Multicriteria Optimization Design for End Effector Mounting Bracket of a High Speed and Heavy Load Palletizing Robot

<table class="fixed-width table-group" id="tab3"><tr><td><table class="table"><colgroup><col style="width:6.31em"/><col style="width:4.78em"/><col style="width:4.93em"/><col style="width:7.48em"/></colgroup><tr><td class="thead-hr" colspan="4"><hr/></td></tr><tr class="thead"><td class="align_left">Modal order</td><td class="align_center"><span style="width: 47.3159ptpx;"><svg height="11.9316pt" id="M11" style="vertical-align:-3.2957pt" version="1.1" viewbox="-0.0498162 -8.6359 47.3159 11.9316" width="47.3159pt" 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="M584 650H137L131 622C214 614 217 612 200 521L125 127C109 41 101 35 23 28L17 0H288L294 28C201 35 193 42 209 128L242 309H348C440 309 442 300 443 226H471L510 422H482C452 354 449 348 357 348H251L295 575C302 609 304 615 338 615H426C502 615 517 604 526 581C534 560 536 524 537 492L565 494C574 554 583 631 584 650Z" id="g113-71"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.786,3.132)"><path d="M303 37L293 66C280 59 256 49 231 49S173 63 173 143V396H281C294 404 298 428 287 439H173V575L158 577L92 509V439H46L17 408L22 396H92V107C92 27 128 -12 192 -12C203 -12 218 -9 237 2L303 37Z" id="g58-114"></path></g><g transform="matrix(.0091,0,0,-0.0091,9.58,3.132)"><path d="M384 109C349 76 310 59 268 59C197 59 123 113 120 245C240 248 367 259 384 262C403 266 407 275 407 296C407 376 338 451 254 451H253C201 451 148 425 105 378C65 334 39 272 39 203C39 88 112 -12 235 -12C266 -12 339 5 402 84L384 109ZM230 411C285 411 319 365 319 313C319 298 313 293 294 293C236 291 180 290 125 290C141 370 185 411 230 411Z" id="g58-99"></path></g><g transform="matrix(.0091,0,0,-0.0091,13.502,3.132)"><path d="M327 325C324 350 312 408 304 433C283 442 256 451 213 451C120 451 59 391 59 320C59 243 125 210 188 181C237 158 266 136 266 93C266 52 231 24 195 24C134 24 93 81 74 148L42 143C42 106 51 37 59 23C78 8 125 -12 177 -12C257 -12 345 35 345 127C345 198 292 234 216 264C171 283 131 305 131 348C131 387 156 416 195 416C243 416 277 378 297 317L327 325Z" id="g58-113"></path></g><g transform="matrix(.0091,0,0,-0.0091,16.877,3.132)"><path d="M303 37L293 66C280 59 256 49 231 49S173 63 173 143V396H281C294 404 298 428 287 439H173V575L158 577L92 509V439H46L17 408L22 396H92V107C92 27 128 -12 192 -12C203 -12 218 -9 237 2L303 37Z" id="g58-114"></path></g><g transform="matrix(.0091,0,0,-0.0091,22.804,3.132)"><path d="M810 0V30C751 35 744 40 744 106V296C744 396 694 451 614 451C583 451 558 438 537 425C512 409 485 391 454 368C434 420 390 451 342 451C309 451 285 439 260 424C228 405 206 387 185 372V453C141 434 90 421 43 413V386C102 378 105 373 105 309V106C105 41 96 35 29 30V0H243V30C194 35 185 41 185 106V335C216 360 256 389 297 389C357 389 384 348 384 275V106C384 41 374 35 312 30V0H530V30C475 35 465 41 465 104V294C465 313 464 324 463 336C502 367 539 389 576 389C637 389 663 346 663 274V109C663 41 653 35 593 30V0H810Z" id="g58-107"></path></g><g transform="matrix(.0091,0,0,-0.0091,30.366,3.132)"><path d="M262 451C169 451 38 375 38 210C38 98 121 -12 261 -12C360 -12 481 64 481 227C481 351 386 451 262 451ZM249 414C337 414 385 320 385 204C385 70 333 25 272 25C189 25 134 117 134 241C134 355 189 414 249 414Z" id="g58-109"></path></g><g