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Perceptual Comparisons of Viscoelastic Renderings: Fractional- vs Integer-Order Models

Gorkem Gemalmaz, Harun Tolasa, Volkan Patoglu

arXiv:2610.21965v1cs.ROcs.SYeess.SY

TL;DR

It is unclear whether fractional-order models’ greater physical fidelity yields a more realistic haptic experience, since system-identification error and model complexity alone cannot establish perceived realism. The paper compares FO-SLS with two integer-order models using physically identified and human-in-the-loop-optimized parameters, finding FO-SLS significantly preferred across both approaches and both reference materials.

  • Problem

    System-identification error and model complexity alone cannot determine which viscoelastic rendering model provides the most realistic haptic experience.

    Source: §I · p. 1

  • Method

    The study compares FO-SLS with 3-parameter IO-SLS and 5-parameter IO-GM using physically identified and population-level human-in-the-loop-optimized parameters.

    Source: §I · p. 2

  • Results

    FO-SLS was significantly preferred over both integer-order models for both parameter-selection approaches and both viscoelastic reference materials.

    Source: §VII · p. 10

  • Takeaways & Limitations

    The findings support using FO-SLS models for realistic viscoelastic virtual environments.

    Source: §VII · p. 10

Abstract

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Realistic haptic rendering of viscoelastic materials requires reproducing time-dependent behaviors such as creep and stress relaxation, which arise from the material's inherent memory effects. Fractional-order viscoelastic models have been shown to capture these dynamics accurately and compactly, but it remains unclear whether their increased fidelity translates into perceptual benefits during haptic interaction. In this work, we investigate whether users perceive fractional-order models as more realistic than conventional integer-order models while rendering viscoelastic materials. We compare the fractional-order standard linear solid (FO-SLS) model with two integer-order alternatives: the 3-parameter SLS model and the 5-parameter generalized Maxwell model. We consider two complementary approaches to optimal parameter selection to evaluate rendering realism from both objective and perceptual perspectives: In the first approach, we determine the model parameters via least-squares based system identification of physical viscoelastic reference materials to establish evaluation baselines. In the second approach, we determine population-level perceptually optimal parameters using a qualitative-feedback-based human-in-the-loop optimization and aggregation method. This method establishes model parameters that maximize perceived realism of viscoelastic behaviors through qualitative evaluations. Human-subject experiments indicate that the FO-SLS model is perceived as significantly more realistic than both integer-order alternatives under both parameter selection approaches. Overall, this study provides experimental evidence that fractional-order viscoelastic models are not only theoretically well-motivated, but also perceptually relevant for realistic haptic rendering of creep- and relaxation-dominated behaviors of viscoelastic materials.

I. INTRODUCTION · II. RELATED WORK · A. Viscoelastic Models for Haptic Rendering

The paper asks whether fractional-order viscoelastic models’ compact representation of memory-dependent behavior yields more realistic haptic interactions than integer-order alternatives. It motivates comparing physical fit and perceived realism separately, then reports that FO-SLS renderings are perceived as more realistic than the integer-order models.

I. INTRODUCTION

  • Viscoelastic materials exhibit history-dependent creep and stress relaxation, making accurate rendering important for physical interaction with soft tissues and medical training simulators, –.Creep is gradual deformation under constant stress, whereas stress relaxation is force decay under constant strain.Source: §I · p. 1, A. Viscoelastic Models for Haptic… · p. 2
  • Physical least-squares fit does not necessarily predict perceived realism, so fair model comparison requires both matching a physical reference and evaluating users’ rendered experience,.Perceived response combines stiffness, damping, and interaction history.Source: §I · p. 1
  • Model-selection trade-offs motivate testing perception: lower-dimensional IO models are easier to tune, whereas FO models use fewer parameters but have coupled, less intuitive perceptual effects –[12].Higher-dimensional IO models can represent richer dynamics but increase the parameter space.Source: §I · p. 1, A. Viscoelastic Models for Haptic… · p. 2
  • The study compares models using both physically identified parameters and population-level perceptually optimal parameters selected through participants’ qualitative feedback and HiL optimization.This tests perceived realism under parameter sets optimized for physical characterization as well as human perception.Source: §I · p. 2
  • Human-subject comparisons indicate that the 4-parameter FO-SLS produces more realistic haptic interactions than the 3-parameter IO-SLS and 5-parameter IO-GM.The comparison uses qualitative participant evaluations under multiple optimal parameter-selection settings.Source: §I · p. 2

