.I. Min P.I. Category Length Category 10 10 1.641026 0.289698 0.14 Lower Upper Mean Value Frequency Bound Bound 0.26 0.40 0.33 0.40 0.54 0.47 0.54 0.68 0.61 0.68 0.82 0.75 0.82 0.96 0.89 0.96 1.10 1.03 12 1.10 1.24 1.17 11 1.24 1.38 1.31 1.38 1.52 1.45 1.52 1.66 1.59 Total 50 Frequency Distribution of P.I. (10 Robots, Speed Ratio: 0.7) (Mean: 1.09, Standard Deviation: 0.28) 25 20 Frequency No. TimeSteps P.I. 347 1.536984 433 1.231717 597 0.893356 529 1.008192 421 1.266825 608 0.877193 481 1.108801 454 1.174743 538 0.991326 10 482 1.106501 11 384 1.388889 12 393 1.357082 13 416 1.282051 14 498 1.07095 15 480 1.111111 16 328 1.626016 17 482 1.106501 18 926 0.575954 19 513 1.039636 20 325 1.641026 21 453 1.177336 22 523 1.019758 23 393 1.357082 24 379 1.407212 25 562 0.948992 26 584 0.913242 27 571 0.934034 28 524 1.017812 29 515 1.035599 30 1841 0.289698 31 483 1.10421 32 564 0.945626 33 669 0.79721 34 451 1.182557 35 1633 0.326597 36 500 1.066667 37 453 1.177336 38 412 1.294498 39 659 0.809307 40 507 1.05194 41 458 1.164483 42 514 1.037613 43 341 1.564027 44 573 0.930774 45 520 1.025641 46 670 0.79602 47 364 1.465201 48 407 1.310401 49 548 0.973236 50 577 0.924321 Mean 1.08886568 Std Dev 0.27597513 15 Data Mean 10 0 0.5 1.5 P.I. Note: P.I. is the non-dimensional performance index. TimeSteps is the number of time steps the robots spent to encircle the target. Category Length = (Max P.I. – Min P.I.) / Number of Categories 118 10 Robots, Speed Ratio: 0.8 Number of Categories Max P.I. Min P.I. Category Length Category 10 10 1.485608 0.296296 0.12 Lower Upper Mean Value Frequency Bound Bound 0.29 0.41 0.35 0.41 0.53 0.47 0.53 0.65 0.59 0.65 0.77 0.71 0.77 0.89 0.83 0.89 1.01 0.95 11 1.01 1.13 1.07 1.13 1.25 1.19 12 1.25 1.37 1.31 1.37 1.49 1.43 Total 50 Frequency Distribution of P.I. (10 Robots, Speed Ratio: 0.8) (Mean: 1.00, Standard Deviation: 0.25) 25 20 Frequency No. TimeSteps P.I. 792 0.673401 455 1.172161 507 1.05194 359 1.485608 406 1.313629 459 1.161946 702 0.759734 582 0.91638 461 1.156905 10 892 0.597907 11 453 1.177336 12 438 1.217656 13 560 0.952381 14 458 1.164483 15 887 0.601278 16 483 1.10421 17 520 1.025641 18 1800 0.296296 19 425 1.254902 20 748 0.713012 21 421 1.266825 22 441 1.209373 23 499 1.068804 24 391 1.364024 25 672 0.793651 26 657 0.811771 27 536 0.995025 28 756 0.705467 29 480 1.111111 30 445 1.198502 31 553 0.964436 32 768 0.694444 33 423 1.260835 34 444 1.201201 35 549 0.971463 36 391 1.364024 37 468 1.139601 38 541 0.985829 39 439 1.214882 40 551 0.967937 41 554 0.962696 42 955 0.558464 43 571 0.934034 44 446 1.195815 45 673 0.792472 46 546 0.976801 47 866 0.615858 48 681 0.783162 49 539 0.989487 50 510 1.045752 Mean 0.99881104 Std Dev 0.24838854 15 Data Mean 10 0 0.5 1.5 P.I. Note: P.I. is the non-dimensional performance index. TimeSteps is the number of time steps the robots spent to encircle the target. Category Length = (Max P.I. – Min P.I.) / Number of Categories 119 10 Robots, Speed Ratio: 0.9 Number of Categories Max P.I. Min P.I. Category Length Category 10 10 1.502347 0.340353 0.12 Lower Upper Mean Value Frequency Bound Bound 0.32 0.44 0.38 0.44 0.56 0.5 0.56 0.68 0.62 0.68 0.80 0.74 0.80 0.92 0.86 13 0.92 1.04 0.98 10 1.04 1.16 1.1 10 1.16 1.28 1.22 1.28 1.40 1.34 1.40 1.52 1.46 Total 50 Frequency Distribution of P.I. (10 Robots, Speed Ratio: 0.9) (Mean: 0.89, Standard Deviation: 0.19) 25 20 Frequency No. TimeSteps P.I. 595 0.896359 696 0.766284 815 0.654397 770 0.692641 629 0.847907 548 0.973236 581 0.917958 903 0.590624 695 0.767386 10 510 1.045752 11 623 0.856073 12 870 0.613027 13 355 1.502347 14 460 1.15942 15 759 0.702679 16 508 1.049869 17 586 0.910125 18 652 0.817996 19 547 0.975015 20 581 0.917958 21 623 0.856073 22 593 0.899382 23 761 0.700832 24 614 0.868621 25 683 0.780869 26 523 1.019758 27 618 0.862999 28 520 1.025641 29 671 0.794834 30 794 0.671704 31 486 1.097394 