Multilateration should not be confused with trilateration, which uses distances or absolute measurements of time-of-flight from three or more sites, or with triangulation, which uses a baseline and at least two angles measured e.g. with receiver antenna diversity and phase comparison.
Multilateration is commonly used in civil and military surveillance applications to accurately locate an aircraft, vehicle or stationary emitter by measuring the time difference of arrival (TDOA) of a signal from the emitter at three or more receiver sites.
If a pulse is emitted from a platform, it will arrive at slightly different times at two spatially separated receiver sites, the TDOA being due to the different distances of each receiver from the platform. In fact, for given locations of the two receivers, a whole series of emitter locations would give the same measurement of TDOA. Given two receiver locations and a known TDOA, the locus of possible emitter locations is a one half of a two-sheeted hyperboloid.
In simple terms, with two receivers at known locations, an emitter can be located onto a hyperboloid. Note that the receivers do not need to know the absolute time at which the pulse was transmitted - only the time difference is needed.
Consider now a third receiver at a third location. This would provide a second TDOA measurement and hence locate the emitter on a second hyperboloid. The intersection of these two hyperboloids describes a curve on which the emitter lies.
If a fourth receiver is now introduced, a third TDOA measurement is available and the intersection of the resulting third hyperboloid with the curve already found with the other three receivers defines a unique point in space. The emitter's location is therefore fully determined in 3D.
In practice, errors in the measurement of the time of arrival of pulses mean that enhanced accuracy can be obtained with more than four receivers. In general, N receivers provide N-1 hyperboloids. When there are N > 4 receivers, the N-1 hyperboloids should, assuming a perfect model and measurements, intersect on a single point. In reality, the surfaces rarely intersect, because of various errors. In this case, the location problem can be posed as an optimization problem and solved using, for example, a least squares method or an extended Kalman filter.
Additionally, the TDOA of multiple transmitted pulses from the emitter can be averaged to improve accuracy.
Consider an emitter at unknown location which we wish to locate. Consider also a multilateration system comprising four receiver sites at known locations: a central site, C, a left site, L, a right site, R and a fourth site, Q.
The travel time (T) of pulses from the emitter at () to each of the receiver locations is simply the distance divided by the pulse propagation rate (c):
Then the time difference of arrival between pulses arriving directly at the central site and those coming via the side sites can be shown to be:
The multilateration system must then solve for the unknown target location in real time. All the other symbols are known.
Note that in civilian air traffic control multilateration systems, the unknown height, , is often directly derived from the Mode C SSR transponder return. In this case, only three sites are required for a 3D solution.
For trilateration or multilateration, calculation is done based on distances, which requires the frequency and the wave count of a received transmission. For triangulation or multiangulation, calculation is done based on angles, which requires the phases of received transmission plus the wave count.
For lateration compared to angulation, the numerical problems compare, but the technical problem is more challenging with angular measurements, as angles require two measures per position when using optical or electronic means for measuring phase differences instead of counting wave cycles.
Trilateration in general is calculating with triangles with known distances/sizes, mathematical a very sound system. In a triangle if the length of three sides is known, the angles can be calculated (congruence), however for calculating the other sides from angles at least one side (baseline) must be known additionally (otherwise just similarity).
In 3D, when four or more angles are in play, locations can be calculated from n+1=4 measured angles plus one known baseline or from just n+1=4 measured sides.
Multilateration is, in general, far more accurate for locating an object than sparse approaches such as trilateration, where with planar problems just three distances are known and computed. Multilateration serves for several aspects:
Accuracy of multilateration is a function of several variables, including:
The accuracy can be calculated by using the Cramér–Rao bound and taking account of the above factors in its formulation.
Fraport Selects Sensis Multilateration System for Expanded Tracking of Ground Movements at Frankfurt Airport.
Jun 16, 2011; Fraport AG, the owner and manager of Frankfurt Airport (FRA), announced that it has selected Sensis Corp. to expand and upgrade...