{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T04:33:25Z","timestamp":1780374805386,"version":"3.54.1"},"reference-count":50,"publisher":"American Association for the Advancement of Science (AAAS)","issue":"81","content-domain":{"domain":["www.science.org"],"crossmark-restriction":true},"short-container-title":["Sci. Robot."],"published-print":{"date-parts":[[2023,8,30]]},"abstract":"<jats:p>Soft robots promise improved safety and capability over rigid robots when deployed near humans or in complex, delicate, and dynamic environments. However, infinite degrees of freedom and the potential for highly nonlinear dynamics severely complicate their modeling and control. Analytical and machine learning methodologies have been applied to model soft robots but with constraints: quasi-static motions, quasi-linear deflections, or both. Here, we advance the modeling and control of soft robots into the inertial, nonlinear regime. We controlled motions of a soft, continuum arm with velocities 10 times larger and accelerations 40 times larger than those of previous work and did so for high-deflection shapes with more than 110\u00b0 of curvature. We leveraged a data-driven learning approach for modeling, based on Koopman operator theory, and we introduce the concept of the static Koopman operator as a pregain term in optimal control. Our approach is rapid, requiring less than 5 min of training; is computationally low cost, requiring as little as 0.5 s to build the model; and is design agnostic, learning and accurately controlling two morphologically different soft robots. This work advances rapid modeling and control for soft robots from the realm of quasi-static to inertial, laying the groundwork for the next generation of compliant and highly dynamic robots.<\/jats:p>","DOI":"10.1126\/scirobotics.add6864","type":"journal-article","created":{"date-parts":[[2023,8,30]],"date-time":"2023-08-30T17:58:19Z","timestamp":1693418299000},"update-policy":"https:\/\/doi.org\/10.34133\/aaas_crossmark","source":"Crossref","is-referenced-by-count":119,"title":["Control of soft robots with inertial dynamics"],"prefix":"10.1126","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9393-6865","authenticated-orcid":true,"given":"David A.","family":"Haggerty","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-4228-8674","authenticated-orcid":true,"given":"Michael J.","family":"Banks","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0921-5548","authenticated-orcid":true,"given":"Ervin","family":"Kamenar","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA."},{"name":"Faculty of Engineering, University of Rijeka, Rijeka, Croatia."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Alan B.","family":"Cao","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106, USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-9238-6352","authenticated-orcid":true,"given":"Patrick C.","family":"Curtis","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2873-9013","authenticated-orcid":true,"given":"Igor","family":"Mezi\u0107","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0420-5025","authenticated-orcid":true,"given":"Elliot W.","family":"Hawkes","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"221","reference":[{"key":"e_1_3_3_2_2","doi-asserted-by":"crossref","unstructured":"I. Ermolov in Industrial Robotics Review (Springer International Publishing 2020) pp.\u00a0195\u2013204.","DOI":"10.1007\/978-3-030-37841-7_16"},{"key":"e_1_3_3_3_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00170-019-03403-z"},{"key":"e_1_3_3_4_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.matpr.2021.09.263"},{"key":"e_1_3_3_5_2","doi-asserted-by":"crossref","unstructured":"S. Bragan\u00e7a E.\u00a0Costa I.\u00a0Castellucci P.\u00a0M.\u00a0Arezes in A Brief Overview of the Use of Collaborative Robots in Industry 4.0: Human Role and Safety (Springer International Publishing 2019) pp.\u00a0641\u2013650.","DOI":"10.1007\/978-3-030-14730-3_68"},{"key":"e_1_3_3_6_2","doi-asserted-by":"crossref","unstructured":"D. Kragic J.\u00a0Gustafson H.\u00a0Karaoguz P.\u00a0Jensfelt R.\u00a0Krug Interactive collaborative robots: Challenges and opportunities in International Joint Conference on Artificial Intelligence (IJCAI 2018) pp.