A Three-Dimensional Mass-Spring Walking Model Could Describe the Ground Reaction Forces

<div>The velocity of the center of mass can be decomposed into three mutually perpendicular velocity components as <span class="nowrap"><svg height="17.2718pt" id="M119" style="vertical-align:-5.282399pt" version="1.1" viewbox="-0.0498162 -11.9894 19.2908 17.2718" width="19.2908pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,-5.768)"><path d="M567 230V280H69V230H567Z"></path></g><g transform="matrix(.013,0,0,-0.013,-1.901,-5.768)"><path d="M901 255C861 290 790 361 716 442L694 427L796 280H69V230H874L901 255Z"></path></g><g transform="matrix(.013,0,0,-0.013,2.347,0)"><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><g 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style="vertical-align:-5.282399pt" version="1.1" viewbox="-0.0498162 -11.9894 22.5138 17.2718" width="22.5138pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,-5.768)"><path d="M567 230V280H69V230H567Z"></path></g><g transform="matrix(.013,0,0,-0.013,-1.901,-5.768)"><path d="M901 255C861 290 790 361 716 442L694 427L796 280H69V230H874L901 255Z"></path></g><g transform="matrix(.013,0,0,-0.013,2.347,0)"><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><g transform="matrix(.0091,0,0,-0.0091,10.578,3.132)"><path d="M482 498C482 606 448 710 328 710C138 710 24 416 24 218C24 99 58 -12 184 -12C373 -12 482 294 482 498ZM395 521C395 483 391 449 383 395H128C158 537 226 671 312 671C388 671 395 566 395 521ZM375 342C348 190 284 25 192 25C128 25 109 116 109 199C109 243 112 292 121 342H375Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,14.955,5.233)"><path d="M394 0V37C301 42 295 50 295 122V635C236 613 178 596 109 583V553L165 551C205 549 209 544 209 495V122C209 50 203 42 111 37V0H394Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,18.166,5.233)"><path d="M394 0V37C301 42 295 50 295 122V635C236 613 178 596 109 583V553L165 551C205 549 209 544 209 495V122C209 50 203 42 111 37V0H394Z"></path></g></svg>;</span> <span class="nowrap"><svg height="17.4153pt" id="M122" style="vertical-align:-5.2823pt" version="1.1" viewbox="-0.0498162 -12.133 19.2908 17.4153" width="19.2908pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,-5.911)"><path d="M567 230V280H69V230H567Z"></path></g><g transform="matrix(.013,0,0,-0.013,-1.901,-5.911)"><path d="M901 255C861 290 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<span class="nowrap"><svg height="15.4241pt" id="M123" style="vertical-align:-3.2911pt" version="1.1" viewbox="-0.0498162 -12.133 18.0997 15.4241" width="18.0997pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,-5.911)"><path d="M567 230V280H69V230H567Z"></path></g><g transform="matrix(.013,0,0,-0.013,-1.901,-5.911)"><path d="M901 255C861 290 790 361 716 442L694 427L796 280H69V230H874L901 255Z"></path></g><g transform="matrix(.013,0,0,-0.013,2.588,0)"><path d="M391 364C391 409 353 448 295 448C249 448 198 426 152 393C65 331 23 225 23 139C23 14 96 -12 146 -12C198 -12 280 9 367 101L351 124C300 78 242 48 194 48C129 48 109 107 109 162V191C208 213 391 266 391 364ZM313 350C313 305 268 261 113 223C132 334 187 381 217 398C227 404 244 405 261 405C290 405 313 385 313 350Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.578,3.132)"><path d="M238 675C244 702 241 710 231 710C220 710 155 682 74 673L70 643H101C147 643 151 635 142 594L39 111C18 11 36 -12 55 -12C89 -12 170 38 223 103L207 129C176 98 130 68 118 68C113 68 108 72 114 102L238 675Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,13.045,3.132)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"></path></g></svg>,</span> and <svg height="17.4153pt" id="M124" style="vertical-align:-5.2823pt" version="1.1" viewbox="-0.0498162 -12.133 22.5138 17.4153" width="22.5138pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,-5.911)"><path d="M567 230V280H69V230H567Z"></path></g><g transform="matrix(.013,0,0,-0.013,-1.901,-5.911)"><path d="M901 255C861 290 790 361 716 442L694 427L796 280H69V230H874L901 255Z"></path></g><g transform="matrix(.013,0,0,-0.013,2.588,0)"><path d="M391 