This work proposes a white-box blade damage model based on Blade Element Theory (BET) which combines the emerging mass and aerodynamic effects of blade damage. The model serves as plug-in to the nominal system model, enables the simulation of any degree of blade damage and does not require costly experimental data from failure cases.

The repository contains Python code for:

• the identification of the airfoil lift and drag coefficient polynomials of drone blades
• the computation of the forces and moments generated by the (damaged) propeller with BET theory

From fault-tolerant control to failure detection, blade damage simulation has been an essential tool for the development and testing of failure resilient modern unmanned aerial vehicles before their entry into service. Previous literature in the field of fault diagnosis have exploited simplifications of the simulation of blade damage. Avram et al. consider quadrotor actuator faults, such as structural damage to the propellers or degradation of the rotors, as partial loss of effectiveness — a partial loss of thrust generated by the damaged rotor. This is simulated by multiplying the commanded rotor angular velocity by a factor lower than one in order to obtain the "true" rotor angular velocity. The main drawback of this approach is that vibrations in the system due to the unbalance of forces and moments are ignored.

Another approach is proposed by Ghalamchi et al., which introduce sinusoids in the force signals to simulate the vibrations caused by the propeller unbalance. The sinusoids only consist of the decomposition of the centrifugal force in the x and y components caused by the displacement of the propeller centre of gravity due to blade damage. Unfortunately, this approach does not consider the vibrations in the moment signals, as well as the vibrations caused by the changed aerodynamics due to the displacement of the centre of pressure.

The development of more accurate blade damage models could contribute to the creation of more realistic simulations that will foster the potential discovery of emerging subtle data features able to improve the current UAV on-board failure detection and diagnosis capabilities. A technique that has been used for the modelling of forces and moments in helicopters, UAVs and wind turbines is Blade Element Theory (BET). Here, the propeller is discretised radially into a finite number of segments of length 𝛿𝑟, each producing a differential thrust and torque. BET is based on the assumption that the wrenches generated by a (rotor) blade can be computed by the addition of the individual contributions of each of its span-wise elements. For this purpose, 2D airfoil characteristics are exploited whereas 3D effects are ignored. Previously, this approach has been used [to model propeller thrust](Propeller Thrust and Drag in Forward Flight). However, it has never been explored for blade damage modelling.

In this repository, a white-box blade damage simulation model based on Blade Element Theory that complements the identified healthy UAV model is implemented. To this end, the developed approach provides the difference in forces and moments with respect to the nominal system. In contrast with existing methods, the effects from both shifts in the centres of gravity and pressure are considered. The approach allows the injection of any level of failure without the need of added costly and dangerous system identification experiments. To the author’s knowledge, this is the first time BET is used for UAV blade damage simulation and the first time mass and aerodynamic effects are modelled together in order to shift research towards more realistic white-box blade damage models. Furthermore, this code also presents a method for identifying the (mostly unknown) UAV blade lift and drag curves with respect to the angle of attack using BET, an approach never tried before in literature.

The project contains the following files:

• main.py: Provides the code for testing the main capabilities of the Blade damage model and the generation of the figures used in the thesis and paper. Basic code to compute the generated moments and forces due to propeller damage.

• user_input.py: Provides all the inputs that the user can modify in one centralised file.

• Propeller.py: Propeller holds all the information related to the propeller assembly as a whole and contains a list with all the Blade objects that define the Propeller. It is used for calling methods applicable to all the blades which are required for the computation of the Propeller center of gravity, as well as the moments and forces generated by the Propeller. Additionally, it updates the rotation state of the propeller; carries out the identification of the lift and drag coefficient polynomials by equating the thrust and torque computed with BEM and the Matlab model identified in the wind tunnel (using an average rotation or a single time instance); computes the uniform and linear inflow field.

• Blade.py: Blade holds all the information related to a single blade and contains a list with all the BladeSection objects that define the Blade. It is used for calling methods applicable to all the BladeSections which are required for the computation of the Blade center of gravity, blade area and mass, as well as the moments and forces generated by the Blade. Additionally, it computes the contribution for the identification of the lift and drag coefficient relative to a single blade.

