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<div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/625d80f0be9ad24cfa814342/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">327 </div><div class="t m1 x2 h3 y2 ff1 fs1 fc0 sc0 ls1 ws0">IEEE </div><div class="t m2 x3 h4 y2 ff2 fs0 fc0 sc0 ls2 ws0">JOURNAL </div><div class="t m3 x4 h2 y2 ff1 fs0 fc0 sc0 ls3 ws0">OF </div><div class="t m4 x5 h5 y2 ff1 fs2 fc0 sc0 ls4 ws0">OCEANIC <span class="_ _0"></span>ENGINEERING, </div><div class="t m0 x6 h6 y2 ff2 fs2 fc0 sc0 ls5 ws0">VOL. </div><div class="t m5 x7 h7 y2 ff1 fs3 fc0 sc0 ls6 ws0">18, </div><div class="t m4 x8 h5 y2 ff1 fs2 fc0 sc0 ls2 ws0">NO. </div><div class="t m6 x9 h7 y2 ff1 fs3 fc0 sc0 ls6 ws0">3, </div><div class="t m7 xa h2 y2 ff1 fs0 fc0 sc0 ls7 ws0">JULY </div><div class="t m8 xb h8 y2 ff1 fs4 fc0 sc0 ls8 ws0">1993 </div><div class="t m9 xc h9 y3 ff2 fs5 fc0 sc0 ls2 ws0">Multivariable <span class="_ _1"></span><span class="ls9">SlidingKMode </span></div><div class="t ma xd h9 y3 ff2 fs5 fc0 sc0 lsa ws0">Control </div><div class="t mb xe ha y4 ff2 fs6 fc0 sc0 lsb ws0">for </div><div class="t m9 xf h9 y4 ff2 fs5 fc0 sc0 ls2 ws0">Autonomous <span class="_ _2"></span>Diving <span class="_ _1"></span><span class="lsa">and <span class="_ _3"></span></span>Steering </div><div class="t mc x10 hb y5 ff2 fs7 fc0 sc0 lsc ws0">of </div><div class="t m9 x11 h9 y5 ff2 fs5 fc0 sc0 lsa ws0">Unmanned <span class="_ _4"> </span>Underwater <span class="_ _4"> </span><span class="lsd">Vehicles </span></div><div class="t m0 x12 hc y6 ff2 fs8 fc0 sc0 lse ws0">Anthony </div><div class="t md x13 hd y6 ff1 fs9 fc0 sc0 lsf ws0">J. </div><div class="t m0 x14 hc y6 ff2 fs8 fc0 sc0 lse ws0">Healey </div><div class="t me x15 hc y6 ff2 fs8 fc0 sc0 ls2 ws0">and <span class="_ _2"> </span>David </div><div class="t m0 x16 hc y6 ff2 fs8 fc0 sc0 ls10 ws0">Lienard </div><div class="t mf x17 he y7 ff1 fsa fc0 sc0 ls11 ws0">Abstruct-A <span class="_ _5"> </span><span class="ls2">six <span class="ls12">degree <span class="_ _0"></span>of <span class="_ _0"></span>freedom <span class="_ _6"></span>model <span class="_ _6"></span></span>for <span class="ls13">the <span class="ls12">maneuvering </span></span></span></div><div class="t mf x2 he y8 ff1 fsa fc0 sc0 ls14 ws0">of <span class="_ _7"> </span><span class="ls13">an <span class="_ _8"></span><span class="ls12">underwater <span class="_ _9"> </span></span>vehicle <span class="_ _6"></span><span class="ls2">is <span class="_ _8"></span><span class="ls12">used <span class="_"> </span>and <span class="_ _9"> </span></span></span>a <span class="_ _8"></span>sliding <span class="_ _8"></span><span class="ls12">mode <span class="_ _8"></span></span>autopilot </span></div><div class="t mf x2 he y9 ff1 fsa fc0 sc0 ls2 ws0">is <span class="_ _6"></span><span class="ls12">designed <span class="_ _0"></span><span class="ls13">for <span class="_ _0"></span></span>the <span class="_ _8"></span>combined </span></div><div class="t mf x2 he ya ff1 fsa fc0 sc0 ls13 ws0">functions. <span class="ls14">In <span class="_"> </span><span class="ls15">flight <span class="_ _7"> </span><span class="ls12">control <span class="_"> </span></span></span></span>a </div><div class="t mf x2 he yb ff1 fsa fc0 sc0 ls13 ws0">arise <span class="_ _8"></span><span class="ls12">because <span class="_ _7"> </span></span>the <span class="_ _9"> </span><span class="ls12">system <span class="_ _2"> </span><span class="ls2">to <span class="_ _9"> </span><span class="ls15">be <span class="_ _1"> </span></span></span></span>controlled <span class="_ _6"></span><span class="ls2">is <span class="_ _9"> </span></span>highly <span class="_ _9"> </span><span class="ls12">nonlinear, </span></div><div class="t mf x2 he yc ff1 fsa fc0 sc0 ls12 ws0">coupled, <span class="_ _a"> </span><span class="ls13">and <span class="_"> </span></span>there <span class="_ _b"> </span><span class="ls2">is <span class="_"> </span><span class="ls13">a <span class="_ _a"> </span>good <span class="_"> </span></span></span>deal <span class="_ _b"> </span><span class="ls14">of <span class="_ _b"> </span></span>parameter <span class="_ _b"> </span>uncertainty </div><div class="t mf x2 he yd ff1 fsa fc0 sc0 ls12 ws0">and <span class="_ _9"> </span>variation <span class="_"> </span>with <span class="_"> </span><span class="ls13">operational </span>conditions. <span class="_"> </span>The <span class="_"> </span><span class="ls13">development </span></div><div class="t m0 x18 hf ye ff1 fsb fc0 sc0 ls16 ws0">of </div><div class="t mf x2 he yf ff1 fsa fc0 sc0 ls12 ws0">variable <span class="_ _b"> </span><span class="ls13">structure <span class="_ _9"> </span></span>control <span class="_ _a"> </span><span class="ls2">in <span class="_ _1"> </span><span class="ls13">the <span class="_ _7"> </span></span></span>form <span class="_ _b"> </span>of <span class="_ _b"> </span><span class="ls13">sliding <span class="_"> </span></span>modes <span class="_ _a"> </span><span class="ls13">has </span></div><div class="t mf x2 he y10 ff1 fsa fc0 sc0 ls12 ws0">been <span class="_"> </span>shown <span class="_ _8"></span><span class="ls13">to <span class="_ _9"> </span></span>provide <span class="_"> </span><span class="ls13">robustness </span>that <span class="_"> </span><span class="ls2">is <span class="_ _8"></span></span>expected <span class="_"> </span><span class="ls13">to <span class="_ _8"></span><span class="ls15">be <span class="_ _7"> </span></span>quite </span></div><div class="t mf x2 he y11 ff1 fsa fc0 sc0 ls12 ws0">remarkable <span class="_"> </span><span class="ls2">for <span class="_ _8"></span><span class="ls15">AUV <span class="_ _1"> </span><span class="ls13">autopilot <span class="_ _0"></span></span></span></span>design. <span class="_ _1"> </span>This <span class="_ _7"> </span>paper <span class="_ _1"> </span><span class="ls13">shows <span class="_ _8"></span>that </span></div><div class="t mf x19 he y12 ff1 fsa fc0 sc0 ls13 ws0">a <span class="_ _0"></span>multivariable <span class="_ _c"></span>sliding <span class="_ _d"></span><span class="ls12">mode <span class="_ _8"></span></span>autopilot <span class="ls12">based <span class="_ _8"></span>on <span class="_ _9"></span></span>state <span class="_ _d"></span><span class="ls12">feedback, </span></div><div class="t mf x19 he y13 ff1 fsa fc0 sc0 ls12 ws0">designed <span class="_ _8"></span><span class="ls13">assuming </span>decoupled <span class="_ _8"></span>modeling, <span class="_ _8"></span><span class="ls2">is <span class="_ _0"></span></span>quite </div><div class="t m10 x1a h10 y14 ff1 fsc fc0 sc0 ls17 ws0">s-ory </div><div class="t mf x1b he y14 ff1 fsa fc0 sc0 ls13 ws0">for </div><div class="t mf x19 he y15 ff1 fsa fc0 sc0 ls12 ws0">the combined <span class="ls13">speed, <span class="_ _c"></span>steering, <span class="_ _e"></span>and <span class="_ _c"></span>diving <span class="_ _e"></span><span class="ls12">response <span class="_ _d"></span>of <span class="_ _6"></span><span class="ls13">a slow <span class="_ _f"></span>speed </span></span></span></div><div class="t mf x19 he y16 ff1 fsa fc0 sc0 ls15 ws0">AUV. <span class="_ _0"></span><span class="ls12">The <span class="ls13">influence <span class="_ _e"></span><span class="ls12">of <span class="_ _d"></span>speed, <span class="_ _0"></span><span class="ls13">modeling <span class="_ _e"></span>nonlinearity, <span class="_ _10"></span>uncertainty, </span></span></span></span></div><div class="t mf x19 he y17 ff1 fsa fc0 sc0 ls12 ws0">and <span class="_ _6"></span><span class="ls13">disturbances, <span class="_ _e"></span>can <span class="_ _d"></span><span class="ls15">be <span class="_ _8"></span></span>effectively <span class="_ _f"></span><span class="ls12">compensated, <span class="_ _8"></span><span class="ls13">even for <span class="_ _d"></span>com- </span></span></span></div><div class="t mf x19 he y18 ff1 fsa fc0 sc0 ls13 ws0">plex <span class="_ _0"></span><span class="ls12">maneuvering. <span class="_"> </span><span class="ls15">Waypoint <span class="_ _7"> </span></span></span>acquisition <span class="ls12">based <span class="_"> </span>on <span class="_ _9"> </span></span>line <span class="_ _8"></span><span class="ls12">of <span class="_ _9"> </span></span>sight </div><div class="t mf x19 he y19 ff1 fsa fc0 sc0 ls12 ws0">guidance <span class="_ _8"></span><span class="ls2">is <span class="_ _0"></span></span>used <span class="_ _8"></span><span class="ls2">to <span class="_ _6"></span><span class="ls13">achieve path <span class="_ _6"></span></span></span>tracking. </div><div class="t m11 x1c h11 y1a ff2 fsd fc0 sc0 ls18 ws0">I. </div><div class="t m12 x1d h11 y1a ff2 fsd fc0 sc0 ls18 ws0">INTRODUCTION </div><div class="t m13 x1e h12 y1b ff1 fse fc0 sc0 ls19 ws0">HIS </div><div class="t m0 x1f h13 y1b ff2 fsf fc0 sc0 ls1a ws0">PAPER <span class="_"> </span><span class="ls1b">proposes <span class="_ _9"> </span><span class="ls2">the <span class="_ _0"></span><span class="ls1c">use <span class="_"> </span></span></span></span>of <span class="_"> </span><span class="ls1b">a <span class="_ _8"></span><span class="ls2">multivariable <span class="_ _0"></span></span>sliding </span></div><div class="t m14 x2 h14 y1c ff1 fs10 fc0 sc0 ls2 ws0">T </div><div class="t m0 x1e h13 y1c ff2 fsf fc0 sc0 ls1b ws0">mode <span class="_ _8"></span><span class="ls2">autopilot <span class="ls1d">for <span class="_ _d"></span></span></span>the <span class="_ _8"></span>combined <span class="_ _9"> </span>control <span class="_ _8"></span><span class="ls1a">of <span class="_"> </span><span class="ls1c">AUV <span class="_"> </span><span class="ls2">steer- </span></span></span></div><div class="t m0 x19 h13 y1d ff2 fsf fc0 sc0 ls1b ws0">ing, <span class="_ _9"></span><span class="ls2">depth, <span class="_ _6"></span></span>and <span class="_ _9"> </span>speed <span class="_ _9"> </span>during <span class="_ _8"></span>complex <span class="_ _9"> </span>flight <span class="_ _8"></span>maneuvers. <span class="_ _8"></span><span class="ls1e">The </span></div><div class="t m0 x19 h13 y1e ff2 fsf fc0 sc0 ls1c ws0">method <span class="_ _8"></span><span class="ls1b">draws <span class="_ _0"></span>upon </span></div><div class="t m15 x20 h15 y1f ff2 fsb fc0 sc0 ls1f ws0">the </div><div class="t m0 x21 h13 y1f ff2 fsf fc0 sc0 ls1b ws0">power <span class="_ _8"></span><span class="ls1a">of <span class="_ _9"> </span></span>sliding <span class="_ _d"></span>modes <span class="_ _8"></span><span class="ls1c">to <span class="_ _0"></span><span class="ls1e">reduce </span></span>the </div><div class="t m0 x19 h13 y20 ff2 fsf fc0 sc0 ls1b ws0">inherent <span class="_"> </span>coupling <span class="_"> </span>between <span class="_"> </span><span class="ls2">the <span class="_ _9"> </span></span>vehicle <span class="_ _1"> </span><span class="ls1e">response <span class="_ _6"></span></span>modes <span class="_"> </span>that </div><div class="t m0 x19 h13 y21 ff2 fsf fc0 sc0 ls1b ws0">naturally <span class="_ _1"> </span><span class="ls2">exist <span class="_ _2"> </span>in <span class="_"> </span>ROV/AUV <span class="_ _9"> </span></span>vehicles. <span class="_ _a"> </span><span class="ls1e">The <span class="_ _2"> </span></span>approach <span class="_ _1"> </span><span class="ls2">leads </span></div><div class="t m0 x19 h13 y22 ff2 fsf fc0 sc0 ls2 ws0">to <span class="_ _a"> </span><span class="ls1b">a <span class="_ _b"> </span>set <span class="_ _1"> </span><span class="ls1a">of <span class="_ _4"> </span></span></span>separate <span class="_"> </span><span class="ls1e">designs <span class="_ _9"> </span></span>for <span class="_ _1"> </span>the <span class="_ _a"> </span>steering, <span class="_"> </span>diving, <span class="_"> </span><span class="ls1b">and </span></div><div class="t m0 x2 h13 y23 ff2 fsf fc0 sc0 ls1b ws0">speed <span class="_ _8"></span>control <span class="_ _2"> </span><span class="ls2">systems, <span class="_ _8"></span><span class="ls20">and <span class="_"> </span></span></span>a <span class="_ _2"> </span><span class="ls2">series <span class="_ _8"></span><span class="ls21">of <span class="_ _b"> </span></span>simulations <span class="_ _6"></span></span>based <span class="_"> </span><span class="ls20">on </span></div><div class="t m0 x19 h13 y24 ff2 fsf fc0 sc0 ls1b ws0">the <span class="_"> </span>dynamics </div><div class="t m16 x22 h16 y25 ff2 fs11 fc0 sc0 ls22 ws0">of </div><div class="t m0 xe h13 y25 ff2 fsf fc0 sc0 ls1b ws0">a <span class="_"> </span>swimmer <span class="_"> </span>delivery <span class="_"> </span>vehicle <span class="_ _1"> </span><span class="ls2">illustrates <span class="_ _9"></span></span>the </div><div class="t m0 x19 h13 y26 ff2 fsf fc0 sc0 ls1b ws0">robustness <span class="_ _8"></span><span class="ls1c">and <span class="_ _0"></span></span>validity <span class="_ _0"></span><span class="ls1a">of <span class="_ _9"> </span></span>the <span class="_ _6"></span><span class="ls2">concept. <span class="ls1e">This <span class="_ _c"></span><span class="ls2">is <span class="_ _0"></span>likely to <span class="_ _6"></span><span class="ls1b">be <span class="_ _0"></span></span>the </span></span></span></div><div class="t m0 x19 h13 y27 ff2 fsf fc0 sc0 ls1b ws0">control <span class="_ _8"></span>method <span class="_ _8"></span><span class="ls1a">of <span class="_"> </span><span class="ls1e">choice <span class="_ _f"></span><span class="ls2">for <span class="_ _8"></span><span class="ls1b">AUV’s <span class="_ _8"></span>in <span class="_ _8"></span></span>the <span class="_ _0"></span>future. </span></span></span></div><div class="t m0 x23 h13 y28 ff2 fsf fc0 sc0 ls1c ws0">Well-behaved <span class="_ _9"></span><span class="ls2">autopilot <span class="_ _d"></span><span class="ls1b">systems <span class="_ _0"></span><span class="ls1e">enable <span class="_ _11"></span><span class="ls1b">the <span class="_ _0"></span>use <span class="_ _8"></span><span class="ls1c">of <span class="_ _8"></span></span>a <span class="_ _0"></span>variety </span></span></span></span></div><div class="t m0 x19 h13 y29 ff2 fsf fc0 sc0 ls1a ws0">of <span class="_ _7"> </span><span class="ls1b">guidance <span class="_ _8"></span>schemes <span class="_ _8"></span><span class="ls2">to <span class="_ _9"></span><span class="ls1e">achieve <span class="_ _c"></span><span class="ls1b">waypoint <span class="_ _2"> </span><span class="ls1c">and <span class="_ _8"></span></span>path <span class="_ _9"> </span>tracking. </span></span></span></span></div><div class="t m0 x19 h13 y2a ff2 fsf fc0 sc0 ls1c ws0">Waypoint <span class="_ _9"> </span><span class="ls1b">acquisition <span class="_ _9"> </span>is <span class="_ _9"> </span><span class="ls2">illustrated <span class="_ _6"></span></span>using <span class="_ _9"></span>proportional <span class="_ _9"> </span><span class="ls2">line <span class="_ _9"> </span><span class="ls1a">of </span></span></span></div><div class="t m0 x19 h13 y2b ff2 fsf fc0 sc0 ls2 ws0">sight <span class="_ _c"></span><span class="ls1b">guidance <span class="ls1c">with <span class="_ _6"></span>and <span class="_ _6"></span></span>without <span class="_ _d"></span>the influence <span class="_ _11"></span><span class="ls1a">of <span class="_ _8"></span><span class="ls1b">ocean <span class="ls2">current. </span></span></span></span></div><div class="t m0 x19 h13 y2c ff2 fsf fc0 sc0 ls1b ws0">The <span class="_ _0"></span><span class="ls1e">paper <span class="_ _d"></span><span class="ls2">contains </span></span></div><div class="t m17 x24 h13 y2d ff2 fsf fc0 sc0 ls2 ws0">a </div><div class="t m0 x25 h13 y2d ff2 fsf fc0 sc0 ls1b ws0">discussion <span class="_ _8"></span><span class="ls1a">of <span class="_"> </span></span>the <span class="_ _8"></span><span class="ls2">vehicle <span class="_ _6"></span></span>modeling; <span class="_ _9"></span>the </div><div class="t m0 x19 h13 y2e ff2 fsf fc0 sc0 ls1b ws0">sliding <span class="_ _9"> </span>mode <span class="_"> </span><span class="ls2">control <span class="_ _9"> </span>design; <span class="_ _8"></span>the <span class="_ _2"> </span><span class="ls1e">guidance <span class="_ _0"></span>scheme; <span class="_ _0"></span><span class="ls1c">and <span class="_ _7"> </span></span></span>the </span></div><div class="t m0 x19 h13 y2f ff2 fsf fc0 sc0 ls1b ws0">results <span class="_ _0"></span><span class="ls1c">of <span class="_ _0"></span></span>computer <span class="_ _8"></span><span class="ls2">simulations <span class="_ _f"></span><span class="ls1b">under <span class="_ _6"></span><span class="ls2">conditions <span class="ls1a">of <span class="_ _8"></span><span class="ls1e">parameter </span></span></span></span></span></div><div class="t m0 x19 h13 y30 ff2 fsf fc0 sc0 ls1b ws0">mismatch. </div><div class="t m18 x1c h17 y31 ff1 fs12 fc0 sc0 ls23 ws0">11. </div><div class="t m0 x26 h13 y31 ff2 fsf fc0 sc0 ls1b ws0">BACKGROUND </div><div class="t m0 x23 h13 y32 ff2 fsf fc0 sc0 ls1e ws0">The <span class="_ _9"></span><span class="ls1b">design </span></div><div class="t m19 x27 h18 y33 ff2 fs13 fc0 sc0 ls24 ws0">of </div><div class="t m0 x28 h13 y33 ff2 fsf fc0 sc0 ls1e ws0">an <span class="_ _9"></span><span class="ls2">autopilot <span class="_ _0"></span>for <span class="_ _2"> </span><span class="ls1b">the <span class="_"> </span>control <span class="_"> </span><span class="ls1a">of <span class="_ _7"> </span></span></span></span>underwater </div><div class="t m0 x19 h13 y34 ff2 fsf fc0 sc0 ls1b ws0">vehicles <span class="_"> </span><span class="ls2">is <span class="_ _9"> </span><span class="ls1a">of <span class="_ _b"> </span></span>interest <span class="_ _9"></span></span>both <span class="_ _2"> </span><span class="ls1e">from <span class="_ _8"></span></span>the <span class="_"> </span><span class="ls1c">view <span class="_ _1"> </span><span class="ls1a">of <span class="_ _b"> </span></span></span>motion <span class="_ _1"> </span>stabi- </div><div class="t mf x19 he y35 ff1 fsa fc0 sc0 ls2 ws0">lization </div><div class="t m1a x29 h19 y35 ff1 fs14 fc0 sc0 ls25 ws0">as </div><div class="t mf x2a he y35 ff1 fsa fc0 sc0 ls26 ws0">well <span class="_ _a"> </span><span class="ls2">as <span class="_"> </span>maneuvering <span class="_ _8"></span><span class="ls15">and <span class="_ _1"> </span><span class="ls13">tracking <span class="_ _9"> </span>performance. </span></span></span></div><div class="t m1b x17 h1a y36 ff2 fs15 fc0 sc0 ls27 ws0">Manuscript <span class="_ _f"></span>received <span class="ls28">November </span></div><div class="t m1c x2b h1b y36 ff2 fs1 fc0 sc0 ls29 ws0">8, </div><div class="t m1b x2c h1a y36 ff2 fs15 fc0 sc0 ls27 ws0">1991; <span class="_ _e"></span>revised <span class="_ _d"></span>April 23, <span class="_ _8"></span><span class="ls2">1993. </span></div><div class="t m1d x2d h1c y36 ff2 fs16 fc0 sc0 ls2a ws0">llus </div><div class="t m1e x2e h19 y36 ff1 fs14 fc0 sc0 ls2b ws0">work </div><div class="t m1b x19 h1a y37 ff2 fs15 fc0 sc0 ls2c ws0">was <span class="_ _0"></span><span class="ls27">supported <span class="_ _f"></span><span class="ls2d">by <span class="_ _9"> </span><span class="ls2e">the <span class="_ _0"></span><span class="ls2">Naval <span class="_ _6"></span><span class="ls27">Postgraduate School <span class="_ _c"></span>Direct Research <span class="ls2">Fund <span class="_ _6"></span>under </span></span></span></span></span></span></div><div class="t m1b x19 h1a y38 ff2 fs15 fc0 sc0 ls2e ws0">the <span class="_ _8"></span><span class="ls27">technical sponsorship </span></div><div class="t m1f x2f h1d y39 ff2 fs17 fc0 sc0 ls2f ws0">of </div><div class="t m1b x30 h1a y39 ff2 fs15 fc0 sc0 ls2e ws0">the <span class="_ _8"> </span>Office </div><div class="t m1f x31 h1d y39 ff2 fs17 fc0 sc0 ls2f ws0">of </div><div class="t m1b x32 h1a y39 ff2 fs15 fc0 sc0 ls27 ws0">Naval <span class="_ _0"></span>Technology. </div><div class="t m1b x17 h1a y3a ff2 fs15 fc0 sc0 ls2 ws0">The <span class="_ _9"> </span><span class="ls27">authors <span class="_ _0"></span><span class="ls2e">are <span class="_"> </span></span>with <span class="_ _8"></span>the <span class="_ _8"></span>Department </span></div><div class="t m20 x33 h1e y3b ff2 fs18 fc0 sc0 ls30 ws0">of </div><div class="t m1b x34 h1a y3b ff2 fs15 fc0 sc0 ls27 ws0">Mechanical <span class="_ _8"></span>Engineering, <span class="_ _6"></span>Naval </div><div class="t m1b x19 h1a y3c ff2 fs15 fc0 sc0 ls27 ws0">Postgraduate </div><div class="t m0 x35 h1f y3d ff2 fs14 fc0 sc0 ls31 ws0">School, </div><div class="t m1b x36 h1a y3d ff2 fs15 fc0 sc0 ls27 ws0">Monterey, </div><div class="t m0 x37 h1f y3d ff2 fs14 fc0 sc0 ls31 ws0">CA </div><div class="t m1b x38 h1a y3d ff2 fs15 fc0 sc0 ls27 ws0">93943. </div><div class="t m21 x17 h20 y3e ff2 fs19 fc0 sc0 ls32 ws0">IEEE </div><div class="t m1b x39 h1a y3e ff2 fs15 fc0 sc0 ls2e ws0">Log <span class="_ _6"></span><span class="ls2">Number <span class="_ _8"></span><span class="ls33">921 <span class="_ _12"></span><span class="ls2">1042. </span></span></span></div><div class="t m0 x3a h13 y3f ff2 fsf fc0 sc0 ls1b ws0">AUV’s <span class="_ _1"> </span><span class="ls1a">of <span class="_ _3"> </span></span>the <span class="_"> </span><span class="ls2">class <span class="_"> </span></span>considered <span class="_"> </span>here, <span class="_ _a"> </span><span class="ls2">fall <span class="_ _2"> </span></span>between <span class="_ _1"> </span>the <span class="_ _1"> </span>two </div><div class="t m0 x3a h13 y40 ff2 fsf fc0 sc0 ls1b ws0">extremes <span class="_ _8"></span><span class="ls1a">of <span class="_"> </span></span>underwater <span class="_ _8"></span>vehicles <span class="_ _8"></span><span class="ls2">(the <span class="_ _6"></span></span>ROV’s <span class="_ _8"></span><span class="ls1c">and <span class="_ _9"></span></span>the <span class="_ _0"></span>torpedo </div><div class="t m0 x3a h13 y41 ff2 fsf fc0 sc0 ls1b ws0">type), <span class="_ _9"> </span><span class="ls1c">and <span class="_ _2"> </span><span class="ls2">are <span class="_ _0"></span></span></span>difficult <span class="_ _8"></span><span class="ls2">to <span class="_ _8"></span></span>control <span class="_ _8"></span>having <span class="_ _9"> </span>highly <span class="_ _9"></span>variable <span class="_ _8"></span>and </div><div class="t m0 x3a h13 y42 ff2 fsf fc0 sc0 ls1b ws0">uncertain <span class="_ _6"></span>dynamics. Recent <span class="ls1c">work <span class="_ _d"></span></span>concerned with <span class="_ _d"></span>the modeling </div><div class="t m0 x3a h13 y43 ff2 fsf fc0 sc0 ls1c ws0">and <span class="_ _9"> </span><span class="ls1b">control </span></div><div class="t m16 x3b h16 y44 ff2 fs11 fc0 sc0 ls22 ws0">of </div><div class="t m0 x3c h13 y44 ff2 fsf fc0 sc0 ls2 ws0">ROV <span class="_ _0"></span><span class="ls1b">vehicles <span class="_ _9"></span></span>includes <span class="_ _d"></span><span class="ls1b">Lewis, <span class="_ _8"></span>Lipscomb, <span class="_ _8"></span><span class="ls1c">and </span></span></div><div class="t m0 x3a h13 y45 ff2 fsf fc0 sc0 ls1b ws0">Thompson <span class="_"> </span><span class="ls2">[14] <span class="_ _0"></span></span>who <span class="_ _2"> </span>described </div><div class="t m22 x3d h21 y46 ff2 fs1a fc0 sc0 ls34 ws0">an </div><div class="t m0 x3e h13 y46 ff2 fsf fc0 sc0 ls1c ws0">ROV <span class="_"> </span><span class="ls1b">simulation <span class="_ _9"></span>program </span></div><div class="t m0 x3a h13 y47 ff2 fsf fc0 sc0 ls1b ws0">using <span class="_ _8"></span><span class="ls2">linear <span class="_ _d"></span></span>hydrodynamic <span class="_ _8"></span>coefficients. <span class="_ _0"></span><span class="ls1c">ROV <span class="_ _8"></span></span>vehicles <span class="_ _8"></span><span class="ls1e">do <span class="_ _6"></span><span class="ls1c">not </span></span></div><div class="t m0 x3a h13 y48 ff2 fsf fc0 sc0 ls1b ws0">possess <span class="_"> </span>hydrodynamically <span class="_ _1"> </span>shaped <span class="_ _9"> </span>profiles <span class="_"> </span><span class="ls1c">and <span class="_"> </span><span class="ls2">the <span class="_ _9"> </span></span></span>hydrody- </div><div class="t m0 x3a h13 y49 ff2 fsf fc0 sc0 ls1b ws0">namic <span class="_ _1"> </span><span class="ls2">forces <span class="_"> </span>are <span class="_"> </span></span>uncertain <span class="_ _a"> </span>and <span class="_ _1"> </span>difficult <span class="_"> </span><span class="ls1c">to <span class="_ _1"> </span></span>predict <span class="_ _3"> </span><span class="ls2">[4]. </span></div><div class="t m23 x3f h18 y4a ff2 fs13 fc0 sc0 ls35 ws0">To </div><div class="t m0 x3a h13 y4b ff2 fsf fc0 sc0 ls1e ws0">overcome <span class="_ _f"></span><span class="ls1b">the <span class="_ _8"></span>widely <span class="_ _0"></span>varying <span class="_ _8"></span><span class="ls1c">and <span class="_ _8"></span></span>uncertain <span class="_ _6"></span><span class="ls1e">behavior <span class="_ _c"></span><span class="ls1a">of <span class="_"> </span><span class="ls1b">these </span></span></span></span></div><div class="t m0 x3a h13 y4c ff2 fsf fc0 sc0 ls1b ws0">vehicles <span class="_ _8"></span>Russel <span class="_ _8"></span>and <span class="_ _6"></span>Bugge <span class="_ _9"> </span><span class="ls2">[22] <span class="ls1c">had <span class="_ _8"></span></span></span>considered <span class="_ _8"></span>the <span class="_ _0"></span>use <span class="_ _8"></span><span class="ls1a">of <span class="_"> </span><span class="ls1c">an </span></span></div><div class="t m0 x3a h13 y4d ff2 fsf fc0 sc0 ls2 ws0">adaptive <span class="_ _f"></span><span class="ls1e">automatic <span class="_ _12"></span>guidance <span class="_ _e"></span><span class="ls1b">system <span class="_ _c"></span><span class="ls2">including <span class="_ _e"></span><span class="ls1b">modeling <span class="_ _d"></span>strong </span></span></span></span></div><div class="t m0 x3a h13 y4e ff2 fsf fc0 sc0 ls1b ws0">measurement <span class="_ _8"></span><span class="ls1e">noise <span class="_ _6"></span>of <span class="_ _0"></span></span>uncertain <span class="_"> </span>spectral <span class="_ _8"></span><span class="ls2">nature, <span class="_ _6"></span><span class="ls20">Yoerger <span class="_"> </span><span class="ls1c">and </span></span></span></div><div class="t m0 x3a h13 y4f ff2 fsf fc0 sc0 ls2 ws0">Slotine <span class="_ _9"> </span>[28], <span class="_ _2"> </span>[30] <span class="_ _8"></span><span class="ls1b">proposed <span class="_"> </span>and <span class="_"> </span>successfully <span class="_"> </span><span class="ls1c">used <span class="_ _1"> </span></span>a <span class="_"> </span>sliding </span></div><div class="t m0 x3a h13 y50 ff2 fsf fc0 sc0 ls1b ws0">mode <span class="ls2">controller <span class="_ _f"></span>for <span class="_ _d"></span><span class="ls1c">an <span class="_ _d"></span>ROV <span class="_ _d"></span><span class="ls1b">maneuvering <span class="_ _d"></span>around </span></span>large objects </span></div><div class="t m0 x3a h13 y51 ff2 fsf fc0 sc0 ls2 ws0">at <span class="_ _c"></span><span class="ls1c">very <span class="_ _8"></span><span class="ls1b">slow <span class="ls2">speed; <span class="_ _e"></span><span class="ls1b">and <span class="_ _d"></span>Goheen, <span class="_ _c"></span>Jefferys, <span class="_ _6"></span><span class="ls1c">and </span>Broome </span></span></span></span></div><div class="t m24 x40 h22 y52 ff3 fs1b fc0 sc0 ls2 ws0">[ </div><div class="t m0 x41 h13 y52 ff2 fsf fc0 sc0 ls36 ws0">121 <span class="_ _c"></span><span class="ls2">de- </span></div><div class="t m0 x3a h13 y53 ff2 fsf fc0 sc0 ls1b ws0">scribed <span class="_ _d"></span>a <span class="_ _6"></span><span class="ls1c">“self <span class="_ _8"></span></span>testing” <span class="ls1e">procedure <span class="_ _e"></span><span class="ls2">for <span class="_ _d"></span><span class="ls1b">evaluation <span class="_ _6"></span><span class="ls1a">of <span class="_ _8"></span></span>the <span class="_ _6"></span>vehicle </span></span></span></div><div class="t m0 x3a h13 y54 ff2 fsf fc0 sc0 ls1b ws0">dynamic <span class="_ _8"></span><span class="ls1e">response <span class="ls1c">and <span class="_ _9"> </span></span></span>a <span class="_ _8"></span>corresponding <span class="_ _8"></span>automatic <span class="_ _9"></span>gain <span class="_ _8"></span>selec- </div><div class="t m0 x3a h13 y55 ff2 fsf fc0 sc0 ls1b ws0">tion. <span class="_"> </span><span class="ls1c">In <span class="_ _1"> </span></span>a <span class="_ _9"> </span>useful <span class="_ _1"> </span>summary <span class="_ _2"> </span><span class="ls1a">of <span class="_ _1"> </span></span>underwater <span class="_"> </span>vehicle <span class="_ _2"> </span>modeling, </div><div class="t m0 x3a h13 y56 ff2 fsf fc0 sc0 ls37 ws0">Yuh <span class="_ _2"> </span><span class="ls2">[32] <span class="_ _f"></span><span class="ls1e">has <span class="ls1b">described the <span class="_ _d"></span><span class="ls2">functional <span class="_ _f"></span><span class="ls1b">form <span class="_ _d"></span><span class="ls1a">of <span class="_ _8"></span></span>vehicle <span class="_ _d"></span>dynamic </span></span></span></span></span></div><div class="t m0 x3a h13 y57 ff2 fsf fc0 sc0 ls1e ws0">equations <span class="ls1a">of <span class="_ _1"> </span><span class="ls1b">motion, <span class="_ _9"> </span><span class="ls2">the <span class="_ _9"></span></span>nature <span class="_ _2"> </span></span>of <span class="_ _7"> </span><span class="ls2">the <span class="_ _9"></span><span