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

As a parametric motion representation, B\ézier 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 B\ézier 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 B\ézier curve control points. In this paper, we present a unifying graph-theoretic perspective for defining and understanding B\ézier curve optimization objectives using a consensus distance of B\ézier control points derived based on their interaction graph Laplacian. In addition to demonstrating how standard physical optimization objectives define a consensus distance between B\ézier 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 B\ézier curve optimization over convex polygonal safe corridors that are automatically constructed around a maximal-clearance minimal-length reference path. We provide an explicit analytical formulation for quadratic optimization of B\ézier curves using B\ézier matrix operations. We conclude that the norm and variance of the finite differences of B\ézier control points lead to simpler and more intuitive interaction graphs and optimization objectives compared to B\ézier derivative norms, despite having similar robot motion profiles.