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Control of Towing Kites for Seagoing Vessels

Michael Erhard, Hans Strauch

arXiv:1202.3641v3eess.SYcs.SY

TL;DR

The paper addresses flight control for a towing kite operating in a highly dynamic regime. It develops a reduced model and cascaded controller, with sea-trial data supporting the turn-rate law and controller performance. It also describes flight-pattern generation, while treating ship motion through an apparent-wind approximation.

  • Problem

    Air-path speed can vary by a factor of 3–5 within seconds and deflection commands can change by more than 60%, complicating controller design for the highly dynamic flight regime.

    Source: §V · p. 5

  • Method

    The paper introduces a three-state controller-design model and separates guidance from control in a cascaded, model-following controller.

    Source: §III · p. 4, §V · p. 5

  • Results

    ±60% of available deflection range is spanned by feedforward during repeating patterns, while feedback hardly needs to correct errors; measured yaw rate also correlates excellently with steering command.

    Source: §IV · p. 4, §VI · p. 7

  • Takeaways & Limitations

    Curve deflections are typically chosen in the 40–70% range of total deflection to minimize curve duration and optimize the performance figure.

    Source: §VIII · p. 8

    The ship is excluded from the treatment, with apparent wind used as an approximation for basic control-system design.

    Source: §IX · p. 9

Abstract

from arXiv · show

In this paper we present the basic features of the flight control of the SkySails towing kite system. After introduction of coordinate definitions and basic system dynamics we introduce a novel model used for controller design and justify its main dynamics with results from system identification based on numerous sea trials. We then present the controller design which we successfully use for operational flights for several years. Finally we explain the generation of dynamical flight patterns.

I. INTRODUCTION

SkySails uses a towing kite to harness wind for ship propulsion, with its latest 320 m2 product generation able to replace up to 2 MW of engine propulsion. The introduction outlines the system components and flight-control setup, then previews the paper’s model, controller, and pattern-generation topics.

I. INTRODUCTION

  • 320 m2 kites can replace up to 2 MW of the main engine’s propulsion power in the latest product generation.The system is presented as a way for cargo ships to use wind power and reduce fuel use, costs, and emissions.Source: §I · p. 1
  • The towing kite is steered by a control pod, and a high-strength synthetic-fiber rope transmits towing force to the ship.A launch and recovery system is installed aboard the ship.Source: §I · p. 1
  • Distributed computers preprocess sensor data at 10 Hz and calculate the steering command sent to the main actuator.Source: §I · p. 1
  • The paper proceeds from system and coordinate definitions to a controller-design model, experimental justification, controller performance, and flight-pattern generation.Source: §I · p. 1

II. BASIC SYSTEM AND COORDINATES

The section describes coordinates and a simplified controller-design model for a constrained kite. A gravity-free symmetry makes the motion equations simpler, supporting largely analytical or semi-analytical feedback and guidance design and straightforward pattern description.

II. BASIC SYSTEM AND COORDINATES

  • The simplified setup substitutes a rigid rod parallel to the kite yaw axis for the flexible rope and neglects all masses.The authors motivate these assumptions by the large line forces during dynamic pattern flight; gust- or wave-related free-flight cases are infrequent and detailed treatment is outside the paper’s scope.Source: §II · p. 2
  • In the gravity-free space-station example, constant steering produces a circular orbit whose smaller diameter means faster flight and greater force.The orbit diameter depends on ψ, which is determined by commanded deflections.Source: §II · p. 2
  • The gravity-free design model gains a symmetry axis that simplifies the equations of motion and supports largely analytical or semi-analytical feedback and guidance design.The model is specifically for controller design; a more sophisticated multibody simulator is used for other development and test purposes.Source: §II · pp. 2–3
  • The controller structure developed from this model worked effectively in numerous sea trials.Source: §II · p. 3
  • For constant line length, the kite state is represented by position angles ϕ and ϑ and orientation angle ψ.The kite axes are designated roll, pitch, and yaw.Source: §II · p. 3

III. SYSTEM DYNAMICS USED FOR DESIGN

The section develops a controller-design model with three state variables and simplified equations of motion. It interprets steady flight and steering dynamics, including how ψ controls flight conditions and how steering deflection sets yaw rate.

