Discrete Globalised Dual Heuristic Dynamic Programming in Control of the Two-Wheeled Mobile Robot

<table>(a) Average of maximal values of the filtered tracking error for wheels 1 (<svg height="11.0849pt" id="M279" style="vertical-align:-4.87474pt" version="1.1" viewbox="-0.0498162 -6.21016 33.8539 11.0849" width="33.8539pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M352 391C352 416 319 448 267 448C236 448 173 423 147 400C107 364 96 332 96 304C96 248 143 210 193 181C241 153 258 124 258 100C258 72 232 38 184 38C151 38 107 66 81 108C77 114 64 116 55 111C34 99 23 84 23 65C23 29 81 -12 134 -12C220 -12 325 61 325 141C325 184 297 215 234 256C194 282 161 309 161 346C161 380 188 401 217 401C255 401 279 380 301 353C308 344 313 341 325 347C341 355 352 371 352 391Z" id="g113-116"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="375" vert-adv-y="375"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,5.013,3.264)"><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><glyph.data ascent="1007" descent="-369" horiz-adv-x="834" vert-adv-y="834"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,12.894,3.264)"><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><glyph.data ascent="1007" descent="-369" horiz-adv-x="449" vert-adv-y="449"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,17.152,3.264)"><path d="M482 0V30C420 38 407 48 370 103L273 248C308 299 331 332 352 357C387 397 401 404 463 409V439H277V409C318 405 324 400 307 369C290 337 270 307 251 279L194 367C175 396 180 402 220 409V439H18V409C74 401 87 395 117 349L204 213C173 168 147 130 119 95C78 46 67 38 5 30V0H195V30C144 38 143 47 162 79C181 112 202 149 225 181L297 70C314 43 307 35 264 30V0H482Z" id="g58-118"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="481" vert-adv-y="481"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,21.786,3.264)"><path d="M297 -163V-131C189 -122 187 -119 187 -41V582C187 660 190 663 297 672V704H122V-163H297Z" id="g50-92"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="358" vert-adv-y="358"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,25.181,3.264)"><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" id="g50-50"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,29.799,3.264)"><path d="M237 -163V704H62V672C170 663 172 660 172 582V-40C172 -118 170 -122 62 -131V-163H237Z" id="g50-94"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="358" vert-adv-y="358"></glyph.data></g></svg>) and 2 (<svg height="11.0849pt" id="M280" style="vertical-align:-4.87474pt" version="1.1" viewbox="-0.0498162 -6.21016 33.8539 11.0849" width="33.8539pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M352 391C352 416 319 448 267 448C236 448 173 423 147 400C107 364 96 332 96 304C96 248 143 210 193 181C241 153 258 124 258 100C258 72 232 38 184 38C151 38 107 66 81 108C77 114 64 116 55 111C34 99 23 84 23 65C23 29 81 -12 134 -12C220 -12 325 61 325 141C325 184 297 215 234 256C194 282 161 309 161 346C161 380 188 401 217 401C255 401 279 380 301 353C308 344 313 341 325 347C341 355 352 371 352 391Z" id="g113-116"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="375" vert-adv-y="375"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,5.013,3.264)"><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><glyph.data ascent="1007" descent="-369" horiz-adv-x="834" vert-adv-y="834"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,12.894,3.264)"><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><glyph.data ascent="1007" descent="-369" horiz-adv-x="449" vert-adv-y="449"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,17.152,3.264)"><path d="M482 0V30C420 38 407 48 370 103L273 248C308 299 331 332 352 357C387 397 401 404 463 409V439H277V409C318 405 324 400 307 369C290 337 270 307 251 279L194 367C175 396 180 402 220 409V439H18V409C74 401 87 395 117 349L204 213C173 168 147 130 119 95C78 46 67 38 5 30V0H195V30C144 38 143 47 162 79C181 112 202 149 225 181L297 70C314 43 307 35 264 30V0H482Z" id="g58-118"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="481" vert-adv-y="481"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,21.786,3.264)"><path d="M297 -163V-131C189 -122 187 -119 187 -41V582C187 660 190 663 297 672V704H122V-163H297Z" id="g50-92"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="358" vert-adv-y="358"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,25.181,3.264)"><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><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,29.799,3.264)"><path d="M237 -163V704H62V672C170 663 172 660 172 582V-40C172 -118 170 -122 62 -131V-163H237Z" id="g50-94"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="358" vert-adv-y="358"></glyph.data></g></svg>), (b) RMSE of the filtered tracking errors <svg height="11.0849pt" id="M281" style="vertical-align:-4.87474pt" version="1.1" viewbox="-0.0498162 -6.21016 17.081 11.0849" width="17.081pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M352 391C352 416 319 448 267 448C236 448 173 423 147 400C107 364 96 332 96 304C96 248 143 210 193 181C241 153 258 124 258 100C258 72 232 38 184 38C151 38 107 66 81 108C77 114 64 116 55 111C34 99 23 84 23 65C23 29 81 -12 134 -12C220 -12 325 61 325 141C325 184 297 215 234 256C194 282 161 309 161 346C161 380 188 401 217 401C255 401 279 380 301 353C308 344 313 341 325 347C341 355 352 371 352 391Z" id="g113-116"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="375" vert-adv-y="375"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,5.013,3.264)"><path d="M297 -163V-131C189 -122 187 -119 187 -41V582C187 660 190 663 297 672V704H122V-163H297Z" id="g50-92"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="358" vert-adv-y="358"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,8.408,3.264)"><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" id="g50-50"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,13.026,3.264)"><path d="M237 -163V704H62V672C170 663 172 660 172 582V-40C172 -118 170 -122 62 -131V-163H237Z" id="g50-94"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="358" vert-adv-y="358"></glyph.data></g></svg> and <svg height="11.0849pt" id="M282" style="vertical-align:-4.87474pt" version="1.1" viewbox="-0.0498162 -6.21016 17.081 11.0849" width="17.081pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M352 391C352 416 319 448 267 448C236 448 173 423 147 400C107 364 96 332 96 304C96 248 143 210 193 181C241 153 258 124 258 100C258 72 232 38 184 38C151 38 107 66 81 108C77 114 64 116 55 111C34 99 23 84 23 65C23 29 81 -12 134 -12C220 -12 325 61 325 141C325 184 297 215 234 256C194 282 161 309 161 346C161 380 188 401 217 401C255 401 279 380 301 353C308 344 313 341 325 347C341 355 352 371 352 391Z" id="g113-116"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="375" vert-adv-y="375"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,5.013,3.264)"><path d="M297 -163V-131C189 -122 187 -119 187 -41V582C187 660 190 663 297 672V704H122V-163H297Z" id="g50-92"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="358" vert-adv-y="358"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,8.408,3.264)"><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><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,13.026,3.264)"><path d="M237 -163V704H62V672C170 663 172 660 172 582V-40C172 -118 170 -122 62 -131V-163H237Z" id="g50-94"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="358" vert-adv-y="358"></glyph.data></g></svg>.</table>

Mathematical Problems in Engineering

fig17

Figure 17

Figure 17: Discrete Globalised Dual Heuristic Dynamic Programming in Control of the Two-Wheeled Mobile Robot 