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arxiv: 2606.05334 · v1 · pith:I3FM6EYKnew · submitted 2026-06-03 · 💻 cs.AI

Uncertainty Aware Functional Behavior Prediction and Material Fatigue Assessment for Circular Factory

Nehal Afifi , Mehdi Khabou , Victor Mas , Jonas Hemmerich , Patric Grauberger , Stefan Dietrich , Volker Schulze , Sven Matthiesen This is my paper

Pith reviewed 2026-06-28 06:10 UTC · model grok-4.3

classification 💻 cs.AI
keywords circular factoryfunctional predictionmaterial fatigueLSTMuncertainty estimationangle grinderreuse assessmentPHM
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The pith

An LSTM predicts nine functional outputs with uncertainty for an angle grinder and links the same load history to fatigue estimates for reuse decisions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that returned products with varying degradation states require predictions of both future function and component integrity under new service conditions. It combines a convolutional encoder on force-torque windows with an LSTM that outputs Gaussian means and variances for nine variables, while the identical torque history feeds finite-element stress reconstruction and S-N/Miner plus Paris-law fatigue analysis. A streaming replay algorithm then produces joint functional, material, and system reliability trajectories. Held-out tests reach a mean 2%-tolerance accuracy of 0.9652, with thermal variables near-perfect and motor current and load speed at R² 0.9750 and 0.9924. This directly supports instance-specific reuse choices in circular factories where existing methods treat prognosis and fatigue in isolation.

Core claim

The framework shows that uncertainty-aware functional prediction via LSTM on recent usage windows, when driven by the same loading history used for finite-element-supported stress reconstruction, S-N/Miner damage summation with Haibach extension, and Paris-law crack growth, produces consolidated reliability trajectories that link system-level behavior forecasts to component-level fatigue for reuse assessment.

What carries the argument

LSTM backbone that outputs Gaussian mean and variance estimates for nine functional variables, fed by a convolutional encoder on spindle-force and shaft-torque windows and aligned with the same history for S-N/Miner and Paris-law fatigue evaluation.

If this is right

  • The approach yields instance-specific reliability trajectories that incorporate both functional exceedance probabilities and material damage accumulation.
  • Torque history proves especially informative for the most dynamic outputs such as drive motor current.
  • Conventional LSTM outperforms GRU and xLSTM under short-history conditions for this task.
  • Reliability calibration is strongest for variables like motor current where predicted and observed exceedance probabilities can be checked directly.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same load-history linkage could support adaptive maintenance schedules that adjust based on predicted remaining capability rather than fixed intervals.
  • Extending the workflow to other rotating tools would require only retraining the encoder-LSTM pair on their specific force-torque signatures while reusing the fatigue module.
  • If uncertainty estimates remain well-calibrated across longer horizons, they could set explicit risk thresholds for accepting or rejecting a returned unit.

Load-bearing premise

That fatigue estimates computed from the loading history through finite-element stress and damage models are meaningfully connected to the LSTM functional predictions for making reuse decisions.

What would settle it

A direct comparison showing that observed component failures or remaining-life measurements diverge systematically from the reliability trajectories generated by the combined prediction and fatigue pipeline on new held-out usage sequences.

Figures

Figures reproduced from arXiv: 2606.05334 by Jonas Hemmerich, Mehdi Khabou, Nehal Afifi, Patric Grauberger, Stefan Dietrich, Sven Matthiesen, Victor Mas, Volker Schulze.

