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ARC: Alignment-based RPM Estimation with Curvature-adaptive Tracking

Weiheng Hua, Changyu Hao

Abstract

Tacho-less rotational speed estimation is critical for vibration-based prognostics and health management (PHM) of rotating machinery, yet traditional methods--such as time-domain periodicity, cepstrum, and harmonic comb matching--struggle under noise, non-stationarity, and inharmonic interference. Probabilistic tracking offers a principled way to fuse multiple estimators, but a major challenge is that heterogeneous estimators produce evidence on incompatible axes and scales. We address this with ARC (Alignment-based RPM Estimation with Curvature-adaptive Tracking) by unifying the observation representation. Each estimator outputs a one-dimensional evidence curve on its native axis, which is mapped onto a shared RPM grid and converted into a comparable grid-based log-likelihood via robust standardization and a Gibbs-form energy shaping. Standard recursive filtering with fixed-variance motion priors can fail under multi-modal or ambiguous evidence. To overcome this, ARC introduces a curvature-informed, state-dependent motion prior, where the transition variance is derived from the local discrete Hessian of the previous log-posterior. This design enforces smooth tracking around confident modes while preserving competing hypotheses, such as octave alternatives. Experiments on synthetic stress tests and real vibration-table data demonstrate stable, physically plausible trajectories with interpretable uncertainty, and ablations confirm that these gains arise from uncertainty-aware temporal propagation rather than per-frame peak selection or ad hoc rules.

ARC: Alignment-based RPM Estimation with Curvature-adaptive Tracking

Abstract

Tacho-less rotational speed estimation is critical for vibration-based prognostics and health management (PHM) of rotating machinery, yet traditional methods--such as time-domain periodicity, cepstrum, and harmonic comb matching--struggle under noise, non-stationarity, and inharmonic interference. Probabilistic tracking offers a principled way to fuse multiple estimators, but a major challenge is that heterogeneous estimators produce evidence on incompatible axes and scales. We address this with ARC (Alignment-based RPM Estimation with Curvature-adaptive Tracking) by unifying the observation representation. Each estimator outputs a one-dimensional evidence curve on its native axis, which is mapped onto a shared RPM grid and converted into a comparable grid-based log-likelihood via robust standardization and a Gibbs-form energy shaping. Standard recursive filtering with fixed-variance motion priors can fail under multi-modal or ambiguous evidence. To overcome this, ARC introduces a curvature-informed, state-dependent motion prior, where the transition variance is derived from the local discrete Hessian of the previous log-posterior. This design enforces smooth tracking around confident modes while preserving competing hypotheses, such as octave alternatives. Experiments on synthetic stress tests and real vibration-table data demonstrate stable, physically plausible trajectories with interpretable uncertainty, and ablations confirm that these gains arise from uncertainty-aware temporal propagation rather than per-frame peak selection or ad hoc rules.

Paper Structure

This paper contains 27 sections, 19 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Proposed Method: Alignment-based RPM Estimation with Curvature-adaptive Tracking
  4. Overview
  5. RPM Grid and Notation
  6. Heterogeneous Evidence Curves
  7. YIN (Lag-domain cost curve)
  8. Cepstrum (Quefrency-domain score curve)
  9. Comb matching (Frequency-domain score curve)
  10. Likelihood Construction from Heterogeneous Curves
  11. Multi-Estimator Fusion
  12. Curvature-Informed State-Dependent Motion Prior
  13. Grid-Based Recursive Filtering
  14. Algorithm Summary
  15. Experiments
  16. ...and 12 more sections

Figures (3)

  • Figure 1: Top row (a--c): synthetic frame with ground-truth speed. Bottom row (d--f): vibration-table frame with tachometer reference (evaluation only). (a,d) Per-estimator C2G evidences on the shared RPM grid. (b,e) Fused likelihood; C2G-Fusion reports MAP. (c,f) ARC posterior update; ARC reports MMSE with uncertainty. Curves are max-normalized. Conflict frames selected by maximum posterior entropy.
  • Figure 2: Tracking ablation: C2G-Fusion (Framewise) vs. ARC. (a) Synthetic step-change (S5) with ground-truth speed. (b) Vibration-table segment with tachometer reference. Shaded band: $\pm\sigma$. Inset: RMSE/P95/Jitter/Max-Jump.
  • Figure 3: Real vibration-table case study. (a) Raw acceleration (0.2 s). (b) Order spectrum. (c) RPM trajectories: YIN/cepstrum/comb (dashed) and Viterbi-STFT vs. ARC (MMSE, with $\pm\sigma$ band); inset: metrics. (d) Tracking error vs. tachometer reference.