  Computing kinodynamically feasible motion plans and repairing them on-the-fly
as the environment changes is a challenging, yet relevant problem in
robot-navigation. We propose a novel online single-query sampling-based motion
re-planning algorithm - PiP-X, using finite-time invariant sets - funnels. We
combine concepts from sampling-based methods, nonlinear systems analysis and
control theory to create a single framework that enables feedback motion
re-planning for any general nonlinear dynamical system in dynamic workspaces.
  A volumetric funnel-graph is constructed using sampling-based methods, and an
optimal funnel-path from robot configuration to a desired goal region is then
determined by computing the shortest-path subtree in it. Analysing and formally
quantifying the stability of trajectories using Lyapunov level-set theory
ensures kinodynamic feasibility and guaranteed set-invariance of the
solution-paths. The use of incremental search techniques and a pre-computed
library of motion-primitives ensure that our method can be used for quick
online rewiring of controllable motion plans in densely cluttered and dynamic
environments.
  We represent traversability and sequencibility of trajectories together in
the form of an augmented directed-graph, helping us leverage discrete
graph-based replanning algorithms to efficiently recompute feasible and
controllable motion plans that are volumetric in nature. We validate our
approach on a simulated 6DOF quadrotor platform in a variety of scenarios
within a maze and random forest environment. From repeated experiments, we
analyse the performance in terms of algorithm-success and length of
traversed-trajectory.
