Stiffness and Elastic Deformation of 4-DoF Parallel Manipulator with Three Asymmetrical Legs for Supporting Helicopter Rotor

<div>Elastic deformations of SPS-type legs and RRPR-type leg under <span class="nowrap"><svg height="13.9952pt" id="M235" style="vertical-align:-5.52893pt" version="1.1" viewbox="-0.0498162 -8.46627 39.4026 13.9952" width="39.4026pt" 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="M627 650H133L124 613C207 607 213 598 199 523L123 130C110 55 99 42 24 37L16 0H339L349 37C253 43 243 52 257 132L289 302H382C475 302 473 297 476 220H513L553 425H515C486 360 482 353 392 353H301L339 562C347 597 355 605 393 605H443C517 605 535 598 549 586C562 573 569 539 574 487L611 489L627 650Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,8.273,3.132)"><path d="M490 97L476 124C442 96 405 70 398 70C392 70 390 78 396 114C419 243 448 379 463 432L457 436C446 436 431 439 418 442C393 447 368 451 343 451C281 451 204 418 155 381C74 320 24 206 24 107C24 23 59 -12 88 -12C118 -12 155 5 191 34C236 70 290 122 328 177H330L312 84C296 0 311 -12 331 -12C355 -12 425 24 490 97ZM374 387C371 367 360 299 347 264C323 202 187 53 142 53C128 53 113 73 113 120C113 224 157 332 221 380C241 395 274 403 303 403C330 403 360 395 374 387Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,13.198,3.132)"><path d="M250 606C250 634 233 656 203 656C168 656 146 618 146 593C146 564 169 545 192 545C227 545 250 573 250 606ZM227 95L212 119C187 98 152 71 135 71C129 71 128 78 134 102L207 373C219 418 217 451 194 451C165 451 92 411 30 351L44 326C77 353 106 371 114 371C124 371 121 357 117 341L55 97C32 5 46 -12 70 -12C108 -12 191 51 227 95Z"></path></g><g transform="matrix(.013,0,0,-0.013,16.27,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"></path></g><g transform="matrix(.013,0,0,-0.013,23.588,0)"><path d="M627 650H133L124 613C207 607 213 598 199 523L123 130C110 55 99 42 24 37L16 0H339L349 37C253 43 243 52 257 132L289 302H382C475 302 473 297 476 220H513L553 425H515C486 360 482 353 392 353H301L339 562C347 597 355 605 393 605H443C517 605 535 598 549 586C562 573 569 539 574 487L611 489L627 650Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,31.861,3.132)"><path d="M387 400C387 425 348 451 303 451C247 451 176 414 132 376C69 322 24 228 24 148C24 43 74 -12 147 -12C211 -12 301 33 363 103L346 128C319 99 249 51 193 51C148 51 112 84 112 165C112 230 130 287 154 330C170 359 199 400 243 400C277 400 304 383 326 354C333 345 343 343 354 348C378 360 387 382 387 400Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,34.855,3.132)"><path d="M400 606C400 634 383 656 353 656C316 656 294 620 294 593C294 564 317 545 343 545C375 545 400 573 400 606ZM366 351C379 413 381 451 356 451C323 451 251 408 183 341L199 313C223 335 267 365 277 365C285 365 284 354 277 312C245 132 222 27 193 -100C182 -148 160 -188 131 -188C113 -188 90 -178 75 -170C64 -164 55 -168 48 -175C38 -185 24 -203 24 -222S48 -257 71 -257C89 -257 131 -241 186 -192C243 -141 286 -46 310 74L366 351Z"></path></g></svg>,</span> and <span class="nowrap"><svg height="12.2323pt" id="M236" style="vertical-align:-3.309319pt" version="1.1" viewbox="-0.0498162 -8.92298 17.0574 12.2323" width="17.0574pt" 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="M671 675H640C620 656 612 650 589 650H155C125 650 115 652 105 675H77C65 620 45 548 21 483H60C83 527 99 557 114 573C136 597 154 605 244 605H290L200 130C186 56 178 42 86 37L76 0H420L428 37C326 42 324 56 338 130L427 605H482C538 605 569 601 587 580C603 560 605 526 605 481L645 483C652 543 663 637 671 675Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,8.586,3.132)"><path d="M387 400C387 425 348 451 303 451C247 451 176 414 132 376C69 322 24 228 24 148C24 43 74 -12 147 -12C211 -12 301 33 363 103L346 128C319 99 249 51 193 51C148 51 112 84 112 165C112 230 130 287 154 330C170 359 199 400 243 400C277 400 304 383 326 354C333 345 343 343 354 348C378 360 387 382 387 400Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,12.329,3.132)"><path d="M219 86C216 168 211 250 206 337C201 410 189 448 163 448C131 448 79 396 43 344L60 322C91 359 110 375 118 375S132 358 136 298C141 238 152 81 155 -12H182C242 62 331 177 390 258C435 321 451 360 451 391C450 424 432 448 408 448C390 448 372 435 366 419C362 410 362 401 366 394C373 383 376 367 376 350C376 283 262 138 221 86H219Z"></path></g></svg>.</span></div>

Journal of Robotics

fig3

Figure 3

Figure 3: Stiffness and Elastic Deformation of 4-DoF Parallel Manipulator with Three Asymmetrical Legs for Supporting Helicopter Rotor 