class="ls1b">loadings <span class="_ _9"> </span>and <span class="_ _9"></span></span>the <span class="_ _8"></span><span class="ls1b">use </span></span></span></div><div class="t m0 x3a h13 y58 ff2 fsf fc0 sc0 ls1a ws0">of <span class="_ _1"> </span><span class="ls1e">adaptive <span class="ls2">control <span class="_ _8"></span>via <span class="_ _8"></span><span class="ls20">on <span class="_"> </span></span>line <span class="_ _8"></span><span class="ls1b">parameter <span class="_ _2"> </span>identification. <span class="_ _8"></span></span></span>The </span></div><div class="t m0 x3a h13 y59 ff2 fsf fc0 sc0 ls1c ws0">work <span class="_ _8"></span><span class="ls1a">of <span class="_ _9"> </span><span class="ls20">Yoerger <span class="_ _8"></span></span></span>and <span class="_ _8"></span><span class="ls2">Slotine <span class="ls1b">with <span class="_ _6"></span>robust <span class="_ _8"></span>control <span class="_ _6"></span>using <span class="_ _8"></span>sliding </span></span></div><div class="t m0 x3a h13 y5a ff2 fsf fc0 sc0 ls1b ws0">modes <span class="_ _6"></span>is most <span class="_ _6"></span><span class="ls1e">encouraging, <span class="_ _12"></span><span class="ls1b">and, <span class="_ _d"></span>although the <span class="_ _6"></span><span class="ls2">extra </span>time taken </span></span></div><div class="t m0 x3a h13 y5b ff2 fsf fc0 sc0 ls2 ws0">to <span class="_ _8"></span><span class="ls1b">perform <span class="_ _9"> </span>the <span class="_ _9"> </span></span>self-test <span class="_ _8"></span><span class="ls1a">of <span class="_"> </span><span class="ls1b">Goheen </span></span></div><div class="t m25 x42 h23 y5c ff4 fse fc0 sc0 ls38 ws0">et </div><div class="t m26 x43 h24 y5c ff4 fs12 fc0 sc0 ls2 ws0">al. </div><div class="t m0 x44 h13 y5c ff2 fsf fc0 sc0 ls20 ws0">may <span class="_"> </span><span class="ls1b">not <span class="_ _9"> </span>always <span class="_ _8"></span>be </span></div><div class="t m0 x3a h13 y5d ff2 fsf fc0 sc0 ls1b ws0">available, <span class="_"> </span><span class="ls2">it <span class="_"> </span>still <span class="_ _9"></span><span class="ls1e">has <span class="_ _9"></span></span></span>merit. <span class="_"> </span>Recently, <span class="_"> </span><span class="ls20">Yoerger <span class="_ _3"> </span></span>[30] <span class="_ _9"> </span>showed </div><div class="t m0 x3a h13 y5e ff2 fsf fc0 sc0 ls1b ws0">that <span class="_ _9"> </span>the <span class="_ _9"> </span><span class="ls1e">dynamics <span class="_ _d"></span><span class="ls1a">of <span class="_ _1"> </span></span></span>torque <span class="_ _9"> </span><span class="ls2">controlled <span class="_ _0"></span></span>thruster <span class="_ _9"> </span><span class="ls1e">elements <span class="ls2">are </span></span></div><div class="t m0 x3a h13 y5f ff2 fsf fc0 sc0 ls1b ws0">problematic <span class="_ _1"> </span>in <span class="_"> </span><span class="ls1c">ROV <span class="_ _1"> </span></span>positioning <span class="_ _a"> </span>because <span class="_"> </span><span class="ls2">lags <span class="_ _2"> </span></span>in <span class="_"> </span><span class="ls2">the <span class="_ _2"> </span></span>thrust </div><div class="t m0 x3a h13 y60 ff2 fsf fc0 sc0 ls1b ws0">response, <span class="_ _9"></span><span class="ls2">if <span class="_ _0"></span><span class="ls1c">not <span class="_"> </span></span></span>taken <span class="_ _8"></span>into <span class="_ _8"></span><span class="ls2">account, <span class="_ _6"></span></span>can <span class="_ _8"></span>lead <span class="_ _9"></span><span class="ls2">to <span class="_ _0"></span>limit <span class="_ _8"></span></span>cycling </div><div class="t m0 x3a h13 y61 ff2 fsf fc0 sc0 ls1b ws0">behavior. <span class="_ _1"> </span>Fossen <span class="_ _a"> </span><span class="ls1c">[SI, <span class="_"> </span>and <span class="_ _7"> </span></span>Fossen <span class="_ _1"> </span><span class="ls1c">and <span class="_ _1"> </span></span>Satagun <span class="_ _1"> </span><span class="ls36">[91 <span class="_ _2"> </span><span class="ls2">describe </span></span></div><div class="t m0 x3a h13 y62 ff2 fsf fc0 sc0 ls1b ws0">the <span class="_ _a"> </span>use <span class="_ _b"> </span><span class="ls1a">of <span class="_ _3"> </span></span>multivariable <span class="_ _3"> </span>sliding <span class="_ _1"> </span>mode <span class="_ _a"> </span><span class="ls2">control <span class="_"> </span></span>in <span class="_ _b"> </span>dynamic </div><div class="t m0 x3a h13 y63 ff2 fsf fc0 sc0 ls1b ws0">positioning </div><div class="t m27 x3b h25 y64 ff2 fs1c fc0 sc0 ls39 ws0">of </div><div class="t m0 x45 h13 y64 ff2 fsf fc0 sc0 ls1b ws0">ROV’s <span class="_ _2"> </span>and <span class="_"> </span><span class="ls1c">show <span class="_"> </span></span>that <span class="_"> </span>this <span class="_"> </span>method <span class="_ _9"> </span><span class="ls1e">has <span class="_ _9"> </span><span class="ls2">great </span></span></div><div class="t m0 x3a h13 y65 ff2 fsf fc0 sc0 ls1b ws0">potential <span class="_ _9"> </span><span class="ls2">for <span class="_ _9"> </span></span>controlling <span class="_ _9"> </span>the <span class="_ _9"> </span><span class="ls1c">ROV <span class="_"> </span><span class="ls2">attitude <span class="_ _8"></span></span>and <span class="_ _9"></span></span>position <span class="_ _2"> </span><span class="ls1c">with </span></div><div class="t m0 x3a h13 y66 ff2 fsf fc0 sc0 ls2 ws0">excellent <span class="_ _11"></span><span class="ls1b">robustness <span class="_ _6"></span><span class="ls2">properties against <span class="_ _c"></span><span class="ls1b">parametric <span class="_ _8"></span>uncertainty. </span></span></span></div><div class="t m0 x46 h13 y67 ff2 fsf fc0 sc0 ls1b ws0">For <span class="_ _a"> </span>higher <span class="_ _3"> </span>speed <span class="_ _1"> </span>vehicles <span class="_ _b"> </span>than <span class="_ _13"> </span>ROV’s, <span class="_ _b"> </span><span class="ls1c">and <span class="_ _13"> </span></span>those <span class="_ _a"> </span><span class="ls1c">with </span></div><div class="t m0 x3a h13 y68 ff2 fsf fc0 sc0 ls1b ws0">more <span class="_"> </span>streamlined <span class="_ _8"></span>hydrodynamic <span class="_ _9"> </span><span class="ls2">characteristics, <span class="_ _d"></span>the <span class="_ _9"> </span></span>situation </div><div class="t m0 x3a h13 y69 ff2 fsf fc0 sc0 ls2 ws0">is <span class="_ _d"></span><span class="ls1b">different, <span class="_ _0"></span><span class="ls1c">and <span class="_ _0"></span><span class="ls1e">previous <span class="_ _e"></span><span class="ls1c">work <span class="_ _8"></span><span class="ls1e">has <span class="_ _c"></span><span class="ls1c">been <span class="_ _8"></span><span class="ls1b">reported <span class="_ _6"></span></span>by <span class="_ _0"></span><span class="ls1b">Lindgren </span></span></span></span></span></span></span></div><div class="t m28 x3a h26 y6a ff5 fs14 fc0 sc0 ls2 ws0">et </div><div class="t m29 x47 h27 y6a ff4 fs1d fc0 sc0 ls2 ws0">al. </div><div class="t m0 x48 h13 y6a ff2 fsf fc0 sc0 ls2 ws0">[17], <span class="_ _8"></span><span class="ls1b">who <span class="_"> </span>addressed <span class="_ _9"> </span>the <span class="_"> </span></span>issues <span class="_ _8"></span><span class="ls1a">of <span class="_ _13"> </span><span class="ls1b">steering <span class="_"> </span><span class="ls1c">and <span class="_"> </span><span class="ls1e">depth </span></span></span></span></div><div class="t m0 x3a h13 y6b ff2 fsf fc0 sc0 ls1b ws0">control <span class="_ _3"> </span><span class="ls1a">of </span></div><div class="t m2a x49 h28 y6c ff1 fs1e fc0 sc0 ls2 ws0">a </div><div class="t m0 x4a h13 y6c ff2 fsf fc0 sc0 ls1b ws0">torpedo <span class="_ _3"> </span>pointing <span class="_ _3"> </span>out <span class="_ _3"> </span>the <span class="_ _3"> </span>importance <span class="_ _3"> </span><span class="ls1a">of <span class="_ _4"> </span></span>the </div><div class="t mf x3a he y35 ff1 fsa fc0 sc0 ls2 ws0">nonlinear </div><div class="t m2b x4b h29 y35 ff6 fsc fc0 sc0 ls3a ws0">hydrodynamic </div><div class="t m2c x4c he y35 ff1 fsa fc0 sc0 ls13 ws0">behavior </div><div class="t m2d x42 h2a y35 ff6 fs1f fc0 sc0 ls3b ws0">and </div><div class="t m2e x4d h2b y35 ff1 fs20 fc0 sc0 ls3c ws0">the </div><div class="t m2f x4e h10 y35 ff1 fsc fc0 sc0 ls3a ws0">stroke <span class="_"> </span><span class="ls2">limits </span></div><div class="t m30 x4f h2c y35 ff2 fs21 fc0 sc0 ls3d ws0">of </div><div class="t m0 x3a h13 y6d ff2 fsf fc0 sc0 ls2 ws0">the <span class="_"> </span>surfaces; <span class="_ _9"> </span><span class="ls20">Young <span class="_ _b"> </span><span class="ls1b">[31] <span class="_ _9"> </span>described <span class="_"> </span></span></span>the <span class="_"> </span><span class="ls1b">stability <span class="_"> </span></span>derivatives </div><div class="t m0 x3a h13 y6e ff2 fsf fc0 sc0 ls1a ws0">of <span class="_ _a"> </span><span class="ls2">the <span class="_ _9"> </span><span class="ls1b">Navy’s <span class="_"> </span></span></span>DSRV <span class="_ _1"> </span><span class="ls1b">vehicle, <span class="_"> </span><span class="ls20">and <span class="_ _a"> </span></span>Smith </span></div><div class="t m31 x50 h2d y6f ff4 fs1c fc0 sc0 ls3e ws0">et </div><div class="t m32 x51 h2e y6f ff4 fsb fc0 sc0 ls2 ws0">al. </div><div class="t m33 x52 h2f y6f ff7 fs22 fc0 sc0 ls3f ws0">[25] </div><div class="t m0 x41 h13 y6f ff2 fsf fc0 sc0 ls1b ws0">gave <span class="_ _2"> </span>a </div><div class="t m0 x3a h13 y70 ff2 fsf fc0 sc0 ls1b ws0">comparison <span class="_ _8"></span><span class="ls1a">of <span class="_"> </span></span>the <span class="_ _6"></span>performance <span class="_ _9"> </span><span class="ls1a">of </span></div><div class="t m34 x53 h13 y71 ff2 fsf fc0 sc0 ls2 ws0">a </div><div class="t m0 x54 h13 y71 ff2 fsf fc0 sc0 ls1b ws0">simulator <span class="_ _6"></span><span class="ls2">for <span class="_ _6"></span></span>a <span class="_ _8"></span>swimmer </div><div class="t m0 x3a h13 y72 ff2 fsf fc0 sc0 ls1b ws0">delivery <span class="_"> </span>vehicle <span class="_"> </span><span class="ls1c">with <span class="_"> </span></span>field <span class="_ _9"> </span><span class="ls1e">response <span class="_ _d"></span></span>data. <span class="_ _9"> </span>Dobeck </div><div class="t m35 x55 h30 y73 ff8 fs1b fc0 sc0 ls2 ws0">et </div><div class="t m32 x56 h2e y73 ff4 fsb fc0 sc0 ls2 ws0">al. </div><div class="t m0 x3f h13 y73 ff2 fsf fc0 sc0 ls2 ws0">[6] </div><div class="t m0 x3a h13 y74 ff2 fsf fc0 sc0 ls1e ws0">provided <span class="_ _9"> </span><span class="ls1c">an <span class="_ _a"> </span><span class="ls1b">unclassified <span class="_ _1"> </span>description <span class="_ _b"> </span><span class="ls1a">of <span class="_ _3"> </span></span>tests <span class="_ _a"> </span>conducted <span class="_ _a"> </span>on </span></span></div><div class="t m1b x57 h1a y75 ff2 fs15 fc0 sc0 ls2 ws0">0364-9059/93$03.00 </div><div class="t m36 x58 h31 y75 ff9 fs18 fc0 sc0 ls2 ws0">0 </div><div class="t m1b x59 h1a y75 ff2 fs15 fc0 sc0 ls27 ws0">1993 </div><div class="t m37 x5a h20 y75 ff2 fs19 fc0 sc0 ls40 ws0">IEEE </div><div class="t m0 x5b h32 y76 ffa fs23 fc1 sc0 ls2 ws0">Authorized licensed use limited to: Yuan Ze University. Downloaded on August 09,2020 at 18:04:36 UTC from IEEE Xplore. Restrictions apply. </div></div><div class="pi" data-data='{"ctm":[1.546195,0.000000,0.000000,1.546195,0.000000,0.000000]}'></div></div>
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<div id="pf2" class="pf w2 h33" data-page-no="2"><div class="pc pc2 w2 h33"><img class="bi x0 y0 w3 h34" alt="" src="https://static.pudn.com/prod/directory_preview_static/625d80f0be9ad24cfa814342/bg2.jpg"><div class="t m38 x5c h35 y77 ff2 fs24 fc0 sc0 ls41 ws0">328 </div><div class="t m1b x5d h1a y78 ff2 fs15 fc0 sc0 ls42 ws0">IEEE <span class="_"> </span><span class="ls43">JOURNAL </span></div><div class="t m39 x5e h36 y78 ff2 fs4 fc0 sc0 ls8 ws0">OF </div><div class="t m1b x5f h1a y78 ff2 fs15 fc0 sc0 ls43 ws0">OCEANIC <span class="_ _8"></span><span class="ls42">ENGINEERING, </span></div><div class="t m4 x60 h6 y78 ff2 fs2 fc0 sc0 ls2 ws0">VOL. </div><div class="t m3a x61 h6 y78 ff2 fs2 fc0 sc0 ls44 ws0">18, </div><div class="t m3b x62 h6 y78 ff2 fs2 fc0 sc0 ls45 ws0">NO </div><div class="t m3c x56 h37 y78 ff2 fs23 fc0 sc0 ls46 ws0">3, </div><div class="t m3d x63 h6 y78 ff2 fs2 fc0 sc0 ls44 ws0">JULY </div><div class="t m3c x64 h37 y78 ff2 fs23 fc0 sc0 ls47 ws0">1993 </div><div class="t m0 x5c h13 y79 ff2 fsf fc0 sc0 ls2 ws0">the <span class="_ _2"> </span>control <span class="_"> </span>systems <span class="_ _8"></span>test <span class="_ _9"> </span>vehicle <span class="_"> </span>(CSTV) <span class="_ _8"></span>under <span class="_"> </span>closed <span class="_ _8"></span>loop </div><div class="t m0 x5c h13 y7a ff2 fsf fc0 sc0 ls2 ws0">computer <span class="_ _0"></span>controlled <span class="_ _0"></span>maneuvering. <span class="_ _8"></span>Humphries </div><div class="t m24 x65 h22 y7b ff3 fs1b fc0 sc0 ls2 ws0">[ </div><div class="t m0 x13 h13 y7b ff2 fsf fc0 sc0 ls36 ws0">101 <span class="_ _8"></span><span class="ls2">describes </span></div><div class="t m0 x5c h13 y7c ff2 fsf fc0 sc0 ls2 ws0">analytical <span class="_ _b"> </span><span class="ls1c">and <span class="_ _3"> </span></span>empirical <span class="_ _a"> </span>considerations <span class="_ _a"> </span>for <span class="_ _13"> </span>the <span class="_ _13"> </span>evaluation </div><div class="t m0 x5c h13 y7d ff2 fsf fc0 sc0 ls36 ws0">of <span class="_ _13"> </span><span class="ls2">hydrodynamic <span class="_ _8"></span><span class="ls1b">coefficients <span class="_ _2"> </span></span>while <span class="_ _9"> </span>Gueler <span class="_"> </span><span class="ls48">1111 <span class="_ _1"> </span></span>studied <span class="_ _8"></span>the </span></div><div class="t m0 x5c h13 y7e ff2 fsf fc0 sc0 ls2 ws0">independent <span class="_ _d"></span><span class="ls1b">use <span class="_ _9"> </span><span class="ls36">of <span class="_ _7"> </span><span class="ls1c">bow <span class="_"> </span></span></span>and <span class="_ _8"></span></span>stern planes <span class="_ _8"></span>in <span class="_ _8"></span>submarine <span class="_ _6"></span>depth </div><div class="t m0 x5c h13 y7f ff2 fsf fc0 sc0 ls2 ws0">control <span class="_ _d"></span>under <span class="_ _0"></span>the <span class="_ _8"></span>action <span class="_ _6"></span><span class="ls48">of <span class="_ _b"> </span></span>wave <span class="_ _8"></span>forces. <span class="_ _6"></span>Richards <span class="_ _0"></span><span class="ls1b">and <span class="_ _8"></span></span>Stoten </div><div class="t m17 x66 h13 y80 ff2 fsf fc0 sc0 ls2 