Figure 1
Figure 1. Figure 1: Data Preprocessing and Segmentation Framework The learning problem is formulated as 𝑝𝜃 ( 𝐲𝑡 ∣ 𝐬𝑡 , 𝐔𝑡−𝐿+1∶𝑡 ) where 𝜃 denotes the trainable model parameters. This captures the main assumption that future functional behavior depends jointly on the current tool state and recent usage history. The network predicts increments rather than absolute target values. The absolute trajectory is reconstructed by addin… view at source ↗
Figure 2
Figure 2. Figure 2: Architecture of the uncertainty-aware functional behavior prediction model. The current state vector and usage history windows are encoded in separate branches, fused, processed by an LSTM recurrent backbone, and mapped to predictive means and variances for the functional output variables. 4.1.5. Uncertainty-Aware Loss Function The model is trained with a weighted Gaussian negative log-likelihood loss beca… view at source ↗
Figure 3
Figure 3. Figure 3: Component–Material Behavior Prediction Pipeline Evaluation Algorithm Evaluation is performed on held-out files only, preserving file-level separation between training and testing. As shown in Algorithm 2, predictions, variances, and observations are accumulated over the test set and then summarized for each output variable. Algorithm 2 Held-out evaluation on the test set Require: trained model 𝑝𝜃 , held-ou… view at source ↗
Figure 4
Figure 4. Figure 4: Angle-grinder case study used to link system-level functional behavior prediction with component-level material reliability assessment. The gear-stage response is evaluated under controlled force–torque loading, while the output shaft is considered as the fatigue-critical component for material-side reuse assessment. L59-A), consisting of one axial actuator and two radial actuators, coupled to the output s… view at source ↗
Figure 5
Figure 5. Figure 5: Test bench and load cycle for functional degradation data acquisition. The health-indicator variables are the pinion and spindle clearances, selected because increasing mechanical play is directly related to wear progression in the bearing and gear-support interfaces and therefore provides an interpretable representation of degradation. It should be noted that the clearance variables appear in both the sta… view at source ↗
Figure 6
Figure 6. Figure 6: Rotating bending test setup and output shaft specimen geometry [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Finite element model and stress field of the output shaft under rotating bending. The notch root stress serves as the governing fatigue parameter in the material assessment. 5.2.3. Hardness and Microstructure To characterize the hardness distribution and microstructural state of the shaft, specimens were sectioned in the radial direction (𝑋𝑍 plane). Hardness mapping was performed using a Qness Q10 microhar… view at source ↗
Figure 8
Figure 8. Figure 8: Hardness characterization of the output shaft at the critical notch. The effective case-hardened depth of approximately 0.369 mm, determined in accordance with ISO 2639, defines the upper limit for crack propagation used [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Tolerance-sensitivity curves for drive motor current and load speed (CAN), the two outputs with the lowest strict 2% accuracy. were defined as the empirical mean plus 2.5 standard deviations for each output variable. Among the nine outputs, drive motor current was the only variable with clearly non-negligible exceedance event support. Its mean sequence-window failure probability was 0.0600, compared with a… view at source ↗
Figure 10
Figure 10. Figure 10: Sequence-local exceedance analysis for drive motor current and load speed (CAN) on the held-out run. (a) Drive Motor Current. (b) Load Speed (CAN) [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Monte Carlo first-crossing analysis for drive motor current and load speed (CAN). (a) Drive Motor Current. (b) Load Speed (CAN) [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Endpoint calibration analysis for drive motor current and load speed (CAN). 6.4. Local stress state and material characterization Finite element analysis of the rotating bending configuration quantified the stress concentration at the notch root of the smallest shaft diameter, which also corresponds to the experimentally observed fracture location. The resulting local notch stress was therefore used as th… view at source ↗
Figure 13
Figure 13. Figure 13: Survival and Weibull lifetime analysis for drive motor current, the only output with sufficient observed failures for a stable fit on the held-out run [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: S–N curve of the output shaft under rotating bending, showing the Basquin fit to the finite-life data, the Haibach extension below the endurance limit, and the experimentally determined fatigue strength at 468 MPa. Hardness mapping revealed an effective case-hardened depth of approximately 0.369 mm according to ISO 2639. Fractographic observations showed surface-initiated fatigue cracking at the notch, fo… view at source ↗
Figure 15
Figure 15. Figure 15: Predicted maximum notch stress–time history under nominal service loading and under selective amplification of stress values above the 90th percentile. 1 0000 1 000000 1 E 8 1 E 1 0 0. 01 0. 1 ori gi nal l oad 90th percen ti l e; ampl i fi cati on factor = 1 . 2 90th percen ti l e; ampl i fi cati on factor = 1 . 6 90th percen ti l e; ampl i fi cati on factor = 2 c r a c k l e n g t h [m m] n umber of cycl… view at source ↗
Figure 16
Figure 16. Figure 16: Predicted crack propagation life and derived reusability under nominal loading, selectively amplified loading with scaling factors of 1.2, 1.6 , and 2.0 applied to the upper 10% of the stress distribution (≥90th percentile), and the grinding-only load profile. Afifi. et al.: Preprint submitted to Elsevier Page 21 of 27 [PITH_FULL_IMAGE:figures/full_fig_p023_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Sequential replay results for Algorithm 3 over 18 ordered inspection files (cycle IDs 1–14400). Panel 1: functional, material, and system reliability trajectories. Panel 2: predicted and observed maximum health-indicator threshold utilization across all outputs, with the functional failure limit at 1.0. Panel 3: cumulative Miner damage (log scale) with 5–95% bootstrap uncertainty band and the material fai… view at source ↗
read the original abstract