\u00a018\u201325.","DOI":"10.24963\/ijcai.2018\/3"},{"key":"e_1_3_3_7_2","doi-asserted-by":"publisher","DOI":"10.1115\/1.4046238"},{"key":"e_1_3_3_8_2","doi-asserted-by":"publisher","DOI":"10.1038\/nature14543"},{"key":"e_1_3_3_9_2","doi-asserted-by":"publisher","DOI":"10.1089\/soro.2013.0001"},{"key":"e_1_3_3_10_2","doi-asserted-by":"publisher","DOI":"10.1089\/soro.2017.0007"},{"key":"e_1_3_3_11_2","doi-asserted-by":"publisher","DOI":"10.1002\/aisy.202100165"},{"key":"e_1_3_3_12_2","doi-asserted-by":"crossref","unstructured":"L. Shi Z.\u00a0Liu K.\u00a0Karydis Koopman operators for modeling and control of soft robotics. arXiv:2301.09708 [cs.RO] (7 February 2023).","DOI":"10.1007\/s43154-023-00099-8"},{"key":"e_1_3_3_13_2","doi-asserted-by":"publisher","DOI":"10.1146\/annurev-control-062322-100607"},{"key":"e_1_3_3_14_2","doi-asserted-by":"crossref","unstructured":"R.\u00a0K.\u00a0Katzschmann C.\u00a0Della Santina Y.\u00a0Toshimitsu A.\u00a0Bicchi D.\u00a0Rus Dynamic motion control of multi-segment soft robots using piecewise constant curvature matched with an augmented rigid body model in 2019 2nd IEEE International Conference on Soft Robotics (RoboSoft) (IEEE 2019) pp.\u00a0454\u2013461.","DOI":"10.1109\/ROBOSOFT.2019.8722799"},{"key":"e_1_3_3_15_2","doi-asserted-by":"crossref","unstructured":"R.\u00a0K.\u00a0Katzschmann M.\u00a0Thieffry O.\u00a0Goury A.\u00a0Kruszewski T.-M. Guerra C.\u00a0Duriez D.\u00a0Rus Dynamically closed-loop controlled soft robotic arm using a reduced order finite element model with state observer in 2019 2nd IEEE International Conference on Soft Robotics (RoboSoft) (IEEE 2019) pp.\u00a0717\u2013724.","DOI":"10.1109\/ROBOSOFT.2019.8722804"},{"key":"e_1_3_3_16_2","doi-asserted-by":"publisher","DOI":"10.1177\/0278364919897292"},{"key":"e_1_3_3_17_2","doi-asserted-by":"publisher","DOI":"10.1109\/LRA.2020.3010738"},{"key":"e_1_3_3_18_2","doi-asserted-by":"publisher","DOI":"10.1109\/LRA.2022.3146903"},{"key":"e_1_3_3_19_2","doi-asserted-by":"crossref","unstructured":"X. Wang N.\u00a0Rojas A data-efficient model-based learning framework for the closed-loop control of continuum robots. arXiv:2204.10454 [cs.RO] (22 April 2022).","DOI":"10.1109\/RoboSoft54090.2022.9762115"},{"key":"e_1_3_3_20_2","doi-asserted-by":"publisher","DOI":"10.1126\/scirobotics.aav1488"},{"key":"e_1_3_3_21_2","doi-asserted-by":"publisher","DOI":"10.1109\/LRA.2020.2976320"},{"key":"e_1_3_3_22_2","doi-asserted-by":"publisher","DOI":"10.1109\/TRO.2018.2878318"},{"key":"e_1_3_3_23_2","doi-asserted-by":"crossref","unstructured":"S. Neppalli B.\u00a0Jones W.\u00a0McMahan V.\u00a0Chitrakaran I.\u00a0Walker M.\u00a0Pritts M.\u00a0Csencsits C.\u00a0Rahn M.\u00a0Grissom Octarm\u2014A soft robotic manipulator in IEEE\/RSJ International Conference on Intelligent Robots and Systems (IEEE 2007) p.\u00a02569.","DOI":"10.1109\/IROS.2007.4399146"},{"key":"e_1_3_3_24_2","doi-asserted-by":"publisher","DOI":"10.1109\/TRO.2007.906248"},{"key":"e_1_3_3_25_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.17.5.315"},{"key":"e_1_3_3_26_2","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-005-2824-x"},{"key":"e_1_3_3_27_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2004.06.015"},{"key":"e_1_3_3_28_2","doi-asserted-by":"publisher","DOI":"10.1137\/15M1013857"},{"key":"e_1_3_3_29_2","doi-asserted-by":"publisher","DOI":"10.1137\/16M1062296"},{"key":"e_1_3_3_30_2","doi-asserted-by":"publisher","DOI":"10.1109\/LRA.2021.3133001"},{"key":"e_1_3_3_31_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00332-017-9423-0"},{"key":"e_1_3_3_32_2","unstructured":"H. Arbabi I.\u00a0Mezic Computation of transient Koopman spectrum using Hankel-dynamic mode decompoisition in APS Division of Fluid Dynamics (Fall) (American Physical Society 2017) p. G1.009."},{"key":"e_1_3_3_33_2","doi-asserted-by":"publisher","DOI":"10.1137\/17M1125236"},{"key":"e_1_3_3_34_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.automatica.2018.03.046"},{"key":"e_1_3_3_35_2","doi-asserted-by":"crossref","unstructured":"D. Bruder B.\u00a0Gillespie C.\u00a0D.\u00a0Remy R.\u00a0Vasudevan Modeling and control of soft robots using the Koopman operator and model predictive control. arXiv:1902.02827 [cs.RO] (1 July 2019).","DOI":"10.15607\/RSS.2019.XV.060"},{"key":"e_1_3_3_36_2","doi-asserted-by":"crossref","unstructured":"D. Bruder C.\u00a0D.\u00a0Remy R.