On the Existence of Static Equilibria of a Cable-Suspended Load with Non-stopping Flying Carriers

TL;DR

The paper investigates whether non-stop flights by multiple fixed-wing carriers can keep a cable-suspended load in a fixed pose. It develops a rigorous model using frames $F_W, F_B$, cable forces $\mathbf{f}$, and a grasp matrix $G$, with the wrench relation $\mathbf{W}=\mathbf{G}\mathbf{f}$ and the internal-force parameterization $\mathbf{f}=\mathbf{G}^†\mathbf{W}+\mathbf{N}\boldsymbol{\lambda}$. The authors prove impossibility for $n=1$ and $n=2$ carriers and establish sufficiency for $n=3$ under a non-degenerate load-orientation condition, providing a constructive sinusoidal family for $\boldsymbol{\lambda}(t)$ that sustains non-stop operation; they also characterize degenerate cases where non-stop operation fails. Numerical simulations in MATLAB-Simulink corroborate the theory, showing sustained non-stop trajectories with three carriers and Stop events in degenerate configurations. The results offer a pathway to energy-efficient, fixed-wing carrier-based cable-suspended manipulation and lay the groundwork for future work on scaling to more carriers and incorporating more realistic dynamics.

Abstract

This work answers positively the question whether non-stop flights are possible for maintaining constant the pose of cable-suspended objects. Such a counterintuitive answer paves the way for a paradigm shift where energetically efficient fixed-wing flying carriers can replace the inefficient multirotor carriers that have been used so far in precise cooperative cable-suspended aerial manipulation. First, we show that one or two flying carriers alone cannot perform non-stop flights while maintaining a constant pose of the suspended object. Instead, we prove that three flying carriers can achieve this task provided that the orientation of the load at the equilibrium is such that the components of the cable forces that balance the external force (typically gravity) do not belong to the plane of the cable anchoring points on the load. Numerical tests are presented in support of the analytical results.

Figures (5)

• Figure 1: Abstract representation of the concept whose theoretical feasibility is studied in this work. A team of non-stop flying carriers regulate the pose of a suspended object to a static forced equilibrium in an ideal application scenario.
• Figure 2: Geometric representation of the cable force, its components, and its derivative. The grey vector $\dot{\bm{f}}_i$ belongs to the plane $P_i$, while its components $T_i\dot{\bm{q}}_i, \dot{T}_i \bm{q}_i \notin P_i$ (in general). $\tilde{\bm{f}}_i$ is contained in the subset $\tilde{P}_i$.
• Figure 3: Representation of the plane including $\dot{\bm{f}}_i$ and its components along the directions parallel and perpendicular to the corresponding cable force. The minimum and maximum values of the angle between $\dot{\bm{f}}_i$ and $\bm{f}_i$ are represented with dotted half-lines, incident in the middle point, and the circle of radius $s$ is the lower bound on the norm of $\dot{\bm{f}}_i$. As a consequence, the norm of the component of $\dot{\bm{f}}_i$ on the direction orthogonal to the cable force, namely along $\dot{\bm{q}}_i$, is lower bounded by a positive quantity $\underline{v}"$.
• Figure 4: On the left: two carriers (black dots) cannot maintain the pose of the object (in grey) while performing non-stop flights; the cables are in black, and colored curves are the carrier's paths. Carriers' velocities and $\dot{\bm{\lambda}}$ are reported in the other two plots.
• Figure 5: Results of three different simulations, one in each column. On the leftmost column, non-stop flights are shown; in the other two columns, cases in which the carriers stop. On the first row of each simulation, a frame in which the 3 flying carriers (black dots) maintain the suspended object at a constant pose. Dotted lines connect the cables' attachment points on the object; the cables are black lines; the path of each carrier is a colored line (blue for carrier 1, red for carrier 2, and yellow for carrier 3); the cross is the object CoM. The values of $\|\dot{\bm{p}}_{R i}\|$ and $\dot{{\lambda}_i}$ are also reported for each simulation.

Theorems & Definitions (8)

• Proposition 1
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• Remark 1
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• Proposition 2
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• proof