Solving the Multilateration Problem without Iteration

The process of positioning, using only distances from control stations, is called trilateration (or multilateration if the problem is over-determined). The observation equation is Pythagoras’s formula, in terms of the summed squares of coordinate differences and, thus, is nonlinear. There is one observation equation for each control station, at a minimum, which produces a system of simultaneous equations to solve. Over-determined nonlinear systems of simultaneous equations are typically solved using iterative least squares after forming the system as a truncated Taylor’s series, omitting the nonlinear terms. This paper provides a linearization of the observation equation that is not a truncated infinite series—it is exact—and, thus, is solved exactly, with full rigor, without iteration and, thus, without the need of first providing approximate coordinates to seed the iteration. However, there is a cost of requiring an additional observation beyond that required by the non-linear approach. The examples and terminology come from terrestrial land surveying, but the method is fully general: it works for, say, radio beacon positioning, as well. The approach can use slope distances directly, which avoids the possible errors introduced by atmospheric refraction into the zenith-angle observations needed to provide horizontal distances. The formulas are derived for two- and three-dimensional cases and illustrated with an example using total-station and global navigation satellite system (GNSS) data.