364C391 409 353 448 295 448C249 448 198 426 152 393C65 331 23 225 23 139C23 14 96 -12 146 -12C198 -12 280 9 367 101L351 124C300 78 242 48 194 48C129 48 109 107 109 162V191C208 213 391 266 391 364ZM313 350C313 305 268 261 113 223C132 334 187 381 217 398C227 404 244 405 261 405C290 405 313 385 313 350Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.578,3.132)"><path d="M482 498C482 606 448 710 328 710C138 710 24 416 24 218C24 99 58 -12 184 -12C373 -12 482 294 482 498ZM395 521C395 483 391 449 383 395H128C158 537 226 671 312 671C388 671 395 566 395 521ZM375 342C348 190 284 25 192 25C128 25 109 116 109 199C109 243 112 292 121 342H375Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,14.955,5.233)"><path d="M394 0V37C301 42 295 50 295 122V635C236 613 178 596 109 583V553L165 551C205 549 209 544 209 495V122C209 50 203 42 111 37V0H394Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,18.166,5.233)"><path d="M394 0V37C301 42 295 50 295 122V635C236 613 178 596 109 583V553L165 551C205 549 209 544 209 495V122C209 50 203 42 111 37V0H394Z"></path></g></svg> represent the unit vectors of <span class="nowrap"><svg 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294 482 498ZM395 521C395 483 391 449 383 395H128C158 537 226 671 312 671C388 671 395 566 395 521ZM375 342C348 190 284 25 192 25C128 25 109 116 109 199C109 243 112 292 121 342H375Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,14.955,5.233)"><path d="M394 0V37C301 42 295 50 295 122V635C236 613 178 596 109 583V553L165 551C205 549 209 544 209 495V122C209 50 203 42 111 37V0H394Z"></path></g></svg>,</span> <span class="nowrap"><svg height="17.2718pt" id="M126" style="vertical-align:-5.282399pt" version="1.1" viewbox="-0.0498162 -11.9894 17.1442 17.2718" width="17.1442pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,-5.768)"><path d="M567 230V280H69V230H567Z"></path></g><g transform="matrix(.013,0,0,-0.013,-1.901,-5.768)"><path d="M901 255C861 290 790 361 716 442L694 427L796 280H69V230H874L901 255Z"></path></g><g transform="matrix(.013,0,0,-0.013,2.347,0)"><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><g transform="matrix(.0091,0,0,-0.0091,10.578,3.132)"><path d="M238 675C244 702 241 710 231 710C220 710 155 682 74 673L70 643H101C147 643 151 635 142 594L39 111C18 11 36 -12 55 -12C89 -12 170 38 223 103L207 129C176 98 130 68 118 68C113 68 108 72 114 102L238 675Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,12.817,5.233)"><path d="M394 0V37C301 42 295 50 295 122V635C236 613 178 596 109 583V553L165 551C205 549 209 544 209 495V122C209 50 203 42 111 37V0H394Z"></path></g></svg>,</span> and <span class="nowrap"><svg height="17.2718pt" id="M127" style="vertical-align:-5.282399pt" version="1.1" viewbox="-0.0498162 -11.9894 22.5138 17.2718" width="22.5138pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,-5.768)"><path d="M567 230V280H69V230H567Z"></path></g><g transform="matrix(.013,0,0,-0.013,-1.901,-5.768)"><path d="M901 255C861 290 790 361 716 442L694 427L796 280H69V230H874L901 255Z"></path></g><g transform="matrix(.013,0,0,-0.013,2.347,0)"><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><g transform="matrix(.0091,0,0,-0.0091,10.578,3.132)"><path d="M482 498C482 606 448 710 328 710C138 710 24 416 24 218C24 99 58 -12 184 -12C373 -12 482 294 482 498ZM395 521C395 483 391 449 383 395H128C158 537 226 671 312 671C388 671 395 566 395 521ZM375 342C348 190 284 25 192 25C128 25 109 116 109 199C109 243 112 292 121 342H375Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,14.955,5.233)"><path d="M394 0V37C301 42 295 50 295 122V635C236 613 178 596 109 583V553L165 551C205 549 209 544 209 495V122C209 50 203 42 111 37V0H394Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,18.166,5.233)"><path d="M394 0V37C301 42 295 50 295 122V635C236 613 178 596 109 583V553L165 551C205 549 209 544 209 495V122C209 50 203 42 111 37V0H394Z"></path></g></svg>,</span> respectively.</div>

Mathematical Problems in Engineering

fig15

Figure 15

Figure 15: A Three-Dimensional Mass-Spring Walking Model Could Describe the Ground Reaction Forces 