32 522 1.021711 33 685 0.778589 34 538 0.991326 35 559 0.954085 36 505 1.056106 37 477 1.118099 38 574 0.929152 39 480 1.111111 40 467 1.142041 41 566 0.942285 42 651 0.819252 43 935 0.57041 44 489 1.090661 45 652 0.817996 46 546 0.976801 47 496 1.075269 48 1567 0.340353 49 852 0.625978 50 713 0.748013 Mean 0.88688796 Std Dev 0.19479622 15 Data 10 Mean 0 0.5 1.5 P.I. Note: P.I. is the non-dimensional performance index. TimeSteps is the number of time steps the robots spent to encircle the target. Category Length = (Max P.I. – Min P.I.) / Number of Categories 120 8.2 Hardware Implementation Results (Chapter 5) Robots: Speed Ratio: 0.2 No. Time Taken (s) 40 100 210 72 93 153 32 120 72 10 170 Mean Mean (Simulation) Std Dev (Simulation) Speed Ratio: 0.3 P.I. 4.00 1.60 0.76 2.22 1.72 1.05 5.00 1.33 2.22 0.94 2.08 2.14 1.22 No. Time Taken (s) 104 72 75 103 69 78 90 89 80 10 74 Mean Mean (Simulation) Std Dev (Simulation) Speed Ratio: 0.4 P.I. 1.54 2.22 2.13 1.55 2.32 2.05 1.78 1.80 2.00 2.16 1.96 1.44 0.93 No. Time Taken (s) 189 236 53 187 88 90 68 109 66 10 90 Mean Mean (Simulation) Std Dev (Simulation) P.I. 0.85 0.68 3.02 0.86 1.82 1.78 2.35 1.47 2.42 1.78 1.70 1.51 0.83 Robots: Speed Ratio: 0.2 No. Time Taken (s) 29 53 35 88 100 25 29 41 58 10 64 Mean Mean (Simulation) Std Dev (Simulation) Speed Ratio: 0.3 P.I. 5.52 3.02 4.57 1.82 1.60 6.40 5.52 3.90 2.76 2.50 3.76 3.18 1.32 No. Time Taken (s) 65 80 92 67 75 83 66 53 103 10 64 Mean Mean (Simulation) Std Dev (Simulation) Speed Ratio: 0.4 P.I. 2.46 2.00 1.74 2.39 2.13 1.93 2.42 3.02 1.55 2.50 2.21 2.10 0.88 No. Time Taken (s) 84 65 60 73 61 69 113 92 72 10 117 Mean Mean (Simulation) Std Dev (Simulation) P.I. 1.90 2.46 2.67 2.19 2.62 2.32 1.42 1.74 2.22 1.37 2.09 1.75 0.61 Note: P.I. is the non-dimensional performance index. 121 Robots: Speed Ratio: 0.3 Speed Ratio: 0.2 No. Time Taken (s) 25 39 33 60 69 46 93 43 73 10 44 Mean Mean (Simulation) Std Dev (Simulation) P.I. 6.40 4.10 4.85 2.67 2.32 3.48 1.72 3.72 2.19 3.64 3.51 3.51 0.97 No. Time Taken (s) 65 62 60 95 77 63 80 51 41 10 59 Mean Mean (Simulation) Std Dev (Simulation) Speed Ratio: 0.4 P.I. 2.46 2.58 2.67 1.68 2.08 2.54 2.00 3.14 3.90 2.71 2.58 2.41 0.84 No. Time Taken (s) 76 63 96 65 58 81 62 110 113 10 124 Mean Mean (Simulation) Std Dev (Simulation) P.I. 2.11 2.54 1.67 2.46 2.76 1.98 2.58 1.45 1.42 1.29 2.02 2.01 0.51 Robots: Speed Ratio: 0.2 No. Time Taken (s) 51 30 37 43 34 40 51 44 53 10 61 Mean Mean (Simulation) Std Dev (Simulation) Speed Ratio: 0.3 P.I. 3.14 5.33 4.32 3.72 4.71 4.00 3.14 3.64 3.02 2.62 3.76 3.60 1.15 No. Time Taken (s) 40 88 65 80 62 76 58 74 44 10 72 Mean Mean (Simulation) Std Dev (Simulation) Speed Ratio: 0.4 P.I. 4.00 1.82 2.46 2.00 2.58 2.11 2.76 2.16 3.64 2.22 2.57 2.58 0.69 No. Time Taken (s) 55 93 73 64 77 90 57 123 80 10 74 Mean Mean (Simulation) Std Dev (Simulation) P.I. 2.91 1.72 2.19 2.50 2.08 1.78 2.81 1.30 2.00 2.16 2.14 2.11 0.65 Note: P.I. is the non-dimensional performance index. 122 [...]... shape of constant diameter like a Reuleaux triangle (see Figure 2.1) rather than a circle is formed In a Reuleaux triangle, arcs ab, bc and ca are drawn with radii equal to D, from the vertices c, a and b Triangle abc is also an equilateral triangle with sides equal to D In another study, Yun, Alptekin, and Albayrak have proposed an algorithm for robots to form a circle under limited sonar range [3] For. .. obstacle-avoidance behaviour has the highest priority followed by target -encirclement, and target- tracking For instance, if Obstacle-avoidance Target- circumnavigation Target- tracking S S S : Suppress Figure 3.6 Subsumption -based coordination of behaviours 16 any of the sonar sensors detects obstacles within the range of TS , regardless of which behaviour the robot is executing at that time, it will always cease... motor-schema behaviour- based system with other navigational