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<span class="vector-toc-numb">2</span> <span>True-range multilateration</span> </div> </a> <ul id="toc-True-range_multilateration-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pseudo-range_multilateration" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Pseudo-range_multilateration"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Pseudo-range multilateration</span> </div> </a> <ul id="toc-Pseudo-range_multilateration-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" 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href="https://bg.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%BB%D0%B0%D1%82%D0%B5%D1%80%D0%B0%D1%86%D0%B8%D1%8F" title="Трилатерация – Bulgarian" lang="bg" hreflang="bg" data-title="Трилатерация" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Trilateraci%C3%B3" title="Trilateració – Catalan" lang="ca" hreflang="ca" data-title="Trilateració" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Trilaterace" title="Trilaterace – Czech" lang="cs" hreflang="cs" data-title="Trilaterace" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Trilateration" title="Trilateration – Danish" lang="da" hreflang="da" data-title="Trilateration" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Lateration" title="Lateration – German" lang="de" hreflang="de" data-title="Lateration" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Trilateraci%C3%B3n" title="Trilateración – Spanish" lang="es" hreflang="es" data-title="Trilateración" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Trilat%C3%A9ration" title="Trilatération – French" lang="fr" hreflang="fr" data-title="Trilatération" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%82%BC%EB%B3%80%EC%B8%A1%EB%9F%89" title="삼변측량 – Korean" lang="ko" hreflang="ko" data-title="삼변측량" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Trilaterazione" title="Trilaterazione – Italian" lang="it" hreflang="it" data-title="Trilaterazione" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A2%E1%83%A0%E1%83%98%E1%83%9A%E1%83%90%E1%83%A2%E1%83%94%E1%83%A0%E1%83%90%E1%83%AA%E1%83%98%E1%83%90" title="ტრილატერაცია – Georgian" lang="ka" hreflang="ka" data-title="ტრილატერაცია" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%BB%D0%B0%D1%82%D0%B5%D1%80%D0%B0%D1%86%D0%B8%D1%8F" title="Трилатерация – Kazakh" lang="kk" hreflang="kk" data-title="Трилатерация" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%BB%D0%B0%D1%82%D0%B5%D1%80%D0%B0%D1%86%D0%B8%D1%98%D0%B0" title="Трилатерација – Macedonian" lang="mk" hreflang="mk" data-title="Трилатерација" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%BB%D0%B0%D1%82%D0%B5%D1%80%D0%B0%D1%86%D0%B8%D1%8F" title="Трилатерация – Russian" lang="ru" hreflang="ru" data-title="Трилатерация" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Trilater%C3%A1cia" title="Trilaterácia – Slovak" lang="sk" hreflang="sk" data-title="Trilaterácia" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%BB%D0%B0%D1%82%D0%B5%D1%80%D0%B0%D1%86%D0%B8%D1%98%D0%B0" title="Трилатерација – Serbian" lang="sr" hreflang="sr" data-title="Трилатерација" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Trilateration" title="Trilateration – Swedish" lang="sv" hreflang="sv" data-title="Trilateration" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%BB%D0%B0%D1%82%D0%B5%D1%80%D0%B0%D1%86%D1%96%D1%8F" title="Трилатерація – Ukrainian" lang="uk" hreflang="uk" data-title="Трилатерація" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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searchaux" style="display:none">Use of distances for determining unknown coordinates of a point</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:3D_Trilateration.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/3D_Trilateration.jpg/210px-3D_Trilateration.jpg" decoding="async" width="210" height="149" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/3D_Trilateration.jpg/315px-3D_Trilateration.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/3D_Trilateration.jpg/420px-3D_Trilateration.jpg 2x" data-file-width="1280" data-file-height="910" /></a><figcaption>Trilateration in three-dimensional geometry</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:GPS-pseudorange.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/GPS-pseudorange.svg/150px-GPS-pseudorange.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/GPS-pseudorange.svg/225px-GPS-pseudorange.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b8/GPS-pseudorange.svg/300px-GPS-pseudorange.svg.png 2x" data-file-width="383" data-file-height="383" /></a><figcaption>Intersection point of three pseudo-ranges</figcaption></figure> <p><b>Trilateration</b> is the use of <a href="/wiki/Distances" class="mw-redirect" title="Distances">distances</a> (or "ranges") for determining the unknown <a href="/wiki/Position_coordinate" class="mw-redirect" title="Position coordinate">position coordinates</a> of a <a href="/wiki/Point_(geometry)" title="Point (geometry)">point</a> of interest, often around Earth (<a href="/wiki/Geopositioning" title="Geopositioning">geopositioning</a>).