III. SYSTEM DYNAMICS USED FOR DESIGN

  • For constant ψ, the kite follows a circle at constant ϑ; ψ also tunes air-path speed and thus the force proportional to v_a^2.At ψ = 0, neutral flight has low force and ϕ can set the flight position.Source: §III · p. 3, Figure 5 · p. 3
  • The proposed steering law makes measured yaw rate proportional to air-path speed and steering deflection, and serves as the cascaded controller’s inner-loop dynamics.The paper says the law can be justified by measured data to a high degree.Source: §III · p. 3
  • Pod-sensor yaw rate differs from coordinate yaw rate by a term involving ϕ motion, which is usually small in the stated flight situations.The authors treat that term as a later controller-design correction.Source: §III · pp. 3–4
  • The novel model uses three state variables and three equations of motion, versus at least four state variables in earlier published models.It neglects acceleration and gravity effects to obtain simple equations suited to controller design.Source: §III · pp. 3–4

IV. JUSTIFICATION OF THE TURN RATE LAW

Bang-bang flights provide a way to vary air-path speed within a single experiment and test the proposed turn-rate law. Measurements support the law across repeated runs, while the authors extend it with a mass term and describe its role in controller design.

IV. JUSTIFICATION OF THE TURN RATE LAW

  • Bang-bang steering alternates ±δ0 at yaw-angle thresholds, producing a figure-eight whose size increases as the kite descends to smaller elevation angles.The human pilot supervises the flight and can take over before overload or water-surface contact.Source: §IV · p. 4
  • Repeated experiments with different δ0 produced similar g values, and time-series measurements showed excellent correlation between steering command and yaw rate divided by g v_a despite perturbations.The measured yaw rate increased with air-path speed during the experiment, supporting the proportional law’s use in the controller.Source: §IV · p. 4, Figure 7 · p. 4, Figure 8 · p. 5
  • Bang-bang experiments vary air-path speed across a range in one short flight, making the proportional law clearer than batch runs across changing conditions.The earlier batch runs were affected by wind gusts, environmental changes, and drifting kite properties.Source: §IV · pp. 4–5
  • The extended turn-rate law adds a gravity-related mass term; at typical dynamic-flight operating points it is neglected in linear feedback and handled through feedforward correction.The mass term represents an unstable effect that requires active stabilization of ψ_m.Source: §IV · p. 5
  • During autopilot dynamic flights, online recursive least squares estimates g and M so parameter changes can be monitored and controller parameters adapted if necessary.The authors say parameter changes may indicate upcoming material failures in advance.Source: §IV · p. 5

V. CONTROLLER DESIGN

The kite’s rapidly varying flight dynamics and modeling difficulties make a single trajectory-to-deflection controller insufficiently robust. The design separates guidance from control and uses a cascaded, model-following controller.

V. CONTROLLER DESIGN

  • Air-path speed can vary by a factor of 3–5 within seconds, while deflection commands can change by more than 60%, complicating robust control.The authors report that a single controller mapping trajectory set-points directly to deflection lacked robustness because of modeling issues.Source: §V · p. 5
  • The overall task is split into guidance, which computes ψs, and control, which tracks that command through a cascaded model-following structure.The cascade reflects the separation between deflection-to-yaw-rate dynamics and yaw-rate-to-angle kinematics.Source: §V · pp. 5–6

VI. CONTROLLER INNER LOOP

The inner loop controls yaw rate with feedforward and feedback, accounting for speed-dependent plant gain, actuator limits, and loop delays. In the illustrated flights, feedforward supplies most steering demand while feedback corrections remain small.

