# Design of an Airship On-Board Crane

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{3}of Helium must be provided within the hull of the airship. The airships targeted for the aforementioned examples will therefore have enormous volumes, and all the obstacles inherent in these large volumes have not yet been overcome. Among these obstacles, we note on the one hand, the high sensitivity of airships to gusts of wind. Various studies have looked into the stabilization of airships, particularly in hovering flight or for tracking trajectories (see, for example, [3,4,5,6,7]). On the other hand, the non-standard dimensions of these devices make the development of landing or handling areas very problematic or costly. This has left the designers of these machines to consider handling operations at altitude. It is in this context that our study takes place. We propose for this task a smart crane capable not only of loading and unloading but also of stabilizing the load during a gust of wind and of arranging the hold of the airship. This crane will be based on a cable-driven parallel manipulator (CDPM).

## 2. Problem Statement and Description of the Smart Crane

^{3}, and it would be difficult to provide them with landing and loading infrastructure. So that these means of transport could be used everywhere, it is essential to provide a means of handling at altitude. This will allow them to be used universally but must be accompanied by various measures, in particular robust control of the airship and a crane adapted to this configuration. It is in this context that we present our smart crane.

#### Presentation of the Smart Crane

- 1.
- A classic container loading and unloading mission enabled by the winches integrated into the cuboid.
- 2.
- Stabilize the suspended containers during a sudden motion of the airship under the effect of a gust of wind by creating accelerations according to six possible degrees of freedom of the cuboid, depending on the nature of the oscillation of the load.
- 3.
- Drop off, collect, and store the containers in the airship hold (as shown in Figure 4).

## 3. Dynamic Modelling

#### 3.1. Modeling of the Cable-Driven Parallel Manipulator

#### 3.1.1. Classical Case: Industrial CDPM

_{i}drives the cylindrical drum in rotation at an angle q

_{i}about its axis of symmetry. This generates a tensile force t

_{i}at the exit point of the cable P

_{i}.

_{g}represents the external forces, such as the weight of the platform and its loading or the force exerted by a gust of wind, $W\left({X}_{p}\right)=-{{\displaystyle J}}^{t}\left({X}_{p}\right)$, M is the mass matrix of the platform, C is the matrix of centrifugal and Coriolis forces. The latter two are defined in detail in Appendix C.

#### 3.1.2. Case of the CDPM on Board an Airship

#### 3.2. Dynamic Modeling of the Multibody System in a Plane Motion

_{1}) and axis (G’X

_{2}), respectively. The system to be modeled is shown in Figure 9. It is composed of an airship, a cuboid, and a load suspended by a cable. The latter, which holds the container, is thick and massive. Unlike the cables that support the cuboid, we will consider here its flexibility as well as its elongation. The cable has a length l

_{c}, a mass per unit of length ρ, a modulus of Young E, and a moment of inertia I

_{y3}. The load has mass m

_{L}and matrix of inertia I

_{L}. The distance between its center of mass and the end of the cable is represented by r

_{L}, and the deformation is denoted by ω.

- i.
- An earth-fixed frame ${R}_{0}\left(O,{X}_{0},{Y}_{0},{Z}_{0}\right)$.
- ii.
- A local frame fixed to the airship ${R}_{1}\left(G,{X}_{1},{Y}_{1},{Z}_{1}\right)$, having as origin the inertia center of the airship.
- iii.
- A local frame fixed to the cuboid ${R}_{2}\left({G}^{\prime},{X}_{2},{Y}_{2},{Z}_{2}\right)$, having as origin the inertia center of the cuboid ${G}^{\prime}$.
- iv.
- A reference frame ${R}_{3}\left({G}^{\prime},{X}_{3},{Y}_{3},{Z}_{3}\right)$, linked to the cable in rotation at an angle $\theta $ with respect to R
_{2}. - v.
- A reference frame ${R}_{4}\left({P}_{c},{X}_{4},{Y}_{4},{Z}_{4}\right)$, attached to any section of the cable located at a distance ${z}_{3}$ from the ${z}_{3}$ axis.
- vi.
- A reference frame ${R}_{5}\left({P}_{L},{X}_{5},{Y}_{5},{Z}_{5}\right)$, at the end of the cable to describe the motion of the load.

