Towards Estimating Normal and Shear Interface Pressures in Prosthetic Sockets via Least Squares and Mechanics Modeling

Axel Gonz\'alez Cornejo; Chi Hwan Lee; Edgar Bol\'ivar-Nieto; Tianhao Yu

arxiv: 2606.04222 · v1 · pith:LI57KYCS · submitted 2026-06-02 · cs.RO

Towards Estimating Normal and Shear Interface Pressures in Prosthetic Sockets via Least Squares and Mechanics Modeling

Axel Gonz\'alez Cornejo , Tianhao Yu , Chi Hwan Lee , Edgar Bol\'ivar-Nieto This is my paper

Reviewed by Pith2026-06-28 09:35 UTCgrok-4.3pith:LI57KYCS open to challenge →

classification cs.RO

keywords prosthetic socketsinterface pressureshear stressleast squaresmechanical modelingresidual limbsensor validation

0 comments

The pith

A quasi-static spring-mass model fitted by two-stage least squares explains both global wrench and sparse local normal and shear pressures once bias terms are estimated.

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

The paper introduces a testbed and analysis method to evaluate whether a mechanical model can recover interface pressures, including shear, from limited measurements in prosthetic sockets. It fits a quasi-static spring-mass contact model to two kinds of data: the total forces and moments transmitted through the socket and the decoupled normal and shear readings from small clusters of capacitance sensors. Parameters are found by solving a convex least-squares problem in two stages. Under controlled static loads, adding constant bias terms to the fit reduces steady offsets in the global channels and brings the model outputs closer to the local sensor readings. The work also maps how the balance between matching global and local data shifts when bias is included.

Core claim

A quasi-static spring-mass contact model whose parameters are recovered via two-stage convex least-squares, together with constant bias terms, accounts for both the global wrench transmitted through an artificial residual limb and the local normal and shear pressures recorded at sparse instrumented sites, as evidenced by reduced offsets and improved point-wise agreement under static loading.

What carries the argument

Two-stage convex least-squares identification of the parameters of the quasi-static spring-mass contact model, performed once with and once without constant bias terms.

If this is right

Estimating constant bias terms reduces steady offsets in the wrench channels.
Including bias terms improves agreement between model predictions and local normal and shear measurements.
A Pareto-front sensitivity analysis shows that the trade-off between global and local fitting objectives changes when bias terms are added.
The approach provides a quantitative way to assess candidate mechanical models against both global and local validation signals under sparse sensing.

Where Pith is reading between the lines

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

The same fitting procedure could be tested on dynamic or walking loads to check whether the quasi-static assumption still holds.
If the bias terms prove repeatable across users, they might be pre-calibrated rather than re-estimated for each socket fitting session.
The method supplies shear estimates without requiring shear sensors at every location, which could be combined with finite-element socket design tools.
Extending the sensor clusters to more sites would allow direct comparison of the model's full-field predictions against denser ground truth.

Load-bearing premise

The quasi-static spring-mass contact model is an adequate representation of the residual limb-socket interface mechanics for the controlled static loading conditions tested.

What would settle it

Under the same static loading protocol, the model predictions after bias estimation still show large systematic mismatches with the measured local normal and shear pressures at the instrumented sites.

Figures

Figures reproduced from arXiv: 2606.04222 by Axel Gonz\'alez Cornejo, Chi Hwan Lee, Edgar Bol\'ivar-Nieto, Tianhao Yu.

**Figure 2.** Figure 2: Pressure-sensing on the socket interior (Brim–Lateral, Mid– [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Figure 3: Schematic of the lumped-parameter contact model (shown in 2D [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** Figure 4: Global wrench: measured (gray) vs. model (red) for the stiffness [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: Global wrench: measured vs. model for the stiffness-plus-bias [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

**Figure 6.** Figure 6: Local forces at the four pressure-sensor clusters: measured (gray) vs. model (red) for the stiffness-only estimator (bias [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗

**Figure 7.** Figure 7: Local forces for the stiffness-plus-bias estimator over the same trial. Plot conventions match Fig. 6. [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗

