Efficient Algorithm for Spatial Trajectories Combined from Double-S Motions of Arbitrary Order Based on a Unified Equation

Abstract

The paper presents an algorithm for generating and optimizing spatial trajectories based on a unified equation applicable to one-dimensional double-S motion profiles of arbitrary order. The proposed algorithm leverages an analytical equation for the time horizon computation, avoiding traditional nested case distinctions, and integrates numerical computation and visualization tools. This framework allows for real-time optimization of time, velocity and higher-order derivatives (with constraints taken into account). High-order trajectories are commonly used in robotics, high-precision systems, or aggressive drone maneuvers. We highlight the generalizability to other kinds of trajectories.