  Multi-robot systems must have the ability to accurately estimate relative
states between robots in order to perform collaborative tasks, possibly with no
external aiding. Three-dimensional relative pose estimation using range
measurements oftentimes suffers from a finite number of non-unique solutions,
or ambiguities. This paper: 1) identifies and accurately estimates all possible
ambiguities in 2D; 2) treats them as components of a Gaussian mixture model;
and 3) presents a computationally-efficient estimator, in the form of a
Gaussian-sum filter (GSF), to realize range-based relative pose estimation in
an infrastructure-free, 3D, setup. This estimator is evaluated in simulation
and experiment and is shown to avoid divergence to local minima induced by the
ambiguous poses. Furthermore, the proposed GSF outperforms an extended Kalman
filter, demonstrates similar performance to the computationally-demanding
particle filter, and is shown to be consistent.