_{tot}is the total kinetic energy of the system, V

_{tot}

^{,}is the total potential energy, and F

_{R}is the Rayleigh dissipation function.

#### 3.2.1. Kinetic Energy of the System

_{A}), the cuboid (T

_{C}), the payload (T

_{L}), and the flexible cable.

_{A}and u

_{C}the speeds of the airship and of the cuboid projected onto the mobile reference frames linked to each of them. In our particular example of a plane case, we have: ${u}_{A}={\dot{x}}_{A}\text{\hspace{0.17em}}and\text{\hspace{0.17em}}{u}_{C}={\dot{x}}_{C}$

**.**

_{C}is the kinetic energy of the cuboid:

_{C}is the mass of the cuboid, and m

_{A}is the mass of the airship including the added mass. For more details concerning the added masses one could consult [28]. The kinetic energy of the load has two terms: the first term which takes into consideration the linear motion of the body with respect to the frame R

_{3}and the second term which takes into consideration the rotation of the load. The expression of the kinetic energy of the load is:

#### 3.2.2. Potential Energy

#### 3.2.3. Dissipation Function

#### 3.2.4. Modal Synthesis

- $\overline{M}\left(X\right)$ the mass matrix;
- $\overline{B}$ the damping matrix;
- $\overline{K}$ the stiffness matrix;
- $\text{\hspace{0.17em}}\overline{G}\left(X\right)$ the vector of forces due to the gravity;
- $\overline{C}\left(X,\dot{X}\right)$ the vector of Coriolis and centrifugal forces;
- $\tau ={{\displaystyle \left({F}_{A},\text{\hspace{0.17em}}{F}_{C},\text{\hspace{0.17em}}0,\text{\hspace{0.17em}}0\right)}}^{t}$ the vector of the forces produced by the actuators and applied on the airship and on the cuboid. (The details of these matrices are given in Appendixes B–D).

#### 3.2.5. Equations of Motion

#### 3.2.6. Simplified Model

- (a)
- The first simplification concerns the shape functions: we have chosen the shape functions ${w}_{i}\left({z}_{3}\right)={z}_{3}^{2},\text{\hspace{1em}}i\in \left\{1,2,\dots ,m\right\}$ and we have taken only one mode (m = 1), the expression of the deformation then becomes:$$w\left({z}_{3},t\right)={z}_{3}^{2}\delta \left(t\right)$$
- (b)
- With the aim of simplifying the dynamic model in order to be able to apply control laws to it, we have analyzed the different terms of the matrix $\overline{M}\left(X\right)$ and succeeded in highlighting the terms whose calculation is complex, but whose absolute value can be neglected compared to the other terms of the matrix. The mass matrix $\overline{M}\left(X\right)$ can thus be subdivided as the sum of two matrices: main matrix noted $\tilde{M}\left(X\right)$ and complementary matrix $\overline{dM}\left(X\right)$:$$\overline{M}\left(X\right)=\tilde{M}\left(X\right)+\overline{dM}\left(X\right)$$

- (c)
- A third simplification is carried out for the Coriolis vector. We replace $\overline{M}$ with $\tilde{M}$ in Equation (A24) (see Appendix D) and thus obtain the following expression of $\overline{C}\left(X,\dot{X}\right)$:$$\overline{C}\left(X,\dot{X}\right)=\dot{\tilde{M}}\text{\hspace{0.17em}}\dot{X}-\frac{1}{2}\frac{\partial}{\partial X}\left({\dot{X}}^{t}\text{\hspace{0.17em}}\tilde{M}\text{\hspace{0.17em}}X\right)$$

**Remark**

**1.**

#### 3.3. Stabilization

## 4. Simulation Results

_{1}axis as we can see in Figure 10:

_{A}, displacement of the cuboid x

_{C}, oscillation and deformation of the cable θ and δ).

