{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:47:01Z","timestamp":1760230021503,"version":"build-2065373602"},"reference-count":44,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,7,5]],"date-time":"2022-07-05T00:00:00Z","timestamp":1656979200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Coordenacao de Aperfeicoamento de Pessoal de N\u00c3 vel Superior (Capes, Brasil)","award":["#88887.478205\/2020-00","#2016\/24561-0"],"award-info":[{"award-number":["#88887.478205\/2020-00","#2016\/24561-0"]}]},{"name":"Fundac\u00e3o de Amparo \u00e1 Pesquisa do Estado de S\u00e3o Paulo (Fapesp, Brazil)","award":["#88887.478205\/2020-00","#2016\/24561-0"],"award-info":[{"award-number":["#88887.478205\/2020-00","#2016\/24561-0"]}]},{"name":"RUDN University Strategic Academic Leadership Program","award":["#88887.478205\/2020-00","#2016\/24561-0"],"award-info":[{"award-number":["#88887.478205\/2020-00","#2016\/24561-0"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>When the length of the tether remains constant, the relative planar motion of the tethered subsatellite with respect to the base satellite in a circular orbit around the Earth, is similar to a simple pendulum motion, i.e., there are two kinds of equilibrium points: local vertical and local horizontal positions, which are center and saddle points, respectively. However, when out-of-plane motion is initially excited, the relative motion of the subsatellite presents symmetric quasi-periodic and chaotic behavior. In the first part of this study, such trajectories are analyzed by means of Poincar\u00e9 sections. In the second part, a non-linear tension force by using a Lyapunov approach is proposed for controlling the coupled pitch-roll motion during the deployment and retrieval phases. The goal of this paper is to guide the relative non-linear motion of the subsatellite to the local upward vertical position. The numerical results show that the non-linear tension control steered the subsatellite close to the local vertical direction very well, reducing the quasi-periodic and chaotic oscillations.<\/jats:p>","DOI":"10.3390\/sym14071381","type":"journal-article","created":{"date-parts":[[2022,7,6]],"date-time":"2022-07-06T21:15:52Z","timestamp":1657142152000},"page":"1381","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Deployment and Retrieval Missions from Quasi-Periodic and Chaotic States under a Non-Linear Control Law"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3599-9745","authenticated-orcid":false,"given":"Francisco J. T.","family":"Salazar","sequence":"first","affiliation":[{"name":"National Institute for Space Research, Av. dos Astronautas-Campus do CTA, S\u00e3o Jos\u00e9 dos Campos 12227-010, SP, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7966-3231","authenticated-orcid":false,"given":"Antonio B. A.","family":"Prado","sequence":"additional","affiliation":[{"name":"National Institute for Space Research, Av. dos Astronautas-Campus do CTA, S\u00e3o Jos\u00e9 dos Campos 12227-010, SP, Brazil"},{"name":"Academy of Engineering, Peoples\u2019 Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Street, 117198 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11071-018-4389-5","article-title":"A review of space tether in new applications","volume":"94","author":"Huang","year":"2018","journal-title":"Nonlinear Dyn."},{"key":"ref_2","first-page":"775","article-title":"Nonlinear dynamics and chaos of two-body tethered satellite systems","volume":"85","author":"Nixon","year":"1993","journal-title":"Adv. Astronaut. Sci."},{"key":"ref_3","unstructured":"Nixon, M.S. (1996). Nonlinear Dynamics and Chaos of Tethered Satellite Systems. [Master\u2019s Thesis, McGill University]."},{"key":"ref_4","first-page":"327","article-title":"Chaotic motion of the tethered satellite system","volume":"16","author":"Peng","year":"1996","journal-title":"Tech. Mech."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1007\/BF01170370","article-title":"Transient chaotic oscillations of a tethered satellite system","volume":"127","author":"Steiner","year":"1998","journal-title":"Acta Mech."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1007\/BF03546319","article-title":"Nonlinear Dynamics of Two-Body Tethered Satellite Systems: Constant Length Case","volume":"49","author":"Misra","year":"2001","journal-title":"J. Astronaut. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1169","DOI":"10.1016\/j.actaastro.2008.06.020","article-title":"Dynamics and control of tethered satellite systems","volume":"63","author":"Misra","year":"2008","journal-title":"Acta Astronaut."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"614","DOI":"10.2514\/1.52399","article-title":"Experimental Verification of Periodic Libration of Tethered Satellite System in Elliptic Orbit","volume":"34","author":"Kojima","year":"2011","journal-title":"J. Guid. Control Dyn."