The Optimal Collision Avoidance Trajectory Planning of Redundant Manipulators in the Process of Grinding Ceramic Billet Surface

<div>Replanning in a given time to obtain multiple collision avoidance trajectories. In the figure, <span class="nowrap"><svg height="11.8174pt" id="M12" style="vertical-align:-3.1815pt" version="1.1" viewbox="-0.0498162 -8.6359 22.8576 11.8174" width="22.8576pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M645 631C614 643 545 666 457 666C215 666 23 519 23 283C23 90 158 -16 337 -16C412 -16 489 2 522 10C543 39 590 127 606 167L580 181C519 89 459 18 348 18C201 18 122 136 122 287C122 464 244 632 435 632C544 632 602 595 608 472L639 475C643 526 645 581 645 631Z" id="g113-68"></path></g><g transform="matrix(.013,0,0,-0.013,8.684,0)"><path d="M699 369C699 549 575 666 407 666C186 666 23 488 23 278C23 101 145 -16 312 -16C535 -16 699 153 699 369ZM600 373C600 210 500 19 321 19C186 19 120 129 120 272C120 450 232 631 399 631C541 631 600 522 600 373Z" id="g113-80"></path></g><g transform="matrix(.0091,0,0,-0.0091,17.745,3.132)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 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transform="matrix(.0091,0,0,-0.0091,17.745,3.132)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" id="g50-51"></path></g></svg>,</span> and <svg height="11.927pt" id="M14" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -8.6359 23.086 11.927" width="23.086pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M645 631C614 643 545 666 457 666C215 666 23 519 23 283C23 90 158 -16 337 -16C412 -16 489 2 522 10C543 39 590 127 606 167L580 181C519 89 459 18 348 18C201 18 122 136 122 287C122 464 244 632 435 632C544 632 602 595 608 472L639 475C643 526 645 581 645 631Z" id="g113-68"></path></g><g transform="matrix(.013,0,0,-0.013,8.684,0)"><path d="M699 369C699 549 575 666 407 666C186 666 23 488 23 278C23 101 145 -16 312 -16C535 -16 699 153 699 369ZM600 373C600 210 500 19 321 19C186 19 120 129 120 272C120 450 232 631 399 631C541 631 600 522 600 373Z" id="g113-80"></path></g><g transform="matrix(.0091,0,0,-0.0091,17.745,3.132)"><path d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 412 102L462 304C492 438 460 451 436 451S388 441 356 423C311 397 229 336 170 256H168L189 340C208 418 200 451 177 451C144 451 82 413 24 352L40 322C65 345 98 369 108 369C114 369 119 363 112 335L28 -3L36 -12C54 -3 83 4 112 10C125 73 138 122 153 174C205 258 328 382 376 382C395 382 399 369 384 305L330 93C312 25 323 -12 351 -12C378 -12 439 21 504 89Z" id="g50-111"></path></g></svg> represent the different obstacles in the environment. <span class="nowrap"><svg height="11.9218pt" id="M15" style="vertical-align:-3.181499pt" version="1.1" viewbox="-0.0498162 -8.7403 14.937 11.9218" width="14.937pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g 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393 665C162 665 42 496 42 319C42 232 74 149 131 87C187 26 281 -9 343 -13C470 -114 547 -148 579 -156C617 -166 676 -172 726 -164L734 -137ZM358 30C254 60 195 186 195 343S263 623 372 623C491 623 571 502 571 303C571 139 512 56 466 37C458 43 449 50 440 56L358 30Z" id="g13-79"></path></g><g transform="matrix(.0091,0,0,-0.0091,9.854,3.132)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" id="g50-51"></path></g></svg>,</span> <span class="nowrap"><svg height="12.0314pt" id="M17" style="vertical-align:-3.2911pt" version="1.1" viewbox="-0.0498162 -8.7403 14.937 12.0314" width="14.937pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M734 -137C662 -100 567 -37 503 8C623 45 724 166 724 333C724 530 575 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xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M734 -137C662 -100 567 -37 503 8C623 45 724 166 724 333C724 530 575 665 393 665C162 665 42 496 42 319C42 232 74 149 131 87C187 26 281 -9 343 -13C470 -114 547 -148 579 -156C617 -166 676 -172 726 -164L734 -137ZM358 30C254 60 195 186 195 343S263 623 372 623C491 623 571 502 571 303C571 139 512 56 466 37C458 43 449 50 440 56L358 30Z" id="g13-79"></path></g><g transform="matrix(.0091,0,0,-0.0091,9.854,3.132)"><path d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 412 102L462 304C492 438 460 451 436 451S388 441 356 423C311 397 229 336 170 256H168L189 340C208 418 200 451 177 451C144 451 82 413 24 352L40 322C65 345 98 369 108 369C114 369 119 363 112 335L28 -3L36 -12C54 -3 83 4 112 10C125 73 138 122 153 174C205 258 328 382 376 382C395 382 399 369 384 305L330 93C312 25 323 -12 351 -12C378 -12 439 21 504 89Z" id="g50-111"></path></g></svg> denote the different collision avoidance trajectories.</div>

Mathematical Problems in Engineering

fig1

Figure 1

Figure 1: The Optimal Collision Avoidance Trajectory Planning of Redundant Manipulators in the Process of Grinding Ceramic Billet Surface 