  In this work, the problem of 4 degree-of-freedom (3D position and heading)
robot-to-robot relative frame transformation estimation using onboard odometry
and inter-robot distance measurements is studied. Firstly, we present a
theoretical analysis of the problem, namely the derivation and interpretation
of the Cramer-Rao Lower Bound (CRLB), the Fisher Information Matrix (FIM) and
its determinant. Secondly, we propose optimization-based methods to solve the
problem, including a quadratically constrained quadratic programming (QCQP) and
the corresponding semidefinite programming (SDP) relaxation. Moreover, we
address practical issues that are ignored in previous works, such as accounting
for spatial-temporal offsets between the ultra-wideband (UWB) and odometry
sensors, rejecting UWB outliers and checking for singular configurations before
commencing operation. Lastly, extensive simulations and real-life experiments
with aerial robots show that the proposed QCQP and SDP methods outperform
state-of-the-art methods, especially in geometrically poor or large measurement
noise conditions. In general, the QCQP method provides the best results at the
expense of computational time, while the SDP method runs much faster and is
sufficiently accurate in most cases.
