  Safe and smooth robot motion around obstacles is an essential skill for
autonomous robots, especially when operating around people and other robots.
Conventionally, due to real-time operation requirements and onboard computation
limitations, many robot motion planning and control methods follow a two-step
approach: first construct a (e.g., piecewise linear) collision-free reference
path for a simplified robot model, and then execute the reference plan via
path-following control for a more accurate and complex robot model. A challenge
of such a decoupled robot motion planning and control method for highly dynamic
robotic systems is ensuring the safety of path-following control as well as the
successful completion of the reference plan. In this paper, we introduce a
novel dynamical systems approach for online closed-loop time parametrization,
called $\textit{a time governor}$, of a reference path for provably correct and
safe path-following control based on feedback motion prediction, where the
safety of robot motion under path-following control is continuously monitored
using predicted robot motion. After introducing the general framework of time
governors for safe path following, we present an example application for the
fully actuated high-order robot dynamics using
proportional-and-higher-order-derivative (PhD) path-following control whose
feedback motion prediction is performed by Lyapunov ellipsoids and Vandemonde
simplexes. In numerical simulations, we investigate the role of reference
position and velocity feedback, and motion prediction on path-following
performance and robot motion.