ws0">[21], </div><div class="t m0 x67 h13 y80 ff2 fsf fc0 sc0 ls2 ws0">modeling <span class="_ _8"></span>the <span class="_ _9"></span>disturbance <span class="_ _6"></span>response <span class="_ _9"></span><span class="ls1b">to <span class="_ _9"> </span></span>waves, <span class="_ _8"></span><span class="ls1b">and <span class="_ _9"> </span></span>later </div><div class="t m0 x5c h13 y81 ff2 fsf fc0 sc0 ls2 ws0">Milliken </div><div class="t m3e x68 h38 y82 ff3 fsc fc0 sc0 ls2 ws0">[ </div><div class="t m0 x69 h13 y82 ff2 fsf fc0 sc0 ls49 ws0">181, <span class="_ _d"></span><span class="ls2">concerned about the <span class="_ _d"></span>yaw-pitch <span class="_ _d"></span>coupling <span class="_ _6"></span>during </span></div><div class="t m0 x5c h13 y83 ff2 fsf fc0 sc0 ls2 ws0">turns, applied <span class="_ _e"></span><span class="ls1e">a <span class="_ _f"></span><span class="ls2">linear <span class="_ _c"></span>model <span class="_ _f"></span>based <span class="_ _c"></span>compensator <span class="_ _f"></span><span class="ls1b">to <span class="_ _d"></span><span class="ls2">the <span class="_ _f"></span>problem </span></span></span></span></div><div class="t m0 x5c h13 y84 ff2 fsf fc0 sc0 ls36 ws0">of <span class="_ _b"> </span><span class="ls2">depth <span class="_ _8"></span>control. <span class="_ _9"></span>Optimized <span class="_ _6"></span>trade-offs <span class="_ _8"></span>between <span class="_ _8"></span>plane <span class="_ _9"> </span>action </span></div><div class="t m0 x5c h13 y85 ff2 fsf fc0 sc0 ls1c ws0">and <span class="_ _1"> </span><span class="ls2">the <span class="_"> </span>depth <span class="_ _9"> </span>error <span class="_ _2"> </span>response <span class="_ _9"> </span>in <span class="_"> </span><span class="ls1b">waves <span class="_"> </span><span class="ls1e">is <span class="_ _2"> </span></span></span>possible. <span class="_"> </span><span class="ls1e">So <span class="_ _9"></span></span>far, </span></div><div class="t m0 x5c h13 y86 ff2 fsf fc0 sc0 ls2 ws0">the <span class="_ _8"></span>issues <span class="_ _8"></span><span class="ls48">of <span class="_ _b"> </span></span>robustness <span class="_ _0"></span><span class="ls1b">had <span class="_ _9"> </span></span>not <span class="_ _8"></span><span class="ls1b">been <span class="_ _9"> </span></span>addressed <span class="_ _6"></span><span class="ls1b">to <span class="_ _9"></span></span>the <span class="_ _9"></span>same </div><div class="t m0 x5c h13 y87 ff2 fsf fc0 sc0 ls2 ws0">degree <span class="ls1c">as <span class="_ _0"></span></span>for the <span class="_ _6"></span><span class="ls1b">ROV <span class="_ _6"></span></span>vehicles. Milliken <span class="_ _d"></span>described the use <span class="_ _6"></span><span class="ls36">of </span></div><div class="t m0 x5c h13 y88 ff2 fsf fc0 sc0 ls1e ws0">a <span class="ls2">linearized <span class="_ _c"></span>model-based <span class="_ _c"></span>compensator <span class="_ _c"></span><span class="ls1b">to <span class="_ _d"></span><span class="ls2">reduce the pitch-yaw </span></span></span></div><div class="t m0 x5c h13 y89 ff2 fsf fc0 sc0 ls2 ws0">coupling <span class="_ _d"></span><span class="ls1b">during <span class="_ _0"></span>turns <span class="_ _0"></span></span>for <span class="_ _8"></span><span class="ls1e">a </span>linear <span class="_ _6"></span>submarine vehicle <span class="_ _d"></span>in <span class="_ _0"></span><span class="ls1b">which </span></div><div class="t m0 x5c h13 y8a ff2 fsf fc0 sc0 ls1e ws0">a <span class="_ _d"></span><span class="ls2">full <span class="_ _0"></span>state <span class="_ _6"></span>observer was <span class="_ _6"></span>employed <span class="_ _d"></span>in <span class="_ _6"></span>the <span class="_ _0"></span>compensator. Some </span></div><div class="t m0 x5c h13 y8b ff2 fsf fc0 sc0 ls2 ws0">reduction </div><div class="t m0 x69 h13 y8c ff2 fs1c fc0 sc0 ls4a ws0">in <span class="_ _0"></span><span class="fsf ls2">pitch <span class="_ _8"></span>induced <span class="_ _0"></span>response <span class="_ _0"></span>was <span class="_ _9"></span>achieved <span class="_ _d"></span>depending </span></div><div class="t m3f x5c h18 y8d ff2 fs13 fc0 sc0 ls2 ws0">on </div><div class="t m0 x39 h13 y8d ff2 fsf fc0 sc0 ls2 ws0">speed. <span class="_ _0"></span><span class="ls1e">A <span class="_ _9"> </span><span class="ls1b">gain <span class="_ _9"> </span></span></span>schedule <span class="_ _6"></span>with <span class="_ _9"> </span><span class="ls1b">two <span class="_ _2"> </span></span>separate <span class="_ _8"></span>speed <span class="_ _0"></span>regimes </div><div class="t m0 x5c h13 y8e ff2 fsf fc0 sc0 ls1b ws0">was <span class="_ _0"></span><span class="ls2">proposed. <span class="_ _8"></span>Consideration </span>was <span class="_ _8"></span><span class="ls2">given <span class="_ _0"></span>neither <span class="_ _0"></span></span>to <span class="_ _8"></span><span class="ls2">the <span class="_ _8"></span>design </span></div><div class="t m40 x5c h18 y8f ff2 fs13 fc0 sc0 ls4b ws0">of </div><div class="t m0 x6a h13 y8f ff2 fsf fc0 sc0 ls1e ws0">a <span class="ls2">command <span class="_ _c"></span>generator <span class="_ _c"></span><span class="ls1b">nor <span class="_ _d"></span><span class="ls2">the feedforward <span class="_ _c"></span>response shaping </span></span></span></div><div class="t m0 x5c h13 y90 ff2 fsf fc0 sc0 ls2 ws0">for <span class="_ _8"></span>depth <span class="_ _0"></span>changing <span class="_ _6"></span>maneuvers. <span class="_ _6"></span>Recently, <span class="_ _0"></span>however, <span class="_ _0"></span>Ruth <span class="_ _8"></span><span class="ls1b">and </span></div><div class="t m0 x5c h13 y91 ff2 fsf fc0 sc0 ls2 ws0">Humphreys </div><div class="t m32 x5 h2e y92 ff4 fsb fc0 sc0 ls4c ws0">[23] </div><div class="t m0 x6b h13 y92 ff2 fsf fc0 sc0 ls2 ws0">discussed <span class="_ _c"></span>the use <span class="ls36">of <span class="_"> </span></span>robust control design <span class="_ _c"></span>us- </div><div class="t m0 x5c h13 y93 ff2 fsf fc0 sc0 ls2 ws0">ing </div><div class="t m41 x6c h39 y94 ff9 fs25 fc0 sc0 ls4d ws0">1-1 </div><div class="t m0 x6d h13 y94 ff2 fsf fc0 sc0 ls2 ws0">synthesis <span class="_ _c"></span>methods in <span class="_ _6"></span>designing the <span class="_ _d"></span>coupled <span class="_ _d"></span><span class="ls1d">speeddepth </span></div><div class="t m0 x5c h13 y95 ff2 fsf fc0 sc0 ls2 ws0">controller <span class="_ _8"></span>for <span class="_ _9"> </span><span class="ls1e">a <span class="_ _8"></span></span>heavy <span class="_ _9"> </span><span class="ls1c">UUV <span class="_ _7"> </span></span>at <span class="_ _2"> </span><span class="ls1b">slow <span class="_"> </span></span>speed <span class="_ _8"></span>where <span class="_ _8"></span>the <span class="_ _9"> </span>plane </div><div class="t m0 x66 h13 y96 ff2 fsf fc0 sc0 ls2 ws0">action </div><div class="t m42 x6e h25 y97 ff2 fs1c fc0 sc0 ls4e ws0">is </div><div class="t m0 x69 h13 y97 ff2 fsf fc0 sc0 ls1b ws0">used </div><div class="t m43 xc h15 y97 ff2 fsb fc0 sc0 ls4f ws0">to </div><div class="t m0 x6f h13 y97 ff2 fsf fc0 sc0 ls2 ws0">control <span class="_ _2"> </span>both <span class="_ _1"> </span>vertical <span class="_"> </span>load <span class="_"> </span>inbalance <span class="_"> </span>and </div><div class="t m0 x66 h13 y98 ff2 fsf fc0 sc0 ls2 ws0">depth. <span class="_ _8"></span>Simulation <span class="_ _d"></span>results <span class="_ _8"></span>indicated <span class="_ _8"></span>that <span class="_ _8"></span><span class="ls1e">a <span class="_ _8"></span></span>classical <span class="_ _0"></span>controller </div><div class="t m0 x66 h13 y99 ff2 fsf fc0 sc0 ls1b ws0">could <span class="_ _0"></span><span class="ls1e">be <span class="_ _0"></span><span class="ls2">improved <span class="_ _6"></span></span></span>using <span class="_ _8"></span><span class="ls2">the <span class="_ _9"></span>10-state <span class="_ _c"></span>compensator described. </span></div><div class="t m0 x5c h13 y9a ff2 fsf fc0 sc0 ls2 ws0">Dougberty <span class="ls1b">and <span class="_ _d"></span>Woolweaver </span></div><div class="t m44 x70 h15 y9b ff2 fsb fc0 sc0 ls4f ws0">[7] </div><div class="t m0 x71 h13 y9b ff2 fsf fc0 sc0 ls2 ws0">have <span class="_ _d"></span>also <span class="_ _6"></span>shown <span class="ls1b">that <span class="_ _0"></span><span class="ls1e">a </span></span>mixed </div><div class="t m0 x66 h13 y9c ff2 fsf fc0 sc0 ls2 ws0">sliding <span class="_ _9"> </span><span class="ls1b">mode <span class="_"> </span></span>control <span class="_ _9"> </span>with <span class="_ _9"> </span><span class="ls1b">an <span class="_"> </span></span>inner <span class="_ _9"> </span>pitch <span class="_"> </span>control <span class="_ _9"> </span><span class="ls1b">and <span class="_"> </span></span>outer </div><div class="t m0 x66 h13 y9d ff2 fsf fc0 sc0 ls2 ws0">depth <span class="_ _9"> </span>control <span class="_ _9"> </span>loop <span class="_"> </span><span class="ls1c">as <span class="_ _7"> </span><span class="ls1b">used <span class="_ _1"> </span></span></span>on <span class="_ _9"> </span>the <span class="_"> </span>MUST <span class="_ _9"> </span>vehicle <span class="_ _9"> </span>provides </div><div class="t m0 x66 h13 y9e ff2 fsf fc0 sc0 ls2 ws0">satisfactory <span class="_"> </span>coupled <span class="_ _1"> </span>behavior, <span class="_ _b"> </span>although <span class="_ _a"> </span>the <span class="_ _13"> </span>details <span class="_ _a"> </span><span class="ls36">of <span class="_ _14"> </span></span>the </div><div class="t m0 x66 h13 y9f ff2 fsf fc0 sc0 ls2 ws0">control <span class="_"> </span>design <span class="_"> </span>are <span class="_"> </span>few. <span class="_"> </span>Others <span class="_"> </span>have <span class="_ _1"> </span>suggested <span class="_ _2"> </span>the <span class="_"> </span>use <span class="_ _a"> </span><span class="ls36">of </span></div><div class="t m0 x66 h13 ya0 ff2 fsf fc0 sc0 ls2 ws0">sliding modes with <span class="_ _6"></span>adaptivity, <span class="ls1c">as <span class="_ _8"></span></span>in <span class="_ _d"></span>Cristi </div><div class="t m3f x72 h3a ya1 ff4 fs13 fc0 sc0 ls50 ws0">et </div><div class="t m45 x57 h3b ya1 ff4 fsf fc0 sc0 ls1e ws0">al. </div><div class="t m46 x13 h3c ya1 ff2 fs12 fc0 sc0 ls51 ws0">131 </div><div class="t m0 x73 h13 ya1 ff2 fsf fc0 sc0 ls2 ws0">where the </div><div class="t m0 x66 h13 ya2 ff2 fsf fc0 sc0 ls2 ws0">sliding <span class="_ _8"></span>surface </div><div class="t m47 x74 h15 ya3 ff2 fsb fc0 sc0 ls2 ws0">is </div><div class="t m0 x1c h13 ya3 ff2 fsf fc0 sc0 ls2 ws0">based </div><div class="t m40 x75 h18 ya3 ff2 fs13 fc0 sc0 ls52 ws0">on </div><div class="t m0 x31 h13 ya3 ff2 fsf fc0 sc0 ls2 ws0">system <span class="_ _8"></span>state <span class="_ _8"></span><span class="ls1b">and <span class="_ _9"> </span></span>state <span class="_ _0"></span>estimators </div><div class="t m0 x66 h13 ya4 ff2 fsf fc0 sc0 ls49 ws0">rathter <span class="_ _7"> </span><span class="ls2">than </span></div><div class="t m40 x76 h18 ya5 ff2 fs13 fc0 sc0 ls2 ws0">on </div><div class="t m0 x77 h13 ya5 ff2 fsf fc0 sc0 ls2 ws0">output <span class="_ _8"></span>error. </div><div class="t m48 x78 h15 ya6 ff2 fsb fc0 sc0 ls53 ws0">It </div><div class="t m19 x79 h18 ya6 ff2 fs13 fc0 sc0 ls2 ws0">is </div><div class="t m0 x7a h13 ya6 ff2 fsf fc0 sc0 ls2 ws0">the <span class="_ _9"></span>robustness <span class="_ _8"></span><span class="ls36">of <span class="_ _a"> </span></span>control <span class="_ _9"> </span><span class="fs22 ls54">of </span></div><div class="t m49 x7b h3d ya6 ff2 fs1f fc0 sc0 ls2 ws0">UUV’s </div><div class="t m0 x7c h13 ya6 ff2 fsf fc0 sc0 ls2 ws0">operating <span class="_ _8"></span>in <span class="_ _8"></span>the </div><div class="t m0 x5c h13 ya7 ff2 fsf fc0 sc0 ls2 ws0">range <span class="_ _8"></span><span class="ls1b">up <span class="_ _2"> </span>to </span></div><div class="t m4a x7d h13 ya8 ff2 fsf fc0 sc0 ls49 ws0">10 </div><div class="t m0 x7e h13 ya8 ff2 fsf fc0 sc0 ls2 ws0">knots <span class="_ _8"></span><span class="ls1b">that <span class="_ _2"> </span>needs </span></div><div class="t m4b x7f h18 ya8 ff2 fs13 fc0 sc0 ls2 ws0">to </div><div class="t m0 x80 h13 ya8 ff2 fsf fc0 sc0 ls1b ws0">be <span class="_ _2"> </span><span class="ls2">addressed, <span class="_ _8"></span></span>and <span class="_ _8"></span><span class="ls1e">is <span class="_ _8"></span><span class="ls2">the </span></span></div><div class="t m0 x66 h13 ya9 ff2 fsf fc0 sc0 ls2 ws0">subject <span class="_ _0"></span><span class="ls36">of <span class="_ _1"> </span><span class="ls1b">this <span class="_ _8"></span></span></span>work. <span class="_ _8"></span>This <span class="_ _0"></span>work <span class="_ _9"> </span>shows <span class="_ _d"></span>that <span class="_ _8"></span><span class="ls1e">a <span class="_ _8"></span></span>comprehensive </div><div class="t m0 x66 h13 yaa ff2 fsf fc0 sc0 ls1c ws0">mulltivariable <span class="ls2">approach <span class="_ _d"></span><span class="ls1b">to <span class="_ _8"></span></span>sliding mode <span class="_ _6"></span>autopilot design leads </span></div><div class="t m0 x66 h13 yab ff2 fsf fc0 sc0 ls2 ws0">to <span class="_ _11"></span>robustly satisfactory <span class="_ _e"></span>results <span class="_ _f"></span>for <span class="ls1b">flight <span class="_ _c"></span><span class="ls2">vehicles, </span></span></div><div class="t m4c x13 h3e yac ff2 fs1e fc0 sc0 ls55 ws0">as </div><div class="t m0 xb h13 yac ff2 fsf fc0 sc0 ls2 ws0">opposed <span class="_ _f"></span><span class="ls1b">to </span></div><div class="t m17 x66 h13 yad ff2 fsf fc0 sc0 ls1e ws0">ROV’s, </div><div class="t m0 x2a h13 yad ff2 fsf fc0 sc0 ls1b ws0">over </div><div class="t m4d x81 h3f yad ff2 fs1d fc0 sc0 ls2 ws0">a </div><div class="t m0 x77 h13 yad ff2 fsf fc0 sc0 ls2 ws0">wide <span class="_ _6"></span>speed range. <span class="_ _0"></span>The <span class="_ _8"></span>sliding <span class="_ _11"></span>mode <span class="_ _6"></span>approach </div><div class="t m0 x66 h13 yae ff2 fsf fc0 sc0 ls1b ws0">used <span class="_ _1"> </span><span class="ls2">here <span class="_ _a"> </span>appears <span class="_"> </span></span>to <span class="_ _b"> </span>offer <span class="_ _a"> </span><span class="ls2">simplicity <span class="_ _9"> </span><span class="ls1e">for <span class="_ _1"> </span></span>vehicle <span class="_"> </span><span class="ls1e">in <span class="_"> </span></span></span>flight </div><div class="t m0 x66 h13 yaf ff2 fsf fc0 sc0 ls2 ws0">conditions <span class="_ _8"></span><span class="ls1b">and </span></div><div class="t m4e xc h3c yb0 ff2 fs12 fc0 sc0 ls56 ws0">is </div><div class="t m0 x25 h13 yb0 ff2 fsf fc0 