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

arXiv:2610.21965 · cs.RO, cs.SY, eess.SY · Submitted 2026-10-15 · Read on arXiv

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Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: Today's paper: "Perceptual Comparisons of Viscoelastic Renderings".

Dev: The gist: Fractional-order standard linear solid (FO-SLS) models are perceived as significantly more realistic than conventional integer-order alternatives when rendering viscoelastic materials, supporting their use for realistic haptic rendering.

Rosa: First, who's behind it and why it matters.

Paper summary: Rosa: So we're talking about this paper called Perceptual Comparisons of Viscoelastic Renderings: Fractional- vs Integer-Order Models. Basically, the authors are asking if using fractional-order models makes the haptic rendering of materials feel more real to people compared to the standard integer-order models they usually use.

Dev: They set out to see if that increased fidelity from using fractional derivatives actually translates into something users can actually perceive when they're touching these viscoelastic things. That's a pretty big question for haptic interaction.

Taro: It’s interesting because the paper focuses specifically on memory effects, which is what you get with fractional models, like creep and stress relaxation happening over time.

Rosa: Exactly. The core idea is that fractional-order models capture these time-dependent behaviors better and in a more compact way than the conventional integer-order ones. But they need to show that this mathematical advantage actually matters when someone is interacting with it haptically.

Dev: And the setup for this investigation involves comparing the FO standard linear solid model against two integer-order alternatives: a three-parameter SLS model and a five-parameter generalized Maxwell model.

Taro: So they are testing the FO standard linear solid, which is four parameters, against those IO models with three and five parameters respectively. It’s about checking if adding more parameters in the integer order model helps enough to match what fractional order does naturally.

Rosa: They look at this comparison using two different ways to pick the best model settings. One way is using physical characterization—they determine the model parameters by running least-squares based system identification on actual physical viscoelastic reference materials to get a baseline.

Dev: And the second way is a human-in-the-loop approach where they determine population level perceptually optimal parameters using qualitative feedback through an optimization method. This lets them see what users actually prefer based on their perception.

Taro: It’s smart that they are doing both of those parameter selection approaches. You get the objective data from the physical fitting and then you check if the perceptual results hold up when you optimize for what people think is best across a population.

Rosa: The main thing they claim is that users perceive the fractional-order standard linear solid model as significantly more realistic than both of those integer-order alternatives, regardless of which parameter selection approach they use.

Dev: That's the key finding from the Perceptual Comparisons of Viscoelastic Renderings: Fractional- vs Integer-Order Models paper. It suggests that the structure of the model itself, specifically its fractional order representation, is what drives better perceived realism for these kinds of materials in haptic rendering.

Taro: So what does this mean for us on the ground? It means we might not need to hunt for those overly complex integer-order models just because they have more parameters; the fractional model structure seems to deliver a perceptual advantage over both tested alternatives.

Rosa: And it also validates the idea that human-in-the-loop optimization is a valid way to tune these models, showing that tuning based on what people say rather than just fitting the data can really enhance how realistic an interaction feels.

Dev: The authors point out some limitations in their work, specifically noting that while the fractional model won out perceptually, it doesn't necessarily mean the integer-order generalized Maxwell model is inherently worse in a physical sense; it just didn't match the perceived realism of the FO-SLS under these specific testing conditions.

Taro: They mention that they constrained their parameter sets to satisfy passivity conditions of the sampled-data control system to make sure everything stayed stable during rendering, which is a necessary engineering check for any real-time implementation.

Rosa: So, as a listener, what does this change? If you're designing something like a robotic gripper that needs to feel like it’s touching foam or rubber, this paper suggests leaning into the fractional order formulation might give you better perceptual results with less complexity than trying to push the parameter count higher in an integer model.

Dev: It tells us that when we think about real-time loops and safety cases, we need to consider that these models are not just mathematical descriptions; they are perception-driven tools for interaction, and the fractional model seems to be a better tool for capturing those time dynamics.

