{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,16]],"date-time":"2026-07-16T14:12:59Z","timestamp":1784211179482,"version":"3.55.0"},"reference-count":91,"publisher":"MIT Press","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Neural Computation"],"published-print":{"date-parts":[[2013,2]]},"abstract":"<jats:p>Nonlinear dynamical systems have been used in many disciplines to model complex behaviors, including biological motor control, robotics, perception, economics, traffic prediction, and neuroscience. While often the unexpected emergent behavior of nonlinear systems is the focus of investigations, it is of equal importance to create goal-directed behavior (e.g., stable locomotion from a system of coupled oscillators under perceptual guidance). Modeling goal-directed behavior with nonlinear systems is, however, rather difficult due to the parameter sensitivity of these systems, their complex phase transitions in response to subtle parameter changes, and the difficulty of analyzing and predicting their long-term behavior; intuition and time-consuming parameter tuning play a major role. This letter presents and reviews dynamical movement primitives, a line of research for modeling attractor behaviors of autonomous nonlinear dynamical systems with the help of statistical learning techniques. The essence of our approach is to start with a simple dynamical system, such as a set of linear differential equations, and transform those into a weakly nonlinear system with prescribed attractor dynamics by means of a learnable autonomous forcing term. Both point attractors and limit cycle attractors of almost arbitrary complexity can be generated. We explain the design principle of our approach and evaluate its properties in several example applications in motor control and robotics.<\/jats:p>","DOI":"10.1162\/neco_a_00393","type":"journal-article","created":{"date-parts":[[2012,11,13]],"date-time":"2012-11-13T18:17:01Z","timestamp":1352830621000},"page":"328-373","source":"Crossref","is-referenced-by-count":1404,"title":["Dynamical Movement Primitives: Learning Attractor Models for Motor Behaviors"],"prefix":"10.1162","volume":"25","author":[{"given":"Auke Jan","family":"Ijspeert","sequence":"first","affiliation":[{"name":"Ecole Polytechnique F\u00e9d\u00e9rale de Lausanne, Lausanne CH-1015, Switzerland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jun","family":"Nakanishi","sequence":"additional","affiliation":[{"name":"School of Informatics, University of Edinburgh, Edinburgh EH8 9AB, U.K."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Heiko","family":"Hoffmann","sequence":"additional","affiliation":[{"name":"Computer Science, Neuroscience, and Biomedical Engineering, University of Southern California, Los Angeles, CA 90089, U.S.A."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Peter","family":"Pastor","sequence":"additional","affiliation":[{"name":"Computer Science, Neuroscience, and Biomedical Engineering, University of Southern California, Los Angeles, CA 90089, U.S.A."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Stefan","family":"Schaal","sequence":"additional","affiliation":[{"name":"Computer Science, Neuroscience, and Biomedical Engineering, University of Southern California, Los Angeles, CA 90089, U.S.A.; Max-Planck-Institute for Intelligent Systems, T\u00fcbingen 72076, Germany; and ATR Computational Neuroscience Laboratories, Kyoto 619-0288, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"281","reference":[{"key":"B1","doi-asserted-by":"publisher","DOI":"10.1109\/5254.867912"},{"key":"B2","volume-title":"The control and regulation of movements","author":"Bernstein N. A.","year":"1967"},{"key":"B3","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-30301-5_60"},{"key":"B4","first-page":"1","volume":"941","author":"Billard A.","year":"2001","journal-title":"Robotics and Autonomous Systems"},{"key":"B5","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-45528-0"},{"key":"B6","doi-asserted-by":"publisher","DOI":"10.1007\/s00422-006-0128-y"},{"key":"B7","doi-asserted-by":"publisher","DOI":"10.1109\/ROBOT.1990.126296"},{"key":"B8","doi-asserted-by":"publisher","DOI":"10.1016\/S0166-4115(08)61915-9"},{"key":"B9","doi-asserted-by":"publisher","DOI":"10.1177\/02783649922066385"},{"key":"B10","doi-asserted-by":"publisher","DOI":"10.1177\/0278364905054929"},{"key":"B11","volume-title":"Theoretical