Source-linked AI summary
Automatic crosswind flight of tethered wings for airborne wind energy: modeling, control design and experimental results
Lorenzo Fagiano, Aldo U. Zgraggen, Manfred Morari, Mustafa Khammash
TL;DR
The paper addresses control of tethered wings for airborne wind energy, targeting figure-eight crosswind paths without measuring wind speed at the wing. It derives and validates a simplified steering model, then uses hierarchical velocity-angle control and guidance; experiments and robustness analysis support the approach across operating conditions and wings.
Problem
Controlling tethered wings for airborne wind energy requires practical figure-eight crosswind flight despite model discrepancies, real-time optimization demands, and unavailable high-altitude wind measurements.
Method
The paper derives a first-principles control-oriented steering model and designs a hierarchical controller with a velocity-angle loop and outer guidance, using one tunable gain Kc.
Results
The control system achieved stable, satisfactory performance across wind conditions and three different wings, with Kc = 0.046 m/rad providing stability over the analyzed parameter ranges.
Takeaways & Limitations
The approach enables figure-eight crosswind control without measuring wind speed at the wing and uses few parameters whose effects can be tuned experimentally.
Takeaways & Limitations
The stability analysis does not explicitly account for actuator input limits, although the corresponding saturations were inactive during operation.
Abstract
from arXiv · showhide
An approach to control tethered wings for airborne wind energy is proposed. A fixed length of the lines is considered, and the aim of the control system is to obtain figure-eight crosswind trajectories. The proposed technique is based on the notion of the wing's "velocity angle" and, in contrast with most existing approaches, it does not require a measurement of the wind speed or of the effective wind at the wing's location. Moreover, the proposed approach features few parameters, whose effects on the system's behavior are very intuitive, hence simplifying tuning procedures. A simplified model of the steering dynamics of the wing is derived from first-principle laws, compared with experimental data and used for the control design. The control algorithm is divided into a low-level loop for the velocity angle and a high-level guidance strategy to achieve the desired flight patterns. The robustness of the inner loop is verified analytically, and the overall control system is tested experimentally on a small-scale prototype, with varying wind conditions and using different wings.
1 Introduction
Airborne wind energy systems use tethered wings or aircraft to harness high-altitude wind, with crosswind wing motion converting tether traction into ground-based electricity. Existing control approaches include advanced nonlinear methods, but practical deployment is complicated by model discrepancies, real-time optimization, and high-altitude wind measurement requirements.
- Airborne wind energy systems harness wind up to 1000 m above ground using tethered wings or aircraft.
- Tethered flexible wings fly roughly perpendicular to the wind, while ground equipment converts tether traction into electricity.
- Electricity is generated through traction and passive phases involving line unwinding, winches, generators, and motors.
- Existing approaches use offline reference trajectories, tracking MPC, adaptive control-Lyapunov methods, or economic MPC.
- Practical deployment is challenged by discrepancies between simplified models and flexible-wing dynamics, real-time nonlinear optimization, and high-altitude wind measurement.
2 System description and model equations
The system models a tethered flexible wing connected to a ground unit by fixed-length lines, using spherical position coordinates and local wind-window frames. A point-mass model describes aerodynamic and gravitational forces, while a simplified steering model links the wing’s velocity angle to control-relevant dynamics and geometric input.
- 2.1 System layout: The ground unit’s inertial frame G = (X, Y, Z) is centered at the unit, with X aligned to nominal wind and Z pointing upward.The wing position is described relative to the ground unit using spherical coordinates θ and φ.
- 2.1 System layout: The prototype uses a flexible wing connected to a ground unit by three lines, with fixed line length r = 30 m.Two steering lines influence trajectory, while the center power line sustains about 70% of the generated load.
- 2.2 Model equations: The wing’s position is parameterized by spherical coordinates θ(t) and φ(t) at fixed radius r, while the local frame L uses north, east, and down axes.The local north axis is tangent to the wind window, and local down points toward the ground unit.
- 2.2 Model equations: Applying Newton’s law yields a point-mass model driven by gravity, aerodynamic forces, and effective wind, with states θ(t), φ(t), ˙θ(t), and ˙φ(t).The aerodynamic force includes wing lift, wing drag, and line drag, with coefficients and reference areas defined in the model.
- 2.2 Model equations: The control-oriented variable is the wing’s velocity angle γ(t), introduced to obtain a simpler model than existing multivariable approaches.The model also represents the wing heading angle ξ(t) from the effective wind projected onto the local horizontal plane.
- 2.3 Input model: The overall steering input combines commanded steering-line displacement δu(t) with a geometry-dependent input δg caused by separated ground-unit attachment points.The geometric input depends on θ(t), φ(t), and attachment distance d, and the simplified steering relationship matches experimental data across several wing sizes and operating angles.
