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The proposed approach utilizes a Lagrangian optimization method that employs the Steepest Descent Method and natural coordinates related by the constraint equations defining the admissible configurations of the mechanism. The proposed method is validated through the synthesis of four mechanisms of varying complexity. The numerical examples shown are slider-crank mechanisms with and without slider eccentricity, a four-bar mechanism, and the Theo Jansen linkage. The results demonstrate a strong agreement between the desired paths and those generated by the synthesized mechanisms, even in highly complex mechanisms such as those of Theo Jansen.\n                  <\/jats:p>","DOI":"10.1007\/s10999-026-09938-w","type":"journal-article","created":{"date-parts":[[2026,6,26]],"date-time":"2026-06-26T07:26:24Z","timestamp":1782458784000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Lagrangian algorithm with steepest descent method applied to the planar mechanism synthesis problem using natural coordinates"],"prefix":"10.1007","volume":"22","author":[{"given":"Hern\u00e1n A. Gonzalez","family":"Rojas","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Enrique E. Zayas","family":"Figueras","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Amelia E. 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