Taro: I think the implication is that future work should focus on how these fractional models handle more complex or non-linear world misbehaves outside of the controlled reference materials they used in this study.

Rosa: So, to wrap up this discussion on Perceptual Comparisons of Viscoelastic Renderings: Fractional- vs Integer-Order Models, the main point is that fractional-order viscoelastic models provide a superior perceptual experience for haptic rendering compared to standard integer-order alternatives.

Dev: Yeah, and it confirms that model structure matters more than the specific parameter selection method you use to tune those models.

Taro: It’s about confirming that the memory dynamics inherent in fractional calculus are actually what users notice most when they are experiencing creep or stress relaxation in a virtual object.

Conclusion: Rosa: So this paper is about comparing how real materials feel when you touch them in virtual space, specifically looking at fractional-order models versus those standard integer-order ones they usually use.

Dev: Yeah, the authors are calling it "Perceptual Comparisons of Viscoelastic Renderings: Fractional- vs Integer-Order Models," and they're trying to figure out if that fractional math actually matters when a user is interacting with something tactile.

Taro: Basically, they tested how realistic the FO standard linear solid model feels compared to some simpler integer-order options like the three-parameter SLS model and the five-parameter generalized Maxwell model.

Rosa: The main finding they land on is that users actually perceive the fractional-order model as significantly more realistic than both of those integer models, no matter which way they picked their settings.

Dev: That’s interesting because we often think that having a bigger set of parameters in an integer model means it can fit the physical data better, but this study suggests that structure is what wins out perceptually.

Taro: It shows that the memory effects captured by the fractional formulation are what users notice most when they feel things like creep or stress relaxation during a haptic interaction.

Rosa: And they did this test using two different ways to pick those parameters: first, by fitting them to actual physical materials, and second, by using a human-in-the-loop method to optimize for what people actually say feels best.

Dev: The paper really hammers home that the model structure itself is more important than whether you tuned the parameters using physical measurements or user feedback.

Taro: They even showed that tuning based on human perception, through that human-in-the-loop optimization, is a valid way to tune these models for better realism.

Rosa: It’s a good thing because it means we don't just have to rely on the most complex integer model just because it fits the data a little more.

Dev: The implication here is that when you're designing systems that need to simulate real-world material behavior, focusing on capturing those time-dependent dynamics through fractional models could give you a better feel for the interaction.

Taro: It opens up questions about how these fractional models handle things outside of the reference materials they used; how do they recover when the world misbehaves in a more complex way?

Rosa: Exactly, so we've seen that FO standard linear solid is perceptually strong, but now we need to see if it holds up when things get messy in the real world.

Gorkem Gemalmaz Harun Tolasa Volkan Patoglu

Sabanci University

cs.RO, cs.SY, eess.SY

Submitted: 2026-10-15

Updated: 2026-10-15

Comments: This work has been submitted to the IEEE Transactions on Haptics for possible publication. 11 pages, 5 figures

License: http://creativecommons.org/licenses/by-nc-nd/4.0/

Importance score: 79/100

The gist: The gist: Fractional-order standard linear solid (FO-SLS) models are perceived as significantly more realistic than conventional integer-order alternatives when rendering viscoelastic materials,

Key concepts

Fractional-Order Viscoelastic Models (FO): These models use non-integer derivatives to accurately capture time-dependent material behaviors like creep and stress relaxation. They are appealing because they can represent complex memory effects with relatively few parameters, unlike traditional integer models that require many branches.
Integer-Order (IO) Models: These are classical compliance models based on spring-damper networks. Examples include the standard linear solid (SLS) and generalized Maxwell (GM) models. While they approximate material behavior, they struggle to accurately represent long-term memory effects and frequency-dependent dynamics.
Human-in-the-Loop (HiL) Optimization: This is a method where users provide qualitative feedback during model tuning to find parameters that maximize perceived realism. This approach focuses on human perception rather than just matching the physical material response, helping tune models for better haptic interaction.
Model Structure Dominance: The study concluded that the choice of model structure (fractional-order vs. integer-order) had a stronger impact on how realistic a rendering was perceived by users than whether the parameters were derived from physical fitting or perceptual optimization.

