An Optimization-Based Impedance Approach for Robot Force Regulation with Prescribed Force Limits

<div>Grasping force <svg height="11.7391pt" id="M269" style="vertical-align:-3.05101pt" version="1.1" viewbox="-0.0498162 -8.68809 16.5015 11.7391" width="16.5015pt" 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="M534 495C529 556 524 633 523 650H42V608C116 601 125 594 125 508V144C125 55 116 49 42 42V0H359V42C270 49 262 55 262 144V296H341C431 296 440 289 450 220H493V424H450C439 360 431 352 341 352H262V565C262 599 265 600 301 600H352C406 600 433 594 452 573C469 555 481 526 492 488L534 495Z" id="g22-68"></path><glyph.data ascent="1041" descent="-369" horiz-adv-x="565" vert-adv-y="565"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,7.345,3.001)"><path d="M226 352V458C164 439 105 428 40 420V387C87 376 95 372 95 322V108C95 49 87 44 31 39V0H310V39C231 45 226 49 226 108V289C243 329 263 341 282 341C299 341 313 334 330 321C337 315 347 312 361 317C388 327 411 355 411 393C411 428 382 457 342 457C308 457 268 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121C515 210 466 288 408 379C457 447 500 504 534 544C574 589 603 600 666 608V650H424V608C482 597 484 588 467 554C443 511 408 457 379 418C338 482 306 535 291 564C280 586 277 596 331 608V650H25V608C94 601 124 582 151 540C203 458 245 390 293 313C234 226 158 129 128 94C99 63 62 47 8 42V0H262V42C198 52 191 58 214 100C232 133 274 200 321 269C394 147 416 112 425 96C442 66 447 52 382 42V0H708Z" id="g22-86"></path><glyph.data ascent="1041" descent="-369" horiz-adv-x="702" vert-adv-y="702"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,9.126,3.001)"><path d="M226 352V458C164 439 105 428 40 420V387C87 376 95 372 95 322V108C95 49 87 44 31 39V0H310V39C231 45 226 49 226 108V289C243 329 263 341 282 341C299 341 313 334 330 321C337 315 347 312 361 317C388 327 411 355 411 393C411 428 382 457 342 457C308 457 268 419 228 352H226Z" id="g22-112"></path><glyph.data ascent="1041" descent="-369" horiz-adv-x="413" vert-adv-y="413"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,13.04,3.001)"><path d="M504 0V39C460 43 432 59 396 115L305 260C341 307 363 339 385 361C421 395 430 399 486 407V445H300V407C336 400 341 393 327 371C316 353 300 326 279 297L233 369C219 392 225 400 257 407V445H17V407C71 400 83 390 111 344L198 202C167 161 142 127 116 98C84 60 61 48 8 39V0H200V39C161 46 159 58 174 84C187 105 205 138 224 164L273 83C291 54 287 44 249 39V0H504Z" id="g22-118"></path><glyph.data ascent="1041" descent="-369" horiz-adv-x="491" vert-adv-y="491"></glyph.data></g></svg> for experiment 4 (<svg height="11.9661pt" id="M271" style="vertical-align:-3.16057pt" version="1.1" viewbox="-0.0498162 -8.80553 69.3754 11.9661" width="69.3754pt" 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="M693 0V42C639 49 615 60 580 153C514 324 454 491 390 665L333 651C266 480 197 310 133 149C98 62 80 49 19 42V0H249V42C177 52 173 63 190 114C230 242 269 346 315 477H319C364 352 418 204 452 109C469 63 464 51 402 42V0H693Z" id="g14-140"></path><glyph.data ascent="3452" descent="-2858" horiz-adv-x="712" vert-adv-y="712"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,9.291,3.001)"><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" id="g50-106"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="280" vert-adv-y="280"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,16.022,0)"><path d="M556 331V384H55V331H556ZM556 140V193H55V140H556Z" id="g54-34"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="611" vert-adv-y="611"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,27.64,0)"><path d="M780 659C713 654 666 640 635 610C611 587 601 562 588 495L578 442C560 343 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vert-adv-y="280"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,44.888,0)"><path d="M556 331V384H55V331H556ZM556 140V193H55V140H556Z" id="g54-34"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="611" vert-adv-y="611"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,56.507,0)"><path d="M389 0V32C297 37 291 46 291 118V635C234 612 176 595 109 583V556L161 554C203 551 207 548 207 497V118C207 46 201 37 110 32V0H389Z" id="g58-47"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,62.838,0)"><path d="M245 635C92 635 37 457 37 312C37 148 92 -12 244 -12C395 -12 449 165 449 312C449 469 394 635 245 635ZM243 598C332 598 358 454 358 312C358 172 334 26 245 26C158 26 128 174 128 313S151 598 243 598Z" id="g58-46"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g></svg>), constrained case.</div>

Mathematical Problems in Engineering

fig10

Figure 10

Figure 10: An Optimization-Based Impedance Approach for Robot Force Regulation with Prescribed Force Limits 