A Neural Network Approach towards Real-Time Knot Point Density Estimates for the Nonlinear Quadrotor

Abstract

Spline representations provide a compact way to approximate continuous trajectories using a finite number of parameters. When approximating trajectories generated by dynamical systems, the efficiency of a spline representation depends strongly on the placement of knot points. Uniform knot distributions often lead to inefficient representations, requiring either unnecessary control points or producing poor approximations in regions where the trajectory varies rapidly. In this work we propose a learning-based method for adaptive knot placement in B-spline approximations of trajectories generated by ordinary differential equations. A neural network learns a mapping from the initial condition of the system to quantities that characterize the local variation of the resulting trajectory. These estimates are used to construct nonuniform knot distributions that approximately equidistribute a local error indicator, reducing the number of control points needed to meet a prescribed accuracy.

arXiv comment8 pages, 5 figures. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. This work is presented at IEEE CDC 2026