Analysis of minima for geodesic and chordal cost for a minimal 2D pose-graph SLAM problem

In this paper, we show that for a minimal pose-graph problem, even in the ideal case of perfect measurements and spherical covariance, using the so-called "wrap function" when comparing angles results in multiple suboptimal local minima. We numerically estimate regions of attraction to these local minima for some numerical examples, and give evidence to show that they are of nonzero measure. In contrast, under the same assumptions, we show that the \textit{chordal distance} representation of angle error has a unique minimum up to periodicity. For chordal cost, we also search for initial conditions that fail to converge to the global minimum, and find that this occurs with far fewer points than with geodesic cost.