the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 17 Oct 2025
Translational Dynamics of Bridled Kites: A Reduced-Order Model in the Course Reference Frame
Abstract. The design and control of airborne wind energy systems requires fast, validated reduced-order models. Because aerodynamic identification of soft, bridled kites is challenging, models that minimise the number of parameters to be identified can be particularly valuable. This paper presents a reduced-order model for the translational dynamics of bridled kites, consisting of a wing supported by multiple bridle lines. The kite is modelled as a point mass in a spherical reference frame aligned with the instantaneous tangential flight direction, referred to as the course reference frame. The angle of attack follows geometrically from a constant angle between the wing chord and the bridle line system, under the assumption that the wing instantaneously aligns with the pull direction, i.e., the rotational dynamics are neglected. The formulation retains gravitational and inertial terms introduced by the curvilinear reference frame and applies a quasi-steady condition of zero path-aligned acceleration, modelling the motion as a sequence of quasi-steady (trimmed) states that relate the trim speed and angle of attack. Model validation is based on public flight datasets from two different soft-wing kites and on dynamic simulations that cover higher wing loadings. Results show that for low wing loadings typical of soft kites, the quasi-steady approximation reproduces the dynamic trajectories with less than 1 % deviation in mean reel-out power. For higher loadings and hard-wing kites, inertia introduces substantial phase lag and amplitude damping, causing power deviations of up to 14 %. Overall, the proposed model provides a computationally efficient framework for analysing the translational dynamics of bridled kites. The formulation is well-suited to trajectory optimisation, parametric studies, and control design in airborne wind energy systems.
Competing interests: At least one of the (co-)authors is a member of the editorial board of Wind Energy Science. R.S. is a co-founder of and advisor for the start-up company Kitepower B.V., which is commercially developing a 100 kW kite power system and provided their test data used in this paper for validation. Both authors were financially supported by the European Union’s MERIDIONAL project, which also provided funding for Kitepower B.V..
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Status: final response (author comments only)
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RC1: 'Comment on wes-2025-205', Anonymous Referee #1, 04 Nov 2025
- AC1: 'Reply on RC1', Oriol Cayon, 16 Jan 2026
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RC2: 'Comment on wes-2025-205', Anonymous Referee #2, 17 Nov 2025
General comments
In the manuscript ‘Translational Dynamics of Bridled Kites: A Reduced-Order Model in the Course Reference Frame’, the authors present a reduced-order model for soft kites and validate it using existing models and data. The study is situated within the broader set of soft-kite modeling approaches, providing a quasi-steady formulation suitable for control and optimization, and it is supported by satisfactory results and explanations.
The two major comments/corrections required are the following:- Subsection 2.1, Longitudinal static stability and trim condition, should be revised to improve clarity. As outlined in the following ‘Specific comments’ and ‘Technical corrections’, the authors are expected to clarify the definitions of the variables, the figure, and some of the equations. Enhancing the flow and depth of this subsection will improve the overall understanding of the physical principles and assumptions underlying the manuscript.
- Subsection 6.3, Dynamic response in crosswind trajectories: The authors introduce here a comparison between soft and rigid wings, while the previous sections are entirely devoted to soft-wing kites and their physics. The comparison and its motivations are clear; however, the lack of a clear description of the rigid-wing case makes this section somewhat unclear for readers without expertise in the field. I suggest improving clarity by adding a more in-depth description of the rigid-wing principles and of the variables/equations needed for the comparison presented here. Alternatively, the authors should consider removing this parallel analysis.
Specific comments
Subsection 2.1 Longitudinal static stability and trim condition:
- Figure 2: Neither the caption nor the text describe all the variables in the drawing. In this section, you don't mention theta_k, r and F_b. Moreover, theta_b should also be represented in the figure. I don't understand, and it is not explained in the text, why the front chord line is perpendicular to the chord line when theta_d is zero. Please adjust the Figure and the caption coherently.
Subsection 4.1 Definition and assumptions:
- Figure 6, caption: Which angle of attack is used for the left image? And which wind velocity? And which velocity is used for the right plot? Moreover, the scale of the acceleration is very large (hundreds of times the gravitation acceleration). This is puzzling me because the difference between the orange and the light blue curve should exactly be equal to g = 9.81 m/s^2, while here it seems equal to 1000 m/s^2, which seems 2 orders of magnitude larger.