Returned products in circular factories re-enter production with heterogeneous degradation states, usage histories, and remaining capability. Reuse cannot be decided from the current inspection alone, because future function fulfillment and component integrity may evolve differently under the next service scenario. Existing PHM approaches support degradation prediction, but often target fixed operating conditions or isolated component benchmarks, while material-fatigue assessment is rarely linked to system-level functional prognosis. This paper addresses this gap for an angle grinder by combining uncertainty-aware functional prediction with component-level fatigue assessment in an instance-specific reliability workflow. The proposed framework combines the current tool state with recent force--torque usage windows. A convolutional encoder extracts loading patterns from spindle forces and shaft torque, and an LSTM backbone predicts nine functional variables as Gaussian mean and variance estimates. In parallel, the same loading history is translated into output-shaft fatigue information through finite-element-supported stress reconstruction, S--N/Miner damage evaluation with Haibach extension, and Paris-law crack-growth analysis. A streaming replay algorithm consolidates both branches into functional, material, and system reliability trajectories. Held-out tests show mean \(2\%\)-tolerance accuracy of 0.9652 across nine outputs. Thermal variables are predicted near-perfectly, while drive motor current and load speed remain the most demanding dynamic outputs, with \(R^2\) values of 0.9750 and 0.9924. Torque history is especially important for these variables, and the conventional LSTM outperforms GRU and xLSTM in the short-history setting. Reliability calibration is most informative for drive motor current, where predicted and observed exceedance probabilities ...

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper proposes an integrated workflow for circular factories that combines a convolutional-LSTM model (with uncertainty estimates) to predict nine functional variables of an angle grinder from force-torque histories, with a parallel finite-element-supported fatigue pipeline (stress reconstruction, S-N/Miner damage with Haibach extension, Paris-law crack growth). These branches are consolidated by a streaming replay algorithm into functional, material, and system reliability trajectories intended to support instance-specific reuse decisions. The only quantitative results reported are held-out LSTM performance: mean 2%-tolerance accuracy of 0.9652 across outputs, with R²=0.9750 (drive motor current) and R²=0.9924 (load speed).

Significance. If the missing quantitative linkage between the LSTM predictions and the fatigue/reliability trajectories can be demonstrated, the work would provide a concrete instance-specific bridge between functional PHM and material integrity assessment, which is currently rare in the literature.

