# Mixed-Integer-Based Path and Morphing Planning for a Tensegrity Drone

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## Abstract

**:**

## 1. Introduction

## 2. State of the Art

#### 2.1. Protective Cage Design for Drones

#### 2.2. Mechanical Design of Tensegrity Structures

#### 2.3. Deformation-Enabled Path Planning

#### 2.4. Motion Planning for Rigid Drones

#### 2.5. Convex and Mixed Integer Optimization in Motion Planning

## 3. Problem Statement

#### 3.1. Static Equilibrium of Tensegrity Structures

#### 3.2. Obstacle-Free Space Description

#### 3.3. Path Planning as a Mixed-Integer Convex Program

## 4. Configurations Dataset

#### 4.1. Configuration Vector

#### 4.2. Dataset Generation

Algorithm 1 Configurations dataset generation algorithm |

Input: ${\mathbf{r}}^{0}$, ${\rho}_{0}$, ${\mathcal{R}}_{a}$Output: Datasetwhile$q\le \mathrm{length}\left({\mathcal{R}}_{a}\right)$do ${\rho}_{a}^{q}=\mathrm{read}\left({\mathcal{R}}_{a}\right)$ ${\rho}^{q}={\rho}_{a}^{q}+{\mathbf{P}}_{a}^{\perp}{\rho}_{0}$ ${\mathbf{r}}^{q}=\mathrm{argmin}\left(\mathrm{\Pi}(\mathbf{r},{\rho}^{q})+\gamma \left|\left|{\mathbf{r}}^{0}-\mathbf{r}\right|\right|\right)$ $\mathrm{Dataset}.\mathrm{append}\left({\mathbf{r}}^{q}\right)$ end while |

## 5. Mixed-Integer Convex Path and Morphing Planner

## 6. Numerical Studies, Results

#### 6.1. Planning Drone Path and Deformation through a Series of Rooms

#### 6.2. Study of the Computational Cost of the Algorithm

## 7. Experimental Study

#### 7.1. Pure Tensegrity Drone

#### 7.2. Tensegrity Drone with Strut Contact

## 8. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MICP | Mixed-Integer Convex Programming |

IRIS | Iterative Regional Inflation by Semi-definite programming |

SDPs | Semidefinite Programs |

MIQP | Mixed-Integer Quadratic Program |

UAV | Unmanned Aerial Vehicle |

HVAC | Heating, Ventilation, and Air-Conditioning |

RRT | Rapidly Exploring Random Tree |

QCQP | Quadratically Constrained Quadratic Programming |

MIP | Mixed-Integer Programming |

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**Figure 1.**Tensodrone: a tensegrity-based quadrotor UAV; (a) flight, (b) crash landing, and (c) take-off.

**Figure 2.**Actuation pattern on the tensegrity drone; static rest length struts are drawn in blue, the actuated struts in green, the nodes in magenta, and the cables in black.

**Figure 3.**Path and sequence of deformation planned with the proposed method; A, C and E are rooms, and B and D are corridors.

**Figure 4.**CPU time as a function of the number of steps, calculated for Gurobi gap values 0.05, 0.02 and 0.005; vertical axis is given in logarithmic scale.

**Figure 5.**Number of iterations as a function of the number of steps, calculated for Gurobi gap values 0.05, 0.02 and 0.005; vertical axis is given in logarithmic scale.

**Figure 6.**Number of optimization variables after simplification as a function of the number of steps.

**Figure 7.**Pure tensegrity drone; ESC stands for Electronic Speed Controller; the flight controller used is PX4.

**Figure 9.**Drone trajectory captured by the visual motion capture system OptiTrack in the horizontal plane. (

**a**) Rigid-frame drone QAV-R ${5}^{\u2033}$ Racer. (

**b**) Second-generation tensegrity drone.

**Figure 10.**Spectrum for a rigid-frame drone QAV-R ${5}^{\u2033}$ Racer collected via PX4 log collector and plotted using PX4 Flight Review.

**Figure 11.**Spectrum for the second-generation tensegrity drone collected via PX4 log collector and plotted using PX4 Flight Review.

**Table 1.**Parameters of the six-bar tensegrity drone path and morphing planning through three rooms connected via tilted shafts, with 5- and 27-step paths.

Parameters | Number of Steps, K | |
---|---|---|

$\mathit{K}=\mathbf{5}$ | $\mathit{K}=\mathbf{27}$ | |

Number of binary variables | 65 | 351 |

Number of linear inequalities | 14,400 | 77,760 |

Solution time | 2.8 s | 733 s |

Parameters | Value |
---|---|

Strut length | 0.5 m |

Strut mass | 0.01 kg |

Cable rest length | 0.305 m |

Cable stiffness | 0.58 N/mm |

Total mass | 0.836 kg |

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**MDPI and ACS Style**

Savin, S.; Al Badr, A.; Devitt, D.; Fedorenko, R.; Klimchik, A. Mixed-Integer-Based Path and Morphing Planning for a Tensegrity Drone. *Appl. Sci.* **2022**, *12*, 5588.
https://doi.org/10.3390/app12115588

**AMA Style**

Savin S, Al Badr A, Devitt D, Fedorenko R, Klimchik A. Mixed-Integer-Based Path and Morphing Planning for a Tensegrity Drone. *Applied Sciences*. 2022; 12(11):5588.
https://doi.org/10.3390/app12115588

**Chicago/Turabian Style**

Savin, Sergei, Amer Al Badr, Dmitry Devitt, Roman Fedorenko, and Alexandr Klimchik. 2022. "Mixed-Integer-Based Path and Morphing Planning for a Tensegrity Drone" *Applied Sciences* 12, no. 11: 5588.
https://doi.org/10.3390/app12115588