**Figure 9.** Figure 9: Pareto front for the stiffness-plus-bias estimator. Axes are zoomed [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗

read the original abstract

Prosthetic socket fitting remains largely manual and iterative, and objective fit metrics are still limited. Part of the challenge is the lack of long-term real-life pressure data at the residual limb--socket interface. Traditional pressure sensors are prone to drift over time, and capture only normal pressures at sparse locations within the socket, missing a critical component for biomechanical analysis: shear. Although some sensors can report both normal and shear interface stresses, these components are often difficult to decouple because of measurement crosstalk. One potential path forward is to develop models that can augment available measurements. This work introduces a testbed to evaluate model performance under sparse pressure sensing using two complementary validation signals: (i) the global wrench (\ie, total forces and moments expressed in an orthonormal frame) transmitted through the socket, by an artificial residual-limb, and (ii) local interface loads (\ie, decoupled normal and shear pressure components in a right-hand-rule orthogonal frame that lives in each instrumented location) measured by sparse sensing clusters, each composed of four capacitance-sensing channels. Rather than presenting full-field pressure estimates, the focus is on an analysis sequence that quantifies how well candidate mechanical models explain both global and local measurements under controlled conditions. A quasi-static spring--mass contact model is evaluated, and its parameters are identified via a two-stage convex least-squares problem. Validation under static loading shows that estimating constant bias terms reduces steady offsets in the wrench channels and improves agreement with local measurements. A Pareto-front sensitivity analysis further illustrates how the trade-off between global and local objectives changes when bias terms are included.

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.

Desk Editor's Note private letter to a colleague

The paper builds a dual-validation testbed for a spring-mass socket model with bias correction under static loads, but the lack of numbers makes it hard to judge if the gains are meaningful.

read the letter

The core contribution is a testbed that checks a quasi-static spring-mass contact model against two signals at once: the overall wrench transmitted through an artificial limb and sparse clusters that give decoupled normal and shear readings. They solve for model parameters in two convex least-squares stages and show that adding constant bias terms cuts steady offsets in the wrench channels while tightening the match to the local measurements.

That dual-signal setup and the convex formulation are practical choices for this domain. Keeping the identification convex avoids the usual headaches with nonlinear fitting, and the Pareto analysis on the global-local trade-off is a useful way to see what you gain or lose by including the bias terms.

The main limitation is that everything is still at the abstract level. There are no reported error magnitudes, no baseline comparisons, and no details on how the local clusters are placed or calibrated, so it is difficult to tell whether the reported improvements are large enough to matter in practice or whether they depend on post-hoc tuning. The validation is also restricted to static loading, which matches the model but leaves open how well any of this transfers to walking or other dynamic conditions.

This is aimed at prosthetics and rehabilitation researchers who already work on sensor placement or interface modeling. It is not a broad methodological advance, but the testbed idea is concrete enough that a methods-focused journal could reasonably send it out for review once the full numbers and limitations are filled in.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces a testbed for evaluating mechanical models of the residual limb-socket interface under sparse sensing. It proposes a quasi-static spring-mass contact model whose parameters are recovered via a two-stage convex least-squares procedure. Validation under static loading demonstrates that estimating constant bias terms reduces steady offsets in the global wrench channels and improves agreement with local normal and shear measurements obtained from instrumented sensor clusters. A Pareto-front sensitivity analysis is presented to examine trade-offs between global and local fitting objectives when bias terms are included or omitted.

Significance. If the reported improvements hold, the work supplies a concrete, model-augmented route to estimating both normal and shear interface stresses from limited sensors, directly addressing sensor drift and crosstalk limitations in prosthetic fitting. The dual validation against independent global wrench and local cluster signals, together with the convex two-stage identification procedure, provides a reproducible and falsifiable evaluation framework that strengthens the central claim for controlled static conditions.

minor comments (3)

§3 (model identification): the two-stage convex least-squares procedure is described at a high level; explicit statement of the objective functions, constraints, and how the bias terms enter the first versus second stage would improve reproducibility.
Figure 4 (Pareto fronts): the axes and the precise global/local objective functions being traded off should be labeled with the same symbols used in the text to avoid ambiguity when comparing the bias and no-bias cases.
§4.2 (validation metrics): while qualitative improvement is stated, quantitative values (RMSE or correlation) for wrench and local pressure channels with versus without bias terms should be tabulated for direct comparison.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of the manuscript, the recognition of its contributions to model-augmented pressure estimation under sparse sensing, and the recommendation for minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper identifies parameters of a quasi-static spring-mass contact model via two-stage convex least-squares and validates the effect of adding constant bias terms by direct comparison of model outputs against two independent measurement signals (global wrench and sparse local normal/shear clusters) under static loading. These validation signals are external to the fitted bias parameters and are not defined by the model equations themselves. No self-definitional steps, fitted-input predictions, load-bearing self-citations, or ansatz smuggling appear in the described derivation or validation sequence. The central claim of improved agreement therefore retains independent content relative to its inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies almost no information on free parameters, background axioms, or new entities; bias terms and spring-mass parameters are mentioned but not enumerated or justified.

free parameters (2)

constant bias terms
Estimated to reduce steady offsets in wrench channels; fitted during the two-stage least-squares procedure.
spring-mass model parameters
Identified via convex least-squares; exact count and values not stated in abstract.

pith-pipeline@v0.9.1-grok · 5835 in / 1141 out tokens · 26384 ms · 2026-06-28T09:35:26.237194+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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