\u00a0Vasudevan Nonlinear system identification of soft robot dynamics using Koopman operator theory in 2019 International Conference on Robotics and Automation (ICRA ) (IEEE 2019) pp.\u00a06244\u20136250.","DOI":"10.1109\/ICRA.2019.8793766"},{"key":"e_1_3_3_37_2","doi-asserted-by":"crossref","unstructured":"D. Bruder X.\u00a0Fu R.\u00a0B.\u00a0Gillespie C.\u00a0D.\u00a0Remy R.\u00a0Vasudevan Koopman-based control of a soft continuum manipulator under variable loading conditions. arXiv:2002.01407 [cs.RO] (4 February 2020).","DOI":"10.1109\/LRA.2021.3095268"},{"key":"e_1_3_3_38_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.mechatronics.2022.102871"},{"key":"e_1_3_3_39_2","first-page":"1","article-title":"Policy learning with embedded Koopman optimal control","volume":"144","author":"Yin H.","year":"2018","unstructured":"H.\u00a0Yin, M.\u00a0Welle, D.\u00a0Kragic, Policy learning with embedded Koopman optimal control. Proc. Mach. Learn. Res. 144, 1\u201314 (2018).","journal-title":"Proc. Mach. Learn. Res."},{"key":"e_1_3_3_40_2","doi-asserted-by":"publisher","DOI":"10.1109\/TRO.2019.2923880"},{"key":"e_1_3_3_41_2","unstructured":"A. Gibson \u201cApplication of Koopman linear quadratic regulator to the control of a spherical microbubble \u201d thesis University of Colorado Colorado Springs (2022)."},{"key":"e_1_3_3_42_2","doi-asserted-by":"publisher","DOI":"10.1109\/TRO.2021.3076581"},{"key":"e_1_3_3_43_2","doi-asserted-by":"publisher","DOI":"10.1109\/TRO.2020.3038693"},{"key":"e_1_3_3_44_2","doi-asserted-by":"publisher","DOI":"10.3390\/math10071180"},{"key":"e_1_3_3_45_2","first-page":"1087","article-title":"Koopman operator, geometry, and learning of dynamical systems","volume":"68","author":"Mezi\u0107 I.","year":"2021","unstructured":"I.\u00a0Mezi\u0107, Koopman operator, geometry, and learning of dynamical systems. Not. Am. Math. Soc. 68, 1087\u20131105 (2021).","journal-title":"Not. Am. Math. Soc."},{"key":"e_1_3_3_46_2","doi-asserted-by":"crossref","unstructured":"Y. Huang M.\u00a0Hofer R.\u00a0D\u2019Andrea Offset-free model predictive control: A ball catching application with a spherical soft robotic arm in 2021 International Conference on Intelligent Robots and Systems (IROS ) (IEEE\/RSJ 2021) pp.\u00a0563\u2013570.","DOI":"10.1109\/IROS51168.2021.9636608"},{"key":"e_1_3_3_47_2","unstructured":"B.\u00a0D.\u00a0Anderson J.\u00a0B.\u00a0Moore Optimal Control: Linear Quadratic Methods (Courier Corporation 2007)."},{"key":"e_1_3_3_48_2","doi-asserted-by":"publisher","DOI":"10.1109\/MRA.2019.2947538"},{"key":"e_1_3_3_49_2","doi-asserted-by":"publisher","DOI":"10.1109\/LRA.2020.2976309"},{"key":"e_1_3_3_50_2","doi-asserted-by":"crossref","unstructured":"L.\u00a0H.\u00a0Blumenschein N.\u00a0S.\u00a0Usevitch B.\u00a0H.\u00a0Do E.\u00a0W.\u00a0Hawkes A.\u00a0M.\u00a0Okamura Helical actuation on a soft inflated robot body in 2018 IEEE International Conference on Soft Robotics (RoboSoft) (IEEE 2018) pp.\u00a0245\u2013252.","DOI":"10.1109\/ROBOSOFT.2018.8404927"},{"key":"e_1_3_3_51_2","doi-asserted-by":"crossref","unstructured":"A. Kazemipour O.\u00a0Fischer Y.\u00a0Toshimitsu K.\u00a0W.\u00a0Wong R.\u00a0K.\u00a0Katzschmann A robust adaptive approach to dynamic control of soft continuum manipulators. arXiv:2109.11388 [cs.RO] (26 February 2021).","DOI":"10.1109\/ICRA46639.2022.9811715"}],"container-title":["Science Robotics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.science.org\/doi\/pdf\/10.1126\/scirobotics.add6864","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,1,16]],"date-time":"2024-01-16T12:34:26Z","timestamp":1705408466000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.science.org\/doi\/10.1126\/scirobotics.add6864"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,30]]},"references-count":50,"journal-issue":{"issue":"81","published-print":{"date-parts":[[2023,8,30]]}},"alternative-id":["10.1126\/scirobotics.add6864"],"URL":"https:\/\/doi.org\/10.1126\/scirobotics.add6864","relation":{},"ISSN":["2470-9476"],"issn-type":[{"value":"2470-9476","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,8,30]]},"assertion":[{"value":"2022-06-28","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2023-08-02","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2023-08-30","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"eadd6864"}}