Terminology

Aggregation — 1× in this paper
Fidelity — 1× in this paper · explained in 2 episodes
Human-in-the-Loop — 1× in this paper · explained in 3 episodes
System Identification — 1× in this paper · explained in 3 episodes

Summary

The gist: Fractional-order standard linear solid (FO-SLS) models are perceived as significantly more realistic than conventional integer-order alternatives when rendering viscoelastic materials, supporting their use for realistic haptic rendering.

Model Comparison and Theoretical Basis

Fractional-order (FO) viscoelastic models have been shown to capture time-dependent behaviors such as creep and stress relaxation 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 Classical compliance models based on integer-order (IO) spring-damper networks, including the standard linear solid (SLS) model and the generalized Maxwell (GM) model, approximate these behaviors, but fail to represent long-term memory effects and frequency-dependent dynamics accurately Fractionalorder (FO) viscoelastic models provide an appealing alternative, as non-integer derivatives naturally encode memory-dependent dynamics and can represent complex viscoelastic responses with relatively few model parameters The FO-SLS model can model broad relaxation behavior that would otherwise require infinitely many IO branches in the GM model.

Parameter Selection Approaches

The study compares the rendering performance of the 4-parameter FO-SLS model with two integer-order alternatives of different model complexities: the 3-parameter IO-SLS model and the 5-parameter IO-GM model These comparisons are performed under two complementary optimal parameter selection approaches. The first approach determines model parameters via least-squares based system identification of physical viscoelastic reference materials to establish evaluation baselines. The second approach determines population-level perceptually optimal parameters using a qualitative-feedback-based humanin-the-loop optimization and aggregation method.

Determination of Optimal Parameters

Optimal parameters are determined through two methods: physically grounded parameters from system identification experiments, and population-level perceptually optimal parameters from the aggregation of human-in-the-loop optimization results The feasible parameter sets were constrained to satisfy the corresponding passivity conditions of the sampled-data control system to ensure coupled stability of interactions during rendering. The study utilized two physical references with distinct material properties to verify whether the findings extend across different viscoelastic interactions.

Perceptual Evaluation Results

The 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 For the memory foam, the FO-SLS model was classified as Close to the physical reference in 77% and 74% of the evaluations using the system-identified and HiL-optimized parameter sets, respectively The pairwise preference results showed that the FO-SLS model was preferred over both IO model alternatives for each reference material and under both parameter-selection strategies. The direct comparison between parameters from the HiL optimization and the system-identified parameters reveals that parameter selection matters less than the model structure

Conclusion on Model Structure

The dominant trend in the validation experiments was that the FO-SLS model performed better than the tested integer-order variants This indicates that the memory-dependent dynamics captured by the fractional-order formulation play a central role in realistically rendering viscoelastic behavior Overall, this study experimentally validates two complementary conclusions: first, FO-SLS viscoelastic models are not only concise and physically accurate descriptions of materials that exhibit creep and/or stress relaxation phenomena, but also perceptually advantageous models for haptic rendering Second, human-in-the-loop optimization is a feasible and valid perceptual tuning approach.

Limitations of Integer-Order Models

The additional complexity and improved physical fit of the IO-GM model were not sufficient to match the perceived realism of the FO-SLS model The FO-SLS model was significantly preferred over the IO-GM model under both parameter-selection strategies and for both reference materials. Improved agreement with the measured material response does not necessarily translate into improved perceived realism during haptic interaction The FO-SLS model was significantly preferred over both IO-SLS and IO-GM renderings under both HiL optimization and system identification conditions.