3 Control design
The proposed control system separates nonlinear guidance from simpler inner loops through three nested feedback levels. The outer loop generates figure-eight guidance from wing position using intuitive target points and filtering, without requiring wind-speed measurement.
- Overall control structure: The controller uses three nested loops: outer guidance computes γref, middle velocity-angle control tracks it through δm,ref, and inner position control commands motor current.This separates the nonlinear controller component from the linear inner controllers, enabling robustness analysis of the middle and inner loops.
- 3.1 Position control loop: The innermost loop uses a motor and linear motion system to track actuator position, with steering-line length difference determined by actuator position.The actuator position is measured using an optical rotary incremental encoder, and current saturation is included in the control scheme.
- 3.2 Velocity angle control loop: The velocity-angle loop uses proportional control with γ as feedback and δm,ref as input, while Kc is its only design parameter.The gain can be tuned using the control-oriented model and experiments, and robust stability is assessed over bounded parameter uncertainty.
- 3.2 Velocity angle control loop: A unique Kc provides robust stability across a wide range of operating conditions, with satisfactory experimental performance across experienced wind speeds and three different wings.The analysis bounds uncertain aerodynamic and flight parameters and checks stability using a quadratic-stability LMI condition.
- 3.3 Outer control loop: The outer loop switches between fixed target points P− and P+ when φ leaves [φ−, φ+], computes a target velocity angle from measured θ and φ, and filters it before feedback.The 2nd-order Butterworth filter uses cutoff frequency ωγ to shape turn sharpness; higher ωγ gives sharper turns and lower ωγ gives wider turns.
- 3.3 Outer control loop: The guidance strategy produces up-loops toward the wind-window zenith and then the opposite target, while avoiding pre-computed reference trajectories and wind-speed measurement.The interval [φ−, φ+] should be centered around the wind direction to maintain crosswind flight.
4 Experimental results
The controller was implemented on a real-time system and tested with three wings, showing robust velocity-angle control and repeatable figure-eight flight across varied wind conditions.
- Experimental setup: The experiments used 6 m2, 9 m2, and 12 m2 wings, with control-loop sampling at 100 Hz for the innermost loop and 50 Hz for the others.The controller ran on a SpeedGoat real-time machine using xPC Target for MATLAB.
- Robustness: The robustness analysis predicted stability for |v| ∈ [2, 80] m/s, Eeq ∈ [2, 8], CL ∈ [0.4, 1], A ∈ [6, 12] m2, ds ∈ [1.8, 3.1] m, and m ∈ [1.7, 3] kg.The tested parameter ranges corresponded to wind speeds from 1 to 10 m/s, and the same Kc = 0.046 m/rad was used for all wind conditions and wings.
- Control performance: The commanded actuator position was typically about 0.1 m, well below the 0.35 m saturation limit, while the velocity-angle loop tracked its reference effectively.The geometric input contributes self-steering toward the wind-window center, so feedback mainly supplies small corrective commands.
- Flight results: Automatic tests produced figure-eight paths with a 9 m2 wing, including a single path at about 2.4 m/s wind and ten consecutive paths with 30 m lines.The reported guidance parameters for these tests were θ+ = θ− = 0.35 rad, φ− = −0.2 rad, φ+ = 0.2 rad, and ωγ = 0.25 Hz.
- Flight results: The controller maintained similar, consistent flight paths for wind speeds from about 2 m/s to 6 m/s, including gusts.Lower wind speeds caused stalling in the employed wings, while higher speeds were avoided to limit stress on the prototype components.
5 Conclusions
The paper develops and validates a simplified steering model and hierarchical controller for fixed-line tethered wings flying figure-eight crosswind paths. Its experiments support robust operation without wind-speed measurements, while the force comparison remains qualitatively consistent with theory.
- Model and validation: A simplified steering model was derived from first principles and assessed against experimental data from three different wings.The model supports the subsequent feedback-controller design for figure-eight crosswind flight.
- Control approach: The controller uses three hierarchical levels and does not require wind-speed measurements at the wing’s altitude or effective wind-speed measurements.The design is based on the wing’s velocity angle and includes a robustness analysis of the inner loop.
- Experimental evaluation: The controller’s effectiveness was demonstrated through extensive experiments with a small-scale prototype and different wings.The experiments targeted figure-eight crosswind paths for airborne wind energy applications.
- Theory–experiment comparison: The measured and theoretically predicted line forces showed general qualitative consistency, with experimental variability linked mainly to uncertainty in wind-speed and aerodynamic estimates.The relevant uncertain quantities include wind speed at wing height, lift coefficient, and equivalent efficiency.