sc0 ls2 ws0">similar <span class="_ _8"></span><span class="ls1b">to <span class="_ _9"> </span>that <span class="_"> </span><span class="ls36">of <span class="_ _b"> </span></span></span>Fossen <span class="_ _2"> </span><span class="ls1b">and <span class="_ _9"> </span></span>Satagun </div><div class="t m4f x82 h40 yb0 ff2 fse fc0 sc0 ls2 ws0">[9] </div><div class="t m0 x66 h13 yb1 ff2 fsf fc0 sc0 ls2 ws0">except </div><div class="t m50 x6e h16 yb1 ff2 fs11 fc0 sc0 ls57 ws0">in </div><div class="t m0 x69 h13 yb1 ff2 fsf fc0 sc0 ls2 ws0">the <span class="_ _8"></span>choice </div><div class="t m51 x1d h3f yb1 ff2 fs1d fc0 sc0 ls58 ws0">of </div><div class="t m0 x11 h13 yb1 ff2 fsf fc0 sc0 ls2 ws0">technique </div><div class="t m52 x83 h16 yb1 ff2 fs11 fc0 sc0 ls57 ws0">for </div><div class="t m0 x84 h13 yb1 ff2 fsf fc0 sc0 ls2 ws0">selection </div><div class="t m51 x85 h3f yb1 ff2 fs1d fc0 sc0 ls58 ws0">of </div><div class="t m0 x14 h13 yb1 ff2 fsf fc0 sc0 ls2 ws0">the <span class="_ _8"></span>sliding </div><div class="t m0 x66 h13 yb2 ff2 fsf fc0 sc0 ls2 ws0">surface <span class="_ _0"></span>parameters </div><div class="t m53 x1c h3d yb3 ff2 fs1f fc0 sc0 ls59 ws0">111. </div><div class="t m0 x86 h13 yb3 ff2 fsf fc0 sc0 ls1b ws0">VEHICLE <span class="_ _15"></span><span class="ls2">MODELING </span></div><div class="t m0 x78 h13 yb4 ff2 fsf fc0 sc0 ls2 ws0">The <span class="_ _2"> </span>three-dimensional <span class="_ _9"></span>equations <span class="_ _2"> </span><span class="ls1a">of <span class="_ _13"> </span><span class="ls1b">motion <span class="_ _1"> </span></span></span>for <span class="_"> </span>hydrody- </div><div class="t m0 x66 h13 yb5 ff2 fsf fc0 sc0 ls2 ws0">namically <span class="_"> </span>shaped <span class="_ _9"> </span>underwater <span class="_ _2"> </span>vehicles <span class="_ _2"> </span>have <span class="_ _2"> </span>been <span class="_"> </span>described </div><div class="t m0 x66 h13 yb6 ff2 fsf fc0 sc0 ls2 ws0">in <span class="_ _b"> </span>general <span class="_ _a"> </span>terms <span class="_ _b"> </span><span class="ls1b">by <span class="_ _3"> </span></span>Abkowitz </div><div class="t m0 x87 h13 yb7 ff2 fsf fc0 sc0 ls5a ws0">[l], <span class="_ _9"> </span><span class="ls2">and <span class="_ _13"> </span>are <span class="_ _b"> </span>most <span class="_ _13"> </span>conve- </span></div><div class="t m0 x66 h13 yb8 ff2 fsf fc0 sc0 ls1b ws0">niently <span class="_"> </span><span class="ls2">developed <span class="_ _8"></span></span>using <span class="_"> </span><span class="ls1e">a <span class="_ _9"></span></span>body <span class="_"> </span>fixed <span class="_ _9"> </span><span class="ls2">coordinate <span class="_ _9"></span>frame <span class="_ _9"> </span></span>and </div><div class="t m0 x66 h13 yb9 ff2 fsf fc0 sc0 ls1e ws0">a <span class="_"> </span><span class="ls2">global <span class="_ _9"> </span>reference <span class="_ _2"> </span>frame. <span class="_"> </span>The <span class="_ _2"> </span><span class="ls1b">body <span class="_ _a"> </span>fixed <span class="_"> </span></span>frame <span class="_ _2"> </span>has <span class="_ _1"> </span>com- </span></div><div class="t m0 x66 h13 yba ff2 fsf fc0 sc0 ls2 ws0">ponents </div><div class="t m54 x68 h13 ybb ff2 fsf fc0 sc0 ls1a ws0">of </div><div class="t m0 x88 h13 ybb ff2 fsf fc0 sc0 ls1b ws0">motion <span class="_ _b"> </span><span class="ls2">given <span class="_"> </span><span class="ls1a">by <span class="_ _4"> </span></span>the <span class="_ _a"> </span>six <span class="_"> </span>velocity <span class="_ _a"> </span>components, </span></div><div class="t m55 x89 h41 ybc ff4 fs1a fc0 sc0 ls5b ws0">[~(t), </div><div class="t m56 x1f h42 ybc ff4 fs1e fc0 sc0 ls5c ws0">w(t), </div><div class="t m57 x76 h43 ybc ff2 fs22 fc0 sc0 ls5d ws0">w(t), </div><div class="t m58 x8a h44 ybc ff4 fs22 fc0 sc0 ls5e ws0">p(t), </div><div class="t m59 x8b h3b ybc ff4 fsf fc0 sc0 ls5f ws0">q(t), </div><div class="t m5a x8c h45 ybc ff4 fsd fc0 sc0 ls60 ws0">~(t)], </div><div class="t m0 x8d h13 ybc ff2 fsf fc0 sc0 ls2 ws0">relative <span class="_ _9"> </span>to <span class="_"> </span><span class="ls1e">a <span class="_ _8"></span></span>constant <span class="_ _9"> </span><span class="ls1b">ve- </span></div><div class="t m0 x66 h13 ybd ff2 fsf fc0 sc0 ls2 ws0">locity <span class="_ _9"> </span>coordinate <span class="_ _8"></span>frame <span class="_ _9"></span>moving <span class="_ _8"></span>with <span class="_ _9"> </span>the <span class="_"> </span>ocean <span class="_ _8"></span>current, </div><div class="t m5b x8e h46 ybe ffb fs2 fc0 sc0 ls61 ws0">U,, </div><div class="t m0 x66 h13 ybf ff2 fsf fc0 sc0 ls1b ws0">and <span class="_ _8"></span><span class="ls2">the <span class="_ _8"></span>velocity <span class="_ _0"></span>vector <span class="_ _0"></span><span class="ls1e">is <span class="_ _0"></span></span>represented <span class="_ _8"></span><span class="ls1c">as </span></span></div><div class="t m0 x8f h13 yc0 ff2 fsf fc0 sc0 ls2 ws0">while <span class="_ _6"></span>the <span class="_ _9"> </span>six <span class="_ _0"></span>components <span class="_ _6"></span><span class="ls36">of <span class="_ _1"> </span></span>position <span class="_ _8"></span>in <span class="_ _8"></span>the <span class="_ _8"></span>global <span class="_ _6"></span>reference </div><div class="t m0 x8f h13 yc1 ff2 fsf fc0 sc0 ls2 ws0">frame <span class="_ _2"> </span>are </div><div class="t m5c x16 h47 yc2 ff4 fs24 fc0 sc0 ls62 ws0">Z’(t) </div><div class="t m5d x90 h48 yc2 ff3 fs21 fc0 sc0 ls2 ws0">= </div><div class="t m5e x91 h49 yc2 ff3 fsa fc0 sc0 ls63 ws0">F(t), </div><div class="t m5f x92 h4a yc2 ff9 fs26 fc0 sc0 ls64 ws0">W), </div><div class="t m60 x93 h27 yc2 ff4 fs1d fc0 sc0 ls65 ws0">Z(t), </div><div class="t m61 x94 h4b yc2 ff9 fs27 fc0 sc0 ls66 ws0">#)(a </div><div class="t m62 x4e h4c yc2 ff4 fs28 fc0 sc0 ls67 ws0">d(t), </div><div class="t m63 x95 h4d yc2 ff2 fs29 fc0 sc0 ls68 ws0">$431. </div><div class="t m64 x96 h3a yc3 ff4 fs13 fc0 sc0 ls69 ws0">(2) </div><div class="t m0 x97 h13 yc4 ff2 fsf fc0 sc0 ls2 ws0">The <span class="_ _0"></span>angles </div><div class="t m55 x98 h41 yc4 ff4 fs1a fc0 sc0 ls6a ws0">#)(t), </div><div class="t m65 x99 h4e yc4 ff9 fsc fc0 sc0 ls6b ws0">O(t), </div><div class="t m55 x9a h41 yc4 ff4 fs1a fc0 sc0 ls6c ws0">$(t), </div><div class="t m0 x9b h13 yc4 ff2 fsf fc0 sc0 ls2 ws0">(azimuth, elevation, <span class="_ _d"></span><span class="ls1b">and <span class="_ _0"></span></span>spin), </div><div class="t m0 x8f h13 yc5 ff2 fsf fc0 sc0 ls2 ws0">are <span class="_ _9"> </span>related <span class="_ _9"></span>through <span class="_ _8"></span>Euler <span class="_"> </span>transformations <span class="_ _0"></span>to <span class="_ _2"> </span>the <span class="_ _2"> </span><span class="ls1b">body <span class="_"> </span></span>yaw, </div><div class="t m0 x8f h13 yc6 ff2 fsf fc0 sc0 ls2 ws0">pitch, <span class="_ _8"></span><span class="ls1b">and <span class="_ _6"></span></span>roll <span class="_ _8"></span>motions. <span class="_ _d"></span>Control <span class="_ _6"></span>inputs <span class="_ _0"></span>from <span class="_ _0"></span>control <span class="_ _0"></span>surfaces, </div><div class="t m0 x8f h13 yc7 ff2 fsf fc0 sc0 ls2 ws0">propeller <span class="_ _0"></span>speeds, <span class="_ _d"></span>thruster <span class="_ _d"></span>forces, <span class="_ _6"></span><span class="ls1b">and <span class="_ _8"></span></span>buoyancy <span class="_ _d"></span>adjustment <span class="ls1e">in </span></div><div class="t m0 x8f h13 yc8 ff2 fsf fc0 sc0 ls2 ws0">general <span class="_ _6"></span><span class="ls1b">may <span class="_ _8"></span><span class="ls1e">be <span class="_ _0"></span></span></span>considered <span class="_ _6"></span>as <span class="_ _8"></span>the <span class="_ _8"></span>vector, </div><div class="t m66 x51 h4c yc9 ff4 fs28 fc0 sc0 ls6d ws0">u(t), </div><div class="t m0 x9c h13 yc9 ff2 fsf fc0 sc0 ls1b ws0">and <span class="_ _8"></span><span class="ls2">are <span class="_ _0"></span>more </span></div><div class="t m0 x8f h13 yca ff2 fsf fc0 sc0 ls1b ws0">specifically <span class="_"> </span><span class="ls2">taken <span class="_ _1"> </span>for <span class="_ _a"> </span></span>this <span class="_ _a"> </span><span class="ls2">work <span class="_ _1"> </span>from <span class="_ _a"> </span><span class="ls1e">a <span class="_ _1"> </span></span>swimmer <span class="_"> </span>delivery </span></div><div class="t m0 x8f h13 ycb ff2 fsf fc0 sc0 ls2 ws0">vehicle, <span class="_ _8"></span>where </div><div class="t m0 x97 h13 ycc ff2 fsf fc0 sc0 ls2 ws0">The <span class="_ _d"></span><span class="ls1b">first </span>four correspond <span class="_ _f"></span><span class="ls1b">to <span class="_ _6"></span><span class="ls2">control surface <span class="_ _11"></span>deflections <span class="_ _e"></span>from </span></span></div><div class="t m0 x8f h13 ycd ff2 fsf fc0 sc0 ls2 ws0">the <span class="_ _11"></span>rudder, <span class="_ _c"></span>port, <span class="_ _d"></span><span class="ls1b">and <span class="_ _d"></span></span>starboard <span class="_ _e"></span><span class="ls1c">bow <span class="_ _d"></span><span class="ls2">planes, <span class="ls1b">and </span>the <span class="_ _d"></span>stem <span class="_ _c"></span>plane, </span></span></div><div class="t m0 x8f h13 yce ff2 fsf fc0 sc0 ls1b ws0">and <span class="_ _6"></span><span class="ls2">the <span class="_ _d"></span>remaining </span>two <span class="_ _8"></span><span class="ls2">arise from <span class="_ _c"></span>propeller <span class="_ _d"></span>rotational rate <span class="_ _6"></span>and </span></div><div class="t m0 x8f h13 ycf ff2 fsf fc0 sc0 ls2 ws0">buoyancy adjustment. <span class="_ _e"></span>The <span class="_ _d"></span>development <span class="_ _c"></span><span class="ls36">of <span class="_ _9"> </span><span class="ls2">the functional <span class="_ _f"></span>form </span></span></div><div class="t m0 x8f h3c yd0 ff2 fs12 fc0 sc0 ls6e ws0">of <span class="_ _4"> </span><span class="fsf ls2">the <span class="_"> </span>hydrodynamic <span class="_ _9"></span>forces <span class="_ _9"> </span>has <span class="_"> </span>been <span class="_"> </span>well <span class="_"> </span>studied <span class="_ _9"> </span>and, <span class="_"> </span>in </span></div><div class="t m0 x8f h13 yd1 ff2 fsf fc0 sc0 ls2 ws0">terms <span class="_ _9"> </span><span class="ls1a">of <span class="_ _13"> </span><span class="ls1b">first <span class="_"> </span></span></span>order <span class="_ _8"></span>variations <span class="_ _9"> </span><span class="ls36">of <span class="_ _3"> </span><span class="ls1b">motion <span class="_ _1"> </span></span></span>components, <span class="_ _8"></span>were </div><div class="t m0 x8f h13 yd2 ff2 fsf fc0 sc0 ls2 ws0">given <span class="_"> </span><span class="ls1a">by <span class="_ _4"> </span></span>Gertler <span class="_ _2"> </span><span class="ls1b">and <span class="_ _a"> </span></span>Hagen <span class="_ _b"> </span><span class="ls48">1111 <span class="_ _b"> </span><span class="ls1b">and <span class="_ _b"> </span></span></span>later <span class="_"> </span><span class="ls1a">by <span class="_ _4"> </span></span>Abkowitz. </div><div class="t m0 x8f h13 yd3 ff2 fsf fc0 sc0 ls2 ws0">Specific <span class="_ _e"></span>values <span class="_ _c"></span><span class="ls36">of <span class="_ _9"></span><span class="ls2">the <span class="_ _f"></span>particular <span class="ls1b">coefficients <span class="_ _c"></span><span class="ls2">depend </span></span></span></span></div><div class="t m40 x41 h18 yd4 ff2 fs13 fc0 sc0 ls52 ws0">on </div><div class="t m0 x9d h13 yd4 ff2 fsf fc0 sc0 ls1b ws0">specific </div><div class="t m0 x8f h13 yd5 ff2 fsf fc0 sc0 ls2 ws0">vehicles, <span class="_ _1"> </span>although <span class="_"> </span>normalization <span class="_ _9"> </span><span class="ls1b">by <span class="_ _3"> </span></span>speed <span class="_ _1"> </span><span class="ls1b">and <span class="_ _b"> </span></span>length <span class="_ _1"> </span><span class="ls1b">can </span></div><div class="t m0 x8f h13 yd6 ff2 fsf fc0 sc0 ls2 ws0">provide <span class="_ _6"></span>some <span class="_ _6"></span>generalized <span class="_ _c"></span>feeling <span class="_ _0"></span>as <span class="_ _d"></span>to <span class="_ _0"></span>their <span class="_ _0"></span>scaling behavior. </div><div class="t m0 x8f h13 yd7 ff2 fsf fc0 sc0 ls2 ws0">The <span class="_ _9"> </span>values <span class="_ _9"></span>used <span class="_ _9"></span>in <span class="_ _9"> </span>this <span class="_ _9"> </span>work <span class="_ _9"> </span>were <span class="_ _9"></span>based <span class="_ _9"> </span><span class="ls1c">on <span class="_"> </span><span class="ls1e">a <span class="_ _9"> </span></span></span>box <span class="_ _2"> </span>shaped </div><div class="t m0 x8f h13 yd8 ff2 fsf fc0 sc0 ls2 ws0">vehicle, <span class="_ _c"></span><span class="ls1e">a <span class="ls1b">simplified <span class="_ _e"></span><span class="ls2">model <span class="ls36">of <span class="_ _8"></span><span class="ls1b">which </span></span>is outlined </span></span></span></div><div class="t m67 x9e h43 yd9 ff2 fs22 fc0 sc0 ls3f ws0">in </div><div class="t m0 x9f h13 yd9 ff2 fsf fc0 sc0 ls2 ws0">the <span class="_ _11"></span>sketch <span class="_ _f"></span>in </div><div class="t m0 x8f h13 yda ff2 fsf fc0 sc0 ls2 ws0">Fig. </div><div class="t m68 xa0 h15 ydb ff2 fsb fc0 sc0 ls53 ws0">1. </div><div class="t m0 x4b h13 ydb ff2 fsf fc0 sc0 ls2 ws0">This <span class="ls1b">was <span class="_ _d"></span></span>chosen <span class="_ _c"></span>mainly <span class="_ _c"></span>because experimental <span class="_ _f"></span>data <span class="_ _c"></span>was </div><div class="t m0 x8f h13 ydc ff2 fsf fc0 sc0 ls1c ws0">in <span class="_ _8"></span><span class="ls2">existence <span class="_ _c"></span>for <span class="_ _6"></span>the <span class="_ _d"></span>verification <span class="_ _c"></span><span class="ls36">of <span class="_ _2"> </span><span class="ls2">the <span class="_ _6"></span>simulation <span class="_ _c"></span>models <span class="_ _d"></span><span class="ls1b">used </span></span></span></span></div><div class="t m0 x8f h13 ydd ff2 fsf fc0 sc0 ls2 ws0">here. <span class="_ _8"></span><span class="ls1e">The <span class="_ _6"></span><span class="ls1b">coefficients <span class="_ _8"></span></span></span>are <span class="_ _0"></span>vehicle <span class="_ _8"></span>dependent <span class="_ _d"></span><span class="ls1b">and <span class="_ _8"></span>would <span class="_ _8"></span></span>need </div><div class="t m0 x8f h13 yde ff2 fsf fc0 sc0 ls1b ws0">modification <span class="_ _0"></span><span class="ls36">if <span class="_ _1"> </span><span