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Active Learning of Fractional-Order Viscoelastic Model Parameters for Realistic Haptic Rendering

Harun Tolasa, Gorkem Gemalmaz, Volkan Patoglu

arXiv:2512.00667v2eess.SYcs.AIcs.ROcs.SY

TL;DR

Fractional-order models can describe viscoelastic memory with few parameters, but choosing parameters for realistic haptic rendering is difficult and individual preferences may not generalize. The paper uses qualitative-feedback human-in-the-loop optimization and aggregates individual perceptual maps to select population-level parameters. Human-subject validation found the aggregate-best memory-foam rendering was preferred over individually optimized best renderings in 72.7% of pairwise comparisons.

  • Problem

    Individual parameter choices may not generalize across users, motivating a population-level account of perceptual preferences for realistic haptic rendering.

    Source: §I · p. 1

  • Method

    The authors use qualitative-feedback human-in-the-loop Bayesian optimization for individual parameter tuning, then aggregate perceptual maps to select population-level parameters.

    Source: §I · pp. 1–2

  • Results

    For memory foam, the aggregate-best rendering was preferred in 72.7% of pairwise comparisons against individually optimized best renderings.

    Source: §VIII · p. 9

  • Takeaways & Limitations

    The authors conclude that the approach offers a systematic, practical route to parameterizing fractional-order viscoelastic models, with potential to enhance perceived realism in medical and haptic training simulators.

    Source: §X · p. 13

Abstract

from arXiv · show

Effective medical simulators necessitate realistic haptic rendering of biological tissues that exhibit viscoelastic material properties, such as creep and stress relaxation. Fractional-order models provide an effective means of describing intrinsically time-dependent viscoelastic dynamics with few parameters, as they naturally capture memory effects. However, due to the unintuitive, frequency-dependent coupling among the order of the fractional element and other parameters, determining appropriate parameter values for fractional-order models that yield high perceived realism remains a significant challenge. In this study, we propose a systematic means of determining the parameters of fractional-order viscoelastic models that optimizes the perceived realism of haptic rendering across general populations. First, we demonstrate that the parameters of fractional-order models can be effectively optimized through active learning, using qualitative feedback-based human-in-the-loop (HiL) optimization, to ensure consistently high realism ratings for each individual. Second, we propose a rigorous method to combine HiL optimization results into an aggregate perceptual map trained on the entire dataset, and demonstrate how to select population-level optimal parameters from this representation that are broadly perceived as realistic across general populations. Finally, we provide evidence of the effectiveness of the generalized fractional-order viscoelastic model parameters for three viscoelastic materials by characterizing their perceived realism through human-subject experiments. Overall, generalized fractional-order viscoelastic models established through the proposed HiL optimization and aggregation approach possess the potential to significantly improve the sim-to-real transition performance of medical training simulators.

I. INTRODUCTION

The paper addresses difficult parameter selection in fractional-order viscoelastic haptic rendering by combining qualitative-feedback active learning with population aggregation. It frames this need around realistic tissue rendering for medical simulators.

A. Viscoelastic Models

  • Fractional-order models capture hereditary, power-law viscoelastic responses with fewer parameters, but frequency-dependent coupling makes realistic parameter selection difficult.The paper contrasts these models with integer-order approaches and emphasizes the challenge of choosing parameters for high perceived realism.Source: §I · p. 1, A. Viscoelastic Models · p. 2

I. INTRODUCTION

  • The authors combine individual perceptual mappings into an aggregate map trained on the full dataset, then use it to select parameters perceived as realistic across the population.The map represents population preferences across feasible parameter sets.Source: §I · pp. 1–2
  • The proposed active-learning framework uses sample-efficient human-in-the-loop Bayesian optimization, iteratively collecting qualitative feedback to identify parameters that maximize perceived realism.The framework targets individualized parameter estimates.Source: §I · p. 2
  • The approach has potential to improve the realism of viscoelastic renderings, including tissue rendering in medical training simulators.The introduction connects simulator realism with medical training and the sim-to-real transition.Source: §I · pp. 1–2

A. Viscoelastic Models

  • Integer-order Maxwell and Kelvin–Voigt models cannot jointly describe creep and stress relaxation accurately; SLS improves both but misses non-exponential responses.The fractional-order SLS replaces the viscous damper with a fractional spring-damper element, enabling power-law-type time responses.Source: A. Viscoelastic Models · p. 2

B. HiL Bayesian Optimization in Haptic Studies

The paper extends qualitative-feedback HiL optimization to fractional-order viscoelastic rendering and develops a population-level perceptual map. It also describes the model and derivative used for rendering.