A. Viscoelastic Models for Haptic Rendering

  • Integer-order spring-damper models are interpretable and computationally convenient, but Kelvin–Voigt and Maxwell each miss one behavior, while SLS has one relaxation timescale and finite-branch GM cannot fully capture history dependence –.SLS represents creep and relaxation more accurately than Kelvin–Voigt or Maxwell, while GM adds branches and complexity to model broader relaxation spectra.Source: A. Viscoelastic Models for Haptic… · p. 2
  • Fractional-order models encode full signal history through fractional derivatives, compactly representing memory-dependent responses and broad relaxation behavior that would require infinitely many GM branches,, –.The GM model converges to FO-SLS as the number of Maxwell branches approaches infinity.Source: A. Viscoelastic Models for Haptic… · p. 2

B. Perceptual Evaluation of Viscoelastic Haptic Rendering · III. PRELIMINARIES · A. Fractional-Order Model of Viscoelasticity

The paper motivates human evaluation as the standard for judging perceived haptic realism and introduces a fractional-order model whose history dependence represents viscoelastic memory. It compares FO-SLS with lower- and higher-parameter integer-order models using physically characterized and perceptually optimized parameters.

B. Perceptual Evaluation of Viscoelastic Haptic Rendering

  • Human judgments of similarity, preference, or naturalness are the gold standard for perceived haptic realism because physical-model accuracy alone is insufficient.Prior evaluations compare interactions with a physical reference object.Source: B. Perceptual Evaluation of… · p. 2
  • Viscoelastic realism depends on time-varying exploratory cues, including stiffness, damping, creep, relaxation, and kinesthetic force-motion relationships.Psychophysics studies provide systematic evaluation methods for perceived similarity or realism, [12], –.Source: B. Perceptual Evaluation of… · p. 2
  • Earlier studies did not test whether viscoelastic model families differ in perceived realism for the same material; this study compares FO-SLS with low- and high-parameter IO models.It uses both physically characterized and population-level perceptually optimized parameters to test whether users perceive FO-model advantages [12],,,.Source: B. Perceptual Evaluation of… · p. 3

A. Fractional-Order Model of Viscoelasticity

  • The study uses the Grünwald–Letnikov fractional derivative because it is well suited to discrete-time implementation of fractional-order models.The supplied passage introduces the derivative definition but does not provide its complete formula.Source: A. Fractional-Order Model of… · p. 3
  • Unlike an integer-order derivative, the fractional-order operator sums weighted values from the entire signal history, providing a direct mechanism for representing memory effects.Source: A. Fractional-Order Model of… · p. 3
  • With zero initial conditions, fractional differentiation maps to multiplication by s^α in the Laplace domain; FO-SLS uses stiffness coefficients K0 and K1, fractional damping B1, and order α.The study compares FO-SLS with the 3-parameter IO-SLS and 5-parameter IO-GM models.Source: A. Fractional-Order Model of… · p. 3

Short-Memory Discretization:

Because exact fractional-order dynamics require infeasible infinite memory, the section uses short-memory discretization to create a practical discrete-time realization while preserving memory-dependent behavior.

Short-Memory Discretization

  • The Grünwald–Letnikov history is truncated to the most recent N samples, whose weights are defined recursively, to approximate the fractional derivative in the z-domain.For α = 1, the formulation reduces to standard backward-difference discretization.Source: Short-Memory Discretization · p. 3
  • Applying the short-memory discretization yields a discrete-time force-position impedance for the FO-SLS model.Source: Short-Memory Discretization · p. 3
  • The study sets N to 101 to balance realistic, passive viscoelastic rendering with computational feasibility, following prior work [12], [45].N controls the retained memory length and therefore affects how accurately the model reproduces time-dependent viscoelastic behavior.Source: Short-Memory Discretization · p. 3

B. Integer-Order Models of Viscoelasticity

The section defines two integer-order baselines for comparison with the FO-SLS model: a low-parameter SLS and a more expressive generalized Maxwell model. Their structures and discretization choices set up a comparison of whether additional integer-order parameters can match the perceptual realism of the compact fractional-order model.

a) IO-SLS Model

  • The 3-parameter IO-SLS is the low-parameter baseline, retaining the FO-SLS spring-damper structure but replacing its fractional-order element with a classical damper.K0 and K1 are stiffness coefficients, B1 is the damping coefficient, and IO models use backward-difference discretization.Source: B. Integer-Order Models of… · p. 3

