Iterative weighted 2D orientation averaging that minimizes arc-length between vectors

The buildup of inaccuracies from frequent and imperfect data averaging can negatively impact system behavior. One potential culprit is improper orientation averaging, such as when combining data from multiple sensors, or reconciling preferences from multiple agents. In practice, the currently prevalent methodology of averaging 2D orientations is that of adding orientation vectors, which minimizes the Euclidean (or chord) distance among the vectors, instead of the geodesic (or arc) distance, resulting in inaccurate or even entirely incorrect averages. While an arc-minimizing alternative exists, it is only defined for angle averaging, posing an issue if orientations also possess a meaningful magnitude within the domain. In this work, we present an iterative weighted 2D orientation arc-based averaging algorithm that minimizes squared arc-lengths between points, incorporates orientation magnitudes as weights, and allows for multiple equally valid averages to be produced whenever applicable. We compare a vector sum approach and the weighted arc-based approach as applied to collaborative transport with obstacle avoidance, and showcase the behavioral advantages of the arc-based weighted averaging.