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Aslanov, V.S., and Ledkov, A.S. (2012). Dynamics of Tethered Satellite Systems, Woodhead Publishing Limited. [1st ed.].","DOI":"10.1533\/9780857096005"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2761","DOI":"10.1007\/s00707-015-1333-3","article-title":"Chaotic motion analysis of a rigid spacecraft dragging a satellite by an elastic tether","volume":"226","author":"Pang","year":"2015","journal-title":"Acta Mech."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2399","DOI":"10.2514\/1.G001460","article-title":"Chaos Behavior of Space Debris During Tethered Tow","volume":"39","author":"Aslanov","year":"2016","journal-title":"J. Guid. Control Dyn."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1007\/s10483-016-2049-9","article-title":"Nonlinear dynamics of flexible tethered satellite system subject to space environment","volume":"37","author":"Yu","year":"2016","journal-title":"Appl. Math. Mech.-Engl. Ed."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1016\/j.actaastro.2017.07.008","article-title":"Chaotic attitude motion of a low-thrust tug-debris tethered system in a Keplerian orbit","volume":"139","author":"Aslanov","year":"2017","journal-title":"Acta Astronaut."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1016\/j.cnsns.2019.04.026","article-title":"Chaotic motion and control of a tethered-sailcraft system orbiting an asteroid","volume":"77","author":"Lian","year":"2019","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1233","DOI":"10.1007\/s11071-020-05844-8","article-title":"Chaos in a tethered satellite system induced by atmospheric drag and Earth\u2019s oblateness","volume":"101","author":"Yu","year":"2020","journal-title":"Nonlinear Dyn."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1183","DOI":"10.1016\/0094-5765(79)90064-X","article-title":"On the deployment dynamics of tether connected two-body systems","volume":"6","author":"Modi","year":"1979","journal-title":"Acta Astronaut."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/0021-8928(95)00020-P","article-title":"Dynamics of retrieval of a space tethered system","volume":"59","year":"1995","journal-title":"J. Appl. Math. Mech."},{"key":"ref_18","unstructured":"Rupp, C.C. (1975). A tether tension control law for tethered subsatellites deployed along local vertical. NASA TMX-64963, Marshall Space Hight Center."},{"key":"ref_19","unstructured":"Baker, P.W., Dunkin, J.A., Galaboff, Z.J., Johnston, K.D., Kissel, R.R., Rheinfurth, M.H., and Siebel, M.P.L. (1976). Tethered Subsatellite Study. NASA TM X-73314, Marshall Space Hight Center."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1333","DOI":"10.1016\/0094-5765(80)90010-7","article-title":"Optimal-Control of the Shuttle-Tethered System","volume":"7","author":"Bainum","year":"1980","journal-title":"Acta Astronaut."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"45","DOI":"10.2514\/3.8543","article-title":"Tethered Satellite Retrieval with Thruster Augmented Control","volume":"7","author":"Banerjee","year":"1984","journal-title":"J. Guid. Control Dyn."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"12","DOI":"10.2514\/3.20059","article-title":"Dynamics of a subsatellite system supported by two tethers","volume":"9","author":"Misra","year":"1986","journal-title":"J. Guid. Control Dyn."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"243","DOI":"10.2514\/3.20397","article-title":"Mission function control for deployment and retrieval of a subsatellite","volume":"12","author":"Fujii","year":"1989","journal-title":"J. Guid. Control Dyn."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"471","DOI":"10.2514\/3.20663","article-title":"Mission function control of tethered subsatellite deployment\/retrieval: In-plane and Out-of-Plane Motion","volume":"14","author":"Fujii","year":"1991","journal-title":"J. Guid. Control Dyn."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"469","DOI":"10.2514\/3.20662","article-title":"Feedback tether deployment and retrieval","volume":"14","author":"Vadali","year":"1991","journal-title":"J. Guid. Control Dyn."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"729","DOI":"10.2514\/3.20706","article-title":"Feedback control of tethered satellites using Lyapunov stability theory","volume":"14","author":"Vadali","year":"1991","journal-title":"J. Guid. Control Dyn."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1292","DOI":"10.2514\/3.21347","article-title":"Deployment\/retrieval control of tethered subsatellite through an optimal path","volume":"17","author":"Fujii","year":"1994","journal-title":"J. Guid. Control Dyn."