Quadrotor Takeoff Trajectory Planning in a One-Dimensional Uncertain Wind-field Aided by Wind-Sensing Infrastructure

TL;DR

This work tackles takeoff trajectory planning for a quadrotor in a one-dimensional convecting wind-field with uncertain wind. It combines a Gaussian Process wind-field model, convected at speed $c$, with noisy upstream wind measurements from wind-sensing infrastructure and ordinary Kriging to estimate the wind at grid points and future times to feed a minimum-time trajectory planner implemented in GPOPS-II. The key contributions are (1) a GP-based wind-field estimation framework using networked anemometers, and (2) a minimum-time takeoff trajectory planning formulation that incorporates the wind estimate as a time-varying disturbance in the quadrotor’s vertical-plane dynamics, solved numerically. Through six simulation trials varying wind intensity, GP hyperparameters, and sensor quality, the study demonstrates that the approach can mitigate wind effects and reach a specified waypoint, with performance degrading as wind strength increases or sensor quality declines, indicating practical viability and limitations for wind-aware UAV planning.

Abstract

This paper investigates optimal takeoff trajectory planning for a quadrotor modeled with vertical-plane rigid body dynamics in an uncertain, one-dimensional wind-field. The wind-field varies horizontally and propagates across an operating region with a known fixed speed. The operating area of the quadrotor is equipped with wind-sensing infrastructure that shares noisy anemometer measurements with a centralized trajectory planner. The measurements are assimilated via Gaussian process regression to predict the wind at unsampled locations and future time instants. A minimum-time optimal control problem is formulated for the quadrotor to take off and reach a desired vertical-plane position in the presence of the predicted wind-field. The problem is solved using numerical optimal control. Several examples illustrate and compare the performance of the trajectory planner under varying wind conditions and sensing characteristics.

Figures (10)

• Figure 1: References frames and quantities used to define the three degree-of-freedom quadrotor model. The wind triangle (upper right) shows the inertial velocity ${\bm v}$ as the sum of the flow-relative velocity ${\bm v}_{{\rm r}}$ and the wind velocity ${\bm v}_\delta$. The quadrotor is drawn with pitch angle $\theta < 0$.
• Figure 2: Wind-field model where coordinate $\beta$ represents the distance from the origin $P$ in the wind frame, $c$ is the wind propagation speed at which the origin $P$ moves to the left, and $d$ is the operating region of the quadrotor. The wind-profile used for simulation (and estimation) has length $\lambda = d +ct_m$, and the horizontal coordinate $\beta \in [0, \lambda]$ is the distance from $P$ in the ${\bm p}_1$ direction.
• Figure 3: Example of the wind-field estimate evolving over the operating environment for three snapshots in time, with GP length scale $L = 1.5$ m, and GP variance $\sigma = 4 \text{ (m/s)}^2$. The red, blue, and black squares represent the anemometers in the environment, and their respective measurements are shown as stars in the same colors (at a sampling frequency of 10 Hz with measurement noise $\sigma_n^2 = 0.6$ (m/s)$^2$). The final estimate is made at time $t_N = 5$ sec. and for times $t > t_N$ the estimated wind-profile is convected downstream.
• Figure 4: Proposed framework for leveraging wind-sensing infrastructure in an optimal control solver to plan a time-optimal trajectory to reach a desired waypoint.
• Figure 5: Simulation results for Trial 1. The left panel depicts the actual and planned trajectories in the vertical plane. The middle three, and bottom right panels show the pitch angle, pitch rate, and $u_{{\rm r}}, w_{{\rm r}}$ flow-relative velocities, respectively. The upper right panel illustrates the control generated by GPOPS-II.