  Fast and accurate safety assessment and collision checking are essential for
motion planning and control of highly dynamic autonomous robotic systems.
Informative, intuitive, and explicit motion trajectory bounds enable
explainable and time-critical safety verification of autonomous robot motion.
In this paper, we consider feedback linearization of nonlinear systems in the
form of proportional-and-higher-order-derivative (PhD) control corresponding to
companion dynamics. We introduce a novel analytic convex trajectory bound,
called $\textit{Vandermonde simplex}$, for high-order companion systems, that
is given by the convex hull of a finite weighted combination of system
position, velocity, and other relevant higher-order state variables. Our
construction of Vandermonde simplexes is built based on expressing the solution
trajectory of companion dynamics in a newly introduced family of
$\textit{Vandermonde basis functions}$ that offer new insights for
understanding companion system motion compared to the classical exponential
basis functions. In numerical simulations, we demonstrate that Vandermonde
simplexes offer significantly more accurate motion prediction (e.g., at least
an order of magnitude improvement in estimated motion volume) for describing
the motion trajectory of companion systems compared to the standard invariant
Lyapunov ellipsoids as well as exponential simplexes built based on exponential
basis functions.
