Fast reliable tether force calculation for drones in real time
Real-Time Bounded Catenary Solver for UAV Tether Modeling
Robotics
Summary
Tethered drones experience forces from the cable that connects them while flying. Simulating these forces quickly and accurately helps control the drone better. The authors improved the math and method used to find these forces, making it much faster and guaranteed to work every time. Their new solver runs consistently fast enough to be used on the drone itself, ensuring smooth flight control even on limited hardware.
What this means in practice
- •For drone control engineers: Implement real-time tether force models directly on flight controllers to improve drone stability and control.
- •For embedded systems developers: Deploy fast, reliable mathematical solvers for physical cable dynamics in resource-limited hardware environments.
Authors
Max Beffert, Andreas Zell
Abstract
For non-stationary tethered multirotor UAVs in real-world conditions, simulating the forces imposed on the drone by the aerodynamic drag of the tether becomes crucial, with online use cases placing a hard bound on the maximum solve time. In previous work, a quasi-analytical catenary tether model reached a mean solve time of 0.51 ms using a general-purpose root finder, but without any worst-case guarantees or proven convergence. In this work, we reformulate the inner solver by reducing the catenary boundary-value problem to a single transcendental equation in one well-conditioned unknown. We derive a closed-form bracket and prove monotonicity and convexity as well as existence and uniqueness of the root, which together guarantee convergence of the solver. We further propose a two-regime initial guess which approximates the true root within 3.4% and reduces the mean iteration count by 68.0% to 2.36 compared to the textbook initialization. Building on the hybrid root-finding method rtsafe (Newton-Raphson with bisection fallback giving bounded iteration counts), we implement a specialized variant that exploits the problem structure to omit unnecessary checks while retaining correctness, which gives up to 1.3 times speedup. With the proposed solver the full tether model achieves a nearly constant solve time of 6.9 us on average and 7.7 us at worst, a 40 times speedup over an optimized re-implementation of the previous method, while agreeing with it to a relative deviation of 8.7e-9. Because the reformulation leaves the underlying physical model untouched, the experimental validation of the previous work carries over unchanged. We further demonstrate its suitability for embedded, resource-constrained platforms with a Lua implementation running directly in ArduPilot on a drone's flight controller, where it stays well inside the scheduling budget with a mean solve time of 0.74 ms.