Source-linked AI summary
Haptic Rendering of Fractional-Order Viscoelasticity: Passivity and Rendering Fidelity
Gorkem Gemalmaz, Harun Tolasa, Volkan Patoglu
TL;DR
Real-time fractional-order viscoelastic rendering faces difficulty in selecting realistic parameters and lacks closed-form passivity bounds for FO-SLS under finite-memory discretization. The paper derives bounds and effective-rendering expressions, then evaluates memory-window effects in human-subject comparisons. Its results support N = 101 as a window with high perceived realism and no added computational cost of a longer window.
Problem
Frequency-dependent coupling complicates tuning FO-model parameters for perceived realism, and closed-form passivity bounds for short-memory FO-SLS rendering were absent.
Source: §I · p. 1Method
The study derives closed-form FO-SLS passivity conditions and effective stiffness and damping expressions under short-memory discretization, and compares virtual renderings with a physical viscoelastic reference.
Source: §I · p. 2Results
N = 101 and N = 151 renderings were judged close to the physical reference and preferred over N = 51, with no significant preference advantage for N = 151 over N = 101.
Source: F. Results and Discussion · pp. 10–11Takeaways & Limitations
Under the study’s experimental conditions, N = 101 is a suitable window because it maintains high perceived realism while avoiding the additional computational cost of a longer window.
Source: §VIII · p. 11
Abstract
from arXiv · showhide
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.
I. INTRODUCTION
The paper addresses challenges in realistically rendering viscoelastic materials: finite-memory FO models complicate passivity analysis, and their parameters are difficult to tune for perceived realism. It presents a theoretical and experimental framework for FO-SLS rendering and examines how model families represent creep and stress relaxation.
I. INTRODUCTION
- Finite-memory discretization complicates analysis of FO dynamics, while frequency-dependent parameter coupling makes realistic parameter selection difficult.The interaction history must be long enough for viscoelastic effects to emerge, yet real-time rendering requires finite memory.Source: §I · p. 1
- Closed-form passivity bounds for short-memory FO-SLS rendering were absent, despite established conditions for integer-order systems and some continuous-time FO models.The paper presents its analysis as addressing this gap for realistic short-memory approximations.Source: §I · pp. 1–2
- The study combines closed-form passivity conditions and effective stiffness and damping expressions with human-subject comparisons of FO-SLS renderings at different memory lengths.The perceptual study compares three representative virtual environments with a physical viscoelastic material.Source: §I · p. 2
- The human-subject experiment compares memory truncation lengths and finds an upper bound beyond which participants do not perceive differences.This evidence supports using truncated models.Source: §I · p. 2
A. Viscoelastic Models
- Kelvin–Voigt captures creep but has limited stress-relaxation range, whereas Maxwell captures stress relaxation but predicts unbounded linear creep.SLS represents both behaviors more appropriately, but its single relaxation time limits fidelity for broader relaxation spectra.Source: A. Viscoelastic Models · p. 2
- Generalized Maxwell models use multiple relaxation times, while FO models offer a compact way to capture memory-dependent viscoelastic behavior.In the limit of infinitely many Maxwell branches, the generalized Maxwell model converges to an FO-SLS representation; FO models can capture intermediate solidlike and fluid-like responses with relatively few parameters.Source: §I · p. 1, A. Viscoelastic Models · pp. 2–3
B. Passivity of Haptic Rendering
Passivity provides an analytical criterion for coupled stability in haptic rendering, and viscoelastic virtual-environment bounds can inform virtual-coupler design. The paper extends prior FO and integer-order work to FO-SLS and describes its fractional derivative as a flexible, history-dependent element.
B. Passivity of Haptic Rendering
- Passivity provides an analytical criterion for coupled stability without modeling the human operator and can reveal how plant parameters and controller gains affect interaction stability.For virtual coupling, passivity of the sampled haptic interface and coupler, together with discrete-time passivity of the simulated environment, is sufficient for coupled stability.Source: B. Passivity of Haptic Rendering · p. 3
- Prior work established passivity bounds for IO-SLS and passivity conditions for FO systems; this study extends the analysis to FO-SLS under short-memory discretization.The authors describe the resulting framework as generalizing previously studied viscoelastic models.Source: B. Passivity of Haptic Rendering · p. 3
A. Fractional-Order Model of Viscoelasticity
- The FO-SLS model adds a fractional derivative to the damping term, with order α ∈ (0, 1), allowing more flexible viscoelastic modeling than classical integer-order models.Its force-position impedance uses elastic stiffness coefficients K0 and K1 and fractional-element coefficient B1.Source: A. Fractional-Order Model of… · p. 3
- The Grunwald–Letnikov derivative is adopted for its compatibility with discrete-time implementations, and its definition uses differentiation order α and step size h.The generalized binomial coefficient is part of this derivative definition.Source: A. Fractional-Order Model of… · p. 3
- Unlike integer-order derivatives, fractional derivatives are nonlocal: the derivative at a given time depends on the function’s history, capturing memory-dependent creep and stress relaxation.Under the stated zero-initial-condition assumption, the Laplace-transform expression reduces to a direct generalization of the integer-order Laplace operator.Source: A. Fractional-Order Model of… · p. 3
B. Short-Memory Discretization
Because exact fractional-order computation requires infinite memory, the paper uses a short-memory approximation that truncates the discrete convolution. The memory-window length controls how accurately the discrete model captures viscoelastic behavior.