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$\Vert \text{\hspace{0.17em}}.\text{\hspace{0.17em}}\Vert $ | is the norm of the vector (.). |

$\dot{A}$ | is the time derivative of A. |

x | is the cross product. |

Diag(R) | is the column matrix of the diagonal components of R. |

A^{t} | is the transpose of the matrix A. |

${w}_{{z}_{3}}$ | is the partial derivative of w with respect to z_{3}. |

$o\left(3\right)$ | are small terms of the third order |

## Appendix A

## Appendix B

## Appendix C

## Appendix D

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Parameter | Symbol | Value |
---|---|---|

Airship mass | ${m}_{A}$ | 500 kg |

Cuboid mass | ${m}_{C}$ | 50 kg |

Load mass | ${m}_{L}$ | 40 kg |

Load radius | ${r}_{L}$ | 0.5 m |

Cable length | ${l}_{c}$ | 15 m |

Linear mass | $\rho $ | 1.5 kg/m |

Young’s modulus | $E$ | ${{\displaystyle 2.10}}^{8}$ Pa |

Moment of inertia of the load | ${I}_{L}$ | $2\text{\hspace{0.17em}}{m}_{L}\text{\hspace{0.17em}}{r}_{L}^{2}/5\text{\hspace{0.17em}}\left(\mathrm{kg}.\text{\hspace{0.17em}}{\mathrm{m}}^{2}\right)$ |

Quadratic moment of the cable | I_{z3} | 10^{−4} m^{4} |

Damping coefficient | k_{e} | 0.1 |

p_{i} | x_{pi} (m) | y_{pi} (m) | z_{pi} (m) |
---|---|---|---|

${p}_{1}$ | 0.500 | −0.507 | 0.555 |

${p}_{2}$ | −0.488 | 0.361 | 0.554 |

${p}_{3}$ | −0.500 | −0.260 | 0.555 |

${p}_{4}$ | 0.503 | 0.342 | 0.548 |

${p}_{5}$ | −0.500 | 0.507 | 0.555 |

${p}_{6}$ | 0.497 | −0.353 | 0.554 |

${p}_{7}$ | 0.499 | 0.260 | 0.549 |

${p}_{8}$ | 0.499 | 0.260 | 0.549 |

vi | x_{vi} (m) | y_{vi} (m) | z_{vi} (m) |

${v}_{1}$ | −7.224 | −5.359 | −5.468 |

${v}_{2}$ | −7.435 | −5.058 | 5.477 |

${v}_{3}$ | −7.425 | 5.196 | 5.486 |

${v}_{4}$ | −7.210 | 5.497 | 5.495 |

${v}_{5}$ | 7.139 | 5.463 | 5.481 |

${v}_{6}$ | 7.440 | 5.158 | 5.494 |

${v}_{7}$ | 7.415 | −5.089 | 5.481 |

${v}_{8}$ | 7.113 | −5.388 | 5.492 |

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**MDPI and ACS Style**

Guesmi, F.; Azouz, N.; Neji, J.
Design of an Airship On-Board Crane. *Aerospace* **2023**, *10*, 290.
https://doi.org/10.3390/aerospace10030290

**AMA Style**

Guesmi F, Azouz N, Neji J.
Design of an Airship On-Board Crane. *Aerospace*. 2023; 10(3):290.
https://doi.org/10.3390/aerospace10030290

**Chicago/Turabian Style**

Guesmi, Fatma, Naoufel Azouz, and Jamel Neji.
2023. "Design of an Airship On-Board Crane" *Aerospace* 10, no. 3: 290.
https://doi.org/10.3390/aerospace10030290