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The work includes the development and validation of a Simscape-based model derived from the experimental platform, followed by controller design using four approaches: Harmony-Search-optimized PID, Harmony-Search-optimized Tilt-Integral-Derivative (TID), self-tuning fuzzy PID, and self-tuning fuzzy TID control. System identification was used to obtain a reduced model suitable for efficient controller tuning, and the identified NLARX model achieved an MSE of 0.02662. Controller performance was evaluated using step, sinusoidal, and repeating-sequence stair inputs. The step-response results show that optimized TID substantially reduces steady-state error compared with optimized PID (0.2982 versus 2.2705 for \u03b8\u2081), although with higher overshoot in some cases. For sinusoidal trajectory tracking, the fuzzy TID controller achieved the lowest or nearly lowest RMSE values in the Cartesian directions, including 0.2602\u00a0mm, 0.3046\u00a0mm, and 0.6548\u00a0mm in X, Y, and Z, respectively. For repeating-sequence stair inputs, fuzzy TID again achieved the lowest RMSE across the three axes. Overall, fuzzy TID provides the best accuracy and smoothness, while optimized TID remains a strong lower-complexity alternative for real-time implementation.<\/jats:p>","DOI":"10.1038\/s41598-026-63558-0","type":"journal-article","created":{"date-parts":[[2026,8,16]],"date-time":"2026-08-16T17:53:17Z","timestamp":1786902797000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A comparative study of HSO-optimized PID and TID controllers with fuzzy adaptation for delta robot control"],"prefix":"10.1038","volume":"16","author":[{"given":"Donia","family":"Saleem","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hasan","family":"Eleashy","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mohamed A.","family":"Shamseldin","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,8,16]]},"reference":[{"key":"63558_CR1","doi-asserted-by":"crossref","unstructured":"Lynch, K. 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Therefore, ethics approval, consent to participate, and consent to publish are not applicable.","order":2,"name":"Ethics","label":"Ethics declarations","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"This study aims to develop and compare four control strategies for a Delta robot: HS-optimized PID, HS-optimized TID, self-tuning fuzzy PID, and self-tuning fuzzy TID. 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