Subsection 6.3 Dynamic response in crosswind trajectories:
- Line 515-518: In the rigid wing case, how do you compute the pitch angle and thus the angle of attack? it is not clear. Moreover, how do you choose the trajectory radius? That would influence the results. You can have a look at Figure 10 of "Trimming a fixed-wing airborne wind system for coordinated circular flights" https://doi.org/10.5194/wes-2025-193 or previous works showing that relation.
- Line 520: This is an interesting conclusion. To strengthen it, you could compare your results with the Loyd mode approximation given in eq. 85 of Trevisi et al. "Flight stability of rigid wing AWES" https://doi.org/10.3390/en14227704 (or with Eq. 87 if you are looking at the linear behavior about the quasi-steady solution). The Loyd mode is indeed approximating the first order differential equation modelling the tendency of AWES to return back to the quasi-steady solution. Note that, since the eigenvalue of the differential equation is function of the mass, it should capture the difference between wing loadings.
Technical corrections
Line 77 "about the bridle point (B)”: Please introduce the Figure 2 before you refer to it.
Line 79: Are x_{cp} and CP (in Figure 2 and line 87) the same point? If yes, please use the same notation.
Line 94 "a constant geometric angle theta_b”: Is this theta_b or lambda_b? If theta_b, the variable is never introduced before, and it is not clear why it is approximated as constant.
Line 102: It is hard to understand the physical reason behind Equations (1) and (2). Please provide a step-by-step explanation.
Line 137: Please define the variables in Equation (3) that are introduced for the first time, before or after the equation itself (e.g. "... and yaw rate psi dot (Erhard..."). Moreover, v_a is shown in Figure 2, but never officially defined.
Figure 5 and line 255: Please describe in the caption the variables introduced for the first time. After the Figure, F_{t,g} appears for the first time in line 255 but it is never explicitly cited in the text. Maybe it is worthy to define it, since it is used also in the subsequent paragraphs. Minor correction: choose between F_{t,g} and F_{tg} for coherence, if you want to indicate the same variable.
Line 240: For the crossflow principle and neglecting viscous terms, only the component of apparent velocity which is perpendicular to the cylinder axis generates aerodynamic drag. These equations (24 to 26) seem to not account for this effect and might lead to an overestimation of the drag acting on the tether.
Line 270 "crosses zero at two points”: In Figure 6, you show only one point. Please explain why two points.
Lines 320-322: The control input u_p is not mentioned here. Was it intentional? Please comment on this.
Line 347 "a validated lifting-line-based model”: In the paragraph before you mention that lifting-line methods and CFD can hardly be used to capture the aerodynamic behavior of these systems. After this discussion, the term "validated" for the lifting line method seems somehow out of place. Could you add a small sentence on how the validation was carried out in the two cited works?
Line 364: Why do you cite here an “onboard turbine”? Aren’t we referring to soft kites?
Line 412 "assumption of straight flight”: Do you mean ‘straight upward’?
Line 436, Equation 44: Do you mean ½ instead of 2? Maybe it is worth adding a small derivation of this equation. Shouldn't v_{r,opt} actually be v_w cosϕcosβ - v_{r,opt}?
Figure 10: You mention in the text the reeling factor, and you show it in the Figure, but this variable was never defined. Can you define what do you mean using this term?
Line 467: Please cite here the two different types of delta_{phi} that you are using, even if the equations and figure suggest that you are differentiating them for circular and figure-eight paths.
Line 487: Missing sentence.
Line 542 "As the kite moves with": Do you mean “As the kite move in”?
Citation: https://doi.org/10.5194/wes-2025-205-RC2 - AC2: 'Reply on RC2', Oriol Cayon, 16 Jan 2026
Data sets
Kite power flight data acquired on 8 October 2019 M. Schelbergen et al. https://doi.org/10.4121/19376174.V1
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Relevant problem setting, good method and strong results.
Minor comments:
* When theta_b is introduced on line 94, the formula is not yet clear, that only comes on line 104. I recommend making a forward reference to Equation (2) and adding theta_b in Figure 2.
* All values in Equation (3) should be defined below it.
* On line 174, "equations of obtained" should be "equations obtained"
* The radial velocity is also set to zero "for practical implementation" on line 296. I understand that this is not used as equilibrium condition, but it is strange to mention this so late in the process, after emphasizing that only the tangential acceleration is set to zero.
* On line 383, the gaps in the curves of Figure 8 are explained, but this could be explained in more detail. What is observed if the model is applied to these situations?
* Missing text on line 487.