class="ls2">applied </span></span></div><div class="t m69 x92 h15 ydf ff2 fsb fc0 sc0 ls2 ws0">to </div><div class="t m0 xa1 h13 ydf ff2 fsf fc0 sc0 ls1b ws0">other <span class="_ _0"></span><span class="ls2">vehicles. <span class="_ _d"></span>The <span class="_ _6"></span>functional <span class="_ _d"></span>form </span></div><div class="t m0 x8f h13 ye0 ff2 fsf fc0 sc0 ls36 ws0">of <span class="_ _8"></span><span class="ls2">the force equations <span class="_ _f"></span>are <span class="_ _11"></span>given <span class="_ _c"></span>in Boncal </span></div><div class="t m32 x4e h2e ye1 ff4 fsb fc0 sc0 ls6f ws0">[2], </div><div class="t m0 xa2 h13 ye1 ff2 fsf fc0 sc0 ls1b ws0">and <span class="_ _d"></span><span class="ls2">are <span class="_ _f"></span>repeated </span></div><div class="t m19 x8f h18 ye2 ff2 fs13 fc0 sc0 ls70 ws0">in </div><div class="t m0 xa3 h13 ye2 ff2 fsf fc0 sc0 ls2 ws0">the Appendix. <span class="_ _11"></span>The model described <span class="_ _f"></span>includes <span class="_ _d"></span><span class="ls1e">a </span>model <span class="ls36">of <span class="_ _9"></span></span>the </div><div class="t m0 x8f h13 ye3 ff2 fsf fc0 sc0 ls2 ws0">cross <span class="_ _6"></span><span class="ls1c">flow <span class="_ _0"></span><span class="ls1b">drag <span class="_ _6"></span></span></span>effects, <span class="_ _d"></span><span class="ls1b">and <span class="_ _8"></span><span class="ls1e">a </span></span>model </div><div class="t m54 xa4 h13 ye4 ff2 fsf fc0 sc0 ls36 ws0">of </div><div class="t m0 xa5 h13 ye4 ff2 fsf fc0 sc0 ls2 ws0">the <span class="_ _6"></span>propulsion <span class="_ _d"></span>system </div><div class="t m0 xa6 h13 ye5 ff2 fsf fc0 sc0 ls1b ws0">and <span class="_ _8"></span><span class="ls2">is <span class="_ _9"></span>therefore <span class="_ _8"></span><span class="ls1e">a <span class="_ _8"></span></span>large <span class="_ _8"></span>departure <span class="_ _d"></span>from <span class="_ _9"></span>the <span class="_ _8"></span>original <span class="_ _0"></span>work <span class="_ _8"></span><span class="ls36">of </span></span></div><div class="t m0 x8f h13 ye6 ff2 fsf fc0 sc0 ls1b ws0">Abko <span class="_ _12"></span>witz </div><div class="t m6a xa7 h4f ye7 ff3 fs1f fc0 sc0 ls2 ws0">. </div><div class="t m0 x97 h13 ye8 ff2 fsf fc0 sc0 ls2 ws0">The vehicle <span class="_ _c"></span>motion <span class="ls1b">may </span></div><div class="t m10 xa8 h50 ye8 ff2 fsc fc0 sc0 ls2 ws0">be </div><div class="t m0 xa9 h13 ye8 ff2 fsf fc0 sc0 ls2 ws0">descnbed <span class="_ _d"></span>in terms </div><div class="t m35 xaa h51 ye8 ff2 fs1b fc0 sc0 ls71 ws0">of </div><div class="t m0 x56 h13 ye8 ff2 fsf fc0 sc0 ls2 ws0">the twelve </div><div class="t m0 x8f h13 ye9 ff2 fsf fc0 sc0 ls2 ws0">nonlinear <span class="_ _9"></span><span class="ls1b">system <span class="_ _8"></span></span>equations </div><div class="t m6b xab h30 yea ff4 fs1b fc0 sc0 ls72 ws0">[32] </div><div class="t m6c xac h52 yeb ff9 fs13 fc0 sc0 ls73 ws0">M(W4t)ldt </div><div class="t m5d xad h48 yeb ff3 fs21 fc0 sc0 ls2 ws0">= </div><div class="t m6d xae h53 yeb ff2 fs2a fc0 sc0 ls74 ws0">f(z(t), </div><div class="t m6e xab h54 yeb ff4 fs2b fc0 sc0 ls75 ws0">4t)l </div><div class="t m6f xaf h55 yeb ff4 fs2c fc0 sc0 ls76 ws0">4t)) </div><div class="t m70 xb0 h56 yeb ff3 fs2d fc0 sc0 ls2 ws0">+ </div><div class="t m71 x4e h57 yeb ff5 fs25 fc0 sc0 ls77 ws0">g(4t), </div><div class="t m72 xb1 h58 yeb ff4 fs2e fc0 sc0 ls78 ws0">4t)Mt) </div><div class="t m73 x49 h45 yec ff4 fsd fc0 sc0 ls79 ws0">dz(t)/dt </div><div class="t m5d xad h48 yec ff3 fs21 fc0 sc0 ls2 ws0">= </div><div class="t m74 xae h59 yec ff9 fs1f fc0 sc0 ls7a ws0">Wt), </div><div class="t m75 xb2 h58 yec ff4 fs2e fc0 sc0 ls7b ws0">4t), </div><div class="t m48 xaf h5a yec ff9 fs2f fc0 sc0 ls7c ws0">uc) </div><div class="t m76 xb3 h15 yed ff2 fsb fc0 sc0 ls2 ws0">(4) </div><div class="t m0 xa6 h13 yee ff2 fsf fc0 sc0 ls2 ws0">in <span class="_ _9"> </span>which <span class="_ _9"> </span>the <span class="_ _9"> </span>coupled <span class="_ _9"></span><span class="ls1b">mass <span class="_ _9"> </span></span>matrix </div><div class="t m77 xb4 h42 yee ff4 fs1e fc0 sc0 ls7d ws0">M(t) </div><div class="t m0 xb5 h13 yee ff2 fsf fc0 sc0 ls2 ws0">includes <span class="_ _8"></span><span class="ls1b">both <span class="_"> </span></span>me- </div><div class="t m0 xa6 h13 yef ff2 fsf fc0 sc0 ls2 ws0">chanical <span class="_ _6"></span>and <span class="_ _0"></span>hydrodynamic <span class="_ _0"></span><span class="ls1b">added <span class="_ _8"></span></span>mass; <span class="_ _0"></span>the <span class="_ _8"></span>functions </div><div class="t m78 x4f h15 yf0 ff2 fsb fc0 sc0 ls2 ws0">f </div><div class="t m0 x64 h13 yf0 ff2 fsf fc0 sc0 ls1b ws0">and </div><div class="t m79 x8f h5b yf1 ff6 fsa fc0 sc0 ls2 ws0">g </div><div class="t m0 x97 h13 yf1 ff2 fsf fc0 sc0 ls2 ws0">are <span class="_ _8"></span>mappings <span class="_ _0"></span><span class="ls36">of <span class="_ _1"> </span></span>the <span class="_ _8"></span>vehicle <span class="_ _8"></span>motions <span class="_ _8"></span>into <span class="_ _0"></span>forces <span class="_ _8"></span>including </div><div class="t m0 xa6 h13 yf2 ff2 fsf fc0 sc0 ls1c ws0">Coriolis, <span class="_ _0"></span><span class="ls2">gravitational, <span class="_ _0"></span><span class="ls1b">and <span class="_ _9"> </span></span>centrifugal <span class="_ _6"></span>forces; <span class="_ _8"></span>the <span class="_ _8"></span>hydrostatic </span></div><div class="t m7a xa6 h5c yf3 ff2 fs25 fc0 sc0 ls2 ws0">and <span class="_ _0"></span><span class="ls7e">hydrodynamic <span class="_ _0"></span>forces </span></div><div class="t m7b xa1 h5c yf3 ff2 fs25 fc0 sc0 ls2 ws0">and </div><div class="t m7a xb6 h5c yf3 ff2 fs25 fc0 sc0 ls7e ws0">moments <span class="_ _8"></span>acting </div><div class="t m7c xb7 h20 yf3 ff2 fs19 fc0 sc0 ls7f ws0">on </div><div class="t m7d xb8 h5b yf3 ffc fsa fc0 sc0 ls80 ws0">the </div><div class="t m0 xb9 h13 yf3 ff2 fsf fc0 sc0 ls2 ws0">vehicle </div><div class="t m0 xa6 h13 yf4 ff2 fsf fc0 sc0 ls2 ws0">in <span class="_ _9"> </span>the <span class="_ _9"> </span><span class="ls1b">body <span class="_"> </span>fixed <span class="_ _9"> </span></span>coordinate <span class="_ _8"></span>frame, <span class="_ _9"> </span><span class="ls1b">with <span class="_"> </span>coefficients </span></div><div class="t m67 x4f h44 yf5 ff8 fs22 fc0 sc0 ls66 ws0">e; </div><div class="t m0 x96 h13 yf5 ff2 fsf fc0 sc0 ls2 ws0">the </div><div class="t m0 xa6 h13 yf6 ff2 fsf fc0 sc0 ls2 ws0">motion <span class="_ _6"></span>dependent influence <span class="_ _c"></span><span class="ls36">of <span class="_ _2"> </span><span class="ls2">control <span class="_ _0"></span>surfaces, thrusters, <span class="ls1b">and </span></span></span></div><div class="t m0 xa6 h13 yf7 ff2 fsf fc0 sc0 ls1b ws0">any <span class="_ _9"> </span><span class="ls2">ballasting; <span class="_ _2"> </span></span>and <span class="_ _2"> </span><span class="ls2">the <span class="_ _9"> </span>function </span></div><div class="t m7e xba h3a yf8 ff8 fs13 fc0 sc0 ls2 ws0">h </div><div class="t m0 xbb h13 yf8 ff2 fsf fc0 sc0 ls2 ws0">includes <span class="_ _9"> </span>the <span class="_ _9"> </span>kinematical </div><div class="t m0 x8f h13 yf9 ff2 fsf fc0 sc0 ls2 ws0">relationships <span class="_ _0"></span>found <span class="_ _8"></span>in <span class="_ _6"></span>performing <span class="_ _8"></span>the <span class="_ _8"></span>coordinate <span class="_ _0"></span>transforma- </div><div class="t m0 xa6 h13 yfa ff2 fsf fc0 sc0 ls2 ws0">tions between <span class="ls1b">body <span class="_ _8"></span>fixed <span class="_ _6"></span>and <span class="_ _0"></span></span>global reference frames <span class="_ _6"></span><span class="ls1b">and <span class="_ _6"></span></span>the </div><div class="t m0 xa6 h13 yfb ff2 fsf fc0 sc0 ls2 ws0">constant <span class="_ _8"></span>ocean <span class="_ _8"></span>current, </div><div class="t m7f xbc h5d yfc ff4 fs30 fc0 sc0 ls2 ws0">U,. </div><div class="t m0 xa3 h13 y6d ff2 fsf fc0 sc0 ls2 ws0">Once <span class="_ _b"> </span><span class="ls1e">a <span class="_ _a"> </span></span>hydrodynamic <span class="_ _a"> </span>design <span class="_ _b"> </span>is <span class="_ _13"> </span>made, <span class="_ _b"> </span>these <span class="_ _b"> </span>functions </div><div class="t m0 xa6 h13 yfd ff2 fsf fc0 sc0 ls2 ws0">can <span class="_ _8"></span><span class="ls1b">be <span class="_ _d"></span></span>’estimated </div><div class="t m69 xbd h15 yfe ff2 fsb fc0 sc0 ls2 ws0">and </div><div class="t m0 xbe h13 yfe ff2 fsf fc0 sc0 ls2 ws0">are <span class="_ _0"></span>known <span class="_ _8"></span>through <span class="_ _9"> </span><span class="ls1e">a <span class="_ _0"></span></span>vehicle <span class="_ _9"></span>dynamic </div><div class="t m0 xa6 h13 yff ff2 fsf fc0 sc0 ls2 ws0">modeling <span class="_ _8"></span><span class="ls1b">to <span class="_ _9"> </span></span>lie <span class="_ _9"> </span>within <span class="_ _9"></span><span class="ls1e">a <span class="_ _8"></span><span class="ls1b">finite <span class="_ _8"></span>bound <span class="_ _2"> </span></span></span>where <span class="_ _8"></span>that <span class="_ _8"></span><span class="ls1b">bound <span class="_ _2"> </span></span>can </div><div class="t m0 xa6 h13 y100 ff2 fsf fc0 sc0 ls1e ws0">be <span class="_ _1"> </span><span class="ls2">established </span></div><div class="t m80 x5e h3d y101 ff2 fs1f fc0 sc0 ls2 ws0">a </div><div class="t m45 xbd h3b y101 ff4 fsf fc0 sc0 ls81 ws0">priori </div><div class="t m0 xbf h13 y101 ff2 fsf fc0 sc0 ls36 ws0">if <span class="_ _4"> </span><span class="ls2">knowledge <span class="_"> </span></span>of <span class="_ _4"> </span><span class="ls2">the <span class="_ _a"> </span>variability <span class="_"> </span></span>of </div><div class="t m0 xa6 h13 y102 ff2 fsf fc0 sc0 ls2 ws0">the <span class="_ _8"></span>vehicle <span class="_ _0"></span>coefficients </div><div class="t m67 xc0 h44 y103 ff8 fs22 fc0 sc0 ls2 ws0">c </div><div class="t m0 xa1 h13 y103 ff2 fsf fc0 sc0 ls1e ws0">is <span class="_ _8"></span><span class="ls2">assumed. <span class="_ _d"></span>The <span class="_ _8"></span>functions </span></div><div class="t m81 xc1 h5e y103 ffb fs1d fc0 sc0 ls2 ws0">f </div><div class="t m0 xc2 h13 y103 ff2 fsf fc0 sc0 ls1b ws0">and </div><div class="t m82 xc3 h26 y103 ff5 fs14 fc0 sc0 ls2 ws0">g </div><div class="t m0 xa6 h13 y104 ff2 fsf fc0 sc0 ls2 ws0">are <span class="_ _8"></span><span class="ls1b">known <span class="_ _2"> </span></span>to <span class="_ _9"> </span><span class="ls1b">be <span class="_"> </span>finite <span class="_ _2"> </span></span>gain <span class="_ _9"> </span>stable. </div><div class="t m83 x4d h3a y104 ff8 fs13 fc0 sc0 ls2 ws0">h </div><div class="t m0 xa5 h13 y104 ff2 fsf fc0 sc0 ls2 ws0">is <span class="_ _9"> </span>known <span class="_ _9"> </span><span class="ls1b">to <span class="_ _9"> </span><span class="ls1e">be <span class="_ _9"> </span></span>finite </span></div><div class="t m0 x1f h32 y76 ffa fs23 fc1 sc0 ls2 ws0">Authorized licensed use limited to: Yuan Ze University. Downloaded on August 09,2020 at 18:04:36 UTC from IEEE Xplore. Restrictions apply. </div><a class="l" rel='nofollow' onclick='return false;'><div class="d m84"></div></a></div><div class="pi" data-data='{"ctm":[1.546195,0.000000,0.000000,1.546195,0.000000,0.000000]}'></div></div>
<div id="pf3" class="pf w4 h5f" data-page-no="3"><div class="pc pc3 w4 h5f"><img class="bi x0 y0 w5 h60" alt="" src="https://static.pudn.com/prod/directory_preview_static/625d80f0be9ad24cfa814342/bg3.jpg"><div class="t m85 x19 h8 y105 ff1 fs4 fc0 sc0 ls82 ws0">“LEY </div><div class="t m3d xc4 h6 y105 ff2 fs2 fc0 sc0 ls83 ws0">AND </div><div class="t m4 x2a h6 y105 ff2 fs2 fc0 sc0 ls2 ws0">LIENARD: </div><div class="t m2 x20 h4 y105 ff2 fs0 fc0 sc0 ls3 ws0">MULTIVARIABLE </div><div class="t m0 xc5 h61 y105 ff2 fs3 fc0 sc0 ls84 ws0">SLIDING </div><div class="t m86 xc6 h61 y105 ff2 fs3 fc0 sc0 ls2 ws0">MODE </div><div class="t m0 xc7 h6 y105 ff2 fs2 fc0 sc0 ls44 ws0">CONTROL </div><div class="t m0 xb9 h62 y106 ff3 fs31 fc0 sc0 ls2 ws0">~ </div><div class="t m87 x3f h7 y107 ff1 fs3 fc0 sc0 ls85 ws0">329 </div><div class="t m88 xc7 h63 y108 ff1 fs25 fc0 sc0 ls2 ws0">Fig. </div><div class="t m89 xc8 h64 y108 ff1 fs16 fc0 sc0 ls86 ws0">1. </div><div class="t m8a x73 h19 y108 ff1 fs14 fc0 sc0 ls2 ws0">Sketch </div><div class="t m28 xc9 h1f y108 ff2 fs14 fc0 sc0 ls87 ws0">of </div><div class="t m8a xca h19 y108 ff1 fs14 fc0 sc0 ls2 ws0">the <span class="_ _8"> </span>vehicle <span class="_ _7"> </span>type. </div><div class="t m0 x19 h13 y109 ff2 fsf fc0 sc0 ls1b ws0">singularity <span class="_ _8"></span><span class="ls20">may <span class="_"> </span></span>be <span class="_ _9"> </span>avoided <span class="_ _8"></span><span class="ls1c">by <span class="_"> </span></span>the </div><div class="t m8b x34 h65 y109 ff2 fs32 fc0 sc0 ls88 ws0">use </div><div class="t m0 x80 h13 y109 ff2 fsf fc0 sc0 ls1a ws0">of <span class="_ _7"> </span><span class="ls1e">quaternions <span class="_ _c"></span><span class="ls1b">in <span class="_ _9"></span><span class="ls2">the </span></span></span></div><div class="t m0 x19 h13 y10a ff2 fsf fc0 sc0 ls1b ws0">rigid body <span class="_ _6"></span>angular motion <span class="_ _d"></span>kinematics <span class="_ _6"></span><span class="ls2">rather </span>than <span class="ls1e">Euler <span class="_ _c"></span><span class="ls1b">angles. </span></span></div><div class="t m0 x19 h13 y10b ff2 fsf fc0 sc0 ls2 ws0">In <span class="_ _8"></span><span class="ls1b">this <span class="_ _8"></span>situation <span class="_ _8"></span>and <span class="_ _9"></span>with </span></div><div class="t m8c x37 h3f y10c ff2 fs1d fc0 sc0 ls89 ws0">(4) </div><div class="t m0 xcb h13 y10c ff2 fsf fc0 sc0 ls2 ws0">as <span class="_ _0"></span>the <span class="_ _8"></span><span class="ls1b">background <span class="_ _8"></span>model <span class="_ _9"> </span></span>for <span class="_ _8"></span><span class="ls1b">a </span></div><div class="t m0 x19 h13 y10d ff2 fsf fc0 sc0 ls1b ws0">general <span class="_ _9"> </span>underwater <span class="_ _9"> </span>vehicle <span class="_ _2"> </span>during <span class="_ _9"></span>maneuvering, <span class="_ _9"></span>the <span class="_ _9"> </span>concept </div><div class="t m0 x19 h13 y10e ff2 fsf fc0 sc0 ls1e ws0">of <span class="_ _d"></span><span class="ls1b">a <span class="_ _8"></span>multivariable <span class="_ _9"></span>sliding <span class="_ _6"></span>mode <span class="_ _8"></span>control <span class="_ _6"></span>will <span class="_ _8"></span></span>be <span class="_ _6"></span><span class="ls1b">developed. </span></div><div