{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,6]],"date-time":"2026-06-06T07:40:27Z","timestamp":1780731627537,"version":"3.54.1"},"reference-count":56,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2018,9,25]],"date-time":"2018-09-25T00:00:00Z","timestamp":1537833600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Robotics"],"abstract":"<jats:p>The problem of the trajectory-tracking and vibration control of highly flexible planar multi-links robot arms is investigated. We discretize the links according to the Hencky bar-chain model, which is an application of the lumped parameters techniques. In this approach, each link is considered as a kinematic chain of rigid bodies, and suitable springs are added in order to model bending resistance. The control strategy employed is based on an optimal input pre-shaping and a feedback of the joint angles to treat the effects of undesired disturbances. Some numerical examples are given to show the potentialities of the proposed control, and a comparison with a standard collocated Proportional-Derivative (PD) control strategy is performed. In particular, we study the cases of a linear and a parabolic trajectory with a polynomial time law chosen to minimize the onset of possible vibrations.<\/jats:p>","DOI":"10.3390\/robotics7040060","type":"journal-article","created":{"date-parts":[[2018,9,25]],"date-time":"2018-09-25T11:12:26Z","timestamp":1537873946000},"page":"60","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":45,"title":["Non-Linear Lumped-Parameter Modeling of Planar Multi-Link Manipulators with Highly Flexible Arms"],"prefix":"10.3390","volume":"7","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0044-9188","authenticated-orcid":false,"given":"Ivan","family":"Giorgio","sequence":"first","affiliation":[{"name":"Department of Mechanical and Aerospace Engineering, SAPIENZA Universit\u00e0 di Roma, via Eudossiana 18, 00184 Rome, Italy"},{"name":"Research center on Mathematics and Mechanics of Complex Systems, Universit\u00e0 degli studi dell\u2019Aquila, 67100 L\u2019Aquila, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6102-2102","authenticated-orcid":false,"given":"Dionisio","family":"Del Vescovo","sequence":"additional","affiliation":[{"name":"Department of Mechanical and Aerospace Engineering, SAPIENZA Universit\u00e0 di Roma, via Eudossiana 18, 00184 Rome, Italy"},{"name":"Research center on Mathematics and Mechanics of Complex Systems, Universit\u00e0 degli studi dell\u2019Aquila, 67100 L\u2019Aquila, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2018,9,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1177\/027836498400300303","article-title":"Initial experiments on the end-point control of a flexible one-link robot","volume":"3","author":"Cannon","year":"1984","journal-title":"Int. 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