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The robot, composed of a U-shaped aluminum elastic beam and lightweight wooden feet, attains steady mobility by synchronizing torsional vibrations with centrifugal forces. Experimental and simulated analyses were performed at three angular velocities (\u03c9\n                    <jats:sub>1<\/jats:sub>\n                    \u2009=\u2009125.66\u00a0rad\/s, \u03c9\n                    <jats:sub>2<\/jats:sub>\n                    \u2009=\u2009251.33\u00a0rad\/s, \u03c9\n                    <jats:sub>3<\/jats:sub>\n                    \u2009=\u2009376.99\u00a0rad\/s) to assess stability, deviation, velocity, and hopping performance. The results indicated that a rise in angular velocity significantly improved locomotion efficiency. At \u03c9\n                    <jats:sub>1<\/jats:sub>\n                    , the robot attained an average body velocity of 85.46\u00a0mm\/s and a hopping distance of 405.88\u00a0mm. At \u03c9\n                    <jats:sub>2<\/jats:sub>\n                    , the velocity rose to 221.24\u00a0mm\/s with a hopping distance of 418.24\u00a0mm, but \u03c9\n                    <jats:sub>3<\/jats:sub>\n                    achieved optimal performance with a velocity of 265.49\u00a0mm\/s and a hopping distance of 424.08\u00a0mm. Deviation responses settled within \u00b1\u20092\u00a0mm after 2.5\u00a0s at \u03c9\n                    <jats:sub>3<\/jats:sub>\n                    , in contrast to more pronounced oscillations at \u03c9\n                    <jats:sub>1<\/jats:sub>\n                    . Simulation results largely aligned with experimental outcomes in hopping distance (error\u2009&lt;\u20093\u00a0mm at \u03c9\n                    <jats:sub>3<\/jats:sub>\n                    ) but routinely underestimated body velocity by 50\u201360%. The findings corroborate the suggested model for predicting vertical displacement, while underscoring the necessity for refinement to accurately capture horizontal velocity dynamics.\n                  <\/jats:p>","DOI":"10.1038\/s41598-025-23496-9","type":"journal-article","created":{"date-parts":[[2025,12,1]],"date-time":"2025-12-01T12:25:36Z","timestamp":1764591936000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Experimental validation and simulation of a U-Shaped elastic beam robot for stable running locomotion"],"prefix":"10.1038","volume":"15","author":[{"given":"Wael","family":"Khalifa","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mahmoud A.","family":"Essam","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"M. 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