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.
Figures & tables
Fig. 1 : Viscoelastic models used in the experiments: (a) standard linear solid (SLS) model with 3 parameters, (b) fractional-order (FO) SLS model with 4 parameters, and (c) generalized Maxwell (GM) model with 5 parameters.
Fig. 2 : (a) Reference materials used in the experiment. (b) System identification setup. (c)–(f) Characterization experiment data and best fits of IO-SLS, IO-GM, and FO-SLS rendering models. The top row shows the results for stress relaxation, while the bottom row shows the results for creep-with-recovery experiments. Experiments in (c) and (d) were conducted with a memory foam reference, while (e) and (f) were conducted with a hard silicone material.
Memory Foam
Hard Silicone
FO-SLS (4-parameter model)
Set
K0
K1
B1
α
NRMSE
K0
K1
B1
α
NRMSE
HiL
−3.18
5.58
7.88
0.18
–
−2.19
4.97
7.59
0.53
–
SysID
−2.89
5.70
5.89
0.20
0.32%
−1.88
4.72
6.35
0.61
0.0172%
Norm. error
–
0.40%
6.20%
2.80%
–
–
0.82%
3.88%
7.89%
–
IO-SLS (3-parameter model)
TABLE I : Parameters determined via system identification (SysID) and population-level perceptually optimized parameters from HiL experiments (HiL). SysID NRMSEs are computed from the stress-relaxation fits, while normalized errors denote parameter-wise distances between the corresponding SysID and HiL parameter values relative to the search-space ranges. The error for K0 is not reported since it is determined via effective stiffness matching.
Fig. 3 : Slices from aggregate posterior GP mappings of perceived realism. Each mapping shows a slice with the population-level optimized rendering parameters in the K1−B1 domain, with other parameters for the FO-SLS and IO-GM models fixed to the HiL-optimized values. Columns represent the rendering models; each row contains mappings from the same reference.
Fig. 4 : (a) The setup with a participant. (b) Two identical haptic interfaces used for the human-subject experiments.
Fig. 5 : Classification and pairwise preference results for the comparison between fractional- and integer-order viscoelastic renderings. The top row presents the classification-rate matrices, while the bottom row presents the corresponding pairwise preference percentages. The evaluations and comparisons for memory foam are shown in (a) with system-identified rendering parameters and in (b) with HiL optimization parameters. Similarly, the results for hard silicone material are shown in (c) for system-identified and in (d) for HiL-optimized parameters.
Material
Rendering Model
Estimate
SE
t -value
p -value
Memory foam
HiL FO
1.04
0.11
9.30
<.001
SysID FO
1.20
0.14
8.65
<.001
HiL IO-SLS
-0.72
0.10
-7.20
1.000
SysID IO-SLS
-0.61
0.09
-6.56
1.000
HiL IO-GM
-0.44
0.08
-5.47
1.000
SysID IO-GM
-0.44
0.09
-4.93
1.000
TABLE II: Binary-logit analysis of the probability that each rendering is classified as Close to the physical reference. Positive coefficients indicate a higher likelihood of a Close classification.
Material
Analysis
Comparison
Estimate
SE
t -value
p -value
Preferred
Memory foam
HiL vs HiL
FO vs. IO-SLS
1.15
0.18
6.29
<0.001
FO
FO vs. IO-GM
1.07
0.14
7.59
<0.001
FO
IO-SLS vs. IO-GM
-0.23
0.13
-1.85
0.065
–
SysID vs SysID
FO vs. IO-SLS
1.30
0.15
8.83
<0.001
FO
FO vs. IO-GM
1.27
0.20
6.22
<0.001
FO
IO-SLS vs. IO-GM
-0.29
0.12
-2.31
0.021
IO-GM
TABLE III: Pairwise preference analysis between the rendering conditions. Positive estimates indicate preference for the first rendering in each comparison, whereas negative estimates indicate preference for the second rendering. The preferred condition is reported only for statistically significant comparisons ( p<0.05 ).
Haptic rendering of viscoelastic materials that exhibit creep and stress relaxation is crucial for many applications, such as medical training with realistic biological tissue models. Fractional-order viscoelastic models provide an effective means of describing intrinsically time-dependent dynamics with few parameters, as these models can naturally capture memory effects. In this study, we present analyses of passivity and rendering performance for fractional-order viscoelastic models under finite-memory discretization. We derive closed-form expressions to ensure the passivity of haptic rendering with a fractional-order (FO) standard linear solid (SLS) model based on Grunwald-Letnikov derivative under short-memory discretization. We also provide symbolic expressions for the effective stiffness and damping of such FO-SLS models. The resulting passivity conditions constitute a unified framework that generalizes previously reported results for integer-order Kelvin-Voigt, Maxwell, and SLS models, since these results are special cases of the newly derived condition. Furthermore, we provide experimental validations of the theoretical passivity bounds and human-subject evaluations of perceived realism of FO-SLS models. Overall, this study establishes a unified theoretical framework and experimental evaluations for FO viscoelastic rendering under short-memory discretization.
Gorkem Gemalmaz, Harun Tolasa, Volkan Patoglu
Faculty of Engineering and Natural Sciences at Sabanci University, Istanbul, Turkiye.
Human softness perception in haptics has mainly been studied using mechanically homogeneous objects, despite the fact that many real-world objects exhibit heterogeneous layered structures with nonuniform stiffness. This study examined how layered heterogeneity modulates haptic softness perception. Sixteen lattice-structured stimuli were fabricated by 3D printing, with the stiffness of the upper four layers systematically varied while the bottom two layers remained fixed. Twenty-two participants evaluated the softness of the stimuli in a psychophysical task, and compression tests were conducted to quantify their mechanical properties. Perceived softness was significantly predicted by displacement under load, however, perceptual ranking did not fully coincide with the physical ranking. Linear mixed-effects analyses showed that the softness of the outermost layer had the greatest impact on the perceived softness. Perceived softness also increased as the number of soft subsurface layers increased, although this contribution decreased with depth. Layers 2 and 3 showed significant effects, whereas Layer 4 did not. These findings suggest that haptic softness perception depends not only on the overall stiffness but also on the depth-dependent distribution of compliance within layered structures.
Yuno Higuchi, Yosuke Iwashita, Yuji Ohgi +1
Faculty of Environment and Information Studies, Keio University, Japan · Graduate School of Media and Governance, Keio University, Japan
Tactile perception is key to dexterous manipulation, yet simulating high-resolution elastomer deformation remains computationally prohibitive. Finite element methods (FEM) deliver high fidelity but demand costly remeshing, while Material Point Methods (MPM) suffer from heavy particle-memory tradeoffs. We propose a {reduced-order neural simulation framework} that couples coarse-grained MPM dynamics with an implicit neural decoder to reconstruct sub-particle tactile details from compact latent states. The framework learns a continuous deformation manifold from paired high- and low-resolution simulations, enabling physically consistent, differentiable inference. Compared to the TacIPC, our method achieves over 65% faster simulation and {40% lower memory usage}, while maintaining better geometric fidelity. In tactile rendering and 3D surface reconstruction, our methods further improve accuracy by 25% and produce realistic depth images and surface mesh within a faster inference speed. These results demonstrate that the proposed reduced-order neural model enables high-detail, physically grounded tactile simulation with substantial efficiency gains for robotic interaction and optimization.
Yuhu Guo, Zhikai Shen, Jiasheng Qu +4
The University of Manchester, UK · The Chinese University of Hong Kong, Hong Kong SAR, China · University of Leeds, UK +1