# Rapid Parametric CAx Tools for Modelling Morphing Wings of Micro Air Vehicles (MAVs)

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## Abstract

**:**

## 1. Introduction

## 2. Geometric Kernel

## 3. Software Architecture

#### 3.1. Airfoil Parameterization

#### 3.1.1. CST Method Theory

#### 3.1.2. CST as a Filter

#### 3.1.3. Actuator Effect Parameterization

#### 3.2. Wing Parameterization

## 4. Software Application

## 5. Bioinspired Configurations

## 6. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ACIS | Geometric modelling kernel developed by Spatial Corporation |

BPO | Bernstein Polynomials Order |

CAD | Computer Aided Design |

CFD | Computational Fluid Dynamics |

CPACS | Common Parametric Aircraft Configuration Schema |

CST | Class-Shape Transformation |

DLR | Deutsches Zentrum für Luft- und Raumfahrt |

IGES | Initial Graphics Exchange Specification |

MAV | Micro Air Vehicle |

MFC | Macro Fibre Composite |

NASA | National Aeronautics and Space Administration |

NURBS | Non-Uniform Rational Basis Spline |

PHIGS | Programmer’s Hierarchical Interactive Graphics System |

RPAS | Remotely Piloted Aircraft System |

STEP | Standard for the Exchange of Product Model Data |

STL | Standard Tessellation Language |

b | Wingspan |

c | Local chord |

$\mathsf{\Lambda}$ | Sweep Angle |

$\mathsf{\Gamma}$ | Dihedral |

${\mathsf{\Theta}}_{twist}$ | Wing twist |

C | B-Spline curve |

${B}_{i}^{k}$ | B-Spline basis functions |

${P}_{i}$ | B-Spline control points |

s | B-Spline surface |

${S}_{f}$ | Profiles interpolation surface |

${S}_{g}$ | Guide curves interpolation surface |

T | Surface that interpolates intersection points between ${S}_{f}$ and ${S}_{g}$ |

S | Shape function |

${A}_{i}$ | Shape function coefficients |

${K}_{i}^{N}$ | Binomial coefficients |

${C}_{N2}^{N1}$ | Class function |

$N1,N2$ | Class function coefficients |

${\left(\right)}_{U}$ | Relative to upper surface |

${\left(\right)}_{L}$ | Relative to lower surface |

$\psi $ | Non-dimensional airfoil station |

$\zeta $ | Non-dimensional airfoil ordinate |

$\Delta \zeta $ | Non-dimensional trailing edge thickness |

${A}_{LE}$ | Leading edge coefficient |

## References

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**Figure 1.**Interpolations necessary to obtain a Gordon surface, $s(u,v)$. ${S}_{f}(u,v)$ interpolates the profiles, ${S}_{g}(u,v)$ is the interpolation of the guide curves and $T(u,v)$ interpolates a network of intersection points of ${S}_{f}(u,v)$ and ${S}_{g}(u,v)$.

**Figure 2.**Comparison between the original CST method and the modified CST method. The green band represents the typical manufacturing accuracy of a wind tunnel model [20]. Subfigure (

**a**) shows how fitting an Eppler 61 airfoil, within tolerances, requires Bernstein polynomials of order 20 (BPO20) for upper and lower surfaces for the original CST method, while subfigure (

**b**) shows that with the modified CST method Bernstein polynomials of order 6 (BPO6) are required to be within tolerances. (

**a**) CST without modification (BPO20); (

**b**) CST with modification (BPO6).

**Figure 4.**Generation process of a double tapered wing surface. Subfigure (

**a**) shows the leading and trailing edge lines. In subfigure (

**b**), the airfoils are placed along the wingspan. Finally, in subfigure (

**c**) the wing surface is generated. (

**a**) Leading and trailing edges; (

**b**) airfoils along the span; and (

**c**) finished wing.

**Figure 10.**MAV airfoil with voltage of 5 V. Figure shows the points measured on test (black) and the CST airfoil adaptation (red).

**Figure 11.**Fitting of the CST coefficients for the upper surface of the test airfoils using a polynomial of degree 3.

**Figure 12.**Fitting of the CST coefficients for the lower surface of the test airfoils using a polynomial of degree 3.

**Table 1.**Mean relative error and mean absolute error between the z/c coordinates of the airfoils obtained with the polynomial curve coefficients and the CST coefficients of the test aerofoils.

0 V | 2 V | 2.8 V | 4 V | 5 V | |
---|---|---|---|---|---|

Mean relative error | 0.076% | 1.167% | 1.419% | 0.576% | 0.164% |

Mean absolute error | 1.625 × 10${}^{-5}$ | 1.897 × 10${}^{-4}$ | 3.069 × 10${}^{-4}$ | 1.862 × 10${}^{-4}$ | 5.386 × 10${}^{-5}$ |

**Table 2.**Execution times to visualize and generate the wing geometry. The .stl file is generated with a linear deviation of 0.005 mm and an angular deviation of 0.5 degrees. These times are calculated on a computer with an Intel Core i5-4210U processor.

Visualization | Generate .stp File | Generate .stl File | |
---|---|---|---|

Execution Time | 0.5 s | 0.6 s | 20 s |

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## Share and Cite

**MDPI and ACS Style**

Rodríguez-Sevillano, Á.A.; Casati-Calzada, M.J.; Bardera-Mora, R.; Nieto-Centenero, J.; Matías-García, J.C.; Barroso-Barderas, E.
Rapid Parametric CAx Tools for Modelling Morphing Wings of Micro Air Vehicles (MAVs). *Aerospace* **2023**, *10*, 467.
https://doi.org/10.3390/aerospace10050467

**AMA Style**

Rodríguez-Sevillano ÁA, Casati-Calzada MJ, Bardera-Mora R, Nieto-Centenero J, Matías-García JC, Barroso-Barderas E.
Rapid Parametric CAx Tools for Modelling Morphing Wings of Micro Air Vehicles (MAVs). *Aerospace*. 2023; 10(5):467.
https://doi.org/10.3390/aerospace10050467

**Chicago/Turabian Style**

Rodríguez-Sevillano, Ángel Antonio, María Jesús Casati-Calzada, Rafael Bardera-Mora, Javier Nieto-Centenero, Juan Carlos Matías-García, and Estela Barroso-Barderas.
2023. "Rapid Parametric CAx Tools for Modelling Morphing Wings of Micro Air Vehicles (MAVs)" *Aerospace* 10, no. 5: 467.
https://doi.org/10.3390/aerospace10050467