Micro Satellite Orbital Boost by Electrodynamic Tethers
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micromachines Article Micro Satellite Orbital Boost by Electrodynamic Tethers Peter Yao and Timothy Sands * Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14850, USA; [email protected] * Correspondence: [email protected] Abstract: In this manuscript, a method for maneuvering a spacecraft using electrically charged tethers is explored. The spacecraft’s velocity vector can be modified by interacting with Earth’s magnetic field. Through this method, a spacecraft can maintain an orbit indefinitely by reboosting without the constraint of limited propellant. The spacecraft-tether system dynamics in low Earth orbit are simulated to evaluate the effects of Lorentz force and torques on translational motion. With 500-meter tethers charged with a 1-amp current, a 100-kg spacecraft can gain 250 m of altitude in one orbit. By evaluating the combined effects of Lorenz force and the coupled effects of Lorentz torque propagation through Euler’s moment equation and Newton’s translational motion equations, the simulated spacecraft-tether system can orbit indefinitely at altitudes as low as 275 km. Through a rare evaluation of the nonlinear coupling of the six differential equations of motion, the one finding is that an electrodynamic tether can be used to maintain a spacecraft’s orbit height indefinitely for very low Earth orbits. However, the reboost maneuver is inefficient for high inclination orbits and has high electrical power requirement. To overcome greater aerodynamic drag at lower altitudes, longer tethers with higher power draw are required. Keywords: actuators; guidance; navigation; and control; cubesats; mini/micro satellites; spacecraft Citation: Yao, P.; Sands, T. Micro maneuvering; dynamics; magnetic field; tether; orbital dynamics; aerodynamic drag Satellite Orbital Boost by Electrodynamic Tethers. Micromachines 2021, 12, 916. https://doi.org/10.3390/mi12080916 1. Introduction Space, particularly in low-earth orbits, is littered with dangerous debris monitored by Academic Editors: Marco Ciarcia and the National Aeronautics and Space Administration (NASA) who together with the United Ruiqi Shen States Space Force issues warnings of impending space collisions [1]. Ecuador’s first and only satellite in orbit, Pegasus, collided with a Soviet-era rocket still in orbit [2]. In 2019 Received: 6 June 2021 the European Space Agency spacecraft dodged potential collision with Starlink satellite Accepted: 28 July 2021 when it maneuvered to avoid collision [3]. Collision avoidance is very challenging when Published: 31 July 2021 the spacecraft at risk has no remaining fuel to perform maneuver, so this manuscript inves- tigates the effectiveness of maneuvering spacecraft in low Earth orbit using electrodynamic Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in tethers. published maps and institutional affil- Spacecraft in low Earth orbit have a limited lifetime due to deceleration from aero- iations. dynamic drag that eventually cause low-earth orbiting satellites to re-enter (but on on- command). Reboost maneuvers can be conducted to keep them on station, however once the propellant is expended the spacecraft loses its maneuvering capability. Spacecraft equipped with electrodynamic tethers are able to take advantage of Earth’s magnetic field and maneuver to avoid collision by generating miss-distance, or alterna- Copyright: © 2021 by the authors. tively, reboost using electrical power, which can be generated from solar arrays. Such Licensee MDPI, Basel, Switzerland. This article is an open access article propellant-less maneuvers could extend spacecraft lifetime significantly; however, the distributed under the terms and effectiveness of maneuvering using electrodynamic tethers depends on several variables, conditions of the Creative Commons including orbit eccentricity, inclination, altitude, and the charge and length of the tether. Attribution (CC BY) license (https:// Various trajectories and tether configurations are illustrated via computer simulation in- creativecommons.org/licenses/by/ vestigating the functional limits of electrodynamic maneuvers in low Earth orbits. The 4.0/). proposed developments should prove especially useful for CubeSat whose extensive use of Micromachines 2021, 12, 916. https://doi.org/10.3390/mi12080916 https://www.mdpi.com/journal/micromachines Micromachines 2021, 12, x 2 of 20 Micromachines 2021, 12, 916 2 of 20 The proposed developments should prove especially useful for CubeSat whose extensive use of commercial off-the shelf components for their subsystems makes their cost a small commercialfraction of the off-the cost shelf of traditional components satellites, for their and subsystems thus likely makes