<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> When more than three distances are involved, it may be called <b>multilateration</b>, for emphasis. </p><p>The distances or ranges might be ordinary <a href="/wiki/Euclidean_distance" title="Euclidean distance">Euclidean distances</a> (<a href="/wiki/Slant_range" title="Slant range">slant ranges</a>) or <a href="/wiki/Spherical_distance" class="mw-redirect" title="Spherical distance">spherical distances</a> (scaled <a href="/wiki/Central_angle" title="Central angle">central angles</a>), as in <i><a href="/wiki/True-range_multilateration" title="True-range multilateration">true-range multilateration</a></i>; or biased distances (<a href="/wiki/Pseudo-range" class="mw-redirect" title="Pseudo-range">pseudo-ranges</a>), as in <i><a href="/wiki/Pseudo-range_multilateration" title="Pseudo-range multilateration">pseudo-range multilateration</a></i>. </p><p>Trilateration or multilateration should not be confused with <i><a href="/wiki/Triangulation" title="Triangulation">triangulation</a></i>, which uses <a href="/wiki/Angle" title="Angle">angles</a> for positioning; and <i><a href="/wiki/Direction_finding" title="Direction finding">direction finding</a></i>, which determines the line of sight <a href="/wiki/Direction_(geometry)" title="Direction (geometry)">direction</a> to a target without determining the <a href="/wiki/Polar_coordinate_system" title="Polar coordinate system">radial distance</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Terminology">Terminology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trilateration&action=edit&section=1" title="Edit section: Terminology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Multiple, sometimes overlapping and conflicting terms are employed for similar concepts – e.g., <i>multilateration</i> without modification has been used for aviation systems employing both true-ranges and pseudo-ranges.<sup id="cite_ref-ICAO_2-0" class="reference"><a href="#cite_note-ICAO-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Wolff_3-0" class="reference"><a href="#cite_note-Wolff-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Moreover, different fields of endeavor may employ different terms. In <a href="/wiki/Geometry" title="Geometry">geometry</a>, <i>trilateration</i> is defined as the process of determining absolute or relative locations of points by measurement of distances, using the geometry of <a href="/wiki/Circle" title="Circle">circles</a>, <a href="/wiki/Sphere" title="Sphere">spheres</a> or <a href="/wiki/Triangle" title="Triangle">triangles</a>. In surveying, <i>trilateration</i> is a specific technique.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="True-range_multilateration">True-range multilateration</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trilateration&action=edit&section=2" title="Edit section: True-range multilateration"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><style data-mw-deduplicate="TemplateStyles:r1066933788">.mw-parser-output .excerpt-hat .mw-editsection-like{font-style:normal}</style><style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/True-range_multilateration" title="True-range multilateration">True-range multilateration</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=True-range_multilateration&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> <p><a href="/wiki/True-range_multilateration" title="True-range multilateration">True-range multilateration</a> (also termed range-range multilateration and spherical multilateration) is a method to determine the location of a movable vehicle or stationary point in space using multiple <a href="/wiki/Ranging" class="mw-redirect" title="Ranging">ranges</a> (<a href="/wiki/Distance" title="Distance">distances</a>) between the vehicle/point and multiple spatially-separated known locations (often termed "stations"). <sup id="cite_ref-True-range_multilateration_Lee_7-0" class="reference"><a href="#cite_note-True-range_multilateration_Lee-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-True-range_multilateration_Grant_8-0" class="reference"><a href="#cite_note-True-range_multilateration_Grant-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Energy waves may be involved in determining range, but are not required. </p><p>True-range multilateration is both a mathematical topic and an applied technique used in several fields. A practical application involving a fixed location occurs in <a href="/wiki/Surveying" title="Surveying">surveying</a>.<sup id="cite_ref-True-range_multilateration_Wirtanen_1969_pp._81–92_9-0" class="reference"><a href="#cite_note-True-range_multilateration_Wirtanen_1969_pp._81–92-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-True-range_multilateration_Escobal_Fliegel_Jaffe_Muller_2013_p._10-0" class="reference"><a href="#cite_note-True-range_multilateration_Escobal_Fliegel_Jaffe_Muller_2013_p.