Analysis of Minima for Geodesic and Chordal Cost for a Minimal 2-D Pose-Graph SLAM Problem

In this letter, we show that for a minimal 2D pose-graph SLAM problem, even in the ideal case of perfect measurements and spherical covariance, using geodesic distance (in 2D, the wrap function) to compare angles results in multiple suboptimal local minima. We numerically estimate regions of attraction to these local minima for some examples, give evidence to show that they are of nonzero measure, and that these regions grow in size as noise is added. In contrast, under the same assumptions, we show that the chordal distance representation of angle error has a unique minimum up to periodicity. For chordal cost, we find that initial conditions failing to converge to the global minimum are far fewer, fail because of numerical issues, and do not seem to grow with noise in our examples.