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Estimating the orientations of nodes in a pose graph from relative angularmeasurements is challenging because the variables live on a manifold productwith nontrivial topology and the maximum-likelihood objective function isnon-convex and has multiple local minima; these issues prevent iterativesolvers to be robust for large amounts of noise. This paper presents anapproach that allows working around the problem of multiple minima, and isbased on the insight that the original estimation problem on orientations isequivalent to an unconstrained quadratic optimization problem on integervectors. This equivalence provides a viable way to compute the maximumlikelihood estimate and allows guaranteeing that such estimate is almost surelyunique. A deeper consequence of the derivation is that the maximum likelihoodsolution does not necessarily lead to an estimate that is "close" to the actualnodes orientations, hence it is not necessarily the best choice for the problemat hand. To alleviate this issue, our algorithm computes a set of estimates,for which we can derive precise probabilistic guarantees. Experiments show thatthe method is able to tolerate extreme amounts of noise (e.g., {\sigma} =30{\deg} on each measurement) that are above all noise levels of sensorscommonly used in mapping. For most range-finder-based scenarios, themulti-hypothesis estimator returns only a single hypothesis, because theproblem is very well constrained. Finally, using the orientations estimateprovided by our method to bootstrap the initial guess of pose graphoptimization methods improves their robustness and makes them avoid localminima even for high levels of noise.

From Angular Manifolds to the Integer Lattice: Guaranteed Orientation Estimation with Application to Pose Graph Optimization