B. Short-Memory Discretization
- The short-memory approximation truncates the Grunwald–Letnikov convolution to a finite window because exact fractional-order computation requires infinite memory.It relies on binomial weights being largest for recent samples and decaying for past data.Source: B. Short-Memory Discretization · p. 4
- In the discrete approximation, T is the sampling period, N is the memory-window length, and coefficients c_i are computed recursively.The approximation is applied to the fractional derivative of a discrete-time signal.Source: B. Short-Memory Discretization · p. 4
- When α = 1, the coefficient recursion reduces to backward-difference discretization; N determines how accurately the model captures stress relaxation and creep.Source: B. Short-Memory Discretization · p. 4
IV. PASSIVITY OF FRACTIONAL-ORDER SLS MODEL UNDER SHORT-MEMORY DISCRETIZATION
The section derives passivity conditions for the short-memory-discretized FO-SLS model by locating the passivity function’s maximum. The bounds distinguish odd and even memory lengths and encompass classical viscoelastic models as special cases.
IV. PASSIVITY OF FRACTIONAL-ORDER SLS MODEL UNDER SHORT-MEMORY DISCRETIZATION
- For odd memory length N, the passivity function reaches its global maximum at the Nyquist frequency, enabling a closed-form passivity condition for the FO-SLS model.The condition is necessary and sufficient for K1 > 0, B1 > 0, 0 < α < 1, and T > 0.Source: §IV · pp. 4–5
- For even N, the passivity-function maximum may occur at an interior frequency ω⋆ < π/T rather than at Nyquist.In the N → ∞ limit, the function is strictly increasing over (0, π/T].Source: §IV · p. 4
- For odd N, the paper also gives a sufficient passivity condition based on a conservative bound for the truncated sum.The bound supplements the necessary-and-sufficient condition for finite odd memory length.Source: §IV · p. 5
- The FO-SLS formulation reduces to classical integer- and fractional-order Kelvin-Voigt, Maxwell, and SLS models under appropriate parameter choices and limits.Table I summarizes the reductions and resulting passivity conditions.Source: §IV · p. 5, Table I · p. 5
V. EFFECTIVE IMPEDANCE OF FO-SLS MODEL
Effective impedance decomposes the closed-loop frequency-dependent impedance into components associated with damping, spring, and inertia. For viscoelastic rendering, the section focuses on effective spring and damping.
V. EFFECTIVE IMPEDANCE OF FO-SLS MODEL
- The effective-impedance decomposition assigns the real positive part of the frequency-dependent impedance to effective damping.The imaginary part is assigned to effective spring and effective inertia according to its phase characteristics.Source: §V · p. 5
- The imaginary part of effective impedance is assigned to effective spring or effective inertia based on its phase characteristics.Source: §V · p. 5
- For viscoelastic rendering, the analysis focuses on the FO-SLS model’s effective spring and effective damping.Source: §V · p. 5
A. Effective Stiffness
The section defines frequency-dependent effective stiffness for the discrete-time FO-SLS environment and gives expressions for finite-memory, asymptotic, and low-frequency cases. The low-frequency case is presented as a practical characterization when human excitation frequencies lie below Nyquist.
A. Effective Stiffness
- Effective stiffness extracts frequency-dependent stiffness from the overall response of a virtual environment containing fractional-order elements.It is useful for understanding the net stiffness of environments combining springs, dampers, and fractional-order elements.Source: A. Effective Stiffness · p. 6
- The frequency-dependent effective stiffness of the discrete-time FO-SLS model is given for finite memory length N.The expression uses the short-memory discretization coefficients of the Grunwald–Letnikov derivative.Source: A. Effective Stiffness · p. 6
- As N →∞, the effective-stiffness expression reduces to the asymptotic form given in Corollary 3.Source: A. Effective Stiffness · p. 6
- Around ω = 0, the finite-memory effective stiffness has a low-frequency expression that can characterize perceived stiffness when human excitations lie well below Nyquist.The passage identifies the typical human-excitation band as 0–10 Hz and notes that the asymptotic case is recovered as N increases.Source: A. Effective Stiffness · p. 6
B. Effective Damping
The section derives frequency-dependent and low-frequency effective damping expressions for the discrete-time FO-SLS model. It distinguishes finite-memory and asymptotic cases and relates them to classical viscoelastic models.
