Flight path optimization

[ufechner7]

I don't have a good plan how to calculate an optimized flight path. Depending on the system configuration and wind speed there are different flight paths to optimize. A quite simple, but useful optimization would be the optimization of a figure of eight for pulling ships at low ship speeds. Questions:

• which model should we use for such an optimization? Our full model with 70 or more states, or a simplified model?
• which input parameters are needed? Idea: Wind speed, kite mass, tether length... What about wind sheer and wind veer?
• what about constraints, like the max speed of the steering controller? Max acceleration of the kite? Max tether force?
• what should be optimized in this scenario? I guess the mean tether force. In addition, the steering power should be minimized.

It would be helpful to define the minimal useful set of features of such an optimizer, and then implement it.

[1-Bart-1]

This optimisation problem can be solved with MPC. Due to the long predict horizon that is necessary, you need to use a simple model. You can take a look at AWEBox, they solve a flight path optimisation problem for rigid wing systems using MPC, with a simple model. ModelPredictiveControl.jl can be used when implementing this in Julia.

at low ship speeds

The optimisation problem doesn't have to take into account the ship speed, as you can just change the apparent wind speed and direction. The difference in the problem for a ship vs other AWE systems is that you want to maximize force in a certain direction, not just total power.

which input parameters are needed?

I would start with using the same states as they use in the simple model in AWEBox.

What about wind sheer and wind veer?

Wind sheer/veer isn't so important, as long as you consider the wind at the height of the kite. The tether drag is relatively small to the aerodynamic forces at the kite.

what about constraints

Constraints can be added easily when you use MPC.

what should be optimized in this scenario?

The work over a cycle, W = F * d. I am not completely sure here, this should be checked. The force would be the component of the winch force that points forwards along the ships direction of motion. The distance d = v * t, and if you assume a constant velocity you get W = F * v * t where v is constant, t is also constant (the time of one power cycle), so you can simply optimize the average force component in the forwards direction of the ship during the last power cycle.

p.s. A concern for this optimization problem is that the VSM underestimates drag, leading to very fast-flying kites which would not be realistic. So this would have to be investigated when we have such an optimization in place.

[1-Bart-1]

A different method would be to parameterize the shape of the kite path, and to create a tuning script that tunes the parameters of the flight path by flying the kite for some cycles. This has to be combined with gain scheduling, to find optimized flight paths for different wind directions and strengths.

[KenzaKHEDACHE]

Very relevant discussion regarding optimized flight path calculation for kite-assisted ship traction. I would like to share some elements based on both literature and my experience in this domain.

1. Choice of Dynamic Model:
   In the literature, most trajectory optimization studies for kites rely on simplified models, primarily the point-mass model, which assumes the kite is a single point subject to aerodynamic forces (see Fagiano et al., 2010; Houska et al., 2010). In some cases, even more simplified zero-mass models have been considered for quasi-static approximations.
   That said, there is room for improvement. Simplified rigid-body models, introducing rotational dynamics while maintaining reasonable complexity, could be a good compromise to better capture the kite's behavior, especially for large kites or when structural effects cannot be neglected. The choice depends on the required accuracy versus the acceptable computational cost, which is always a trade-off in trajectory optimization.

2. Optimization Algorithms and Tools:
   Regarding the numerical resolution, many different frameworks and algorithms have been successfully applied in the literature:

• Direct multiple shooting, collocation methods, and direct transcription are widely used.
• Solvers such as IPOPT, SNOPT, or ACADO are frequently chosen for solving the resulting nonlinear programming problems.
• Open-source tools like CasADi have become very popular due to their flexibility and symbolic differentiation capabilities.
• Other academic or proprietary toolboxes have been developed, for example within GPOPS, ICLOCS, MUSCOD, among others.

The specific choice depends on problem complexity, real-time requirements, and available computational resources. For offline trajectory optimization, a wide range of methods are applicable.

3. Parameters and Constraints:
   As for input parameters, classical ones include:

• Wind speed and direction : Wind shear is most of the time considered in the literature, as it adds realism without significantly complicating the model.
  Including wind veer would be a very interesting extension. It adds an extra level of realism to the wind field representation and could influence the optimal trajectory, especially in offshore or complex environments.
• Kite mass, aerodynamic coefficients, and tether length,
• Ship speed and heading.

Regarding constraints, realistic limitations must be considered:

• Maximum tether force (to avoid structural failure),
• Maximum steering rates and kite acceleration (actuator limits and safety),
• Possibly energy or power limitations for onboard systems.

4. Cost Function and Objectives:
   In traction applications for ships, the mean tether force projected along the ship's longitudinal axis (X-axis) is traditionally used as the primary objective, as it directly relates to the pulling force transmitted by the kite system. However, your point regarding minimizing the steering power consumption is extremely relevant and often underestimated. Onboard power is limited, and excessive steering demands can reduce overall system efficiency. Therefore, a multi-objective formulation combining traction maximization and steering power minimization would be a realistic and energy-efficient approach.

