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The Geometry of Coordinated Trajectories for Non-stop Flying Carriers Holding a Cable-Suspended Load

Pieter van Goor, Chiara Gabellieri, Antonio Franchi

Abstract

This work considers the problem of using multiple aerial carriers to hold a cable-suspended load while remaining in periodic motion at all times. Using a novel differential geometric perspective, it is shown that the problem may be recast as that of finding an immersion of the unit circle into the smooth manifold of admissible configurations. Additionally, this manifold is shown to be path connected under a mild assumption on the attachment points of the carriers to the load. Based on these ideas, a family of simple linear solutions to the original problems is presented that overcomes the constraints of alternative solutions previously proposed in the literature. Simulation results demonstrate the flexibility of the theory in identifying suitable solutions.

The Geometry of Coordinated Trajectories for Non-stop Flying Carriers Holding a Cable-Suspended Load

Abstract

This work considers the problem of using multiple aerial carriers to hold a cable-suspended load while remaining in periodic motion at all times. Using a novel differential geometric perspective, it is shown that the problem may be recast as that of finding an immersion of the unit circle into the smooth manifold of admissible configurations. Additionally, this manifold is shown to be path connected under a mild assumption on the attachment points of the carriers to the load. Based on these ideas, a family of simple linear solutions to the original problems is presented that overcomes the constraints of alternative solutions previously proposed in the literature. Simulation results demonstrate the flexibility of the theory in identifying suitable solutions.
Paper Structure (12 sections, 5 theorems, 19 equations, 1 figure)

This paper contains 12 sections, 5 theorems, 19 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Problem Description
  4. Geometric Description of Admissible Force Configurations
  5. Admissible Forces
  6. Periodic Trajectories as Immersions
  7. A Linear Solution
  8. Simulations
  9. Setup
  10. Results
  11. Discussion
  12. Conclusions

Key Result

Theorem 1

Define the manifold where $P_i \in \mathbb{R}^{3\times 3n}$ is given by Then a configuration of forces is admissible if and only if $f = G^\dag w + N \lambda$ for some $\lambda \in \mathcal{M}_\lambda$.

Figures (1)

  • Figure 1: Results of two simulations. A representation of the system shows the colored orbits of the carriers, on which the carriers are represented in grey; black solid lines are the cables and black dashed lines connect the cable attachment points on the load, which center of mass is a black cross. $p_L^{X,Y,Z}$ are the X-, Y-, and Z- coordinated of the position of $\{O\}$ expressed in $\{W\}$, while $\phi_L,\theta_L,\psi_L$ the roll, pitch, and yaw angles extracted from $R$.

Theorems & Definitions (11)

  • Definition 1
  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Corollary 1
  • proof
  • Theorem 2
  • proof
  • Lemma 2
  • ...and 1 more