Graph-Theoretic Bézier Curve Optimization over Safe Corridors for Safe and Smooth Motion Planning

As a parametric motion representation, Bézier curves have significant applications in polynomial trajectory optimization for safe and smooth motion planning of various robotic systems, including flying drones, autonomous vehicles, and robotic manipulators. An essential component of Bézier curve optimization is the optimization objective, as it significantly influences the resulting robot motion. Standard physical optimization objectives, such as minimizing total velocity, acceleration, jerk, and snap, are known to yield quadratic optimization of Bézier curve control points. In this paper, we present a unifying graph-theoretic perspective for defining and understanding Bézier curve optimization objectives using a consensus distance of Bézier control points derived based on their interaction graph Laplacian. In addition to demonstrating how standard physical optimization objectives define a consensus distance between Bézier control points, we also introduce geometric and statistical optimization objectives as alternative consensus distances, constructed using finite differencing and differential variance. To compare these optimization objectives, we apply Bézier curve optimization over convex polygonal safe corridors automatically constructed around a maximal-clearance minimal-length reference path. We provide an explicit analytical formulation for quadratic optimization of Bézier curves using Bézier matrix operations. We conclude that the norm and variance of the finite differences of Bézier control points lead to simpler and more intuitive interaction graphs and optimization objectives compared to Bézier derivative norms, despite having similar robot motion profiles.