{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T20:02:52Z","timestamp":1767211372168,"version":"build-2065373602"},"reference-count":40,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,4,9]],"date-time":"2025-04-09T00:00:00Z","timestamp":1744156800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"European Union\u2014NextGenerationEU and Romanian Government","award":["121\/15.11.2022"],"award-info":[{"award-number":["121\/15.11.2022"]}]},{"name":"Romanian Ministry of Research, Innovation and Digitalization","award":["121\/15.11.2022"],"award-info":[{"award-number":["121\/15.11.2022"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>Robotic-assisted motor rehabilitation has seen significant development over the past decade, driven by the distinct advantages that robots offer in this medical task. A key aspect of robotic-assisted rehabilitation is ensuring that the performed rehabilitation exercises are safely planned (i.e., without the risk of patient injury or anatomic joint strain). This paper presents a new numerical approach to trajectory planning for a LegUp parallel robot designed for lower limb rehabilitation. The proposed approach generates S-shaped motion profiles, also called S-curves, with precise control over all kinematic parameters, resulting in smooth acceleration and deceleration. This approach ensures the safety and effectiveness of rehabilitation exercises by minimizing strain on the patient\u2019s anatomical joints. The mathematical models employed (numerical integration and differentiation) are well-established and computationally efficient for real-time implementation in the robot\u2019s control hardware. Experimental tests using LegUp validate the effectiveness of the proposed trajectory-smoothing approach.<\/jats:p>","DOI":"10.3390\/math13081241","type":"journal-article","created":{"date-parts":[[2025,4,9]],"date-time":"2025-04-09T12:05:56Z","timestamp":1744200356000},"page":"1241","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Numerical Approach for Trajectory Smoothing for LegUp Rehabilitation Parallel Robot"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4026-2318","authenticated-orcid":false,"given":"Iosif","family":"Birlescu","sequence":"first","affiliation":[{"name":"CESTER\u2014Research Center for Industrial Robots Simulation and Testing, Technical University of Cluj-Napoca, Memorandumului 28, 400114 Cluj-Napoca, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8752-6747","authenticated-orcid":false,"given":"Vlad","family":"Mihaly","sequence":"additional","affiliation":[{"name":"CESTER\u2014Research Center for Industrial Robots Simulation and Testing, Technical University of Cluj-Napoca, Memorandumului 28, 400114 Cluj-Napoca, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2822-9790","authenticated-orcid":false,"given":"Calin","family":"Vaida","sequence":"additional","affiliation":[{"name":"CESTER\u2014Research Center for Industrial Robots Simulation and Testing, Technical University of Cluj-Napoca, Memorandumului 28, 400114 Cluj-Napoca, Romania"}]},{"given":"Andrei","family":"Caprariu","sequence":"additional","affiliation":[{"name":"CESTER\u2014Research Center for Industrial Robots Simulation and Testing, Technical University of Cluj-Napoca, Memorandumului 28, 400114 Cluj-Napoca, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5660-8259","authenticated-orcid":false,"given":"Paul","family":"Tucan","sequence":"additional","affiliation":[{"name":"CESTER\u2014Research Center for Industrial Robots Simulation and Testing, Technical University of Cluj-Napoca, Memorandumului 28, 400114 Cluj-Napoca, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4917-2474","authenticated-orcid":false,"given":"Jose","family":"Machado","sequence":"additional","affiliation":[{"name":"CESTER\u2014Research Center for Industrial Robots Simulation and Testing, Technical University of Cluj-Napoca, Memorandumului 28, 400114 Cluj-Napoca, Romania"},{"name":"MEtRICs Research Center, Campus of Azur\u00e9m, University of Minho, 4800-058 Guimar\u00e3es, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7014-9431","authenticated-orcid":false,"given":"Doina","family":"Pisla","sequence":"additional","affiliation":[{"name":"CESTER\u2014Research Center for Industrial Robots Simulation and Testing, Technical University of Cluj-Napoca, Memorandumului 28, 400114 Cluj-Napoca, Romania"},{"name":"Technical Sciences Academy of Romania, B-dul Dacia, 26, 030167 Bucharest, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1186\/1743-0003-11-111","article-title":"Training modalities in robot-mediated upper limb rehabilitation in stroke: A framework for classification based on a systematic review","volume":"10","author":"Basteris","year":"2014","journal-title":"J. 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