Nonlinear Control of a Double Pendulum Electrodynamic Tether System

Abstract

T HE electrodynamic tether (EDT) system uses the Lorentz force as a thruster generated by the interference between the Earth’s magnetic field and the electric current along the tether. The main advantage of this system is that its required electric current can be produced by solar power and electric particles such as Xenon, the mass of which is much smaller than those of traditional chemical propellants. As a result, this system is expected to be employed as a new space technology that will be useful in many types of space applications [1] such as reboosting of the international space station [2]; deorbiting of dysfunction satellites [3] (space debris removal system); the momentum exchange electrodynamic reboost (MXER) [4] system, which could be used to boost payloads from a low Earth orbit (LEO) to a geosynchronous transfer orbit (GTO); and scientific missions that could include the observation of meteors, the highaltitude atmosphere, and the aurora. The motion of the EDT system consists of librations that are similar to themotion of a pendulum and vibrations of the tether that are similar to the vibrations of a string. The librational motion of the tether system in an elliptic orbit is known to be chaotic [5,6]. The librational and vibrational motion of the tethered satellite system must be stabilized to successfully perform space operations such as observation of the atmospheric environment at high altitude. For the purpose of stabilization, a great number of control schemes have been proposed for various aspects of control [7], such as stationkeeping, deployment and retrieval control, generation of optimal trajectories [8], suppression of tether vibrations by electric current variation [9], periodic motion of the electrodynamic tether systemon inclined orbits [10], and effect of the electrodynamic force on orbit [11]. However, to the best knowledge of the authors, except delayed feedback controls, application of nonlinear control schemes to the multipendulum type of electrodynamic tether systems for achieving constellation missions has not yet been widely studied. The tether system treated in this Note consists of two subsatellites and a mother satellite, such as the space shuttle, connected in series via conductive massless tethers. Although this model seems simple, because the mass and flexibility of the tether are ignored, it can be used to investigate the behavior of the in-plane motion of an electrodynamic tether system in an elliptic orbit without significant computational effort and to investigate the applicability of the nonlinear controllers to the simple model. In this study, two nonlinear control methods to stabilize the inplane librational motion of an electrodynamic tether system in an elliptic orbit are investigated. The nonlinear controllers treated in this Note are a decoupling control method [6,12] and amodel-followingdecoupling control method [6,12]. The decoupling control can be used to control each tether attitude independently, and this independent motion is suitable for achieving satellite constellations. If this motion is realized for a three-mass tethered satellite system, various scientific missions will be possible, including the observation of planets with a magnetic field and the observation of the aurora from more than two directions simultaneously. The model-following-decoupling control method is introduced to achieve a periodic motion by which the Earth atmosphere at the specific altitude can be studied periodically. In this study, the periodic in-plane motion of a tethered satellite system in a circular orbit is employed as the reference trajectory for tracking by the actual tethered satellite system in an elliptic orbit. The results of numerical simulations show that the nonlinear control schemes considered herein can stabilize the in-plane librational motion of a double pendulum electrodynamic tether system in an elliptic orbit.