Why Three Measurements are not Enough for Trilateration-based Localisation

In this paper, we analyse the problem of simultaneous trilateration, i.e. when three ranging sensors retrieve their distance from a target, and we found that is ill-posed from an algebraic view-point. Then, we explain this fact using the concept of "delayed" trilateration, i.e. we consider three ranging sensors that measure their distance from the agent at three distinct time steps, while the agent has moved through the environment. We prove with a counterexample that, in a general case, there are multiple trajectories that are compliant with the motion of the target and with the ranging measurements collected by the anchor of known position. Therefore, we claim here that three measurements are not sufficient to localise a moving target in a given environment even in ideal conditions. In particular, we claim that the classic trilateration problem assumes an additional implicit information besides the three ranging measurements, that is that the three measurements are taken with respect to the same point in space or, in the most general and most probable case of a moving target, simultaneously.