Integer-Order Models of Viscoelasticity

  • The comparison uses the IO-SLS and IO-GM as low- and high-parameter baselines, respectively, to assess a perceptual advantage of the FO-SLS model.The IO-GM has two Maxwell branches, whereas the IO-SLS substitutes a classical damper for the fractional-order element.Source: B. Integer-Order Models of… · p. 3

b) IO-GM Model

  • The 5-parameter IO-GM adds a second Maxwell branch in parallel with static stiffness, enabling multiple relaxation time scales beyond the SLS baseline.Its additional parameters test whether a more expressive integer-order model can match the perceptual realism of the compact FO-SLS representation.Source: B. Integer-Order Models of… · p. 4

IV. DETERMINATION OF MODEL PARAMETERS FOR VISCOELASTICITY RENDERING

The section compares two ways of selecting viscoelastic rendering parameters: least-squares system identification and population-level perceptual optimization from human-in-the-loop feedback. It also describes controls and reference materials used to focus the comparison on time-dependent behavior across distinct viscoelastic interactions.

IV. DETERMINATION OF MODEL PARAMETERS FOR VISCOELASTICITY RENDERING

  • System-identified parameters quantitatively match measured responses, but may not maximize perceived realism; population-level HiL optimization instead aggregates qualitative feedback, [12],.Source: §IV · p. 4
  • All feasible parameter sets satisfy sampled-data passivity conditions for coupled stability, and effective stiffness is matched at low frequencies to isolate time-dependent behavior [45].The comparison therefore emphasizes creep and stress relaxation rather than differences in overall stiffness.Source: §IV · p. 4
  • Two physical references span distinct behaviors: memory foam shows history-dependent deformation and recovery, while hard silicone produces stiffness-dominated compression and release.The materials represent two extremes within a broad range of viscoelastic behaviors; Fig. 2(a) shows the references.Source: §IV · p. 4
  • The study uses least-squares system identification and aggregated human-in-the-loop feedback to determine physically grounded and population-level perceptually optimal parameters, respectively.Table I reports the parameters from both approaches and describes SysID fit errors and normalized parameter-wise distances.Source: Table I · p. 5

A. Determining Optimal Parameters via System Identification · B. Determining Perceptually Optimal Parameters

System identification and human-in-the-loop optimization provided two approaches to selecting viscoelastic rendering parameters; creep-with-recovery validation found lower NRMSE for FO-SLS than for both integer-order models.

A. Determining Optimal Parameters via System Identification

  • System identification used stress-relaxation measurements to fit IO-SLS, IO-GM, and FO-SLS to physical reference behavior, with FO-SLS short-memory length set to N = 101.Stress relaxation was measured under a constant 5 mm deformation for 3 s; creep-with-recovery used a 3 N step force for 3 s, then 0.5 N.Source: A. Determining Optimal Parameters via… · p. 5
  • The system-identified FO-SLS parameters included negative K0 to compensate for the fractional element’s large positive stiffness, and the resulting model was passive and physically realizable [45].Source: A. Determining Optimal Parameters via… · p. 5
  • FO-SLS had the lowest creep-with-recovery NRMSE for memory foam (3.63% vs. 5.28% for both IO models) and hard silicone (3.21% vs. 5.73% IO-SLS, 5.74% IO-GM).The system-identified rendering parameters were validated against creep-with-recovery experiments.Source: A. Determining Optimal Parameters via… · p. 5

B. Determining Perceptually Optimal Parameters

  • Six population-level perceptually optimized parameter sets covered three rendering models and two reference materials; FO-SLS sets came from prior HiL experiments [12], while four new experiments determined IO-SLS and IO-GM sets.Source: B. Determining Perceptually Optimal… · p. 5
  • Participants rated rendered interactions as Close, Similar, or Different; ordinal feedback iteratively updated a Gaussian process model of perceived realism to select subsequent parameters.The procedure is detailed in Section V-C.Source: B. Determining Perceptually Optimal… · p. 5
  • The population-level optimum maximized the aggregate GP posterior mean after participant-specific models were combined using a Bayesian Committee Machine, yielding group-level rather than individual-specific parameters.Fig. 3 depicts slices of the aggregate perceived-realism mappings and optimized parameters.Source: B. Determining Perceptually Optimal… · p. 5, Figure 3 · p. 6

A. Determining Optimal Parameters via System Identification

V. HUMAN-SUBJECT EXPERIMENTS · A. Apparatus · B. Participant Familiarization

The human-subject experiments first used HiL optimization to set population-level perceptual parameters, then compared model realism using those and system-identified parameters. Participants evaluated renderings against physical viscoelastic references using paired haptic interfaces after a warm-up.