5. Conclusion:
   An effective and scientifically grounded optimization strategy would therefore be based on:

• A simplified but representative dynamic model (point-mass, or rigid-body with reasonable complexity),
• Inclusion of relevant constraints and physical limits,
• A flexible optimization framework (CasADi, ACADO, or others),
• A multi-objective formulation considering both traction maximization and energy minimization.

Such an approach is well aligned with both the current state-of-the-art and practical system limitations.

Happy to hear any suggestions or additional thoughts on this topic.

[1-Bart-1]

Julia has it's own, well-established optimization framework called JuMP.jl. ModelPredictiveControl.jl uses JuMP under the hood. So I would suggest using these packages, as we have most of our code in Julia already, and their performance is very good.

[ufechner7]

Thanks for your detailed reply!

Some of my current thoughts:

1. **Choice of Dynamic Model**:

I don't think a point-mass model is a good choice. For example the tether drag can be significant, in particular if you have more than one line. I would like to leverage one of our existing models like KPS4 or KPS5, preferably with a quasy-steady tether (to reduce the number of states) and without using the VSM model, a two or three plate model should be sufficient. With this kind of model we can achieve 100x real-time speed, so a 100s flight (one figure of eight plus initialization) can be simulated in 1s using one core.

2. **Optimization Algorithms and Tools**:
* Direct multiple shooting, collocation methods, and direct transcription are widely used.

I don't know much about these methods, must read about them.

* Solvers such as IPOPT, SNOPT, or ACADO are frequently chosen for solving the resulting nonlinear programming problems.

This toolbox looks promising: https://github.com/control-toolbox/OptimalControl.jl . I met the author, prof. Jean-Baptiste-Caillau from the Université de Cote-D'azur on the JuliaCon in Eindhoven last year. He is also supervising Antonin Bavoil who also gave a talk on the WESC conference. This package comes with very nice tutorials, e.g.: https://control-toolbox.org/Tutorials.jl/stable/tutorial-mpc.html

* Open-source tools like CasADi have become very popular due to their flexibility and symbolic differentiation capabilities.

While CasADI is quite powerful, it is also quite limited compared to similar Julia tools, see https://docs.sciml.ai/ModelingToolkit/stable/comparison/#Comparison-Against-CASADI

3. **Parameters and Constraints:**

To start with, I would use:

• wind speed and direction
• the ship speed and direction
  Well, I think you can combine them to the relative speed and direction, to sure, though.
  And as you wrote

* Kite mass, aerodynamic coefficients, and tether length

The inertia of the kite is also relevant (but constant).

Regarding constraints, realistic limitations must be considered:
I agree with your suggestions.

4. Cost Function and Objectives:

I agree.

5. Conclusion:

In my PhD thesis I optimized the figure of eight manually using three parameters:

• average elevation angle
• width of the figure of eight
• height of the figure of eight

As a next step I would like to implement a similar approach, but using more parameters and automated
optimization. Just using a parameterizable figure of eight with perhaps eight parameters. The https://github.com/bbopt/NOMAD.jl solver should be able to find a good solution for 8 parameters using 1000 simulations. So finding a result should be possible using a model with about 20 states within 30 minutes.

[1-Bart-1]

I would like to leverage one of our existing models like KPS4 or KPS5

It would be nice to put some effort into making the optimization modular enough such that it will work with a system with multiple tethers as well, so we can easily switch to using different models, and such that it will work with the SymbolicAWEModel.

[ufechner7]

It would be nice to put some effort into making the optimization modular enough such that it will work with a system with multiple tethers as well, so we can easily switch to using different models, and such that it will work with the SymbolicAWEModel.

Well, the optimization of one figure of eight should work with any model that is supported by KiteModels.jl and by SimpleKiteControllers.jl .

[baptistelabat]

Hi, I am just rising a few questions on assumptions:
If we consider ships with high fraction of kite power, we might have to consider both kite and ship trajectory. The boat is typically following the kite as it moves to avoid huge lateral forces. This would allow to deal with manoeuvres as well (but adds complexity on ship model). A simpler approach valid for lower acceleration of ship with kite would be to consider a constant boat course and speed.
Considering only apparent wind speed (AWS@10m) and angle (AWA@10m) might be an option if we neglect the power used for steering (relative to ship extra power due to kite) but also won't allow to consider wind gradient. Wind gradient impact is low when going upwind but might be large when going downwind. Seakite was said to have headwind at boat height while kite was downwind up in the sky. With small kiteboats we also notice that we have either to fly the kite high in the sky or fast close to ground (no slow quasi static flight close to ground) when downwind. I think Eckman spiral (wind veer) can be neglected at small scale but don't know for kilometer long lines.

[1-Bart-1]

we might have to consider both kite and ship trajectory.

We are planning on making some kind of dynamical ship model for exactly this purpose.

[ufechner7]

I started with some experiments at https://github.com/OpenSourceAWE/OptimalFlightPaths.jl

[ocayon]

Hi all,

Thanks for the stimulating discussion — I’d like to share a few points based on my current work and related modeling efforts.

