Sub-Frame Contact-Onset Estimation in a Self-Calibrated BJT Thermal Pixel Array Using a Four-Frame erfc Template

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—252 ms at the 4 Hz rate of the 16 × 16 array studied here—well 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 (τ1), solved by a grid-initialized semi-analytic Levenberg–Marquardt scheme (Path A) at deterministic single-pass cost. On 42 contacts from five subjects, the per-contact Cramér–Rao lower bound for τ1 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—ramp 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.