The purpose of this work is to highlight the paramount importance of representing and quantifying uncertainty to correctly report the associated confidence of the robot's location estimate at each time step along its trajectory and therefore decide the correct course of action in an active SLAM mission. We analyze the monotonicity property of different decision-making criteria, both in 2-D and 3-D, with respect to the representation of uncertainty and of the orientation of the robot's pose. Monotonicity, the property that uncertainty increases as the robot moves, is essential for adequate decision making. We analytically show that, by using differential representations to propagate spatial uncertainties, monotonicity is preserved for all optimality criteria, A-opt, D-opt, and E-opt, and for Shannon's entropy. We also show that monotonicity does not hold for any criteria in absolute representations using Roll-Pitch-Yaw and Euler angles. Finally, using unit quaternions in absolute representations, the only criteria that preserve monotonicity are D-opt and Shannon's entropy.

In this paper we investigate the monotonicity of various optimality criteria during the exploration phase of an active SLAM algorithm. Optimality criteria such as A-opt, D-opt or E-opt are used in active SLAM to account for uncertainty in the map or the robot's pose, and these criteria are usually part of utility functions which help active SLAM algorithms decide where the robot should move next. The monotonicity of the optimality criteria is of utmost importance. During the exploration phase, i.e. when the robot is traversing new territory or cannot perform a loop closure, the most common way of estimating the pose of the robot is through dead-reckoning. Correctly accounting for the uncertainty is important for an active SLAM algorithm and in particular for a dead-reckoning scenario, where by definition the uncertainty in the robot's pose grows. If monotonicity does not hold in this scenario, active SLAM algorithms can execute actions under the false belief that the uncertainty has reduced. We show analytically and experimentally some conditions in which the A-opt and E-opt criteria lose monotonicity in a dead-reckoning scenario, where the propagation of the robot's pose is done using a linearized framework. We also show analytically and experimentally that under the same conditions the D-opt does not lose monotonicity and, in general for the linearized framework under consideration, D-opt does not break monotonicity.