VI. CONTROLLER INNER LOOP

  • The deflection-to-yaw-rate plant is proportional, but its gain K ˙ψ = g va varies with air-path speed.The feedforward term computes the deflection needed for the commanded rate; feedback corrects residual error from disturbances or unmodeled dynamics.Source: §VI · p. 6
  • The feedforward block inverts the turn-rate law and uses a steering-pod model with rate and command limiting, while accounting for loop delays.About 60% of the deflection range is reserved for feedforward and the remaining 40% for feedback.Source: §VI · p. 6, Figure 10 · p. 6
  • Yaw-rate feedback uses a PI controller whose output is divided by K ˙ψ to account for velocity dependence.A low-pass filter is added for measurement-noise rejection.Source: §VI · p. 7, Figure 11
  • Feedforward deflection moves between ±60% of the available range, while feedback hardly needs to correct errors from unmodeled plant dynamics.The figure presents repeating flight patterns and is described as further evidence of the turn-rate law’s good fit.Source: §VI · p. 7, Figure 12 · p. 7

VII. CONTROLLER OUTER LOOP

The outer loop tracks the commanded kite angle using proportional feedback and feedforward shaping. Its internal steering-pod model turns even jump-like angle commands into flyable references and rate patterns.

VII. CONTROLLER OUTER LOOP

  • Because the outer-loop plant is integrating, proportional feedback augmented by a low-pass filter and limiter is sufficient.The feedforward block includes an internal feedback loop that models the steering pod.Source: §VII · p. 7, Figure 14 · p. 7
  • The internal feedback loop shapes ψs into a flyable ψc and supplies the corresponding required rate pattern to the inner loop.The shaping uses the steering-pod model and limiters to keep the commanded evolution within the response capabilities of the pod and kite.Source: §VII · p. 7
  • Separating deflection-to-rate dynamics from rate-to-angle kinematics lets limiters shape commands to the constraints of the software-to-kite chain.The controller’s feedforward/feedback separation also decouples nonlinear elements, such as the mass term, from feedback design.Source: §VII · pp. 7–8
  • With fast response handled by feedforward, closed-loop bandwidth selection for the two feedback loops focuses more on stability margin than response speed.The feedback loops retain only proportional or integral dynamics that can be handled using classical linear control theory.Source: §VII · p. 8

VIII. PATTERN GENERATION

Pattern generation alternates the commanded yaw direction at selected points to produce figure-eight flight, with the controller shaping the commands into flyable references. Flight tests illustrate the pattern and its high air-path speed relative to measured wind.

VIII. PATTERN GENERATION

  • The figure-eight alternates between ψs = +ψ0 and ψs = −ψ0 when the kite reaches the two specified ϕ thresholds.The resulting sequence is shaped by the feedforward block into ψc.Source: §VIII · p. 8, Figure 17 · p. 8
  • The controller accepts square-wave ψs commands and shapes them into ψc using its internal model of the control pod.The model uses the pod’s curve deflection and design speed; the optimal curve deflection or turning radius is outside the paper’s scope.Source: §VIII · p. 8
  • Curve deflections are typically set to 40–70% of the total range to minimize curve duration and optimize the performance figure.The authors state that selecting an optimal deflection or turning radius depends on geometric and aerodynamic characteristics.Source: §VIII · p. 8
  • The pattern-size parameter ϕa and center parameters ϕ0 and ϑ0 define the trajectory; ψ0 tunes air-path speed and thus force.ϕ0 can be selected within a range to optimize the force component pointing forward along the vessel.Source: §VIII · p. 8
  • An outer-loop force controller evaluates force peaks during figure-eight flight and feeds back a value for ψ0.This approach is used because predicting the exact wind situation at flight altitude is cumbersome.Source: §VIII · p. 8
  • With a 160 m2 kite on a 300 m towing line, the tested dynamical flight mode produced air-path speed 3–4 times the estimated wind speed at flight altitude.That estimated wind speed was significantly higher than the wind speed measured aboard the ship.Source: Figure 18 · p. 9

IX. DISCUSSION AND FURTHER CHALLENGES

The discussion identifies omitted ship and environmental effects, sensor-processing dependencies, and operational settings that constrain the controller’s scope. It also notes behavior in gust-driven free flight and leaves stall response as ongoing work.

