Lateration
Lateration (lat. Lateral = to the side) or trilateration is a measuring method for determining the position of a point. While the triangulation is based on the measurement of three angles, the trilateration is based on distance or distance measurements to three points.
If you only know the distance to a known point, your own location (when viewed from a plane) lies on a circle, in 3D space on a spherical shell around this point. With two known points, the location is on the intersection of the two spherical shells, i.e. on a circular line . The figure illustrates the situation in one plane. Points A and B are equidistant from P1 and P2 .
Terrestrial surveys are often made in a model plane and then corrections for the height are taken into account.
Trilateration and multilateration
One often speaks of trilateration , since only knowing the distance to three known points enables a clear determination of the position in space. Although three spherical shells have two mutually symmetrical intersections, one of them can usually be ruled out on the basis of plausibility considerations.
With the electromagnetic rangefinders designed for large distances for national surveying , distances of up to 100 km are accurate to a decimeter. The electro-optical range finders with ranges of up to 60 km achieve accuracies better than ± 1 mm / km.
In the 1960s, lateration replaced triangulation as the main method for measuring triangular meshes, since triangulation also requires the length measurement of a base. The fixed point network of the German national survey was therefore completely re-measured and calculated in the 1960s and 70s.
In the 1980s, measuring methods for determining routes were developed using the GPS satellite navigation method . No lengths, but transit times of the radio signals are measured. The distances derived from this are called pseudo- distances . If more than three pseudo-routes are available for evaluation, one speaks of multilateration . It includes methods (for example Kalman filtering ) in order to optimally reduce error-prone measured variables of the overdetermined system .
If the receiver cannot synchronize its clock with that of the transmitter, possible positions are no longer on spherical shells. Instead, the observer measures time differences between the signals he receives from different transmitters. The points of equal time differences lie on hyperboloids, their intersection points provide their own position. Hyperbola navigation is often referred to as multilateration.
Outdoor applications
Outdoor applications mostly rely on the coincidence of the three distance dimensions in a pixel in a plane, as indicated in the principle diagram. The general measuring method with satellite support is the GPS system. In physics, this only applies with errors. The smaller the errors can be kept, the better the result can be used.
The error budget of a usable multilateration must, for example, take into account the following deviations:
- Different heights of the reference points and the target point: Compensation can also be achieved in three-dimensional space with n + 1 reference points
- Stochastic errors due to measurement noise and transmission noise, different depending on the measurement method: Compensation can be achieved through longer measurement times
- Systematic errors due to missing calibration of a reference signal: Compensation can be achieved through continuous calibration
- Geometric errors due to multiple reflections: Compensation by formation of tracks with changing error patterns and by longer measuring times
- Systematic errors due to a lack of discrimination against multipath propagation: Compensation through track formation with changing error patterns, particularly short measurement signals below the secondary path extension
- numerical errors through finitely accurate calculation method: compensation through overdetermination and compensation calculation
- Measurement uncertainty due to several subsystems involved: Compensation can be partially achieved through bidirectional measurement (patents IEEE 802.15.4a CSS, Nanotron measurement method, see Chirp Spread Spectrum )
Field tests with GPS , among others, show that the errors for two different, neighboring locations are not sufficiently coupled to quickly determine the direct distance with indirect individual measurements. The direct measurement in line of sight should always provide the shortest distance. A guarantee of recognizing this shortest distance is only provided by a (quasi) optical measuring method without multipath propagation.
Applications in buildings
Similar concepts are often used for tasks in buildings. For this purpose, the radio-technical view with quasi-optical propagation of radio waves is also used through light materials instead of the unobstructed optical view of the light propagation. The simplest solutions measure levels and not run times.
The error balance of the solutions is very complex due to the attenuation in walls, due to phase cancellation and due to multiple propagation with multiple reflections. As long as the signal propagation is used for data transmission and measurement at the same time, these interference effects can hardly be controlled. Many solutions are therefore described without any accuracy that can be reliably achieved. The error budget can only be easily mastered in the immediate vicinity.
See also
literature
- Wolfgang Torge : Geodesy. de Gruyter, Berlin New York 2003, ISBN 3-11-017545-2
- Hans Zetsche: Electronic distance measurement. Konrad Wittwer Verlag, Stuttgart 1979, ISBN 3-87919-127-1
- Jürgen Bollmann and Wolf Günther Koch (eds.): Lexicon of cartography and geomatics. Spectrum Academic Publishing House, Heidelberg Berlin 2001, ISBN 3-8274-1055-X
Individual evidence
- ↑ RSSI measurement method ( Memento from March 4, 2011 in the Internet Archive )