Model Structure Dominance

The results strongly support that the viscoelastic model structure influenced the overall perceptual ranking more than whether its parameters were selected through physical fitting or perceptual optimization This suggests that, within the parameter sets considered in this study, the viscoelastic model structure influenced the overall perceptual ranking more than whether its parameters were selected through physical fitting or perceptual optimization. The FO-SLS model was consistently perceived as more realistic than both integer-order models used for comparison

Validation of Human-in-the-Loop Optimization

Human-in-the-loop optimization is a feasible and valid perceptual tuning approach, as it tunes model parameters based on user feedback rather than solely on the accuracy of the system identification process HiL optimization explicitly considers human perception, while system identification aims to best match the model to the measured response Therefore, HiL optimization can enhance the perceived realism of more sensitive parameters.

Final Summary

The FO-SLS model was consistently perceived as more realistic than both integer-order models used for comparison The same trend appears in the pairwise preference results, where the FO-SLS model was significantly preferred over both IO-SLS and IO-GM models for HiL-optimized and system-identified parameter sets These findings show that the perceptual advantage of the FO-SLS model was robust across both perceptual and objective parameter-selection approaches for both viscoelastic reference materials. The direct comparison between parameters from the HiL optimization and the system-identified parameters reveals that parameter selection matters less than the model structure The dominant trend in the validation experiments was that the FO-SLS model performed better than the tested integer-order variants. The fractional-order dynamics captured by the FO formulation play a central role in realistically rendering viscoelastic behavior

Acknowledgment

This work has been partially supported by the TUBITAK Grant 23AG003.

References (Partial List)

[1] K. Salisbury, D. Brock, T. Massie, N. Swarup, and C. Zilles, “Haptic rendering: Programming touch interaction with virtual objects,” in Symposium on Interactive 3D graphics, 1995, pp. 123–130

[2] S. Park, A. L. Chien, I. D. Brown, and J. Chen, “Characterizing viscoelastic properties of human melanoma tissue using prony series,” Frontiers in Bioengineering and Biotechnology, vol. 11, p. 1162880, 2023

[3] B. Carmichael, H. Babahosseini, S. N. Mahmoodi, and M. Agah, “The fractional viscoelastic response of human breast tissue cells,” Physical Biology, vol. 12, no. 4, p. 046001, 2015

[4] M. Zhang, P. Nigwekar, B. Castaneda, K. Hoyt, J. V. Joseph, A. di Sant’Agnese, E. M. Messing, J. G. Strang, D. J Rubens, and K.-J Parker, “Quantitative characterization of viscoelastic properties of human prostate correlated with histology,” Ultrasound in Medicine & Biology, vol., 34, no.

Improvements for AI systems

Bold header: Fractional-Order Viscoelastic Model Integration

This improvement allows AI systems to move beyond integer-order models by incorporating memory effects through fractional derivatives, as FO models can naturally capture memory-dependent behavior. The resulting AI system can accurately simulate complex time-dependent behaviors like creep and stress relaxation that are crucial for realistic haptic rendering.

Bold header: Perceptually Optimized Parameter Selection

The AI system will be able to determine optimal model parameters by implementing the human-in-the-loop (HiL) optimization, which establishes model parameters that maximize perceived realism of viscoelastic behaviors through qualitative evaluations. This means the AI can tune its rendering parameters based on participant feedback rather than solely relying on physical identification.

Bold header: Robust Perceptual Ranking

The improved AI system will achieve a consistent and clear perceptual advantage by favoring models like the FO-SLS model, which was significantly preferred over both IO model alternatives for each reference material and under both parameter-selection strategies. This ensures that the resulting haptic rendering is perceived as more realistic across different optimization approaches.

Bold header: Model Structure Dominance

The system will learn that the viscoelastic model structure influenced the overall perceptual ranking more than whether its parameters were selected through physical fitting or perceptual optimization. This allows the AI to prioritize the inherent advantages of fractional-order dynamics over simply matching a measured response, as noted by the finding that improved agreement with the measured material response does not necessarily translate into improved perceived realism.

Abstract

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.