behaviours so that a robotic team can reach navigational goal, avoid hazards and simultaneously maintain in their intended formation The other common architecture for behaviour- based robotic control system is called subsumption architecture This architecture was developed by Rodney Brooks [13] It is a purely reactive behaviour- based and layered control... cease the current behaviour and launch obstacle-avoidance behaviour 3.4 Encirclement Strategy The switching between target- tracking behaviour and target- circumnavigation behaviour plays a very important role to attain the desired encirclement (see Figure 3.7) When the target- circumnavigation behaviour is triggered, the robot is actually moving tangentially to the virtual circle around the target The robot... behaviours so that the robots can encircle the target successfully In the next chapter, we will present the simulation program we have developed to verify our encirclement algorithm Target Figure 3.7 Switching between target- tracking behaviour and target- circumnavigation behaviour 18 Target Figure 3.8 Obstacle-avoidance behaviour makes robots distribute more evenly around the target 19 Chapter 4 VALIDATION... validates the encirclement algorithm via a simulation program we have written The individual robotic behaviour and the process of encirclement implemented on simulation are shown here using snapshots of the program After discussing the experimental setup, we will analyse the simulation results by using a non-dimensional performance index A general law governing the performance and the speed ratio (target. .. process of encirclement, some parameters are critical to the overall performance of the results like the number of pursuer/hunter and the speed ratio of the pursuer/hunter to the target In this project, we will replicate the encirclement problem using multiple mobile robots in simulation and hardware experiments We will develop an algorithm for multiple mobile robots to encircle a dynamic target and finally... robotic behaviours we have designed for encirclement These three behaviours are obstacle-avoidance, target- tracking, and target- circumnavigation respectively After that, we give introduce a neural controller we have designed to execute the required robotic behaviours Finally, how the coordination of the behaviours can achieve encirclement will be explained 3.1 3.1.1 Three Basic Robotic Behaviours Obstacle-Avoidance... Fredslund and Mataric have suggested a general algorithm for robot formations using local sensing and minimal communication [4] Thus, no global positioning system is required In their algorithm, the robots are provided with information of the total number of participating robots A conductor/leader robot that will then decide on the type and the heading of the formation while the rest of the robots need... chapters are summarized below Chapter 2 surveys related research works related to the encirclement problem These works are divided into two sections: circle formation of distributed mobile robots and behaviour- based control of multiple robots Chapter 3 will first introduce the three basic robotic behaviours used in our project These three behaviours are obstacle-avoidance, target- tracking, and targetcircumnavigation . other navigational behaviours so that a robotic team can reach navigational goal, avoid hazards and simultaneously maintain in their intended formation. The other common architecture for behaviour- based. architecture that can coordinate the three behaviours in such a way that an encirclement of target will be completed. Chapter 4 validates the encirclement algorithm via a simulation program we have written bc and ca are drawn with radii equal to D, from the vertices c, a and b. Triangle abc is also an equilateral triangle with sides equal to D. In another study, Yun, Alptekin, and Albayrak have