Inexact Loops in Robotics Problems

Loops are pervasive in robotics problems; appearing in mapping and localization; where one is interested in finding loop closure constraints to better approximate robot poses or other estimated quantities; as well as planning and prediction; where one is interested in the homotopy classes of the space through which a robot is moving. We generalize the standard topological definition of a loop to cases where a trajectory passes close to itself; but doesn't necessarily touch; giving a definition that is more practical for real robotics problems. This relaxation leads to new and useful properties of inexact loops; such as their ability to be partitioned into topologically connected sets closely matching the concept of a "loop closure"; and the existence of simple and nonsimple loops. Building from these ideas; we introduce several ways to measure properties and quantities of inexact loops on a trajectory; such as the trajectory's "loop area" and "loop density'"; and use them to compare strategies for sampling representative inexact loops to build constraints in mapping and localization problems.