Euclidean and non-Euclidean Trajectory Optimization Approaches for Quadrotor Racing

TL;DR

This paper addresses the challenge of computing high‑fidelity racelines for quadrotor racing by presenting two geometry‑based trajectory optimization methods: a Euclidean formulation and a novel non‑Euclidean formulation that moves with the centerline. Both approaches leverage full drone dynamics, avoid gateway approximations, and support obstacle avoidance, with the non‑Euclidean method augmented by a safe flight corridor to handle dense obstacle fields. The authors demonstrate approximately 100× faster compute times than prior methods and show improved convergence, including successful obstacle avoidance on complex scenarios. The work significantly reduces planning time while maintaining dynamic feasibility, enabling more practical and scalable raceline optimization for competitive quadrotor racing.

Abstract

We present two quadrotor raceline optimization approaches which differ in using Euclidean or non-Euclidean geometry to describe vehicle position. Both approaches use high-fidelity quadrotor dynamics and avoid the need to approximate gates using waypoints. We demonstrate both approaches on simulated racetracks with realistic vehicle parameters where we demonstrate 100x faster compute time than comparable published methods and improved solver convergence. We then extend the non-Euclidean approach to compute racelines in the presence of numerous static obstacles.

Figures (19)

• Figure 1: Schematic of a non-Euclidean coordinate system for racing. For trajectory optimization the racetrack is discretized along a curvilinear coordinate, with variable time for each finite element. Lateral coordinate limits of each element shrink to avoid obstacles and pass through gates.
• Figure 2: Non-Euclidean coordinate system. A curve parameterized by $s$ is augmented by a frame that moves along this curve, with relative coordinates $y$ and $n$.
• Figure 3: Frames of reference used in our approach. Superscript $g$ denotes an inertial Euclidean frame of reference also referred to as the global frame. Superscript $b$ denotes the body-fixed Euclidean frame and $c$ denotes a non-Euclidean frame which varies along the centerline $\boldsymbol{x}^c$. Note that $\boldsymbol{x}^c(s)$ may not be an arc length parameterization of the centerline.
• Figure 4: Euclidean Approach: Phases group finite elements from each gate to the next. Gate interior constraints at either end are shrunk for the diameter of the drone. Each element of a phase is given the same, variable time duration.
• Figure 5: Non-Euclidean Approach: Finite elements are given fixed $s$ coordinate corresponding naturally to points along the centerline. Every finite element is given its own time duration and regularity constraints.

Theorems & Definitions (5)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5