neuroscience","author":"Dayan P.","year":"2001"},{"key":"B12","doi-asserted-by":"publisher","DOI":"10.1007\/BF00198467"},{"key":"B13","doi-asserted-by":"publisher","DOI":"10.1037\/0096-1523.29.2.343"},{"key":"B14","doi-asserted-by":"publisher","DOI":"10.1523\/JNEUROSCI.05-07-01688.1985"},{"key":"B15","doi-asserted-by":"publisher","DOI":"10.1016\/S0959-4388(01)00265-3"},{"key":"B16","volume-title":"Control system design: An introduction to state-space methods","author":"Friedland B.","year":"1986"},{"key":"B17","doi-asserted-by":"publisher","DOI":"10.1007\/s10514-009-9118-y"},{"key":"B18","first-page":"101","volume-title":"Neural control of rhythmic movements in vertebrates","author":"Getting P. A.","year":"1988"},{"key":"B19","doi-asserted-by":"publisher","DOI":"10.1523\/JNEUROSCI.13-02-00467.1993"},{"key":"B20","doi-asserted-by":"publisher","DOI":"10.1126\/science.272.5258.117"},{"key":"B21","doi-asserted-by":"publisher","DOI":"10.1007\/s004220050329"},{"key":"B22","doi-asserted-by":"publisher","DOI":"10.1177\/0278364910376251"},{"key":"B23","first-page":"1179","volume-title":"Handbook of physiology: The nervous system","volume":"2","author":"Grillner S.","year":"1981"},{"key":"B24","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1140-2"},{"key":"B25","doi-asserted-by":"publisher","DOI":"10.1016\/0893-6080(90)90022-D"},{"key":"B26","doi-asserted-by":"publisher","DOI":"10.1080\/00222895.1996.9941728"},{"key":"B27","doi-asserted-by":"publisher","DOI":"10.1109\/ROBOT.2009.5152423"},{"key":"B28","first-page":"139","volume":"106","author":"Hollerbach J. M.","year":"1984","journal-title":"Transactions of the ASME"},{"key":"B29","doi-asserted-by":"publisher","DOI":"10.1007\/s004220000211"},{"key":"B30","doi-asserted-by":"publisher","DOI":"10.1016\/j.neunet.2008.03.014"},{"key":"B31","doi-asserted-by":"publisher","DOI":"10.1177\/105971239900700202"},{"key":"B32","doi-asserted-by":"publisher","DOI":"10.1109\/IRDS.2002.1041514"},{"key":"B33","doi-asserted-by":"publisher","DOI":"10.1109\/ROBOT.2002.1014739"},{"key":"B34","first-page":"1547","volume-title":"Advances in neural information processing systems","volume":"15","author":"Ijspeert A. J.","year":"2003"},{"key":"B35","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511623974"},{"key":"B36","doi-asserted-by":"publisher","DOI":"10.1126\/science.1091277"},{"key":"B37","doi-asserted-by":"publisher","DOI":"10.1162\/0899766054026684"},{"key":"B38","first-page":"225","volume-title":"Advances in motor learning and control","author":"Kawato M.","year":"1996"},{"key":"B39","volume-title":"Dynamic patterns: The self-organization of brain and behavior","author":"Kelso J. A. S.","year":"1995"},{"key":"B40","doi-asserted-by":"publisher","DOI":"10.1016\/0375-9601(88)90537-3"},{"key":"B41","doi-asserted-by":"publisher","DOI":"10.1109\/IROS.2010.5651259"},{"key":"B42","doi-asserted-by":"publisher","DOI":"10.1177\/027836498600500106"},{"key":"B43","doi-asserted-by":"publisher","DOI":"10.1109\/ROBOT.2001.933274"},{"key":"B44","doi-asserted-by":"publisher","DOI":"10.1109\/ROBOT.2009.5152577"},{"key":"B45","doi-asserted-by":"publisher","DOI":"10.1109\/ROBOT.1987.1088038"},{"key":"B46","volume-title":"Information, natural law, and the self-assembly of rhythmic movement","author":"Kugler P. N.","year":"1987"},{"key":"B47","doi-asserted-by":"publisher","DOI":"10.1109\/TRO.2011.2163863"},{"key":"B48","volume-title":"Control of human movement","author":"Latash