Trilateration

Lateration ( lat = lateral side ) or trilateration is a measurement method for determining the position of a point. While triangulation is based on the measurement of three angles, trilateration based on distance and distance measurements to three points.

If one knows only the distance to a known point, the own site is ( in flat viewing ) on a circle in 3D space on a spherical shell around this point. At two known points, the location of the points of intersection of the two spherical shells, so on a circular line. The figure illustrates the situation in a plane. Points A and B have the same distance to P1 and P2.

Terrestrial surveys are often carried out in a model plane and then corrections for the height to be considered.

Trilateration and multilateration

Often also referred to as trilateration, as only the knowledge of the distance to three known points allows an unambiguous determination of the position in space. Although three spherical shells have two symmetrical intersections, but one of them can be eliminated by plausibility considerations usually.

With the designed for long distances electromagnetic rangefinders for the land surveying distances up to 100 km distance accuracy is achieved in the decimeter at. The electro-optical distance meter with ranges up to 60 km achieve accuracies better than / - 1 mm / km.

The lateration has replaced in the 1960s, the triangulation as the main method for triangular mesh measurement as well as triangulation requires the length measurement of a base. The fixed-point network of the German State Survey was therefore completely measured and recalculated in the 1960s and 70s.

In the 1980s, measurement methods have been developed for distance determination using the GPS satellite navigation method. There are no lengths, but running times of the radio signals measured. The derived distances are called pseudoranges. If more than 3 pseudo-paths available for evaluation, it is called multilateration. It includes methods (such as Kalman filtering) with an order to reduce erroneous measures of certain system optimal.

If the receiver can not synchronize its clock with that of the transmitter, are no longer possible positions on spherical shells. Instead, the observer measures time differences between the signals received from different transmitters. The points of equal time differences lie on hyperboloids, their intersections provide their own position. Often the hyperbolic is called multilateration.

Outdoor applications

Outdoor applications usually set to the specified image in principle meeting of the three distance dimensions in an image point in a plane. The general measurement procedure with satellite support is the GPS system. This is true only with errors in physics. The smaller the error can be kept, the better the result can be used.

The error budget of a suitable use multilateration must take into account for example the following deviations:

• Different heights of the reference points of the target point: Compensation can be achieved by n 1 points of reference in the three-dimensional space
• Stochastic errors due to measurement noise and transmission noise, varies depending on the measurement method: Compensation can be achieved by longer measuring time
• Systematic errors due to lack of calibration of a reference signal: Compensation can be reached by continuous calibration
• Geometric errors due to multiple reflection: compensation by rutting with varying error image and longer measurement time
• Systematic errors due to lack of discrimination against multipath propagation: compensation by rutting with varying error image, especially short measuring signals below the secondary Laufwegverlängerung
• Numerical errors due to finite precision arithmetic method: compensation by over-determination and compensation calculation
• Uncertainty by multiple subsystems involved: reachable compensation partially offset by bidirectional measurement

Field tests show among others, with GPS, that the error for two different, adjacent locations are not sufficiently coupled to determine the direct distance with indirect single measurement quickly. The direct measurement in line of sight should always yield the shortest distance. A guarantee to recognize this shortest distance offers only a ( quasi-) optical measuring method without multipath propagation.

Applications in buildings

Similar concepts are often used for tasks in buildings. These usually the harmful vision in quasi- optical propagation of radio waves is also used by lightweight materials throughout instead of unobstructed optical view of light propagation. Simple solutions will measure level and no maturities.

The error budget of the solutions is very complex due to the attenuation in walls, due to phase cancellation and as a result of multiple expansions with multiple reflections. As long as the signal propagation is simultaneously used for data transmission and for the measurement, these interference effects are hard to control. Many solutions will therefore be described without any reliable accuracy to be achieved. Only in the vicinity of the direct view of the error budget is easy to master.