IX. DISCUSSION AND FURTHER CHALLENGES

  • The treatment excludes the ship, using apparent wind as an approximation; pattern-parameter choice, towing-force optimization, and wave effects are not presented as operational-autopilot improvements.The omitted topics are identified as possible improvements to an operational kite autopilot.Source: §IX · p. 9
  • Sensor setup and preprocessing contribute significantly to controller performance and robustness, while ψ estimation uses inertial navigation and flight-altitude wind estimates that can vary over minutes.Further discussion of these technical topics is deferred to future publications.Source: §IX · p. 9
  • The analysis assumes constant, long line lengths; launch and recovery at shorter lengths use analytically computed receding-horizon feedback to provide ψs.Winching introduces additional dynamic effects treated as correction terms.Source: §IX · p. 9
  • During gust-driven free flight, the kite’s kinematics change, yet the chosen coordinates keep controller inputs favorable and internal-mode disturbances are effectively suppressed despite being absent from the design model.The passage characterizes free flight as an exceptional occurrence for the usually constrained system.Source: §IX · p. 9
  • Stall avoidance and response remain a challenge, with systematic experimental tests described as hardly feasible and the topic still under research and development.This is the section’s stated ongoing challenge.Source: §IX · p. 9

X. SUMMARY

The paper presents a reduced dynamical model for constrained kites, uses it to develop and test a cascaded controller, and outlines flight-pattern generation. It states that the underlying equations extend beyond the SkySails system to constrained-kite dynamics generally.

X. SUMMARY

  • The controller-design model reduces constrained-kite aerodynamics to a law involving one or two quantities, and flight data support that this reduction describes reality to a high degree.The paper presents the simplified dynamics as a basis for controller design.Source: §X · p. 9
  • A cascaded controller based on the model-following principle was developed, and flight-test data are reported as demonstrating its effectiveness.The paper describes its operational use over several years in the abstract.Source: §X · p. 9
  • The paper gives the main principles of flight-pattern generation and states that its underlying equations are commonly valid for constrained-kite dynamics beyond SkySails.It identifies kite-based electricity generation as an upcoming application area.Source: §X · pp. 9–10

APPENDIX A DERIVATION OF SYSTEM DYNAMICS

The appendix derives system dynamics using coordinate and airflow relations, with simplifying assumptions including neglected gravity. It relates airflow to external wind and kite motion, and invokes a glide-ratio condition in describing the aerodynamics.

APPENDIX A DERIVATION OF SYSTEM DYNAMICS

  • Neglecting gravity makes the problem independent of ϕ, permitting analysis at ϕ = 0 without loss of generality.The passage states that this yields the associated basis vectors.Source: Appendix A · p. 10
  • The airflow vector is described as the sum of external wind and components due to roll and pitch motion relative to their basis vectors.These kinematic terms are identified as contributions to the flying system’s airflow.Source: Appendix A · p. 10
  • The aerodynamic simplification assumes airflow lies in the roll-yaw plane and its direction there is given by the glide ratio E = L/D.The cited passage introduces these as two conditions based on a simple airfoil model.Source: Appendix A · p. 10
  • The appendix uses geometric and kinematic relations to derive the kite’s angular dynamics, including a differential equation for ϑ and a motion equation for ϕ.The displayed ϕ equation relates its rate to roll- and pitch-direction velocities and ϑ.Source: Appendix A · p. 10

APPENDIX B EQUATIONS OF MOTION

Appendix B summarizes the motion equations and explains how airflow speed enters them, distinguishing direct measurement in operation from estimation in simulation. Appendix C derives a feedforward shaping relation that limits initial steering deflection to avoid overshoot under stated assumptions.

APPENDIX B EQUATIONS OF MOTION

  • In the motion equations, airflow speed va depends on ambient wind speed v0 and ϑ, but operational control can measure va directly to avoid estimation inaccuracies.The passage notes that simulation instead computes va from v0.Source: Appendix B · pp. 10–11

APPENDIX C FUNCTION f(x) FOR ψ-FEEDFORWARD BLOCK

  • Appendix C derives f(x) for the ψ-feedforward block by relating an initial deflection δi, steered to zero at constant velocity, to the resulting change in ψ.The derivation integrates the modeled change in ψ during the deflection ramp.Source: Appendix C · p. 11
  • For a given ψ deviation from set point, the derived δi is the maximum model deflection that avoids overshoot under constant ψs and K ψ̇ assumptions.This δi is used as an input to the model-shaping limiter.Source: Appendix C · p. 11
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