V. HUMAN-SUBJECT EXPERIMENTS

  • HiL optimization sessions determined population-level perceptual parameters before performance comparisons used those and system-identified parameter sets to evaluate model realism.Source: §V · p. 6
  • Participants had no known sensory-motor impairments and provided written informed consent under Sabanci University IRB Protocol No. FENS-2025-17.Source: §V · p. 6

A. Apparatus

  • The apparatus paired two identical haptic interfaces: one held the physical reference material, while the other rendered a viscoelastic model with selected parameters.Source: A. Apparatus · p. 6, Figure 4 · p. 6
  • Participants used their dominant hands, palms inward, to interact with the paired devices using their index fingers.Source: A. Apparatus · p. 6
  • Each interface used a linear actuator and 2000-counts/inch optical encoder; open-loop impedance control ran in real time at 1 kHz using MATLAB RTE.The handle and support structure were 3D printed for comfortable grasping and repeatable interaction.Source: A. Apparatus · p. 6

B. Participant Familiarization

C. HiL Optimization Sessions … D. Performance-Comparison Experiments

The study used human-in-the-loop optimization to select integer-order rendering parameters for two physical reference materials, then compared FO-SLS, IO-SLS, and IO-GM renderings using optimized and system-identified parameters.

C. HiL Optimization Sessions

  • Four HiL optimization experiments used IO-SLS and IO-GM across the two reference materials to select parameters maximizing perceived realism against the physical material.Source: C. HiL Optimization Sessions · p. 6

1) Participants:

  • The HiL sessions recruited 24 participants for memory foam and 8 for hardsilicone, and all participants completed their corresponding sessions.Source: 1) Participants · p. 7

2) Task and Experimental Design:

  • Participants interacted with the physical reference and then a HiL-selected test rendering for a total of 8 seconds, rating its perceptual similarity to the reference.Source: 2) Task and Experimental Design · p. 7
  • Ratings used three ordinal categories: Close meant the same or highly similar, Similar meant distinguishable but moderately resembling, and Different meant clearly different from the physical reference.Source: 2) Task and Experimental Design · p. 7
  • Responses updated a GP model of perceived realism over the feasible parameter space, and each HiL optimization session ran for 25 trials.Source: 2) Task and Experimental Design · p. 7

D. Performance-Comparison Experiments

1) Participants: … A. Comparison of Perceived Realism of Viscoelastic Models

Across memory foam and hard silicone, FO-SLS renderings were judged more realistic than both integer-order alternatives under both parameter-selection approaches. Participants explored each physical reference and two virtual candidates, classified each candidate, and chose the closer rendering.

1) Participants:

  • 28 participants completed the memory-foam comparison experiment, while 16 completed the hard-silicone experiment.Source: 1) Participants · p. 7

2) Task and Experimental Design:

  • Participants compared FO-SLS, IO-SLS, and IO-GM renderings with the physical reference in separate HiL-optimized and system-identified conditions.All model pairings were tested four times per condition, yielding 24 comparisons per participant; condition-block order was randomized.Source: 2) Task and Experimental Design · p. 7

3) Experimental Procedure:

  • Participants explored the physical reference, then anonymously alternated between two virtual candidates for 12 seconds before rating each and selecting the closer rendering.They classified each as Close, Similar, or Different, judged overall haptic interaction—including stiffness, damping, and time-dependent behavior—ignored non-haptic cues, and received no correctness feedback.Source: 3) Experimental Procedure · p. 7, 2) Task and Experimental Design · p. 7