class="t m0 x3a h13 y10f ff2 fsf fc0 sc0 ls1b ws0">will <span class="_"> </span>also <span class="_ _2"> </span>imply </div><div class="t m8d xcc h11 y110 ff2 fsd fc0 sc0 ls18 ws0">(9) </div><div class="t m0 x3a h13 y111 ff2 fsf fc0 sc0 ls1b ws0">Global <span class="_ _0"></span>asymptotic <span class="_ _9"> </span>stability <span class="_ _0"></span><span class="ls1a">of <span class="_ _9"> </span><span class="ls2">the </span></span></div><div class="t m8e xb6 h23 y111 ff8 fse fc0 sc0 ls2 ws0">sliding </div><div class="t m8f xcd h66 y111 ffb fse fc0 sc0 ls8a ws0">sut$uce </div><div class="t m8e x52 h23 y111 ff8 fse fc0 sc0 ls8b ws0">dynamics </div><div class="t m0 xc2 h13 y111 ff2 fsf fc0 sc0 ls2 ws0">is </div><div class="t m0 x3a h13 y112 ff2 fsf fc0 sc0 ls1b ws0">guaranteed <span class="_ _8"></span>through <span class="_ _0"></span><span class="ls2">consideration <span class="_ _c"></span><span class="ls1a">of </span></span></div><div class="t m90 xce h42 y113 ff4 fs1e fc0 sc0 ls8c ws0">u(%(t), </div><div class="t m91 xcf h23 y113 ff4 fse fc0 sc0 ls8d ws0">2(t)) </div><div class="t m0 xd0 h13 y113 ff2 fsf fc0 sc0 ls2 ws0">in <span class="_ _6"></span><span class="ls1b">terms <span class="_ _0"></span><span class="ls1a">of </span></span></div><div class="t m0 x3a h13 y114 ff2 fsf fc0 sc0 ls1b ws0">a <span class="_ _8"></span><span class="ls1c">Lyapunov <span class="_ _9"> </span><span class="ls2">function </span></span></div><div class="t m92 xbd h3f y115 ff2 fs1d fc0 sc0 ls8e ws0">V(a, </div><div class="t m55 xd1 h41 y115 ff4 fs1a fc0 sc0 ls6c ws0">t), </div><div class="t m0 xd2 h13 y115 ff2 fsf fc0 sc0 ls1b ws0">yielding </div><div class="t m55 xd3 h41 y116 ff4 fs1a fc0 sc0 ls8f ws0">V(a(t)) </div><div class="t m93 xd4 h4f y116 ff3 fs1f fc0 sc0 ls2 ws0">= </div><div class="t m60 xd5 h5e y116 ffb fs1d fc0 sc0 ls90 ws0">0.5a’(f(t), </div><div class="t m94 xab h41 y116 ff8 fs1a fc0 sc0 ls91 ws0">Z(t))*a(%(t), </div><div class="t m95 xd6 h67 y116 ff5 fs22 fc0 sc0 ls92 ws0">Z(t)). </div><div class="t md xc1 h68 y116 ff6 fs9 fc0 sc0 ls93 ws0">(IO) </div><div class="t m19 xd7 h18 y117 ff2 fs13 fc0 sc0 ls2 ws0">is </div><div class="t m0 x49 h13 y117 ff2 fsf fc0 sc0 ls1b ws0">positive </div><div class="t m19 xd8 h18 y117 ff2 fs13 fc0 sc0 ls2 ws0">definite </div><div class="t m69 xd9 h15 y117 ff2 fsb fc0 sc0 ls2 ws0">and </div><div class="t m96 xda h40 y117 ff2 fse fc0 sc0 ls2 ws0">unbounded* </div><div class="t m0 xdb h13 y117 ff2 fsf fc0 sc0 ls1b ws0">global </div><div class="t m97 xdc h69 y118 ff4 fs32 fc0 sc0 ls94 ws0">%(t), </div><div class="t m98 xdd h6a y118 ff5 fsb fc0 sc0 ls95 ws0">Z(t) </div><div class="t m99 x92 h6b y118 ff3 fs2f fc0 sc0 ls96 ws0">-+ </div><div class="t m9a xde h40 y118 ff2 fse fc0 sc0 ls2 ws0">0 </div><div class="t m0 xdf h13 y118 ff2 fsf fc0 sc0 ls2 ws0">as </div><div class="t m9b xce h6c y118 ff4 fs33 fc0 sc0 ls2 ws0">t </div><div class="t m9c xe0 h6d y118 ff3 fs24 fc0 sc0 ls97 ws0">-+ </div><div class="t m9d xe1 h6e y118 ff9 fs34 fc0 sc0 ls98 ws0">00. </div><div class="t m26 x27 h17 y119 ff1 fs12 fc0 sc0 ls99 ws0">IV. </div><div class="t m9e x7e h6f y119 ff1 fsf fc0 sc0 ls1a ws0">SLIDING </div><div class="t m0 xe2 h13 y119 ff2 fsf fc0 sc0 ls1c ws0">MODE </div><div class="t m96 xe3 h40 y119 ff2 fse fc0 sc0 ls9a ws0">CONCEPTS </div><div class="t m0 xe4 h13 y11a ff2 fsf fc0 sc0 ls48 ws0">We <span class="_ _1"> </span><span class="ls1b">define <span class="_ _0"></span>sliding <span class="_ _8"></span>surfaces <span class="_ _0"></span>in <span class="_ _8"></span>the <span class="_ _8"></span><span class="ls2">state error <span class="_ _8"></span></span>space <span class="_ _6"></span><span class="ls1c">with <span class="_ _9"> </span></span>the </span></div><div class="t m0 x19 h13 y11b ff2 fsf fc0 sc0 ls1e ws0">element <span class="_ _c"></span><span class="ls2">inputs that <span class="_ _8"></span><span class="ls1c">will <span class="_ _8"></span><span class="ls1e">guarantee <span class="_ _e"></span><span class="ls2">global <span class="_ _0"></span>stability <span class="ls1a">of <span class="_ _9"> </span><span class="ls1b">the <span class="_ _8"></span></span></span>state </span></span></span></span></div><div class="t m0 x19 h13 y11c ff2 fsf fc0 sc0 ls1b ws0">variable <span class="ls2">errors <span class="_ _c"></span><span class="ls1b">and provide adequate performance <span class="_ _6"></span>under <span class="ls1e">closed </span></span></span></div><div class="t m0 x19 h13 y11d ff2 fsf fc0 sc0 ls1e ws0">loop <span class="_ _e"></span><span class="ls2">conditions. <span class="_ _e"></span><span class="ls1a">To <span class="_ _d"></span><span class="ls1b">this <span class="_ _d"></span><span class="ls2">end, <span class="ls1c">we <span class="_ _d"></span></span></span>define a sliding <span class="_ _d"></span><span class="ls2">surface </span></span></span></span></div><div class="t m9f xe5 h13 y11e ff2 fsf fc0 sc0 ls1b ws0">so </div><div class="t m0 x2e h13 y11e ff2 fsf fc0 sc0 ls1b ws0">that </div><div class="t m0 x19 h13 y11f ff2 fsf fc0 sc0 ls1b ws0">object <span class="ls1c">of </span>finding <span class="_ _c"></span>a <span class="_ _d"></span><span class="ls2">sufficient <span class="_ _12"></span>relationship <span class="_ _e"></span><span class="ls1d">for <span class="_ _e"></span><span class="ls1e">each <span class="_ _e"></span>of <span class="_ _c"></span><span class="ls1b">the <span class="ls2">control </span></span></span></span></span></div><div class="t m0 x3a h40 y120 ff2 fse fc0 sc0 ls2 ws0">Since </div><div class="t m0 x3a h13 y121 ff2 fsf fc0 sc0 ls1b ws0">asymptotic </div><div class="t ma0 x91 h61 y121 ff2 fs3 fc0 sc0 ls9b ws0">Of </div><div class="t ma1 xe6 h45 y121 ff8 fsd fc0 sc0 ls9c ws0">a(t) </div><div class="t ma2 xe7 h18 y121 ff2 fs13 fc0 sc0 ls4b ws0">be </div><div class="t m16 xe8 h16 y121 ff2 fs11 fc0 sc0 ls9d ws0">assured </div><div class="t ma3 xcf h3c y121 ff2 fs12 fc0 sc0 ls9e ws0">if </div><div class="t ma4 xe9 h23 y122 ff4 fse fc0 sc0 ls9f ws0">dV(a) </div><div class="t ma5 x99 h4f y122 ff3 fs1f fc0 sc0 ls2 ws0">= </div><div class="t ma6 x4c h70 y122 ffb fs2e fc0 sc0 lsa0 ws0">cia </div><div class="t ma7 xa1 h71 y122 ff3 fs35 fc0 sc0 ls2 ws0">< </div><div class="t m9e xab h13 y122 ff2 fsf fc0 sc0 ls2 ws0">0 </div><div class="t ma8 xea h72 y122 ff9 fs2c fc0 sc0 lsa1 ws0">vt </div><div class="t ma7 xe1 h71 y122 ff3 fs35 fc0 sc0 ls2 ws0">> </div><div class="t m96 x60 h12 y122 ff1 fse fc0 sc0 ls8b ws0">0. </div><div class="t ma9 xc1 h73 y122 ff1 fs13 fc0 sc0 lsa2 ws0">(11) </div><div class="t maa xeb h41 y123 ff4 fs1a fc0 sc0 ls6c ws0">dt </div><div class="t m0 xca h13 y124 ff2 fsf fc0 sc0 ls36 ws0">If <span class="_ _b"> </span><span class="ls1c">we <span class="_"> </span><span class="ls1b">define <span class="_ _8"></span><span class="ls2">positive <span class="_ _8"></span></span>functions </span></span></div><div class="t m97 xab h69 y124 ff4 fs32 fc0 sc0 lsa3 ws0">~;(t) </div><div class="t m0 xea h13 y124 ff2 fsf fc0 sc0 ls1b ws0">then <span class="_ _9"> </span><span class="ls2">global <span class="_ _8"></span></span>asymptotic </div><div class="t m0 x3a h13 y125 ff2 fsf fc0 sc0 ls2 ws0">stability for <span class="_ _0"></span><span class="ls1b">the <span class="_ _8"></span>dynamics <span class="_ _0"></span><span class="ls1a">of <span class="_"> </span><span class="ls1e">each </span></span></span></div><div class="t m0 xb6 h13 y126 ffb fs1c fc0 sc0 lsa4 ws0">ui(t) <span class="_ _16"></span><span class="ff2 fsf ls1b">will <span class="_ _0"></span>be <span class="_ _8"></span>given <span class="_ _0"></span><span class="ls1c">by </span></span></div><div class="t mab xec h74 y127 ff5 fs12 fc0 sc0 lsa5 ws0">(5) </div><div class="t mac xed h75 y127 ff4 fs8 fc0 sc0 lsa6 ws0">&i(%(t), </div><div class="t mad xee h41 y127 ff8 fs1a fc0 sc0 lsa7 ws0">Z(t)) </div><div class="t mae xef h76 y127 ff3 fs12 fc0 sc0 ls2 ws0">= </div><div class="t m0 xf0 h13 y127 ffd fs25 fc0 sc0 ls4d ws0">-vi <span class="ff2 fsf ls1b">sgn </span></div><div class="t maf xf1 h69 y127 ff4 fs32 fc0 sc0 ls94 ws0">(c~(%(t), </div><div class="t mb0 x42 h41 y127 ff8 fs1a fc0 sc0 lsa8 ws0">Z(t)) </div><div class="t mb1 xd6 h15 y127 ff2 fsb fc0 sc0 ls2 ws0">i </div><div class="t ma5 xf2 h4f y127 ff3 fs1f fc0 sc0 ls2 ws0">= </div><div class="t m64 x9e h73 y127 ff1 fs13 fc0 sc0 lsa9 ws0">1, </div><div class="t m54 x9f h6f y127 ff1 fsf fc0 sc0 ls81 ws0">6. </div><div class="t m0 xc1 h12 y127 ff1 fse fc0 sc0 lsaa ws0">(12) </div><div class="t m0 x19 h13 ydb ff2 fsf fc0 sc0 ls1e ws0">each <span class="ls1b">passes <span class="_ _9"></span>through <span class="_ _8"></span>the <span class="_ _9"></span><span class="ls2">origin <span class="_ _6"></span><span class="ls1a">of <span class="_"> </span></span></span>the <span class="_ _9"> </span><span class="ls2">state <span class="_ _0"></span>error <span class="_ _8"></span>space. <span class="_ _6"></span>State </span></span></div><div class="t m0 x19 h13 y128 ff2 fsf fc0 sc0 ls2 ws0">errors </div><div class="t mb2 xf3 h3f y129 ff2 fs1d fc0 sc0 ls58 ws0">are </div><div class="t m0 x6e h13 y129 ff2 fsf fc0 sc0 ls1b ws0">defined <span class="_ _8"></span><span class="ls1a">by </span></div><div class="t mb3 xf4 h77 y12a ff1 fs36 fc0 sc0 lsab ws0">corn </div><div class="t m0 xca h13 y12b ff2 fsf fc0 sc0 ls48 ws0">We <span class="_"> </span><span class="ls1c">find <span class="_ _6"></span><span class="ls2">it better, <span class="_ _c"></span><span class="ls1b">in <span class="_ _d"></span><span class="ls2">fact, <span class="ls1c">to <span class="_ _6"></span></span></span>use <span class="_ _6"></span>a <span class="_ _d"></span>continuous <span class="ls2">function <span class="_ _11"></span>to <span class="ls1b">define </span></span></span></span></span></div><div class="t m0 x3a h13 y12c ff2 fsf fc0 sc0 ls1b ws0">“practical” <span class="_ _9"> </span>sliding <span class="_ _8"></span>surface <span class="_ _8"></span>dynamics <span class="_ _8"></span>using <span class="_ _9"></span>a <span class="_ _8"></span>“tanh” <span class="_ _0"></span><span class="ls2">function, </span></div><div class="t mb4 x3a h78 y12d ff3 fs37 fc0 sc0 lsac ws0">-- </div><div class="t m6a xf5 h4f y12d ff3 fs1f fc0 sc0 ls2 ws0">. </div><div class="t mb5 x59 h79 y12d ff3 fs38 fc0 sc0 ls2 ws0">. </div><div class="t m0 x19 h13 y12e ff2 fsf fc0 sc0 ls1b ws0">where <span class="_ _0"></span>the <span class="_ _8"></span>commands <span class="_ _0"></span>are <span class="_ _0"></span>derived <span class="_ _8"></span>from <span class="_ _6"></span>a <span class="_ _8"></span><span class="ls2">consistent </span>command </div><div class="t m0 x19 h13 y12f ff2 fsf fc0 sc0 ls1b ws0">generation <span class="_ _9"> </span>system <span class="_ _8"></span><span class="ls1d">or <span class="_ _d"></span></span>planned <span class="_ _9"> </span><span class="ls2">path, <span class="_ _8"></span><span class="ls1d">or <span class="_ _d"></span></span></span>from <span class="_ _9"> </span>a <span class="_ _9"> </span><span class="ls2">series <span class="_ _6"></span><span class="ls1c">of <span class="_"> </span>way- </span></span></div><div class="t mb6 xca h7a y130 ff1 fs1b fc0 sc0 ls71 ws0">as </div><div class="t m0 xf5 h13 y130 ff2 fsf fc0 sc0 ls1b ws0">in </div><div class="t m0 xf6 h13 y131 ff2 fsf fc0 sc0 ls1b ws0">points <span class="_ _9"></span>corresponding <span class="_ _8"></span><span class="ls2">to <span class="_ _8"></span>desired <span class="_ _6"></span></span>values <span class="_ _9"></span><span class="ls1a">of <span class="_"> </span></span>vehicle <span class="_ _9"></span>velocity <span class="_ _9"></span><span class="ls2">or </span></div><div class="t m0 xf6 h13 y132 ff2 fsf fc0 sc0 ls1b ws0">position <span class="_ _6"></span>(posture). <span class="ls1e">The <span class="_ _c"></span><span class="ls1b">set <span class="ls1a">of <span class="_ _8"></span></span>sliding surfaces in <span class="_ _d"></span>the <span class="ls2">error <span class="ls1e">space </span></span></span></span></div><div class="t mb7 x59 h23 y133 ff4 fse fc0 sc0 ls8d ws0">d-i(Z(t), </div><div class="t mb8 xf7 h75 y133 ff4 fs8 fc0 sc0 lsad ws0">Z(t)) </div><div class="t mae xf8 h76 y133 ff3 fs12 fc0 sc0 ls2 ws0">= </div><div class="t mb9 xad h4c y133 ff4 fs28 fc0 sc0 lsae ws0">-~i(t) </div><div class="t m0 xbc h13 y133 ff2 fsf fc0 sc0 ls1b ws0">tanh </div><div class="t mba xe7 h45 y133 ff4 fsd fc0 sc0 lsaf ws0">(c~i(Z(t), </div><div class="t mbb xda h24 y133 ff4 fs12 fc0 sc0 lsb0 ws0">Z(t))/#i). </div><div class="t m0 xc1 h12 y134 ff1 fse fc0 sc0 ls2 ws0">(13) </div><div class="t m0 xf6 h13 y135 ff2 fsf fc0 sc0 ls1b ws0">are <span class="_ _2"> </span>then </div><div class="t m0 xf6 h13 y136 ff2 fsf fc0 sc0 ls1b ws0">where </div><div class="t mbc x27 h75 y137 ff4 fs8 fc0 sc0 lsb1 ws0">a(t) </div><div class="t m8e x24 h66 y137 ff6 fse fc0 sc0 ls2 ws0">E </div><div class="t mbd xf9 h7b y137 ff3 fs3 fc0 sc0 ls2 ws0">726x1; </div><div class="t mbe x38 h7c y137 ff2 fs37 fc0 sc0 lsb2 ws0">SI, </div><div class="t mbf x32 h7d y137 ff4 fs1 fc0 sc0 lsb3 ws0">s2 </div><div class="t m8e xfa h66 y137 ff6 fse fc0 sc0 ls2 ws0">E </div><div class="t mc0 xfb h7e y137 ff3 fs13 fc0 sc0 ls24 ws0">7Px6. </div><div class="t m0 xe4 h13 y138 ff2 fsf fc0 sc0 ls1b ws0">Notice <span class="_ _8"></span>that <span class="_ _6"></span>in <span class="_ _0"></span>this <span class="_ _0"></span>work <span class="_ _8"></span><span class="ls2">as <span class="_ _6"></span></span>opposed <span class="_ _6"></span><span class="ls2">to <span class="_ _6"></span></span>the <span class="_ _8"></span>work <span class="_ _6"></span><span class="ls1a">of <span class="_ _9"> </span><span class="ls1c">Yoerger </span></span></div><div class="t m0 xf6 h13 y139 ff2 fsf fc0 sc0 ls1c ws0">and <span class="_ _d"></span><span class="ls2">Slotine </span></div><div class="t m16 xfc h7f y13a ff1 fs11 fc0 sc0 lsb4 ws0">[28], </div><div class="t m0 x7d h13 y13a ff2 fsf fc0 sc0 ls2 ws0">the <span class="ls1b">sliding surfaces <span class="_ _11"></span>are <span class="_ _c"></span>based on <span class="ls2">state <span class="_ _f"></span><span class="ls1b">variable </span></span></span></div><div class="t m0 xf6 h13 y13b ff2 fsf fc0 sc0 ls2 ws0">errors <span class="_ _e"></span>rather <span class="ls1b">than <span class="_ _d"></span><span class="ls1e">output <span class="_ _10"></span><span class="ls2">errors. <span class="_ _f"></span><span class="ls1e">For <span class="ls1b">flight <span class="_ _f"></span>vehicles <span class="_ _d"></span>in <span class="_ _6"></span><span class="ls1c">which <span class="_ _6"></span></span>the </span></span></span></span></span></div><div class="t m0 xf6 h13 y13c ff2 fsf fc0 sc0 ls1b ws0">modes <span class="_ _8"></span><span