transform="matrix(.0091,0,0,-0.0091,35.171,3.132)"><path d="M526 56L493 58C457 61 448 68 448 119V710C411 697 344 682 288 675V646C362 641 367 639 367 573V439C344 448 315 451 300 451C164 451 40 342 40 202C40 61 147 -12 227 -12C239 -12 266 -7 305 16L367 53V-12C427 10 505 21 526 26V56ZM367 88C342 69 306 54 271 54C207 54 133 111 133 229C133 373 219 409 264 409C301 409 342 394 367 359V88Z" id="g58-98"></path></g><g transform="matrix(.0091,0,0,-0.0091,40.067,3.132)"><path d="M442 39L431 68C420 63 406 57 394 57C377 57 359 71 359 118V301C359 353 349 393 314 424C291 443 261 451 230 451C171 438 107 401 79 381C59 366 46 355 46 340C46 316 72 295 91 295C106 295 117 303 122 319C131 350 139 371 152 384C163 397 178 404 197 404C246 404 281 365 281 293V278C260 262 186 232 124 213C69 194 41 161 41 112C41 47 92 -12 163 -12C195 -12 243 25 283 51C288 35 293 22 308 7C319 -4 339 -12 354 -12L442 39ZM281 87C262 68 226 51 199 51C169 51 129 74 129 126C129 162 151 181 190 198C211 208 259 229 281 240V87Z" id="g58-95"></path></g><g transform="matrix(.0091,0,0,-0.0091,44.189,3.132)"><path d="M244 0V30C178 35 170 41 170 107V710C136 697 72 680 18 675V647C83 642 89 639 89 572V107C89 43 80 35 16 30V0H244Z" id="g58-106"></path></g></svg></span></td><td class="align_center"><span style="width: 49.4168ptpx;"><svg height="11.9316pt" id="M12" style="vertical-align:-3.2957pt" version="1.1" viewbox="-0.0498162 -8.6359 49.4168 11.9316" width="49.4168pt" 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="M584 650H137L131 622C214 614 217 612 200 521L125 127C109 41 101 35 23 28L17 0H288L294 28C201 35 193 42 209 128L242 309H348C440 309 442 300 443 226H471L510 422H482C452 354 449 348 357 348H251L295 575C302 609 304 615 338 615H426C502 615 517 604 526 581C534 560 536 524 537 492L565 494C574 554 583 631 584 650Z" id="g113-71"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.786,3.132)"><path d="M56 439L27 407L32 396H103V106C103 41 96 35 31 30V0H271V29C191 37 184 40 184 110V396H293C304 404 309 428 303 439H184V473C183 558 195 604 207 623S234 653 261 653C293 653 322 635 341 616C350 607 360 606 370 614C381 622 386 631 389 638C394 652 393 664 384 675C366 696 339 708 303 710C263 705 225 686 190 655C137 608 121 561 114 538S103 487 103 456V439H56Z" id="g58-100"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.274,3.132)"><path d="M384 109C349 76 310 59 268 59C197 59 123 113 120 245C240 248 367 259 384 262C403 266 407 275 407 296C407 376 338 451 254 451H253C201 451 148 425 105 378C65 334 39 272 39 203C39 88 112 -12 235 -12C266 -12 339 5 402 84L384 109ZM230 411C285 411 319 365 319 313C319 298 313 293 294 293C236 291 180 290 125 290C141 370 185 411 230 411Z" id="g58-99"></path></g><g transform="matrix(.0091,0,0,-0.0091,14.223,3.132)"><path d="M810 0V30C751 35 744 40 744 106V296C744 396 694 451 614 451C583 451 558 438 537 425C512 409 485 391 454 368C434 420 390 451 342 451C309 451 285 439 260 424C228 405 206 387 185 372V453C141 434 90 421 43 413V386C102 378 105 373 105 309V106C105 41 96 35 29 30V0H243V30C194 35 185 41 185 106V335C216 360 256 389 297 389C357 389 384 348 384 275V106C384 41 374 35 312 30V0H530V30C475 35 465 41 465 104V294C465 313 464 324 463 336C502 367 539 389 576 389C637 389 663 346 663 274V109C663 41 653 35 593 30V0H810Z" id="g58-107"></path></g><g