Semianalytical Solutions of Relative Motions with Applications to Periodic Orbits about a Nominal Circular Orbit

<div>The rotating coordinate system centered at the Earth and the coordinate system centered at the equilibrium point specified by <span class="nowrap"><svg height="9.49473pt" id="M16" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -9.28833 6.59789 9.49473" width="6.59789pt" 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="M475 507C475 612 440 712 326 712C139 712 23 420 23 215C23 96 58 -12 180 -12C369 -12 475 293 475 507ZM391 522C391 486 387 448 379 394H126C155 538 222 677 310 677C386 677 391 571 391 522ZM373 346C344 193 283 22 189 22C126 22 106 114 106 196C106 243 111 293 118 346H373Z" id="g113-230"></path></g></svg>.</span> The position vectors of the follower satellite in these two coordinate systems are denoted by <svg height="11.6478pt" id="M17" style="vertical-align:-3.18153pt" version="1.1" viewbox="-0.0498162 -8.46627 13.7365 11.6478" width="13.7365pt" 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="M668 26C646 31 631 35 610 64C593 87 572 122 543 176L484 306C568 331 646 391 646 489C646 553 620 593 575 617C533 644 471 650 409 650H139L131 613C216 609 222 604 206 520L134 135C116 53 101 42 25 37L16 0H331L338 37C261 42 249 54 264 136L291 281H318C352 281 365 267 375 235C392 185 416 130 438 87C471 25 501 -8 656 -8L668 26ZM502 485C502 431 477 379 426 347C395 329 361 323 322 323H298L343 558C347 577 351 587 357 593C368 602 384 605 403 605C454 605 502 575 502 485Z" id="g5-83"></path></g><g transform="matrix(.0091,0,0,-0.0091,8.69,3.132)"><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" id="g50-50"></path></g></svg> and <span class="nowrap"><svg height="6.25863pt" id="M18" style="vertical-align:-0.1933298pt" version="1.1" viewbox="-0.0498162 -6.0653 5.99765 6.25863" width="5.99765pt" 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="M422 431C411 443 392 454 370 454C321 454 263 368 219 295H214L223 337C242 422 231 454 202 454C161 454 88 406 26 350L41 318C59 334 99 354 107 354C116 354 114 338 107 302C87 198 66 95 41 -4L49 -12C78 -4 129 5 169 9C177 62 196 155 207 207C212 225 238 269 265 306C287 336 301 338 312 338C326 338 344 331 358 318C362 314 374 314 384 323C396 334 406 348 419 368C433 390 435 417 422 431Z" id="g5-115"></path></g></svg>.</span></div>

Mathematical Problems in Engineering

fig1

Figure 1

Figure 1: Semianalytical Solutions of Relative Motions with Applications to Periodic Orbits about a Nominal Circular Orbit 