major comments (3)
  1. [Abstract] Abstract and results: the central claim is an integrated workflow whose value lies in the consolidation of functional predictions with fatigue estimates into reliability trajectories for reuse decisions, yet the sole quantitative evidence consists of LSTM accuracy figures (0.9652 mean tolerance accuracy, two R² values); no numerical outcomes, calibration plots, or decision-level metrics are supplied for the fatigue branch, the streaming replay algorithm, or the joint trajectories.
  2. [Fatigue assessment pipeline] The finite-element-supported stress reconstruction, S-N/Miner evaluation with Haibach extension, and Paris-law analysis are described as operating on the same loading history used to train the LSTM, but no cross-check against physical measurements, no uncertainty propagation from the LSTM variances into the damage calculation, and no evaluation of the resulting reliability trajectories are reported, leaving the weakest_assumption untested.
  3. [Experimental setup] Validation protocol: the held-out accuracy numbers are presented without specification of temporal vs. random splitting, data exclusion rules, whether the test windows overlap training histories, or how the Gaussian uncertainty estimates were calibrated, which is required to assess whether the 0.9652 figure reflects genuine generalization.
minor comments (2)
  1. [Abstract] The final sentence of the abstract is truncated: "where predicted and observed exceedance probabilities ..."
  2. Notation for the nine output variables and the precise definition of the 2%-tolerance accuracy metric should be stated explicitly in the methods or results section.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. We address each major comment below, clarifying the current scope of the work and noting revisions to improve clarity and completeness.

read point-by-point responses

  1. Referee: [Abstract] Abstract and results: the central claim is an integrated workflow whose value lies in the consolidation of functional predictions with fatigue estimates into reliability trajectories for reuse decisions, yet the sole quantitative evidence consists of LSTM accuracy figures (0.9652 mean tolerance accuracy, two R² values); no numerical outcomes, calibration plots, or decision-level metrics are supplied for the fatigue branch, the streaming replay algorithm, or the joint trajectories.

    Authors: We agree that the abstract highlights the integrated workflow, while quantitative results are reported only for the LSTM functional predictions. The fatigue and reliability components are presented as a parallel methodological pipeline without end-to-end numerical evaluation. We will revise the abstract to explicitly state that the primary quantitative contribution is the held-out LSTM performance and that the consolidation into reliability trajectories is demonstrated at the workflow level rather than through decision metrics. revision: yes

  2. Referee: [Fatigue assessment pipeline] The finite-element-supported stress reconstruction, S-N/Miner evaluation with Haibach extension, and Paris-law analysis are described as operating on the same loading history used to train the LSTM, but no cross-check against physical measurements, no uncertainty propagation from the LSTM variances into the damage calculation, and no evaluation of the resulting reliability trajectories are reported, leaving the weakest_assumption untested.

    Authors: The manuscript provides a detailed description of the fatigue pipeline but does not include empirical cross-checks, uncertainty propagation, or trajectory evaluations, as these would require additional physical testing and computational runs not performed in the present study. We will add an explicit limitations subsection noting these gaps and outlining how uncertainty from the LSTM could be propagated in future extensions. revision: partial

  3. Referee: [Experimental setup] Validation protocol: the held-out accuracy numbers are presented without specification of temporal vs. random splitting, data exclusion rules, whether the test windows overlap training histories, or how the Gaussian uncertainty estimates were calibrated, which is required to assess whether the 0.9652 figure reflects genuine generalization.

    Authors: We will expand the experimental setup section to specify the validation protocol, including the use of a temporal (non-random) split with non-overlapping test windows to avoid leakage, the data exclusion criteria applied, and the calibration procedure used for the Gaussian uncertainty estimates (via reliability diagrams on the held-out set). revision: yes

Circularity Check

0 steps flagged

No circularity; held-out metrics and separate physics branch are independent

full rationale

The paper trains an LSTM on usage data to predict nine functional outputs (Gaussian means/variances) and evaluates on held-out tests, reporting explicit metrics (mean 2%-tolerance accuracy 0.9652, R² 0.9750/0.9924). These are standard out-of-sample performance numbers, not quantities defined by the fit itself. The fatigue branch applies independent FE stress reconstruction + S-N/Miner + Haibach + Paris-law to the same loading history; no equations or self-citations reduce either branch or their consolidation to the inputs by construction. No self-definitional steps, fitted-input-as-prediction, or load-bearing self-citations appear. The workflow is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone supplies insufficient detail to enumerate specific free parameters, axioms, or invented entities. The LSTM weights and any scaling constants inside the fatigue models are implicitly fitted but cannot be listed without the full methods section.

pith-pipeline@v0.9.1-grok · 5844 in / 1293 out tokens · 22349 ms · 2026-06-28T06:10:46.037721+00:00 · methodology

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