  The main contribution of this paper is the proof of the convexity of the
omni-directional tethered robot workspace (namely, the set of all
tether-length-admissible robot configurations), as well as a set of
distance-optimal tethered path planning algorithms that leverage the workspace
convexity. The workspace is proven to be topologically a simply-connected
subset and geometrically a convex subset of the set of all configurations. As a
direct result, the tether-length-admissible optimal path between two
configurations is proven exactly the untethered collision-free locally shortest
path in the homotopy specified by the concatenation of the tether curve of the
given configurations, which can be simply constructed by performing an
untethered path shortening process in the 2D environment instead of a path
searching process in the pre-calculated workspace.
  The convexity is an intrinsic property to the tethered robot kinematics, thus
has universal impacts on all high-level distance-optimal tethered path planning
tasks: The most time-consuming workspace pre-calculation (WP) process is
replaced with a goal configuration pre-calculation (GCP) process, and the
homotopy-aware path searching process is replaced with untethered path
shortening processes. Motivated by the workspace convexity, efficient
algorithms to solve the following problems are naturally proposed: (a) The
optimal tethered reconfiguration (TR) planning problem is solved by a locally
untethered path shortening (UPS) process, (b) The classic optimal tethered path
(TP) planning problem (from a starting configuration to a goal location whereby
the target tether state is not assigned) is solved by a GCP process and $n$ UPS
processes, where $n$ is the number of tether-length-admissible configurations
that visit the goal location, (c) The optimal tethered motion to visit a
sequence of multiple goal locations, referred to as

Robotics