  We introduce an information measure, termed clarity, motivated by information
entropy, and show that it has intuitive properties relevant to dynamic coverage
control and informative path planning. Clarity defines the quality of the
information we have about a variable of interest in an environment on a scale
of [0, 1], and has useful properties for control and planning such as: (I)
clarity lower bounds the expected estimation error of any estimator, and (II)
given noisy measurements, clarity monotonically approaches a level q_infty < 1.
We establish a connection between coverage controllers and information theory
via clarity, suggesting a coverage model that is physically consistent with how
information is acquired. Next, we define the notion of perceivability of an
environment under a given robotic (or more generally, sensing and control)
system, i.e., whether the system has sufficient sensing and actuation
capabilities to gather desired information. We show that perceivability relates
to the reachability of an augmented system, and derive the corresponding
Hamilton-Jacobi-Bellman equations to determine perceivability. In simulations,
we demonstrate how clarity is a useful concept for planning trajectories, how
perceivability can be determined using reachability analysis, and how a Control
Barrier Function (CBF) based controller can dramatically reduce the
computational burden.
