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

Chiara Gabellieri, Antonio Franchi

TL;DR

The paper addresses whether $n\ge 3$ aerial carriers can sustain non-stop, energy-efficient flights while holding a cable-suspended load in a fixed pose. It introduces a rigorous methodology that combines selecting $n$ internal-force directions from the nullspace of the grasp matrix using a Hamiltonian cycle on the load-attachment graph, with constructing elliptical, periodic trajectories in corresponding 2D affine subspaces via graph coloring. A complete algorithm is provided to generate coordinated non-stop trajectories, and a formal existence proof is established for all $n\ge 3$, complemented by simulations and laboratory experiments validating the approach. The work demonstrates the feasibility and practical potential of persistent carrier motion for aerial manipulation tasks, enabling long-duration operations with controlled load pose without stopping.

Abstract

Multirotor UAVs have been typically considered for aerial manipulation, but their scarce endurance prevents long-lasting manipulation tasks. This work demonstrates that the non-stop flights of three or more carriers are compatible with holding a constant pose of a cable-suspended load, thus potentially enabling aerial manipulation with energy-efficient non-stop carriers. It also presents an algorithm for generating the coordinated non-stop trajectories. The proposed method builds upon two pillars: (1)~the choice of $n$ special linearly independent directions of internal forces within the $3n-6$-dimensional nullspace of the grasp matrix of the load, chosen as the edges of a Hamiltonian cycle on the graph that connects the cable attachment points on the load. Adjacent pairs of directions are used to generate $n$ forces evolving on distinct 2D affine subspaces, despite the attachment points being generically in 3D; (2)~the construction of elliptical trajectories within these subspaces by mapping, through appropriate graph coloring, each edge of the Hamiltonian cycle to a periodic coordinate while ensuring that no adjacent coordinates exhibit simultaneous zero derivatives. Combined with conditions for load statics and attachment point positions, these choices ensure that each of the $n$ force trajectories projects onto the corresponding cable constraint sphere with non-zero tangential velocity, enabling perpetual motion of the carriers while the load is still. The theoretical findings are validated through simulations and laboratory experiments with non-stopping multirotor UAVs.

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

TL;DR

The paper addresses whether aerial carriers can sustain non-stop, energy-efficient flights while holding a cable-suspended load in a fixed pose. It introduces a rigorous methodology that combines selecting internal-force directions from the nullspace of the grasp matrix using a Hamiltonian cycle on the load-attachment graph, with constructing elliptical, periodic trajectories in corresponding 2D affine subspaces via graph coloring. A complete algorithm is provided to generate coordinated non-stop trajectories, and a formal existence proof is established for all , complemented by simulations and laboratory experiments validating the approach. The work demonstrates the feasibility and practical potential of persistent carrier motion for aerial manipulation tasks, enabling long-duration operations with controlled load pose without stopping.

Abstract

Multirotor UAVs have been typically considered for aerial manipulation, but their scarce endurance prevents long-lasting manipulation tasks. This work demonstrates that the non-stop flights of three or more carriers are compatible with holding a constant pose of a cable-suspended load, thus potentially enabling aerial manipulation with energy-efficient non-stop carriers. It also presents an algorithm for generating the coordinated non-stop trajectories. The proposed method builds upon two pillars: (1)~the choice of special linearly independent directions of internal forces within the -dimensional nullspace of the grasp matrix of the load, chosen as the edges of a Hamiltonian cycle on the graph that connects the cable attachment points on the load. Adjacent pairs of directions are used to generate forces evolving on distinct 2D affine subspaces, despite the attachment points being generically in 3D; (2)~the construction of elliptical trajectories within these subspaces by mapping, through appropriate graph coloring, each edge of the Hamiltonian cycle to a periodic coordinate while ensuring that no adjacent coordinates exhibit simultaneous zero derivatives. Combined with conditions for load statics and attachment point positions, these choices ensure that each of the force trajectories projects onto the corresponding cable constraint sphere with non-zero tangential velocity, enabling perpetual motion of the carriers while the load is still. The theoretical findings are validated through simulations and laboratory experiments with non-stopping multirotor UAVs.

Paper Structure

This paper contains 18 sections, 2 theorems, 25 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Kinematics
  4. Dynamics
  5. Statics
  6. Formal Derivation of the Proposed Methodology
  7. Preliminaries on Hamiltonian Cycles
  8. Decision of the null-space matrix N
  9. Force of the $i$-th cable for the chosen N
  10. Relation between force derivative and carrier velocity
  11. Decision of the time-varying coefficients ${\lambda}(t)$
  12. Algorithm and Qualitative Remarks
  13. Algorithm to generate Coordinated Trajectories for Non-stop Flying Carriers Holding a Cable-Suspended Load
  14. Qualitative Remarks about the Choice of Free Parameters in the Algorithm
  15. Numerical Results
  16. ...and 3 more sections

Key Result

Lemma 1

After arbitrarily selecting one of the Hamiltonian cycles of the complete graph with $n$ vertices, denoted with $H$, construct a matrix $\bm{N}(H)\in\mathbb{R}^{3n\times n}$ as described in eq:choice_of_N. Then, the following conditions are sufficient to have $\|\dot{\bm{p}}_{R i}(t)\|>0$

Figures (5)

  • Figure 1: A team of $n\geq3$ non-stop flying carriers maintains a cable-suspended load in a fixed position to enable a potential construction scenario, while all carriers continue on their respective non-zero speed flight paths
  • Figure 2: Geometric visualization of the force of the $i$-th carrier and its components.
  • Figure 3: Simulation for the 3 Hamiltonian cycles of a 4-carrier system. The cycles are in the first plot on the left; the carriers' velocities for each of the 3 cycles are reported, in the same order, in the other plots. The minimum of the carriers' velocities in the first cycle is indicated by a dashed line in the three plots. As expected, it is greater than the minimum carrier's velocity when the other two cycles are selected.
  • Figure 4: Four carriers (top row) 6 carriers (bottom row) manipulating a load. In the first column, a snapshot of an animation, where the carriers follow the colored planned paths, the cross indicates the load CoM, and dotted lines indicate the edges of the Hamiltonian cycle (connecting the cable anchoring point to the carrier on the corresponding graph vertex). In the second column, $|\dot{\bm{p}}_{R i}(t)|>0$. In the third and fourth columns, the position and attitude of the load are reported.
  • Figure 5: Results of 5-carrier system. Each column corresponds to one of the 12 Hamiltonian cycles, displayed in the first row, where the cable anchoring points of the load are displayed. The second row shows the values of $\bm{e}_{pL}$ and the third of $\bm{e}_{RL}$.

Theorems & Definitions (9)

  • Example 1
  • Remark 1
  • Example 2
  • Remark 2
  • Lemma 1
  • Proposition 1
  • Remark 3
  • Remark 4
  • Remark 5