TETHERED SYSTEM MOTION STABILITY

Анотація

This paper investigates the motion stability of a payload suspended on a tether, in a formulation suitable for the analysis of tethered suspension systems of unmanned aerial vehicles (UAVs) during maneuvers. The payload is subject to an aerodynamic force: a wind load of an arbitrarily given direction. An air drag proportional to the payload velocity is taken into account.

The theoretical framework is based on the classical spherical pendulum model as a baseline approximate tether–payload subsystem.

Nonlinear equations of motion are derived in Lagrangian form as a system of two nonlinear fourth-order differential equations. The equilibrium position is determined, and its stability is further investigated. The nonlinear system of motion is represented in normal form and linearized. The first-approximation system decouples into two independent subsystems. Based on the condition that the real parts of the roots of the characteristic equation of each subsystem shall be negative, conditions for the asymptotic stability of the system motion are obtained.

The analytical results are validated by numerical computations. A numerical solution to the nonlinear system of differential equations is obtained using the eighth-order Runge–Kutta method. The numerical results are in agreement with the analytical ones, thus validating the latter.

The obtained results and physical limitations allow one to determine the admissible range of the payload mass in which the motion of the system is stable.

The obtained results have practical applications. They may be used in the design of tethered payload suspension systems for UAVs — in particular, for determining the admissible range of the payload mass in which the motion of the system is stable under wind loading. The derived stability criteria may be employed in UAV automatic control systems for real-time assessment of the suspension state and in the certification of tethered cargo delivery systems under wind conditions.

In contrast to classical formulations of the spherical pendulum problem, this paper presents, for the first time, a model that simultaneously accounts for: a wind force of arbitrary three-dimensional direction, an aerodynamic drag, and a physical limitation on tether tension. This made it possible, for the first time in this formulation, to analytically express the minimum and the maximum admissible payload mass in explicit form in terms of the wind load parameters and the tensile strength of the tether material.

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