1. On the Dynamic Model

I’d recommend aiming for a simplified model that still captures the key dynamics. A point-mass or simplified rigid-body kite model should be sufficient for most optimization tasks. For the tether, a quasi-static straight-segment approximation might be enough to account for both weight and drag, while keeping the computational cost low.

One advantage of a point-mass model is that it requires fewer parameters to estimate — in particular, you avoid the need to identify rotational inertias and aerodynamic moments, which are often difficult to obtain accurately for soft kites. This makes it well-suited for early-stage design, model-based optimization, or parameter sweeps.

I’m currently working on a point-mass model for soft kites that explicitly resolves the angle of attack, so that aerodynamic polars can be used as input. This adds realism while keeping the model compact.

In my current model, the required inputs are::

• Wind profile (speed and direction with height)
• Kite area and mass
• Aerodynamic polars (lift, drag vs angle of attack)
• Maximum steering
• Tether diameter and density
• For ship propulsion, possibly add ship speed (to compute apparent wind)

In addition to the previously mentioned constraints I would add a minimum elevation angle.

Regarding drag estimation, as noted earlier, lifting-line models (VSM) and even CFD methods tend to underestimate drag, especially for soft kites. In my case, I’ve applied corrections based on experimental data. However, these corrections are likely kite-type dependent — ideally, characterization data should be available for different kite types (e.g., ram-air, inflatable, hybrid) to ensure reliable simulations.

2. On the Optimization Method

For trajectory optimization, I’d recommend parameterizing the flight path (e.g., figure-eight or circular loops) using a small number of shape parameters (between 2 and 10 for reel-out or ship-propulsion scenarios).

With this approach, you can formulate a direct optimal control problem, where the path shape is parameterized and the steering inputs are solved as part of the NLP. The main motivation is to avoid developing a dedicated closed-loop controller: the solver computes both the optimal control actions and the resulting trajectory, given the system dynamics and constraints.

3. On Wind Profile and Veer

In ship propulsion scenarios, wind shear and veer may play a smaller role than in classic AWE systems, especially at low elevation angles. That said, including a simple wind profile and veer model is feasible and adds robustness to the optimization without much overhead — so I’d recommend including it when possible.

Let me know if there’s anything critical I’m missing — I’d be glad to refine this further. Also happy to share some numerical examples once my current model is running robustly.

[omidhe89]

In response to your concerns about calculating optimal trajectories, I’d like to share some thoughts based on my thesis experience.
The optimization of mechanical systems is a broad topic, and the strategy, problem formulation, models, and tools must be carefully chosen to meet the specific optimization goals. For example, finding an optimal trajectory when you can still modify kite parameters such as dimensions is different from a scenario where the kite design is fixed, and you have a clear understanding of the Airborne Wind Energy (AWE) system’s limitations and capabilities.
In the latter case, you might seek an optimal trajectory to maximize power—such as lift or traction if you are pulling a ship—while accounting for changing conditions like wind speed or waves. This optimization would be subject to the ship’s travel path, and within the AWE controller and actuators limitation. From now on, I will talk about the trajectories which are going to be used as reference for the controller.
It is clear that environmental parameters change continuously, meaning the optimal trajectory must also adapt in real-time to satisfy kite constraints while maximizing traction force. Therefore, the optimization process needs to be fast enough to generate trajectories on demand.
Given the importance of this challenge, some research suggests precomputing a bank of optimal trajectories and selecting the most appropriate one based on current operational conditions. Other studies, however, focus on developing methods to compute the optimal trajectory in real time, within an acceptable timeframe.
To compute an optimal trajectory with a certain level of accuracy within an acceptable timeframe, the model and solver should be carefully selected and tailored to the problem. As a general approach, I recommend starting with the simplest model and gradually increasing its complexity as long as the computational budget allows.
There are also more structured methods, such as homotopy approaches, which systematically increase problem complexity. These might be of interest if you haven’t explored them yet.
In addition to the model, Problem formulation (Direct Multiple Shooting, Direct Collocation Method), the optimization algorithm (solver) and even the hardware play significant roles in overall performance. A review of the literature shows that gradient-based optimization algorithms are widely used, as they are well-suited for high-dimensional and constrained optimization problems. Their practicality and robustness are further enhanced by the availability of analytical or automatic gradients through tools like CasADi (in MATLAB and Python) or JuMP (in Julia).
Choosing a gradient-based solver, such as Interior Point OPTimizer IPOPT or Sequential Quadratic Programming (SQP), is a key factor in achieving both robustness and computational efficiency. As far as I know, IPOPT is already well-integrated into the Julia ecosystem and should be relatively straightforward to use for your applications.
Finally, it’s worth emphasizing that finding an optimal trajectory for pulling ships—even at low speeds—is a non-trivial task. Clearly defining the optimization objective and carefully selecting the appropriate tools, models, and solution strategies are essential to tackling this problem effectively.