-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Applications involving vehicle location are termed <a href="/wiki/Navigation" title="Navigation">navigation</a> when on-board persons/equipment are informed of its location, and are termed <a href="/wiki/Surveillance" title="Surveillance">surveillance</a> when off-vehicle entities are informed of the vehicle's location. </p><p>Two <i><a href="/wiki/Slant_range" title="Slant range">slant ranges</a></i> from two known locations can be used to locate a third point in a two-dimensional Cartesian space (plane), which is a frequently applied technique (e.g., in surveying). Similarly, two <i><a href="/wiki/Spherical_range" class="mw-redirect" title="Spherical range">spherical ranges</a></i> can be used to locate a point on a sphere, which is a fundamental concept of the ancient discipline of <a href="/wiki/Celestial_navigation" title="Celestial navigation">celestial navigation</a> — termed the <i>altitude intercept</i> problem. Moreover, if more than the minimum number of ranges are available, it is good practice to utilize those as well. This article addresses the general issue of position determination using multiple ranges. </p><p>In <a href="/wiki/2D_geometric_model" title="2D geometric model">two-dimensional geometry</a>, it is known that if a point lies on two circles, then the circle centers and the two radii provide sufficient information to narrow the possible locations down to two – one of which is the desired solution and the other is an ambiguous solution. Additional information often narrow the possibilities down to a unique location. In three-dimensional geometry, when it is known that a point lies on the surfaces of three spheres, then the centers of the three spheres along with their radii also provide sufficient information to narrow the possible locations down to no more than two (unless the centers lie on a straight line). </p> True-range multilateration can be contrasted to the more frequently encountered <a href="/wiki/Pseudo-range_multilateration" title="Pseudo-range multilateration">pseudo-range multilateration</a>, which employs range differences to locate a (typically, movable) point. Pseudo range multilateration is almost always implemented by measuring <a href="/wiki/Time_of_arrival" title="Time of arrival">times-of-arrival</a> (TOAs) of energy waves. True-range multilateration can also be contrasted to <a href="/wiki/Triangulation_(surveying)" title="Triangulation (surveying)">triangulation</a>, which involves the measurement of <a href="/wiki/Angle" title="Angle">angles</a>.</div></div> <div class="mw-heading mw-heading2"><h2 id="Pseudo-range_multilateration">Pseudo-range multilateration</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trilateration&action=edit&section=3" title="Edit section: Pseudo-range multilateration"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1066933788"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Pseudo-range_multilateration" title="Pseudo-range multilateration">Pseudo-range multilateration</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Pseudo-range_multilateration&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> <p><a href="/wiki/Pseudo-range_multilateration" title="Pseudo-range multilateration">Pseudo-range multilateration</a>, often simply multilateration (MLAT) when in context, is a technique for <a href="/wiki/Geopositioning" title="Geopositioning">determining the position</a> of an unknown point, such as a vehicle, based on measurement of biased <i><a href="/wiki/Time_of_flight" title="Time of flight">times of flight</a></i> (TOFs) of energy waves traveling between the vehicle and multiple stations at known locations. TOFs are biased by synchronization errors in the difference between <i><a href="/wiki/Time_of_arrival" title="Time of arrival">times of arrival</a></i> (TOA) and <i><a href="/wiki/Time_of_transmission" class="mw-redirect" title="Time of transmission">times of transmission</a></i> (TOT): <i>TOF=TOA-TOT</i>. <i><a href="/wiki/Pseudo-range" class="mw-redirect" title="Pseudo-range">Pseudo-ranges</a></i> (PRs) are TOFs multiplied by the wave propagation speed: <i>PR=TOF ⋅ s</i>. In general, the stations' clocks are assumed synchronized but the vehicle's clock is desynchronized. </p><p>In MLAT for <a href="/wiki/Surveillance" title="Surveillance">surveillance</a>, the waves are transmitted by the vehicle and received by the stations; the TOT is unique and unknown, while the TOAs are multiple and known. When MLAT is used for <a href="/wiki/Navigation" title="Navigation">navigation</a> (as in <i><a href="/wiki/Hyperbolic_navigation" title="Hyperbolic navigation">hyperbolic navigation</a></i>), the waves are transmitted by the stations and received by the vehicle; in this case, the TOTs are multiple but known, while the TOA is unique and unknown. In navigation applications, the vehicle is often termed the "user"; in surveillance applications, the vehicle may be termed the "target". </p><p>The vehicle's clock is considered an additional unknown, to be estimated along with the vehicle's position coordinates. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> is the number of physical dimensions being considered (e.g., 2 for a plane) and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> is the number of signals received (thus, TOFs measured), it is required that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\geq d+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>≥</mo> <mi>d</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\geq d+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2037f748abf9ea784e34c8caceb699f201d84255" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.358ex; height:2.343ex;" alt="{\displaystyle m\geq d+1}"></span>. </p><p>Processing is usually required to extract the TOAs or their differences from the received signals, and an <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> is usually required to solve this set of equations. An algorithm either: (a) determines numerical values for the TOT (for the receiver(s) clock) and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> vehicle coordinates; or (b) ignores the TOT and forms <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecbbd201e0d8f1ccc91cb46362c4b72fa1bbe6c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.043ex; height:2.343ex;" alt="{\displaystyle m-1}"></span> (at least <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span>) <a href="/wiki/Time_difference_of_arrival" class="mw-redirect" title="Time difference of arrival">time difference of arrivals</a> (TDOAs), which are used to find the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> vehicle coordinates. Almost always, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c28d2048804ba61b4fc8761e42223f561e0a7ce6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.477ex; height:2.176ex;" alt="{\displaystyle d=2}"></span> (e.g., a plane or the surface of a sphere) or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58a4a6128b5cc1e25fcba3a16cbd9126b2d3e2fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.477ex; height:2.176ex;" alt="{\displaystyle d=3}"></span> (e.g., the real physical world). Systems that form TDOAs are also called <i>hyperbolic</i> systems,<sup id="cite_ref-Pseudo-range_multilateration_Proc_11-0" class="reference"><a href="#cite_note-Pseudo-range_multilateration_Proc-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> for reasons discussed below. </p><p>A multilateration <i><a href="/wiki/Navigation" title="Navigation">navigation</a></i> system provides vehicle position information to an entity "on" the vehicle (e.g., aircraft pilot or GPS receiver operator). A multilateration <i><a href="/wiki/Surveillance" title="Surveillance">surveillance</a></i> system provides vehicle position to an entity "not on" the vehicle (e.g., air traffic controller or cell phone provider). By the reciprocity principle, any method that can be used for navigation can also be used for surveillance, and vice versa (the same information is involved). </p> Systems have been developed for both TOT and TDOA (which ignore TOT) algorithms. In this article, TDOA algorithms are addressed first, as they were implemented first. Due to the technology available at the time, TDOA systems often determined a vehicle location in two dimensions. TOT systems are addressed second. They were implemented, roughly, post-1975 and usually involve satellites. Due to technology advances, TOT algorithms generally determine a user/vehicle location in three dimensions. However, conceptually, TDOA or TOT algorithms are not linked to the number of dimensions involved.</div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trilateration&action=edit&section=4" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFEngineers1994" class="citation book cs1">Engineers, A.S.C. (1994). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jPVxSDzVRP0C&pg=PA548"><i>Glossary of the Mapping Sciences</i></a>. American Society of Civil Engineers. p. 548. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7844-7570-6" title="Special:BookSources/978-0-7844-7570-6"><bdi>978-0-7844-7570-6</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-11-07</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Glossary+of+the+Mapping+Sciences&rft.pages=548&rft.pub=American+Society+of+Civil+Engineers&rft.date=1994&rft.isbn=978-0-7844-7570-6&rft.aulast=Engineers&rft.aufirst=A.S.C.