B. HiL Bayesian Optimization in Haptic Studies

C. Aggregation Methods

  • Equal-weight averaging of individual GP models can pull latent scores toward priors and reduce variance in unsampled regions, impairing generalization.Source: C. Aggregation Methods · p. 3
  • The Bayesian Committee Machine combines participant-specific GP posteriors into an aggregate perceptual map that represents shared population patterns and supports parameter selection.Source: C. Aggregation Methods · p. 3

III. FRACTIONAL-ORDER MODEL OF VISCOELASTICITY

  • The fractional-order SLS generalizes the standard linear solid with a derivative of order 0 to 1; its parameters include stiffnesses K0 and K1, order α, and coefficient B1.Source: §III · pp. 3–4
  • The non-local Grunwald–Letnikov derivative captures memory-dependent viscoelastic dynamics and was selected because it adapts readily to discrete-time realizations.Source: §III · p. 3

Short-Memory Discretization:

Short-memory discretization approximates the fractional derivative with a finite history window, enabling practical computation while making the window length an important modeling choice.

Short-Memory Discretization:

  • Short-memory discretization truncates the fractional derivative’s infinite history to a finite window, using recent samples whose binomial coefficients are largest and older samples whose coefficients decay.Source: Short-Memory Discretization · p. 4
  • The discretized derivative’s z-transform uses sampling time T, window length N, and recursively computed coefficients.Source: Short-Memory Discretization · p. 4
  • Window length N affects how well the discrete model captures creep and stress relaxation; at α = 1, the formula becomes backward-difference discretization.Source: Short-Memory Discretization · p. 4

IV. CHARACTERIZATION OF THE REFERENCE MATERIALS

The study characterizes reference materials using creep-with-recovery and stress-relaxation tests, and models perceived realism from participants’ ordinal feedback. For memory foam, it selects a 101-sample window and reports the identified model parameters.

IV. CHARACTERIZATION OF THE REFERENCE MATERIALS

  • Reference characterization mechanically coupled two identical direct-drive haptic interfaces; it repeated 3 N creep-with-recovery tests, observing creep for 3 s, and 5 mm stress-relaxation tests monitored for 3 s, 16 times each.Source: §IV · p. 4
  • At N = 101, NRMSE was 3.92% for creep with recovery and 0.32% for stress relaxation, with only marginal accuracy gains from larger windows.Source: §IV · p. 5
  • Memory foam parameters were K0 = −2.89 N/mm, K1 = 5.70 N/mm, B1 = 5.89 N·sα/mm, and α = 0.203; the resulting model was passive.Source: §IV · p. 5

1) Gaussian Process Model for Perceived Realism:

  • For perceived realism, a Gaussian-process model uses parameter settings and ordinal participant feedback to infer a latent score, with a Laplace approximation to the posterior.Source: 1) Gaussian Process Model for… · p. 5

2) HiL Bayesian Optimization:

HiL Bayesian optimization begins with space-filling parameter trials, then uses an acquisition function to select further trials and updates the Gaussian-process posterior after each outcome.

2) HiL Bayesian Optimization:

  • The algorithm uses initial space-filling trials, then acquisition-function selections, updates the GP posterior after each outcome, and returns the parameter with the highest posterior mean.Source: 2) HiL Bayesian Optimization · p. 5

3) GP Hyperparameters, Search Space, and Sampling Strategy:

The HiL search uses fixed GP and acquisition settings over a passivity-constrained, reduced parameter space, while Bayesian Committee Machine aggregation combines individual perceptual models into a population prediction. A post-hoc check found similar optima under empirically selected and marginal-likelihood-optimized GP hyperparameters.

