Modeling of UAV Tether Aerodynamics for Real-Time Simulation

TL;DR

This paper tackles the challenge of enabling truly continuous operation for UAVs by powering them from a ground tether and needing real-time tether force estimation under wind and base motion. It introduces two complementary approaches: a fast analytical catenary model with uniform drag and a more flexible CasADi/IPOPT-based numerical discretization with lumped masses, both designed for real-time performance. The analytical method achieves a mean solve time around $0.51$ ms, while the numerical method reaches about $5$ ms with warm-start initialization; real-world load-cell tests show tension estimates agree within about $0.04$ N (under 1% of total force), validating both approaches. The framework offers a lightweight, extensible tool for offline optimization and online tasks such as simulation, control, and trajectory planning, with future work aimed at reintroducing dynamic effects to improve transient accuracy without sacrificing speed.

Abstract

One of the main limitations of multirotor UAVs is their short flight time due to battery constraints. A practical solution for continuous operation is to power the drone from the ground via a tether. While this approach has been demonstrated for stationary systems, scenarios with a fast-moving base vehicle or strong wind conditions require modeling the tether forces, including aerodynamic effects. In this work, we propose two complementary approaches for real-time quasi-static tether modeling with aerodynamics. The first is an analytical method based on catenary theory with a uniform drag assumption, achieving very fast solve times below 1ms. The second is a numerical method that discretizes the tether into segments and lumped masses, solving the equilibrium equations using CasADi and IPOPT. By leveraging initialization strategies, such as warm starting and analytical initialization, real-time performance was achieved with a solve time of 5ms, while allowing for flexible force formulations. Both approaches were validated in real-world tests using a load cell to measure the tether force. The results show that the analytical method provides sufficient accuracy for most tethered UAV applications with minimal computational cost, while the numerical method offers higher flexibility and physical accuracy when required. These approaches form a lightweight and extensible framework for real-time tether simulation, applicable to both offline optimization and online tasks such as simulation, control, and trajectory planning.

Figures (9)

• Figure 1: A photo of the tethered drone showing the cable curvature.
• Figure 2: Example where the start and end points are almost vertical and there is no wind. The fixed segment length of the discrete approach limits the minimum bend radius.
• Figure 3: Comparison of solve times for the analytical approach, numerical approach with analytical initialization, numerical approach with warm start, and the numerical approach initialized with a straight line. The time to calculate the numerical initialization is not included. Note that the y-axis is on a logarithmic scale. The numerical method was run for 60 segments. It shows that the numerical approach is roughly 10 times slower than the analytical one, and using unsuitable initialization can further increase the solve time by a factor of 8.
• Figure 4: Examples of the shape and tension of different configurations solved via the analytical and numerical (CasADi) approach. \ref{['example_a']} and \ref{['example_b']} show typical cable behaviour for a tethered UAV, while \ref{['example_c']} highlights the high forces on tight cables. The comparison between \ref{['example_d']} and \ref{['example_e']}, \ref{['example_f']} highlight that the analytical solution is less accurate for large horizontal distances and strong winds. Note that the color bars are scaled differently.
• Figure 5: Comparison between the tension from the analytical and numerical (CasADi) method on GPS data from a real flight. Start force (red) on the ground, end force (blue) on the drone. The tension was measured on the drone (black). The airspeed is shown in green. It demonstrates close agreement between the analytical and numerical method. The measurements are in the same order of magnitude but show some discrepancies due to system dynamics and GPS accuracy.