Euclidean geometry | Elementary geometry
Pseudo-range multilateration
Pseudo-range multilateration, often simply multilateration (MLAT) when in context, is a technique for determining the position of an unknown point, such as a vehicle, based on measurement of the times of arrival (TOAs) of energy waves traveling between the unknown point and multiple stations at known locations. When the waves are transmitted by the vehicle, MLAT is used for surveillance; when the waves are transmitted by the stations, MLAT is used for navigation (hyperbolic navigation). In either case, the stations' clocks are assumed synchronized but the vehicle's clock is not. Prior to computing a solution, the common time of transmission (TOT) of the waves is unknown to the receiver(s), either on the vehicle (one receiver, navigation) or at the stations (multiple receivers, surveillance). Consequently, also unknown is the wave times of flight (TOFs) – the ranges of the vehicle from the stations multiplied by the wave propagation speed. Each pseudo-range is the corresponding TOA multiplied by the propagation speed with the same arbitrary constant added (representing the unknown TOT). In navigation applications, the vehicle is often termed the "user"; in surveillance applications, the vehicle may be termed the "target". For a mathematically exact solution, the ranges must not change during the period the signals are received (between first and last to arrive at a receiver). Thus, for navigation, an exact solution requires a stationary vehicle; however, multilateration is often applied to the navigation of moving vehicles whose speed is much less than the wave propagation speed. If is the number of physical dimensions being considered (thus, vehicle coordinates sought) and is the number of signals received (thus, TOAs measured), it's required that . Then, the fundamental set of measurement equations is: TOAs ( measurements) = TOFs ( unknown variables embedded in expressions) + TOT (one unknown variable replicated times). Processing is usually required to extract the TOAs or their differences from the received signals, and an algorithm 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 vehicle coordinates; or (b) ignores the TOT and forms (at least ) time difference of arrivals (TDOAs), which are used to find the vehicle coordinates. Almost always, (e.g., a plane or the surface of a sphere) or (e.g., the real physical world). Systems that form TDOAs are also called hyperbolic systems, for reasons discussed below. A multilateration navigation system provides vehicle position information to an entity "on" the vehicle (e.g., aircraft pilot or GPS receiver operator). A multilateration surveillance 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). 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. (Wikipedia).