3) GP Hyperparameters, Search Space, and Sampling Strategy:

VI. AGGREGATION OF GAUSSIAN PROCESS MODELS

  • Bayesian Committee Machine combines participant GP posteriors into a population prediction, weighting more confident, lower-variance models more strongly.Source: §VI · p. 6

VII. HUMAN SUBJECT EXPERIMENTS

Participants judged how closely a virtual viscoelastic rendering matched a physical reference through paired haptic interfaces. The study used separate cohorts for optimization and validation, and a setup designed to support palpation-like interaction.

A. Task

  • Participants compared the physical reference material with a virtual viscoelastic model presented through a second haptic interface, judging similarity based on stiffness, damping, creep, and stress relaxation.Source: A. Task · p. 6

B. Participants

  • The memory-foam optimization and validation sessions recruited 24 and 20 participants, respectively; 16 participants were assigned across silicone experiments, and eight additional participants completed aggregate-best validation for all three materials.Source: B. Participants · p. 6
  • Participants reported no sensory-motor deficiencies and had limited prior experience with haptic interfaces or psychophysical experiments.Source: B. Participants · p. 6

C. Apparatus

  • Two mechanically identical interfaces served as test and reference devices; participants held them sideways and used index fingers in a motion intended to simulate diagnostic palpation.Source: C. Apparatus · p. 7, Figure 3 · p. 7
  • The test interface rendered models using open-loop impedance control at 1 kHz, with linear actuators, 2000-counts/inch encoders, and low-friction guides.Source: C. Apparatus · p. 7

D. Experimental Protocol

Participants compared virtual renderings with physical reference materials under a standardized, feedback-based protocol. Separate sessions optimized individual parameters and validated selected renderings using forced-choice similarity judgments.

D. Experimental Protocol

  • Participants used their dominant hands to explore the systems freely, with auditory and visual cues minimized and the physical material serving as the reference.Source: 2) Procedure · p. 7

2) Procedure:

  • Each trial presented the reference before the virtual rendering, allowed eight seconds of interaction, and collected Close, Similar, or Different feedback for GP updates.Source: 2) Procedure · p. 7

E. Sessions

1) Viscoelastic Model Optimization:

2) Model Validation:

  • Model validation compared best, mid, and worst posterior-selected renderings; memory-foam sets came from the aggregate GP, while silicone sets came from individual GPs.Source: 2) Model Validation · pp. 7–8
  • Figure 4 shows posterior-model progression across trials in K1 and B1 slices at α = 0.19, with non-passive regions omitted from sampling.Source: Figure 4 · p. 8
  • Validation used forced-choice comparisons and Close, Similar, or Different classifications over 12 trials per participant, with each rendering type shown eight times.Source: 2) Model Validation · p. 8

3) Aggregate-Best Validation:

The validation results support using an aggregate perceptual map to identify population-level parameters that are broadly perceived as realistic. For memory foam, participants preferred the aggregate-best rendering over individually optimized best renderings, and its parameters closely matched system identification.

VIII. RESULTS

  • The results presented here cover memory foam; corresponding optimization and validation results for the two silicone materials are in the supplementary document.The paper states that results for medium- and high-stiffness silicones are available in.Source: §VIII · p. 8
  • Participants rated an average of 5.96 ± 4.47 of the 25 trials as close to the reference.Figure 5 also tracks each participant’s current best parameter set against their final best set using normalized error.Source: §VIII · pp. 8–9, Figure 5 · p. 8
  • The best set was preferred over the mid and worst sets in 95% and 100% of comparisons, while the mid set was preferred over the worst set in 90%.The user responses showed a consistent ordering among the three parameter sets.Source: §VIII · p. 9, Figure 6 · p. 9

3) Aggregate-Best Validation:

  • 72.7% of pairwise comparisons favored the aggregate-best rendering over individually optimized best renderings for memory foam.The preference was significantly more frequent in the statistical analysis.Source: §VIII · p. 9, Table IV · p. 9

VIII. RESULTS

  • The aggregate model’s best parameters had 4% NRMSE against system identification, compared with a 24% median for individually trained models (IQR 13%–41%).Errors were normalized with respect to the search space.Source: Table I · p. 9, §VIII · p. 10
  • The close-dominant intervals were K1 ∈ [4.57, 6.09] N/mm, B1 ∈ [4.33, 15.09] N·sα/mm, and α ∈ [0.128, 0.327], evaluated only in feasible passive regions.The aggregate optimum lies near the passivity boundary; higher K1 values violate passivity.Source: §VIII · p. 10

IX. DISCUSSION

The discussion reviews the hypotheses and examines sensitivity to parameter changes.

