  In this article we consider the problem of tether entanglement for tethered
robots. In many applications, such as maintenance of underwater structures,
aerial inspection, and underground exploration, tethered robots are often used
in place of standalone (i.e., untethered) ones. However, the presence of a
tether also introduces the risk for it to get entangled with obstacles present
in the environment or with itself. To avoid these situations, a
non-entanglement constraint can be considered in the motion planning problem
for tethered robots. This constraint can be expressed either as a set of
specific tether configurations that must be avoided, or as a quantitative
measure of a `level of entanglement' that can be minimized. However, the
literature lacks a generally accepted definition of entanglement, with existing
definitions being limited and partial. Namely, the existing entanglement
definitions either require a taut tether to come into contact with an obstacle
or with another tether, or they require for the tether to do a full loop around
an obstacle. In practice, this means that the existing definitions do not
effectively cover all instances of tether entanglement. Our goal in this
article is to bridge this gap and provide new definitions of entanglement,
which, together with the existing ones, can be effectively used to qualify the
entanglement state of a tethered robot in diverse situations. The new
definitions find application mainly in motion planning for tethered robot
systems, where they can be used to obtain more safe and robust
entanglement-free trajectories. The present article focuses exclusively on the
presentation and analysis of the entanglement definitions. The application of
the definitions to the motion planning problem is left for future work.