M. L.","year":"1993"},{"key":"B49","doi-asserted-by":"publisher","DOI":"10.1109\/70.782030"},{"key":"B50","doi-asserted-by":"publisher","DOI":"10.1016\/S0005-1098(98)00019-3"},{"key":"B51","doi-asserted-by":"publisher","DOI":"10.1162\/089976602760407955"},{"key":"B52","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(91)90129-W"},{"key":"B53","doi-asserted-by":"publisher","DOI":"10.1016\/j.brainresrev.2007.08.006"},{"key":"B54","doi-asserted-by":"publisher","DOI":"10.1016\/S0893-6080(96)00043-3"},{"key":"B55","doi-asserted-by":"publisher","DOI":"10.1109\/CIRA.1997.613842"},{"key":"B56","doi-asserted-by":"publisher","DOI":"10.1016\/S0959-4388(99)00029-X"},{"key":"B57","doi-asserted-by":"publisher","DOI":"10.1016\/j.robot.2004.03.003"},{"key":"B58","doi-asserted-by":"publisher","DOI":"10.1109\/ROBOT.2002.1014741"},{"key":"B59","doi-asserted-by":"publisher","DOI":"10.1016\/j.neunet.2004.08.005"},{"key":"B60","doi-asserted-by":"publisher","DOI":"10.1109\/ROBOT.2009.5152385"},{"key":"B62","doi-asserted-by":"publisher","DOI":"10.1016\/j.neunet.2008.02.003"},{"key":"B63","doi-asserted-by":"publisher","DOI":"10.1109\/IROS.2005.1545257"},{"key":"B64","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2006.02.009"},{"key":"B65","doi-asserted-by":"publisher","DOI":"10.1109\/70.163777"},{"key":"B66","doi-asserted-by":"publisher","DOI":"10.1109\/ROBOT.1994.350893"},{"key":"B67","doi-asserted-by":"publisher","DOI":"10.1016\/S0166-2236(98)01260-0"},{"key":"B68","doi-asserted-by":"publisher","DOI":"10.1109\/TASSP.1978.1163055"},{"key":"B69","doi-asserted-by":"publisher","DOI":"10.1016\/S1364-6613(99)01327-3"},{"key":"B70","first-page":"160","volume-title":"Advances in neural information processing systems, 6","author":"Schaal S.","year":"1994"},{"key":"B71","doi-asserted-by":"publisher","DOI":"10.1162\/089976698300016963"},{"key":"B72","doi-asserted-by":"publisher","DOI":"10.1098\/rstb.2002.1258"},{"key":"B73","doi-asserted-by":"publisher","DOI":"10.1016\/S0079-6123(06)65027-9"},{"key":"B74","volume-title":"Proceedings of the International Conference on Computational Intelligence in Neuroscience","author":"Schaal S.","year":"1998"},{"key":"B75","doi-asserted-by":"publisher","DOI":"10.1038\/nn1322"},{"key":"B76","doi-asserted-by":"publisher","DOI":"10.1007\/BF00203449"},{"key":"B77","doi-asserted-by":"publisher","DOI":"10.1126\/science.3281253"},{"key":"B79","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4471-0449-0"},{"key":"B80","volume-title":"Encyclopedia of nonlinear science","author":"Scott A.","year":"2005"},{"key":"B81","volume-title":"Applied nonlinear control","author":"Slotine J. J. E.","year":"1991"},{"key":"B82","doi-asserted-by":"publisher","DOI":"10.1080\/00222895.1996.9941750"},{"key":"B83","volume-title":"Nonlinear dynamics and chaos: With applications to physics, biology, chemistry, and engineering","author":"Strogatz S. H.","year":"1994"},{"key":"B85","doi-asserted-by":"publisher","DOI":"10.1007\/BF00198086"},{"key":"B86","doi-asserted-by":"crossref","DOI":"10.7551\/mitpress\/2524.001.0001","volume-title":"A dynamical systems approach to the development of cognition and action","author":"Thelen E.","year":"1994"},{"issue":"11","key":"B87","first-page":"3137","volume":"2010","author":"Theodorou E.","year":"2010","journal-title":"Journal of Machine Learning Research"},{"key":"B88","doi-asserted-by":"publisher","DOI":"10.1038\/nn1309"},{"key":"B89","doi-asserted-by":"publisher","DOI":"10.1109\/TSMCC.2002.807273"},{"key":"B90","doi-asserted-by":"publisher","DOI":"10.1037\/0003-066X.45.8.938"},{"key":"B91","doi-asserted-by":"publisher","DOI":"10.1109\/TRO.2010.2065430"},{"key":"B92","doi-asserted-by":"publisher","DOI":"10.1016\/j.neunet.2003.11.009"},{"key":"B93","doi-asserted-by":"publisher","DOI":"10.1016\/S1364-6613(97)01070-X"},{"key":"B94","doi-asserted-by":"publisher","DOI":"10.1109\/AT-EQUAL.2009.32"}],"container-title":["Neural Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mitpressjournals.org\/doi\/pdf\/10.1162\/NECO_a_00393","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,20]],"date-time":"2025-04-20T20:51:16Z","timestamp":1745182276000},"score":1,"resource":{"primary":{"URL":"https:\/\/direct.mit.edu\/neco\/article\/25\/2\/328-373\/7850"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,2]]},"references-count":91,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2013,2]]}},"alternative-id":["10.1162\/NECO_a_00393"],"URL":"https:\/\/doi.org\/10.1162\/neco_a_00393","relation":{},"ISSN":["0899-7667","1530-888X"],"issn-type":[{"value":"0899-7667","type":"print"},{"value":"1530-888X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,2]]}}}