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1016\/0094-5765(94)00284-S","article-title":"On the dynamics and control of the relative motion between two spacecraft","volume":"35","author":"Yu","year":"1995","journal-title":"Acta Astronaut."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1016\/S0093-6413(97)00021-9","article-title":"A new tension control law for deployment of tethered satellites","volume":"24","author":"Pradeep","year":"1997","journal-title":"Mech. Res. Commun."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1016\/S0093-6413(98)00071-8","article-title":"Strategies for three dimensional deployment of tethered satellites","volume":"25","author":"Kumar","year":"1998","journal-title":"Mech. Res. Commun."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"849","DOI":"10.1016\/S0094-5765(00)00133-8","article-title":"Tethered Satellite System Analysis(1)\u2014Two Dimensional Case and Regular Dynamics","volume":"47","author":"Yu","year":"2000","journal-title":"Acta Astronaut."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1177\/107754603030747","article-title":"Various Methods of Controlling the Deployment of a Tethered Satellite","volume":"9","author":"Barkow","year":"2003","journal-title":"J. Vib. Control"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1023\/A:1022956002484","article-title":"Optimal control of deployment of a tethered subsatellite","volume":"31","author":"Steindl","year":"2003","journal-title":"Nonlinear Dyn."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1007\/s11071-006-9021-4","article-title":"Optimal Control of a Tethered Subsatellite of Three Degrees of Freedom","volume":"46","author":"Jin","year":"2006","journal-title":"Nonlinear Dyn."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"501","DOI":"10.1007\/s11071-007-9240-3","article-title":"Optimal feedback control of the deployment of a tethered subsatellite subject to perturbations","volume":"51","author":"Wen","year":"2008","journal-title":"Nonlinear Dyn."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"915","DOI":"10.1080\/00207160701582085","article-title":"Three-dimensional optimal deployment of a tethered subsatellite with an elastic tether","volume":"85","author":"Wen","year":"2008","journal-title":"Int. J. Comput. Math."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"845","DOI":"10.1016\/j.actaastro.2010.05.013","article-title":"Deployment and retrieval of tethered satellite system under J2 perturbation and heating effect","volume":"67","author":"Yu","year":"2010","journal-title":"Acta Astronaut."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"2057","DOI":"10.2514\/1.G000496","article-title":"Fractional-Order Tension Control Lawfor Deployment of Space Tether System","volume":"37","author":"Sun","year":"2014","journal-title":"J. Guid. Control Dyn."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"580","DOI":"10.2514\/1.G006364","article-title":"Suppression of Chaotic Motion of Tethered Satellite Systems Using Tether Length Control","volume":"45","author":"Salazar","year":"2021","journal-title":"J. Guid. Control Dyn."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"22","DOI":"10.2174\/1877611611202010022","article-title":"A Review of Space Tether Technology","volume":"2","author":"Williams","year":"2012","journal-title":"Recent Patents Space Technol."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"278","DOI":"10.2514\/3.56168","article-title":"Deployment and Retrieval of a Subsatellite Connected by a Tether to the Space Shuttle","volume":"5","author":"Misra","year":"1982","journal-title":"J. Guid. Control Dyn."},{"key":"ref_42","unstructured":"Wiggins, S. (2003). Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer. [2nd ed.]."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"012116","DOI":"10.1088\/1742-6596\/744\/1\/012116","article-title":"Chaotic motions of a tethered satellite system in circular orbit","volume":"744","author":"Jin","year":"2016","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"663","DOI":"10.2514\/3.20162","article-title":"Thruster-Augmented Active Control of a Tethered Subsatellite System During its Retrieval","volume":"9","author":"Xu","year":"1986","journal-title":"J. Guid. Control Dyn."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/7\/1381\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:42:57Z","timestamp":1760139777000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/7\/1381"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,5]]},"references-count":44,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2022,7]]}},"alternative-id":["sym14071381"],"URL":"https:\/\/doi.org\/10.3390\/sym14071381","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,7,5]]}}}