IX. DISCUSSION

  • The discussion assesses the hypotheses introduced in Section VII-F and considers sensitivity to parameter changes.The supplied passage identifies these topics but gives no findings.Source: §IX · p. 10

A. Hypothesis 1

HiL optimization found high-realism parameters for most individuals, while aggregation produced a population-level parameter set that was broadly perceived as realistic. Validation favored the aggregate-best set over individually optimized sets and showed that aggregation outperformed simple averaging.

A. Hypothesis 1

  • Some participants' GP models did not fully converge within 25 trials, but convergence across every individual was not required for a reliable population-level estimate.Variable feedback and the three-dimensional search space made full individual convergence more challenging.Source: A. Hypothesis 1 · p. 11
  • For most participants, active learning identified fractional-order SLS parameters with high perceived realism relative to the reference.GP models generally approached an optimal or near-optimal region within 25 trials, though some did not fully converge.Source: A. Hypothesis 1 · p. 11

B. Hypothesis 2

  • Aggregate-best normalized errors were 1.8% for K0, 0.4% for K1, 6.2% for B1, and 2.8% for α, below the corresponding errors from averaging individual best parameters.The aggregate-best parameters yielded 4% NRMSE relative to system-identification parameters.Source: B. Hypothesis 2 · p. 11
  • The aggregate model captured perceived realism and ordered best, mid, and worst parameter sets consistently with human judgments.Regression analyses also found significant effects for the best and worst parameter sets in comparisons with the reference and mid set.Source: B. Hypothesis 2 · p. 11
  • The aggregate-best memory-foam rendering was preferred in 72.7% of pairwise comparisons, significantly favoring a population-level solution over individual optima.The preference was statistically significant at p < 0.001.Source: B. Hypothesis 2 · p. 11

C. Hypothesis 3

The aggregate GP model reached a stable population-level optimum even when individual models had not fully converged, and the approach also showed evidence of generalizing across viscoelastic materials. Its perceptual map identifies locally acceptable parameter ranges, while the conclusions remain bounded by feedback consistency and the three-parameter evaluation.

C. Hypothesis 3

  • The aggregate GP optimum stabilized as trials increased, even when individual participants' models had not fully converged.Aggregate-model NRMSE decreased rapidly during the first 11 trials and then approached a plateau.Source: C. Hypothesis 3 · p. 12

D. Hypothesis 4

  • The approach generalized to medium- and high-stiffness silicone materials, provided the materials could be faithfully represented by the fractional-order SLS model.Population-level aggregate-best renderings remained favorable in comparisons with individually optimized renderings.Source: D. Hypothesis 4 · p. 12

E. Analysis of Perceptual Sensitivity

  • The aggregate perceptual map identifies a local region of acceptable renderings, with moderate B1 and α changes better tolerated than K1 changes.The K1 range is more restricted because its feasible values are constrained by the passivity boundary.Source: E. Analysis of Perceptual Sensitivity · p. 12
  • These sensitivity intervals describe model-parameter sensitivities, not a full decomposition of viscoelastic perception into independent attributes.Separating stiffness, damping, creep, and stress-relaxation contributions would require dedicated psychophysical experiments.Source: E. Analysis of Perceptual Sensitivity · p. 12

X. CONCLUSION AND FUTURE WORK

  • The proposed active-learning approach determines fractional-order model parameters for realistic haptic rendering, and human-subject experiments supported the generalized population-level parameters.The framework was demonstrated for three parameters; further efficiency work may be needed as optimization dimensions increase.Source: §X · p. 13
  • A systematic evaluation of perceived realism across models of increasing complexity remains to be conducted.The planned comparison would examine optimization costs and perceptual performance across models with different numbers of parameters.Source: §X · p. 13

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