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

Vandermonde Trajectory Bounds for Linear Companion Systems