{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,25]],"date-time":"2026-06-25T16:44:46Z","timestamp":1782405886405,"version":"3.54.5"},"reference-count":34,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,7,18]],"date-time":"2025-07-18T00:00:00Z","timestamp":1752796800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"European Union","award":["1058 23\/06\/2022"],"award-info":[{"award-number":["1058 23\/06\/2022"]}]},{"name":"European Union","award":["ECS_00000043"],"award-info":[{"award-number":["ECS_00000043"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Robotics"],"abstract":"<jats:p>Underactuated robotic systems are appealing for industrial use due to their reduced actuator number, which lowers energy consumption and system complexity. Underactuated systems are, however, often affected by residual vibrations. This paper addresses the challenge of generating energy-optimal trajectories while imposing theoretical null residual (and yet practical low) vibration in underactuated systems. The trajectory planning problem is cast as a constrained optimal control problem (OCP) for a two-degree-of-freedom revolute\u2013revolute planar manipulator. The proposed method produces energy-efficient motion while limiting residual vibrations under motor torque limitations. Experiments compare the proposed trajectories to input shaping techniques (ZV, ZVD, NZV, NZVD). Results show energy savings that range from 12% to 69% with comparable and negligible residual oscillations.<\/jats:p>","DOI":"10.3390\/robotics14070098","type":"journal-article","created":{"date-parts":[[2025,7,18]],"date-time":"2025-07-18T12:09:53Z","timestamp":1752840593000},"page":"98","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Minimum-Energy Trajectory Planning for an Underactuated Serial Planar Manipulator"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6308-9474","authenticated-orcid":false,"given":"Domenico","family":"Dona\u2019","sequence":"first","affiliation":[{"name":"Dipartimento di Tecnica e Gestione dei Sistemi Industriali, Universit\u00e0 degli Studi di Padova, 36100 Vicenza, Italy"},{"name":"Dipartimento di Psicologia Generale, Universit\u00e0 degli Studi di Padova, 35131 Padova, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6556-1955","authenticated-orcid":false,"given":"Jason","family":"Bettega","sequence":"additional","affiliation":[{"name":"Dipartimento di Tecnica e Gestione dei Sistemi Industriali, Universit\u00e0 degli Studi di Padova, 36100 Vicenza, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0333-4420","authenticated-orcid":false,"given":"Iacopo","family":"Tamellin","sequence":"additional","affiliation":[{"name":"Dipartimento di Ingegneria per la Medicina di Innovazione, Universit\u00e0 degli Studi di Verona, 37134 Verona, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8032-5272","authenticated-orcid":false,"given":"Paolo","family":"Boscariol","sequence":"additional","affiliation":[{"name":"Dipartimento di Tecnica e Gestione dei Sistemi Industriali, Universit\u00e0 degli Studi di Padova, 36100 Vicenza, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0728-1303","authenticated-orcid":false,"given":"Roberto","family":"Caracciolo","sequence":"additional","affiliation":[{"name":"Dipartimento di Tecnica e Gestione dei Sistemi Industriali, Universit\u00e0 degli Studi di Padova, 36100 Vicenza, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"805","DOI":"10.1038\/s41893-019-0352-9","article-title":"Six transformations to achieve the sustainable development goals","volume":"2","author":"Sachs","year":"2019","journal-title":"Nat. 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