ubiquitously their cost lack a small propulsion fraction ofsystems the cost to of maintain traditional low satellites, cost. and thus likely ubiquitously lack propulsion systems to maintainPropellant-free low cost. space maneuvers [4] are currently a hot research topic usually per- formedPropellant-free by angular momentum space maneuvers storage [devices4] are currently such as reaction a hot research wheels or topic control usually moment per- formedgyroscopes by angular [5]. The momentum gyroscopes storage are complicated devices such by assingular reaction mathematics wheels or control [6], which moment was gyroscopesonly recently [5 ].solved The gyroscopes [7]. Further are complicatin complicatedg using by singular momentum mathematics exchange [devices6], which is wastheir onlylimited recently capacity, solved and [7 that]. Further capacity complicating soon becomes using saturated momentum when exchange they are devicesrequired is to their ab- limitedsorb spacecraft capacity, disturbance and that capacity torques soon caused becomes by (aerodynamic saturated when drag they[8], gravity are required gradient to ab-[9], sorbsolar spacecraft wind [9], disturbanceand electro-magnetic torques caused disturba by (aerodynamicnces [10,11]). dragThe prop [8], gravityosals in gradient this manu- [9], solarscript wind reverse [9], the and paradigm electro-magnetic by instead disturbances seeking to [10 utilize,11]). these The proposals external momentum in this manuscript source reverseto intentionally the paradigm modify by spacecraft instead seeking orbits, to particularly utilize these by external orienting momentum an electric sourcetether as to de- in- tentionallypicted in Figure modify 1. spacecraftUtilization orbits, of electric particularly propulsion by orienting was recently an electric proposed tether for as depictedChipSats in[12], Figure and1 the. Utilization dynamics of were electric studied propulsion for CubeSats was recently [13,14], proposed including for electric ChipSats translational [ 12], and thepropulsion dynamics [15–17] were in studied addition for to CubeSats tether-assis [13,ted14], propulsion including electric[18] in the translational inaugural propul-issue of sionAerospace [15–17. ]This in addition manuscript to tether-assisted will study utilization propulsion of [18 electric] in the inauguraltethers to issuegenerate of Aerospace angular. This manuscript will study utilization of electric tethers to generate angular momentum, momentum, which propagates through the six coupled nonlinear equations of motion which propagates through the six coupled nonlinear equations of motion forming the basis forming the basis for modern spacecraft attitude control and guidance proposed in 2007 for modern spacecraft attitude control and guidance proposed in 2007 [19], formulated [19], formulated in 2009 [20], and experimentally validated in 2012 [21]. The method will in 2009 [20], and experimentally validated in 2012 [21]. The method will be evaluated for be evaluated for microsatellites. Electromagnetic tethers were recently proposed for microsatellites. Electromagnetic tethers were recently proposed for spacecraft docking [22], spacecraft docking [22], and many other examples were just articulated in the review by and many other examples were just articulated in the review by O’Reilly et al., 2021 [23], O’Reilly et al., 2021 [23], which was already demonstrated by NASA to produce transla- which was already demonstrated by NASA to produce translational propulsion forces [24]. tional propulsion forces [24]. This manuscript will instead evaluate the resultant transla- This manuscript will instead evaluate the resultant translational effects of coupled motion tional effects of coupled motion due to externally applied forces and torques through teth- due to externally applied forces and torques through tethers. Multisatellite tether systems ers. Multisatellite tether systems are not addressed. are not addressed. (a) (b) FigureFigure 1.1. ((aa)) Tether Tether orientation orientation for for reboost. reboost. (b ()b Les) Les Johnson, Johnson, a scie a scientistntist at Marshall at Marshall Space Space Flight Flight Cen- Centerter inspects inspects nonconducting nonconducting part part of a of tether a tether [25]. [25 ]. DevelopmentsDevelopments presentedpresented inin thisthis manuscriptmanuscript followfollow thethe naturenature ofof thethe 19981998 NASANASA studystudy [[10],10], whichwhich focusedfocused onon translationaltranslational thrustthrust generationgeneration usingusing LorentzLorentz forceforce combinedcombined withwith thethe recentrecent augmentationaugmentation byby WeisWeis andand PeckPeck[ [12],12], which whichevaluated evaluated attitudeattitude controlcontrol byby LorentzLorentz torquestorques determined determined by