Editing True-range multilateration (section)

=== Three Cartesian dimensions, three measured slant ranges ===
[[File:3D Trilat Scenario 2019-0116.jpg|thumb|Fig. 2 3-D True-Range Multilateration Scenario. C1, C2 and C3 are known centers of spheres in the x,y plane. P is point whose (x,y,z) coordinates are desired based on its ranges to C1, C2 and C3.]]
[[File:3D Trilateration.jpg|thumb|right|220px|3-D '''Tri'''lateration limits the potential positions amount to two (here A or B)]]

There are multiple algorithms that solve the 3-D Cartesian true-range multilateration problem directly (i.e., in closed-form) – e.g., Fang.<ref name="Fang1">"Trilateration and extension to global positioning system navigation", B.T. Fang, ''Journal of Guidance, Control, and Dynamics'', vol. 9 (1986), pp 715–717.</ref>  Moreover, one can adopt closed-form algorithms developed for pseudo range [[multilateration]].<ref name="Sirola">"[https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=06995137cf1a12d6adf75b6e2246e44fe6c5aa6d Closed-form Algorithms in Mobile Positioning: Myths and Misconceptions]", Niilo Sirola, ''Proceedings of the 7th Workshop on Positioning, Navigation and Communication 2010 (WPNC'10)'', March 11, 2010.</ref><ref name="Geyer">{{cite book|last=Geyer|first=Michael|date=June 2016|title=Earth-Referenced Aircraft Navigation and Surveillance Analysis|url=https://rosap.ntl.bts.gov/view/dot/12301|location=U.S. DOT National Transportation Library|publisher=U.S. DOT John A. Volpe National Transportation Systems Center}}</ref> Bancroft's algorithm<ref name="Bancroft">[https://ieeexplore.ieee.org/search/searchresult.jsp?searchWithin=p_Authors:.QT.Bancroft,%20S..QT.&searchWithin=p_Author_Ids:37296953500&newsearch=true "An Algebraic Solution of the GPS Equations"], Stephen Bancroft, ''IEEE Transactions on Aerospace and Electronic Systems'', Volume: AES-21, Issue: 7 (Jan. 1985), pp 56–59.</ref> (adapted) employs vectors, which is an advantage in some situations.

The simplest algorithm corresponds to the sphere centers in Fig. 2. The figure 'page' is the plane containing '''C1''', '''C2''' and '''C3'''. If '''P''' is a 'point of interest' (e.g., vehicle) at <math>(x,y,z)</math>, then Pythagoras's theorem yields the slant ranges between '''P''' and the sphere centers:

: <math> \begin{align}
r_1^2 & = x^2 + y^2 + z^2  \\[4pt]
r_2^2 & = (x-U)^2 + y^2 + z^2  \\[4pt]
r_3^2 & = (x-V_x)^2 + (y-V_y)^2 + z^2
\end{align} </math>

Thus, the coordinates of '''P''' are:

{{NumBlk|::|
<math> \begin{align}
x & = \frac { r_1^2 - r_2^2 + U^2 } {2 U}   \\[4pt]
y & = \frac { r_1^2 - r_3^2 + V_x^2 + V_y^2 - 2 V_x x } {2 V_y}  \\[4pt]
z & = \pm \sqrt{r_1^2 - x^2 - y^2}
\end{align} </math>
|{{EquationRef|2}}}}

The plane containing the sphere centers is a plane of symmetry. The correct and ambiguous solutions are perpendicular to it and equally distant from it, on opposite sides.

Many applications of 3-D true-range multilateration involve short ranges—e.g., precision manufacturing.<ref name="Schneider" /> Integrating range measurement from three or more radars (e.g., FAA's [[ERAM]]) is a 3-D aircraft surveillance application. 3-D true-range multilateration has been used on an experimental basis with GPS satellites for aircraft navigation.<ref name="Zhang">[https://www.ucalgary.ca/engo_webdocs/GL/97.20112.ZNZhang.pdf ''Impact of Rubidium Clock Aiding on GPS Augmented Vehicular Navigation''], Zhaonian Zhang; University of Calgary; December, 1997.</ref> The requirement that an aircraft be equipped with an atomic clock precludes its general use. However, GPS receiver clock aiding is an area of active research, including aiding over a network. Thus, conclusions may change.<ref name="Langley2">[https://www.gpsworld.com/innovation-reducing-the-jitters/ "How a Chip-Scale Atomic Clock Can Help Mitigate Broadband Interference"]; Fang-Cheng Chan, Mathieu Joerger, Samer Khanafseh, Boris Pervan, and Ondrej Jakubov; ''GPS World --  Innovations''; May 2014.</ref> 3-D true-range multilateration was evaluated by the International Civil Aviation Organization as an aircraft landing system, but another technique was found to be more efficient.<ref name="Evans">[https://ieeexplore.ieee.org/document/5005109 "Microwave Landing System"]; Thomas E. Evans; ''IEEE Aerospace and Electronic Systems Magazine''; Vol. 1, Issue 5; May 1986.</ref> Accurately measuring the altitude of aircraft during approach and landing requires many ground stations along the flight path.