Efficient Collision-Avoidance Constraints for Ellipsoidal Obstacles in Optimal Control: Application to Path-Following MPC and UAVs

TL;DR

The paper presents a modular optimal-control framework for collision avoidance with ellipsoidal obstacles in 3D, applied to model predictive path-following control for UAVs. It introduces a differentiable ellipsoidal collision test based on an auxiliary ellipsoid with a parameter $\lambda \in [0,1]$ and the condition $K(\lambda)<0$, linking to Minkowski-sum interpretations through a tight over-approximation. To address numerical hardness in the resulting OCP, a two-stage optimization is employed: first solve for a fixed $\bar{\lambda}$ by minimizing $K(\bar{\lambda}_{k|t},\mathbf{x}_{k|t})$, then solve the OCP with these fixed parameters, iterating as needed. This approach enables real-time performance in 3D, and is experimentally validated on the Crazyflie 2.1 UAV, including a moving obstacle scenario, marking the first real-time hardware MPC demonstration of this kind in three dimensions. The work advances practical collision avoidance for UAVs by combining differentiable collision tests with a robust, staged optimization strategy.

Abstract

This article proposes a modular optimal control framework for local three-dimensional ellipsoidal obstacle avoidance, exemplarily applied to model predictive path-following control. Static as well as moving obstacles are considered. Central to the approach is a computationally efficient and continuously differentiable condition for detecting collisions with ellipsoidal obstacles. A novel two-stage optimization approach mitigates numerical issues arising from the structure of the resulting optimal control problem. The effectiveness of the approach is demonstrated through simulations and real-world experiments with the Crazyflie quadrotor. This represents the first hardware demonstration of an MPC controller of this kind for UAVs in a three-dimensional task.

Figures (8)

• Figure 1: Illustration of the function $K(\lambda)$ for different positions of a robot (color coded) relative to the obstacle (green dashed outline).
• Figure 2: Crazyflie 2.1 quadrotor with OptiTrack markers for position and attitude estimation during real-world experiments.
• Figure 3: Crazyflie (orange) navigating through a complex environment with a priori known obstacles (grey) and a locally detected, a priori unknown ellipsoidal obstacle (green). The blue dashed lines indicate the reference path $\mathcal{P}$.
• Figure 4: Comparison of two different values $\hat{\lambda}$ in the collision-avoidance test.
• Figure 5: Results of the real-world experiment. Visualized for different time points in \ref{['fig:2DPath']}. A snapshot of the Crazyflie during the experiment is shown in \ref{['fig:2DPathExperiment']}.

Theorems & Definitions (2)

• Remark 1
• Remark 2