A. Comparison of Perceived Realism of Viscoelastic Models

  • 77% and 74% of memory-foam FO-SLS evaluations were Close under system-identified and HiL-optimized parameters, respectively.The corresponding Close rates were 35% and 33% for IO-SLS and 39% for IO-GM under both approaches.Source: A. Comparison of Perceived Realism of… · p. 7, Figure 5 · p. 8
  • 68% and 69% of hard-silicone FO-SLS evaluations were Close under system-identified and HiL-optimized parameters, respectively.The corresponding rates were 27% and 36% for IO-SLS, and 52% and 45% for IO-GM.Source: A. Comparison of Perceived Realism of… · p. 7, Figure 5 · p. 8
  • For memory foam and hard silicone, HiL-optimized and system-identified FO-SLS renderings had Close-classification coefficients above the reference probability; no IO rendering showed evidence of exceeding that reference.Memory-foam coefficients were β = 1.04 (HiL-optimized) and β = 1.20 (system-identified); hard-silicone coefficients were β = 0.79 (HiL-optimized) and β = 0.75 (system-identified), all p < 0.001.Source: A. Comparison of Perceived Realism of… · pp. 7–8
  • FO-SLS was significantly preferred over both integer-order models across both materials and parameter-selection approaches.Preference rates were 71–79% over IO-SLS and 62–78% over IO-GM; the preference advantage held for system-identified and HiL-optimized parameters.Source: Figure 5 · p. 8, A. Comparison of Perceived Realism of… · pp. 8–9, Table III · p. 9

B. Comparison of Integer-Order Models

Adding a second Maxwell branch improved integer-order physical fit, but did not ensure greater perceived realism: FO-SLS was preferred over IO-GM across parameter-selection strategies and materials.

B. Comparison of Integer-Order Models

  • IO-GM reduced NRMSE versus IO-SLS from 1.42% to 0.17% for memory foam and from 0.062% to 0.018% for hard silicone.The improvement followed the addition of a second Maxwell branch.Source: B. Comparison of Integer-Order Models · p. 9
  • Under system identification, IO-GM was significantly preferred over IO-SLS for memory foam (β = −0.29, p = 0.021) and hard silicone (β = −0.86, p < 0.001), while HiL comparisons showed no significant preference (p = 0.065 and p = 0.096, respectively).The HiL p-values correspond to memory foam and hard silicone, respectively.Source: B. Comparison of Integer-Order Models · p. 9
  • FO-SLS was significantly preferred over IO-GM under both parameter-selection strategies for both materials, including 78% of system-identified memory-foam comparisons despite IO-GM’s lower NRMSE.For that condition, IO-GM’s NRMSE was 0.17% versus 0.32% for FO-SLS, showing that better physical fit did not necessarily yield greater perceived realism.Source: B. Comparison of Integer-Order Models · p. 9

C. Comparison of Parameter-Selection Strategies · VII. CONCLUSIONS

Across both parameter-selection strategies, FO-SLS was perceived as more realistic than the tested integer-order models. The results also suggest that model structure mattered more than whether parameters were selected by physical fitting or human feedback.

C. Comparison of Parameter-Selection Strategies

  • Although system identification fits measured physical responses and HiL optimization uses participants’ realism evaluations, their perceptual results showed broadly similar trends.The parameter sets were not necessarily identical, but the main perceptual trends remained consistent.Source: C. Comparison of Parameter-Selection… · p. 9
  • For memory foam, FO-SLS Close rates were 77% (system-identified) and 74% (HiL-optimized), versus IO-SLS at 35% and 33%, and IO-GM at 39% under both conditions.For hard silicone, FO-SLS Close rates were 68% and 69% for system-identified and HiL-optimized parameters, respectively; pairwise comparisons preserved the same model ordering.Source: C. Comparison of Parameter-Selection… · p. 9

VII. CONCLUSIONS

  • FO-SLS was rated Close to the physical reference significantly more often than the 50% reference and significantly preferred over both integer-order models across both parameter sets.These trends held for both viscoelastic reference materials.Source: §VII · pp. 9–10
  • Across the validation experiments, model structure influenced perceptual ranking more than parameter-selection approach, with FO-SLS outperforming the tested integer-order variants.HiL can improve perceived realism for more sensitive parameters, but the dominant trend favored FO-SLS.Source: C. Comparison of Parameter-Selection… · p. 9, §VII · p. 10
  • The fractional-order formulation’s memory-dependent dynamics play a central role in realistically rendering viscoelastic behavior.The study presents FO-SLS as a concise, physically accurate description of materials exhibiting creep and/or stress relaxation, with perceptual advantages for haptic rendering.Source: §VII · p. 10
  • Human-in-the-loop optimization is a feasible and valid perceptual tuning approach because it selects model parameters using user feedback rather than only system-identification accuracy.The findings support using FO-SLS models in realistic viscoelastic virtual environments; fractional-order dynamics offered advantages that the tested integer-order models did not fully recover under either tuning approach.Source: §VII · p. 10
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