transform="matrix(.0091,0,0,-0.0091,24.9,3.132)"><path d="M810 0V30C751 35 744 40 744 106V296C744 396 694 451 614 451C583 451 558 438 537 425C512 409 485 391 454 368C434 420 390 451 342 451C309 451 285 439 260 424C228 405 206 387 185 372V453C141 434 90 421 43 413V386C102 378 105 373 105 309V106C105 41 96 35 29 30V0H243V30C194 35 185 41 185 106V335C216 360 256 389 297 389C357 389 384 348 384 275V106C384 41 374 35 312 30V0H530V30C475 35 465 41 465 104V294C465 313 464 324 463 336C502 367 539 389 576 389C637 389 663 346 663 274V109C663 41 653 35 593 30V0H810Z" id="g58-107"></path></g><g transform="matrix(.0091,0,0,-0.0091,32.462,3.132)"><path d="M262 451C169 451 38 375 38 210C38 98 121 -12 261 -12C360 -12 481 64 481 227C481 351 386 451 262 451ZM249 414C337 414 385 320 385 204C385 70 333 25 272 25C189 25 134 117 134 241C134 355 189 414 249 414Z" id="g58-109"></path></g><g transform="matrix(.0091,0,0,-0.0091,37.267,3.132)"><path d="M526 56L493 58C457 61 448 68 448 119V710C411 697 344 682 288 675V646C362 641 367 639 367 573V439C344 448 315 451 300 451C164 451 40 342 40 202C40 61 147 -12 227 -12C239 -12 266 -7 305 16L367 53V-12C427 10 505 21 526 26V56ZM367 88C342 69 306 54 271 54C207 54 133 111 133 229C133 373 219 409 264 409C301 409 342 394 367 359V88Z" id="g58-98"></path></g><g transform="matrix(.0091,0,0,-0.0091,42.163,3.132)"><path d="M442 39L431 68C420 63 406 57 394 57C377 57 359 71 359 118V301C359 353 349 393 314 424C291 443 261 451 230 451C171 438 107 401 79 381C59 366 46 355 46 340C46 316 72 295 91 295C106 295 117 303 122 319C131 350 139 371 152 384C163 397 178 404 197 404C246 404 281 365 281 293V278C260 262 186 232 124 213C69 194 41 161 41 112C41 47 92 -12 163 -12C195 -12 243 25 283 51C288 35 293 22 308 7C319 -4 339 -12 354 -12L442 39ZM281 87C262 68 226 51 199 51C169 51 129 74 129 126C129 162 151 181 190 198C211 208 259 229 281 240V87Z" id="g58-95"></path></g><g transform="matrix(.0091,0,0,-0.0091,46.285,3.132)"><path d="M244 0V30C178 35 170 41 170 107V710C136 697 72 680 18 675V647C83 642 89 639 89 572V107C89 43 80 35 16 30V0H244Z" id="g58-106"></path></g></svg></span></td><td class="align_center">Relative error%</td></tr><tr><td class="thead-hr" colspan="4"><hr/></td></tr><tr><td class="align_left">1</td><td class="align_center">229.11</td><td class="align_center">228.34</td><td class="align_center">0.3404</td></tr><tr><td class="align_left">2</td><td class="align_center">448.41</td><td class="align_center">449.75</td><td class="align_center">0.3011</td></tr><tr><td class="align_left">3</td><td class="align_center">757.03</td><td class="align_center">732.32</td><td class="align_center">3.2628</td></tr><tr><td class="align_left">4</td><td class="align_center">789.54</td><td class="align_center">768.58</td><td class="align_center">2.6547</td></tr><tr><td class="align_left">5</td><td class="align_center">932.24</td><td class="align_center">942.74</td><td class="align_center">1.1263</td></tr><tr class="table-tr"><td colspan="4"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

<div>Comparison between the test model and the FEM model for the first five modal frequencies (Hz).</div>

Mathematical Problems in Engineering

tab3

Table 3

Table 3: Multicriteria Optimization Design for End Effector Mounting Bracket of a High Speed and Heavy Load Palletizing Robot 