class="ls2">are <span class="_ _0"></span></span>highly <span class="_ _9"> </span>coupled, </div><div class="t mc1 x2b h21 y13d ff2 fs1a fc0 sc0 lsb5 ws0">we </div><div class="t m0 xfd h13 y13d ff2 fsf fc0 sc0 ls1c ws0">find <span class="_ _8"></span><span class="ls1e">@is <span class="_ _f"></span><span class="ls1b">approach <span class="_ _8"></span><span class="ls1c">to <span class="_ _9"> </span></span>be <span class="_ _8"></span>more </span></span></div><div class="t m0 xf6 h13 y13e ff2 fsf fc0 sc0 ls1b ws0">flexible. <span class="_ _8"></span><span class="ls1e">The <span class="_ _6"></span><span class="ls2">coefficient <span class="_ _d"></span></span></span>matrices </div><div class="t mc2 x32 h80 y13f ff6 fs39 fc0 sc0 lsb6 ws0">SI </div><div class="t m0 xfa h13 y13f ff2 fsf fc0 sc0 ls1b ws0">and </div><div class="t mc3 xc6 h81 y13f ff8 fs0 fc0 sc0 ls7 ws0">S2 </div><div class="t m0 x72 h13 y13f ff2 fsf fc0 sc0 ls2 ws0">are <span class="_ _8"></span><span class="ls1b">assumed <span class="_ _9"></span></span>to </div><div class="t m0 xf6 h13 y140 ff2 fsf fc0 sc0 ls1e ws0">be <span class="ls1c">known <span class="_ _0"></span><span class="ls2">at <span class="ls1b">this </span>point <span class="_ _d"></span><span class="ls1b">in <span class="_ _d"></span>the <span class="_ _6"></span>development, <span class="_ _0"></span>although <span class="_ _6"></span></span>as <span class="ls1b">will </span></span></span>be </div><div class="t m0 xf6 h13 y141 ff2 fsf fc0 sc0 ls1b ws0">seen, <span class="_ _6"></span><span class="ls2">are </span>not <span class="_ _0"></span><span class="ls2">arbitrary. <span class="ls1c">In <span class="_ _0"></span></span>fact, <span class="_ _0"></span></span>system <span class="_ _d"></span><span class="ls1e">closed-loop <span class="_ _10"></span>response <span class="_ _f"></span><span class="ls2">is </span></span></div><div class="t m0 xf6 h13 y142 ff2 fsf fc0 sc0 ls1b ws0">dependent <span class="_ _8"></span>on <span class="_ _8"></span><span class="ls1e">values <span class="_ _d"></span></span>selected <span class="_ _9"> </span><span class="ls1c">and <span class="_ _9"> </span><span class="ls2">at <span class="_ _8"></span>least <span class="_ _6"></span>part <span class="_ _8"></span><span class="ls1a">of <span class="_ _7"> </span></span></span></span>any <span class="_ _8"></span>design </div><div class="t m0 xf6 h13 y143 ff2 fsf fc0 sc0 ls1e ws0">procedure <span class="_ _d"></span><span class="ls1b">using <span class="_"> </span><span class="ls2">sliding <span class="_ _0"></span></span></span>mode <span class="_ _8"></span><span class="ls1b">methods </span></div><div class="t mc4 x7b h18 y144 ff2 fs13 fc0 sc0 lsb7 ws0">is </div><div class="t m0 xf4 h13 y144 ff2 fsf fc0 sc0 ls2 ws0">to <span class="_ _8"></span><span class="ls1e">properly <span class="_ _8"></span></span>select </div><div class="t mf xf6 he y145 ff1 fsa fc0 sc0 ls2 ws0">surfaces <span class="_ _6"></span><span class="ls26">yielding <span class="_ _8"></span><span class="ls13">stable <span class="_ _8"></span><span class="lsb8">closed-loop </span></span></span></div><div class="t mc5 x6 h63 y145 ff1 fs25 fc0 sc0 lsb9 ws0">error </div><div class="t mf xc6 he y145 ff1 fsa fc0 sc0 ls26 ws0">dynamics, namely, </div><div class="t m0 xf6 h13 y146 ff2 fsf fc0 sc0 ls1b ws0">that <span class="_ _0"></span><span class="ls2">for <span class="_ _8"></span></span>all </div><div class="t mc6 x7a h69 y146 ff4 fs32 fc0 sc0 ls2 ws0">t </div><div class="t m0 x4 h13 y146 ff2 fsf fc0 sc0 ls1c ws0">and </div><div class="t mc6 x36 h69 y146 ff4 fs32 fc0 sc0 lsba ws0">[~(t), </div><div class="t m0 xfe h58 y146 ff4 fs2e fc0 sc0 lsbb ws0">z(t)] <span class="_ _17"></span><span class="ff2 fsf ls1b">lying <span class="_ _0"></span>in <span class="_ _8"></span>the <span class="_ _8"></span>space <span class="_ _0"></span><span class="ls1a">of <span class="_"> </span></span>maneuvers </span></div><div class="t m0 xf6 h13 y147 ff2 fsf fc0 sc0 ls1c ws0">to <span class="_ _9"></span><span class="ls1e">be <span class="_ _8"></span><span class="ls1b">accomplished, <span class="_ _8"></span><span class="ls2">the <span class="_ _8"></span>conditions </span></span></span></div><div class="t mc7 xff h82 y148 ff9 fsa fc0 sc0 ls2 ws0">b(jE(t), </div><div class="t m97 x100 h69 y148 ff4 fs32 fc0 sc0 lsbc ws0">%(t)) </div><div class="t mc8 x101 h78 y148 ff3 fs37 fc0 sc0 lsbd ws0">-, </div><div class="t mc9 x70 h43 y148 ff2 fs22 fc0 sc0 ls2 ws0">0 </div><div class="t m0 x34 h13 y148 ff2 fsf fc0 sc0 ls2 ws0">as </div><div class="t mca x102 h41 y148 ff4 fs1a fc0 sc0 ls2 ws0">t </div><div class="t mc8 x80 h78 y148 ff3 fs37 fc0 sc0 lsbd ws0">-, </div><div class="t mcb xf4 h83 y148 ff3 fs3a fc0 sc0 lsbe ws0">00 </div><div class="t mcc xec h84 y148 ff3 fs1c fc0 sc0 ls4e ws0">(7) </div><div class="t m0 xca h13 y149 ff2 fsf fc0 sc0 ls1e ws0">The <span class="_ _9"></span><span class="ls1b">powerful <span class="_"> </span>result <span class="_ _1"> </span><span class="ls1a">of </span></span></div><div class="t mcd x103 h3c y149 ff2 fs12 fc0 sc0 lsbf ws0">(13) </div><div class="t m0 xa1 h13 y149 ff2 fsf fc0 sc0 ls1b ws0">means <span class="_"> </span>that <span class="_"> </span><span class="ls1a">if <span class="_ _13"> </span></span>the <span class="_ _18"> </span>are <span class="_ _2"> </span><span class="ls2">large </span></div><div class="t m0 xca h13 y14a ff2 fsf fc0 sc0 ls1b ws0">enough, then in <span class="_ _d"></span><span class="ls2">spite </span></div><div class="t m16 x104 h16 y14b ff2 fs11 fc0 sc0 ls22 ws0">of </div><div class="t m0 x99 h13 y14b ff2 fsf fc0 sc0 ls1b ws0">modeling <span class="_ _d"></span>uncertainty, nonlinear terms, </div><div class="t m0 xca h13 y14c ff2 fsf fc0 sc0 ls1b ws0">and <span class="_"> </span>disturbances, <span class="_ _2"> </span><span class="ls2">the <span class="_"> </span></span>system <span class="_ _9"> </span><span class="ls1e">response <span class="_ _6"></span></span>will <span class="_"> </span><span class="ls1e">be <span class="_ _8"></span></span>governed <span class="_"> </span><span class="ls1a">by </span></div><div class="t m0 xca h13 y14d ff2 fsf fc0 sc0 ls1b ws0">the <span class="_"> </span>response <span class="_"> </span><span class="ls1a">of <span class="_ _b"> </span></span>the </div><div class="t mce xd8 h23 y14e ff4 fse fc0 sc0 lsc0 ws0">ai(t) </div><div class="t m0 x92 h13 y14e ff2 fsf fc0 sc0 ls1c ws0">and <span class="_"> </span>by <span class="_ _1"> </span><span class="ls1b">the <span class="_"> </span><span class="ls1e">choice <span class="_ _8"></span><span class="ls21">of <span class="_ _13"> </span><span class="ls2">the <span class="_"> </span></span></span></span>sliding </span></div><div class="t m0 xca h13 y14f ff2 fsf fc0 sc0 ls2 ws0">surface <span class="ls1b">parameters: <span class="_ _9"> </span></span>it <span class="_ _0"></span>is <span class="_ _8"></span>less <span class="_ _6"></span><span class="ls1b">influenced <span class="_ _0"></span><span class="ls1c">by <span class="_ _9"> </span></span>the <span class="_ _0"></span>parameters <span class="_ _9"> </span><span class="ls1a">of </span></span></div><div class="t m0 xca h13 y150 ff2 fsf fc0 sc0 ls1b ws0">the <span class="_ _9"> </span>vehicle <span class="_"> </span>dynamics <span class="_ _9"> </span><span class="ls2">as <span class="_ _9"> </span>is <span class="_ _8"></span></span>more <span class="_ _2"> </span>usual <span class="_ _9"> </span>with <span class="_ _2"> </span><span class="ls2">linear <span class="_ _8"></span><span class="ls1e">feedback </span></span></div><div class="t m0 xca h13 y151 ff2 fsf fc0 sc0 ls2 ws0">controllers. <span class="_ _9"> </span><span class="ls1e">The </span></div><div class="t mcf x105 h85 y152 ff3 fs3b fc0 sc0 ls2 ws0">&, </div><div class="t m0 x106 h13 y152 ff2 fsf fc0 sc0 ls1b ws0">not <span class="_ _1"> </span><span class="ls1c">to <span class="_ _1"> </span><span class="ls1e">be <span class="_"> </span></span></span>confused <span class="_ _1"> </span><span class="ls1c">with <span class="_ _a"> </span><span class="ls2">the <span class="_"> </span></span></span>vehicle’s </div><div class="t m0 xca h13 y153 ff2 fsf fc0 sc0 ls1b ws0">roll <span class="_ _1"> </span>motion <span class="_"> </span><span class="ls1e">Euler <span class="_ _9"> </span>angle </span></div><div class="t m97 x107 h69 yf1 ff4 fs32 fc0 sc0 lsbc ws0">#(t), </div><div class="t m0 x108 h13 yf1 ff2 fsf fc0 sc0 ls2 ws0">are <span class="_"> </span><span class="ls1b">sliding <span class="_"> </span>surface <span class="_"> </span>boundary </span></div><div class="t m0 xca h13 yf2 ff2 fsf fc0 sc0 ls2 ws0">layer <span class="_ _c"></span><span class="ls1e">parameters <span class="_ _e"></span><span class="ls1c">used <span class="_ _0"></span>to <span class="_ _0"></span><span class="ls1b">retain <span class="_ _6"></span>continuity <span class="_ _6"></span><span class="ls21">of <span class="_ _7"> </span></span>control <span class="_ _d"></span><span class="ls2">as </span>motion </span></span></span></div><div class="t m0 xca h13 y154 ff2 fsf fc0 sc0 ls2 ws0">trajectories <span class="_ _8"></span><span class="ls1b">cross <span class="_ _2"> </span>the <span class="_"> </span>sliding <span class="_ _2"> </span></span>surface <span class="_ _8"></span><span class="ls1c">and <span class="_"> </span><span class="ls1b">prevent <span class="_ _2"> </span></span></span>chattering. </div><div class="t m0 xca h13 y155 ff2 fsf fc0 sc0 ls2 ws0">It <span class="_ _8"></span><span class="ls1b">remains <span class="_ _8"></span><span class="ls1c">to <span class="_ _9"> </span><span class="ls1e">compute <span class="_ _11"></span><span class="ls1b">the <span class="_ _8"></span>control <span class="_ _8"></span>law <span class="_ _8"></span><span class="ls2">to <span class="_ _8"></span></span>provide <span class="_ _8"></span><span class="ls2">the <span class="_ _8"></span></span>desired </span></span></span></span></div><div class="t md0 xca h86 y156 ff8 fs2 fc0 sc0 ls2 ws0">B </div><div class="t m0 x109 h13 y156 ff2 fsf fc0 sc0 ls1b ws0">dynamics <span class="_ _9"> </span><span class="ls1c">and <span class="_"> </span><span class="ls2">select </span></span></div><div class="t md1 x10a h87 y156 ff6 fs3c fc0 sc0 lsc1 ws0">S1, </div><div class="t md2 xc0 h7b y156 ff3 fs3 fc0 sc0 lsc2 ws0">S2 </div><div class="t md3 x108 h3e y156 ff2 fs1e fc0 sc0 lsc3 ws0">so </div><div class="t m0 x9b h13 y156 ff2 fsf fc0 sc0 ls1b ws0">that <span class="_"> </span>stable <span class="_ _8"></span>dynamics </div><div class="t md4 x10b h88 y156 ff1 fs28 fc0 sc0 lsc4 ws0">of </div><div class="t m4a xc2 h89 y156 ffb fsf fc0 sc0 ls2 ws0">a </div><div class="t m0 xca h13 y157 ff2 fsf fc0 sc0 ls2 ws0">results <span class="_ _d"></span><span class="ls1b">in <span class="_ _8"></span></span>stability <span class="ls21">of <span class="_ _7"> </span><span class="ls1b">the <span class="_ _0"></span>tracking <span class="_ _8"></span></span></span>errors. <span class="ls1b">Use </span></div><div class="t md5 x4e h40 y157 ff2 fse fc0 sc0 lsc5 ws0">of </div><div class="t m0 x10c h13 y157 ff2 fsf fc0 sc0 ls2 ws0">the <span class="ls1b">boundary </span></div><div class="t m0 xca h13 y158 ff2 fsf fc0 sc0 ls1b ws0">layer <span class="_ _b"> </span><span class="ls1c">to <span class="_ _a"> </span></span>prevent <span class="_ _b"> </span>high <span class="_ _b"> </span>frequency <span class="_ _13"> </span>chattering <span class="_ _b"> </span><span class="ls21">of <span class="_ _19"> </span></span>the <span class="_ _a"> </span><span class="ls2">control, </span></div><div class="t m0 xca h13 y159 ff2 fsf fc0 sc0 ls1b ws0">however, <span class="_"> </span>means <span class="_ _9"> </span>that <span class="_"> </span>in <span class="_"> </span>the <span class="_ _8"></span><span class="ls1e">presence </span></div><div class="t m16 x43 h16 y15a ff2 fs11 fc0 sc0 ls22 ws0">of </div><div class="t m0 xcd h13 y15a ff2 fsf fc0 sc0 ls1b ws0">uncertainty <span class="_ _2"> </span><span class="ls1c">we <span class="_"> </span><span class="ls1e">can </span></span></div><div class="t mf xca he y15b ff1 fsa fc0 sc0 ls26 ws0">only <span class="_ _0"></span><span class="ls12">guarantee </span></div><div class="t md6 x3c h7a y15b ff1 fs1b fc0 sc0 lsc6 ws0">global </div><div class="t mf xbd he y15b ff1 fsa fc0 sc0 ls13 ws0">bounded </div><div class="t md7 x10d h8a y15b ff1 fs1f fc0 sc0 lsc7 ws0">stability </div><div class="t m2f x10e h10 y15b ff1 fsc fc0 sc0 ls3a ws0">for </div><div class="t mf xa5 he y15b ff1 fsa fc0 sc0 ls13 ws0">both </div><div class="t md8 x61 h23 y15b ff8 fse fc0 sc0 lsc8 ws0">a(t) </div><div class="t md9 x40 h8b y15b ff6 fs20 fc0 sc0 lsc9 ws0">and </div><div class="t mda x10f h8b y15b ff6 fs20 fc0 sc0 ls3c ws0">the </div><div class="t m0 xca h13 y15c ff2 fsf fc0 sc0 ls2 ws0">state <span class="_ _d"></span>errors <span class="_ _d"></span><span class="ls1b">rather <span class="_ _8"></span>than <span class="_ _8"></span></span>global <span class="_ _d"></span><span class="ls1b">asymptotic <span class="_ _8"></span>stability <span class="_ _0"></span></span>for </div><div class="t mdb x40 h2a y15c ffb fs1f fc0 sc0 lsca ws0">a@). </div><div class="t m0 xed h13 y15d ff2 fsf fc0 sc0 ls1b ws0">Assuming <span class="_ _2"> </span>that </div><div class="t mdc xd4 h8c y15d ff7 fs4 fc0 sc0 lscb ws0">S1,2 </div><div class="t m0 xe6 h13 y15d ff2 fsf fc0 sc0 ls2 ws0">are <span class="_ _8"></span>established <span class="_ _8"></span><span class="ls1b">(by <span class="_ _9"> </span>the <span class="_ _9"> </span>method <span class="_ _2"> </span></span>to <span class="_ _8"></span><span class="ls1b">be </span></div><div class="t m0 xca h13 y15e ff2 fsf fc0 sc0 ls1b ws0">described <span class="_ _8"></span>here), <span class="_ _9"> </span>substitution <span class="_ _8"></span><span class="ls1a">of </span></div><div class="t m0 x108 h12 y15f ff1 fse fc0 sc0 ls2 ws0">(4) <span class="ff2 fsf ls1b">and </span></div><div class="t m9f x10e h13 y15f ff2 fsf fc0 sc0 ls1c ws0">(6) </div><div class="t m0 x60 h12 y15f ff2 fsf fc0 sc0 ls1b ws0">into <span class="_ _8"></span><span class="ff1 fse lsaa">(13) <span class="_ _6"></span></span>yields <span class="_ _8"></span>a </div><div class="t m0 xca h13 y160 ff2 fsf fc0 sc0 ls1b ws0">solution <span class="_ _9"></span><span class="ls2">for <span class="_ _8"></span></span>the <span class="_ _9"> </span><span class="ls2">control <span class="_ _8"></span>law. </span></div><div class="t m0 x5b h32 y76 ffa fs23 fc1 sc0 ls2 ws0">Authorized licensed use limited to: Yuan Ze University. Downloaded on August 09,2020 at 18:04:36 UTC from IEEE Xplore. Restrictions apply. </div></div><div class="pi" data-data='{"ctm":[1.546195,0.000000,0.000000,1.546195,0.000000,0.000000]}'></div></div>
