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As a parametric motion representation, B\'ezier curves have significantapplications in polynomial trajectory optimization for safe and smooth motionplanning of various robotic systems, including flying drones, autonomousvehicles, and robotic manipulators. An essential component of B\'ezier curveoptimization is the optimization objective, as it significantly influences theresulting robot motion. Standard physical optimization objectives, such asminimizing total velocity, acceleration, jerk, and snap, are known to yieldquadratic optimization of B\'ezier curve control points. In this paper, wepresent a unifying graph-theoretic perspective for defining and understandingB\'ezier curve optimization objectives using a consensus distance of B\'eziercontrol points derived based on their interaction graph Laplacian. In additionto demonstrating how standard physical optimization objectives define aconsensus distance between B\'ezier control points, we also introduce geometricand statistical optimization objectives as alternative consensus distances,constructed using finite differencing and differential variance. To comparethese optimization objectives, we apply B\'ezier curve optimization over convexpolygonal safe corridors that are automatically constructed around amaximal-clearance minimal-length reference path. We provide an explicitanalytical formulation for quadratic optimization of B\'ezier curves usingB\'ezier matrix operations. We conclude that the norm and variance of thefinite differences of B\'ezier control points lead to simpler and moreintuitive interaction graphs and optimization objectives compared to B\'ezierderivative norms, despite having similar robot motion profiles.

Graph-Theoretic B\'ezier Curve Optimization over Safe Corridors for Safe and Smooth Motion Planning