{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,29]],"date-time":"2026-06-29T06:45:14Z","timestamp":1782715514559,"version":"3.54.5"},"reference-count":30,"publisher":"MDPI AG","issue":"13","license":[{"start":{"date-parts":[[2026,6,26]],"date-time":"2026-06-26T00:00:00Z","timestamp":1782432000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Low-cost bipolar-junction-transistor (BJT) thermal pixel arrays provide robust, force-free contact sensing for tactile skins, but their slow frame rate confines contact-timing resolution to the inter-frame interval\u2014252 ms at the 4 Hz rate of the 16 \u00d7 16 array studied here\u2014well below the needs of contact-aware control. We propose a four-frame complementary-error-function (erfc) template, derived from one-dimensional semi-infinite heat conduction, that jointly estimates the contact amplitude, the thermal-diffusion parameter, and the sub-frame contact-onset offset (\u03c41), solved by a grid-initialized semi-analytic Levenberg\u2013Marquardt scheme (Path A) at deterministic single-pass cost. On 42 contacts from five subjects, the per-contact Cram\u00e9r\u2013Rao lower bound for \u03c41 is 16.2 ms, and the empirical cross-contact dispersion is 83.5 ms; both are internal, model-derived quantities, since no synchronised external timing reference was available. A two-layer rejection pipeline separates 19\/19 valid contacts from 2\/2 hardware faults; transfers to four held-out subjects (23\/23) without retuning; attains an overall AUC of 0.878 on a five-class synthetic disturbance library\u2014ramp and saturating-exponential remain acknowledged failure modes; and rejects 5\/6 disturbance trials in a real-airflow stress session. Larger independent cohorts and externally synchronised timing validation remain parameters for future work.<\/jats:p>","DOI":"10.3390\/s26134074","type":"journal-article","created":{"date-parts":[[2026,6,29]],"date-time":"2026-06-29T05:49:33Z","timestamp":1782712173000},"page":"4074","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Sub-Frame Contact-Onset Estimation in a Self-Calibrated BJT Thermal Pixel Array Using a Four-Frame erfc Template"],"prefix":"10.3390","volume":"26","author":[{"given":"Yinglei","family":"Ma","sequence":"first","affiliation":[{"name":"College of Smart Materials and Future Energy, Fudan University, Shanghai 200433, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fei","family":"Xiao","sequence":"additional","affiliation":[{"name":"College of Smart Materials and Future Energy, Fudan University, Shanghai 200433, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,6,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Fraden, J. 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