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DjPVxSDzVRP0C%26pg%3DPA548&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrilateration" class="Z3988"></span></span> </li> <li id="cite_note-ICAO-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-ICAO_2-0">^</a></b></span> <span class="reference-text">"Multilateration (MLAT) Concept of use", International Civil Aviation Organization, 2007</span> </li> <li id="cite_note-Wolff-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Wolff_3-0">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.radartutorial.eu/02.basics/rp52.en.html">"Radar Basics"</a>, Christian Wolff, undated</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.britannica.com/EBchecked/topic/605329/trilateration">Encyclopædia Britannica</a></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.diracdelta.co.uk/science/source/t/r/trilateration/source.html">diracdelta</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100812144807/http://www.diracdelta.co.uk/science/source/t/r/trilateration/source.html">Archived</a> 2010-08-12 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.thefreedictionary.com/trilateration">free dictionary</a></span> </li> <li id="cite_note-True-range_multilateration_Lee-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-True-range_multilateration_Lee_7-0">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://rosap.ntl.bts.gov/view/dot/12134"><i>Accuracy limitations of range-range (spherical) multilateration systems</i></a>, Harry B. Lee, Massachusetts Institute of Technology, Lincoln Laboratory, Report Number: DOT/TSC-RA-3-8-(1) (Technical note 1973-43), Oct. 11, 1973</span> </li> <li id="cite_note-True-range_multilateration_Grant-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-True-range_multilateration_Grant_8-0">^</a></b></span> <span class="reference-text">"<a rel="nofollow" class="external text" href="https://journals.lib.unb.ca/index.php/ihr/article/download/23824/27609">Rho-Rho Loran-C Combined with Satellite Navigation for Offshore Surveys</a>". S.T. Grant, <i>International Hydrographic Review</i>, undated</span> </li> <li id="cite_note-True-range_multilateration_Wirtanen_1969_pp._81–92-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-True-range_multilateration_Wirtanen_1969_pp._81–92_9-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWirtanen1969" class="citation journal cs1">Wirtanen, Theodore H. (1969). "Laser Multilateration". <i>Journal of the Surveying and Mapping Division</i>. <b>95</b> (1). American Society of Civil Engineers (ASCE): <span class="nowrap">81–</span>92. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1061%2Fjsueax.0000322">10.1061/jsueax.0000322</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0569-8073">0569-8073</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Surveying+and+Mapping+Division&rft.atitle=Laser+Multilateration&rft.volume=95&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E81-%3C%2Fspan%3E92&rft.date=1969&rft_id=info%3Adoi%2F10.1061%2Fjsueax.0000322&rft.issn=0569-8073&rft.aulast=Wirtanen&rft.aufirst=Theodore+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrilateration" class="Z3988"></span></span> </li> <li id="cite_note-True-range_multilateration_Escobal_Fliegel_Jaffe_Muller_2013_p.-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-True-range_multilateration_Escobal_Fliegel_Jaffe_Muller_2013_p._10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEscobalFliegelJaffeMuller2013" class="citation journal cs1">Escobal, P. R.; Fliegel, H. F.; Jaffe, R. M.; Muller, P. M.; Ong, K. M.; Vonroos, O. H. (2013-08-07). <a rel="nofollow" class="external text" href="https://ntrs.nasa.gov/citations/19730002255">"A 3-D Multilateration: A Precision Geodetic Measurement System"</a>. <i>JPL Quart. Tech. Rev</i>. <b>2</b> (3)<span class="reference-accessdate">. Retrieved <span class="nowrap">2022-11-06</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=JPL+Quart.+Tech.+Rev.&rft.atitle=A+3-D+Multilateration%3A+A+Precision+Geodetic+Measurement+System&rft.volume=2&rft.issue=3&rft.date=2013-08-07&rft.aulast=Escobal&rft.aufirst=P.+R.&rft.au=Fliegel%2C+H.+F.&rft.au=Jaffe%2C+R.+M.&rft.au=Muller%2C+P.+M.&rft.au=Ong%2C+K.+M.&rft.au=Vonroos%2C+O.+H.&rft_id=https%3A%2F%2Fntrs.nasa.gov%2Fcitations%2F19730002255&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrilateration" class="Z3988"></span></span> </li> <li id="cite_note-Pseudo-range_multilateration_Proc-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-Pseudo-range_multilateration_Proc_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFProc2021" class="citation web cs1">Proc, Jerry (2021). <a rel="nofollow" class="external text" href="http://www.jproc.ca/hyperbolic/">"Hyperbolic Radionavigation Systems"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">April 11,</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Hyperbolic+Radionavigation+Systems&rft.date=2021&rft.aulast=Proc&rft.aufirst=Jerry&rft_id=http%3A%2F%2Fwww.jproc.ca%2Fhyperbolic%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrilateration" class="Z3988"></span></span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Trilateration&oldid=1226607139">https://en.wikipedia.org/w/index.php?title=Trilateration&oldid=1226607139</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Geometry" title="Category:Geometry">Geometry</a></li><li><a href="/wiki/Category:Geopositioning" title="Category:Geopositioning">Geopositioning</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Articles_with_excerpts" title="Category:Articles with excerpts">Articles with excerpts</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 31 May 2024, at 18:44<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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