• BladeSection.py: Provides the BladeSection, class for the aerodynamic model identification and computation of forces and moments. BladeSection holds all the information related to a single blade element according to BET theory. It is used for the computation of the angle of attack and velocity seen by each BladeSection. Additionally, it computes the contribution of the BladeSection lift and drag to the thrust force and torque.

• helper_func.py: Provides the helper functions, workhorse of the whole blade damage implementation It contains functions that carry out simple mathematical/geometrical computations, implements the gray-box aerodynamic matlab model, implements the airfoil lift and drag coefficient identification, the computation of forces and moments as a function of time, and all the plotters.

• Gradient_descent_efficiency.py: Provides the procedural code that demonstrates the superiority of the custom made gradient-descent approach when compared to Nelder-Mead optimization.

• aero_data.py: Provides aerodynamic data of the Bebop2 gray-box aerodynamic model identified in the wind tunnel at TUDelft. It contains the aerodynamic parameters for the Matlab model computations

• ClCd_plotter.py: Provides the procedural code to plot the airfoil lift and drag coefficient curves with respect to the angle of attack.

• Polar_plots.py: Provides the procedural code that show the effect of drone moving velocity on the angle of attack of the blade (advancing vs retreating blades), the effects of the induced velocity (model) on the angle of attack and the effect of the propeller rotational velocity on the angle of attack.

• V-T_plot.py: Provides the procedural code to generate the Velocity vs Thrust plot that demonstrates the correctness of thein-house developed gradient descend algorithm for computing the induced velocity.

To start using the code you can download the required Python libraries stored within requirements.txt. For that purpose, it is as simple as running the following command within the command line:

pip install -r requirements.txt

Then adjust the parameters that you deem necessary in user_input.py and run main.py.

You can also run it within Google Colab. For that you only copy-paste two lines. First:

!git clone https://github.com/joigalcar3/Blade_damage

This will clone the repository. Then you can open the user_input.py file and alter the user input. Second, run:

!python Blade_damage/main_general.py

The results of this work can be found in the author's Master thesis and paper:

1. Master thesis: "From Data to Prediction: Vision-Based UAV Fault Detection and Diagnosis". Chapters 9.1-9.3, Appendices A-D.
2. Paper: "Blade Element Theory Model for UAV Blade Damage Simulation". Everything except "VII. Model Validation."

These documents explain:

• Force and moment contribution of the mass effects
• Force and moment contribution of the aerodynamic effects
• Airfoil lift and drag coefficient identification
• Effects of the linear induced velocity model

Next are listed the assumptions taken for the drag and lift coefficient coefficients polynomial curve identification and the computation of the moments and forces of the propeller.

Assumptions:

• Homogeneous mass along the blade: the centroid equals the location of the cg
• The Bebop 2 blades are simplified as two trapezoids with parallel sides connected by the long parallel side
• The twist decreases linearly from the root to the tip
• The airfoil is constant throughout the blade
• The cross flow along the span of the blade is ignored
• Aeroelasticity effects are ignored
• The root and tip losses are ignored
• The induced velocity is computed with the simplified linear induced inflow
• The nonlinear effects between (damaged) blades are not considered
• The nonlinear aerodynamic effects between propellers are not considered.
• The nonlinear aerodynamic effects between the propellers and the body frame are not considered
• The data used for the cl cd identification is obtained from the Matlab model that provides the propeller thrust
• The blade is cut parallel to the edge of the propeller such that the remaining polygon is still a trapezoid

Hope you enjoy the code!! For any questions, comments or suggestions, please reach out to me at [email protected]. Also, please consider citing our research work below when using this repository.

@inbook{deAlvearCardenasBET2024,
    author = {José Ignacio de Alvear Cárdenas and Coen C. de Visser},
    title = {Blade Element Theory Model for UAV Blade Damage Simulation},
    booktitle = {AIAA SCITECH 2024 Forum},
    chapter = {},
    pages = {},
    doi = {10.2514/6.2024-2816},
    URL = {https://arc.aiaa.org/doi/abs/10.2514/6.2024-2816}
}