B. Effective Damping
- The FO-SLS model’s frequency-dependent effective damping is given for finite-memory discretization.The expression is for the discrete-time FO-SLS model; its short-memory coefficients are the Grunwald–Letnikov coefficients.Source: B. Effective Damping · p. 6
- As N →∞, the frequency-dependent effective damping reduces to the asymptotic expression.The finite-memory expression at arbitrary N is also characterized around ω = 0.Source: B. Effective Damping · pp. 6–7
- At low frequencies, the finite-memory effective damping has an expression for arbitrary N, while the asymptotic case is recovered as N increases.The general finite-memory series is complex, and evaluating damping around ω = 0 may be sufficient for some applications.Source: B. Effective Damping · p. 7
- The effective impedance formulation reduces to classical IO and FO Kelvin-Voigt, Maxwell, and SLS models under appropriate parameter selections and limits.Table II summarizes the reductions and the resulting effective stiffness and damping expressions.Source: B. Effective Damping · p. 7, Table II · p. 7
C. Interpretation of Effective Stiffness and Damping
The effective stiffness and damping expressions admit an equivalent impedance-network interpretation. The fractional element’s magnitude and phase depend on sampling time and fractional order, shaping energy storage and dissipation across frequencies.
C. Interpretation of Effective Stiffness and Damping
- As ωT →0, the discrete-time fractional operator approaches continuous-time fractional-derivative behavior.The discrete-time response remains bounded at high frequencies due to discretization.Source: C. Interpretation of Effective… · p. 7
- The expressions correspond to a series branch comprising stiffness K1 and frequency-dependent complex term Bfo(ω), in parallel with static stiffness K0.The real part of the series combination contributes to elastic energy storage, while its imaginary part, scaled by 1/ω, quantifies dissipation.Source: C. Interpretation of Effective… · p. 7
- Decreasing T raises Bfo(ω)’s high-frequency magnitude proportionally to T^-α, while α governs magnitude growth and phase rotation.These effects shape the trade-off between energy storage and dissipation across frequencies.Source: C. Interpretation of Effective… · p. 7
VI. VALIDATIONS OF COUPLED STABILITY
The section introduces experimental evidence intended to validate the theoretical passivity bounds.
VI. VALIDATIONS OF COUPLED STABILITY
- The study provides experimental evidence to help validate the theoretical passivity bounds.Source: §VI · p. 7
A. Identification of Plant Parameters
The experiments identify the haptic plant, test FO-SLS passivity bounds, and evaluate model fitting for viscoelastic rendering. The reported passivity boundaries agree closely with analytical predictions, while longer memory windows improve fitting accuracy.
A. Identification of Plant Parameters
- A first-order plant model, Zplant(s) = ms + b, was fitted from a 1–10 Hz chirp, yielding m = 73.4 g and b = 0.0025 N/(mm·s) (R2 = 0.90).The chirp ran for 15 s, commanded in real time at 1 kHz.Source: A. Identification of Plant Parameters · p. 8
B. Experimental Verifications of Passivity Bounds
- The experimental parameters agree closely with analytical passivity predictions (NRMSE 1.61%), while coupled-stability tests extend the observed passivity region by approximately 15%.The experiments used impacts and judged stability over five trials per parameter set; friction and hysteresis are identified as possible sources of the discrepancy.Source: B. Experimental Verifications of… · p. 8, Figure 4 · p. 8
VII. VALIDATIONS OF HAPTIC RENDERING FIDELITY
- The human-subject study compares FO-SLS renderings with physical viscoelastic material across memory lengths N ∈{51, 101, 151}.Source: §VII · p. 8
A. Determination of Parameters of the FO-SLS Model
- For memory-foam identification, stress-relaxation and creep-with-recovery responses were measured and fitted using the FO-SLS model, with parameters constrained by analytical passivity criteria.The reference sample was tested using mechanically coupled haptic interfaces.Source: A. Determination of Parameters of the… · p. 8, Figure 5 · p. 9
- Increasing N improves model accuracy; N = 101 was selected, with 0.324% NRMSE on stress relaxation, to balance accuracy and computational effort.The tested memory lengths were N ∈{51, 101, 151}.Source: A. Determination of Parameters of the… · p. 9
- Across fitted models with different memory lengths, low-frequency effective stiffness is approximately the same, while larger N primarily improves dynamic fidelity.The identified K0 values are negative because K0 compensates for stiffness introduced by the fractional-order element; the fitted parameters satisfy passivity.Source: A. Determination of Parameters of the… · p. 9, Table III · p. 9
B. Task
The study compared virtual FO-SLS renderings with a physical viscoelastic reference through subjective haptic comparisons. Twenty participants explored the devices and judged the similarity of model pairs.
B. Task
- Participants compared each virtual rendering with a physical viscoelastic reference and judged which felt more similar, using haptic feedback alone.They also rated each model independently as close, similar, or different from the reference.Source: B. Task · p. 9, 2) Cross-Comparison · p. 10
C. Participants
- Twenty right-handed participants took part; none reported sensory–motor impairments or extensive prior experience with haptic interfaces or psychophysical experiments.Their mean age was 22.1 years.Source: C. Participants · p. 9
D. Apparatus
- Two identical interfaces let participants compare a physical reference with virtual FO-SLS renderings using memory lengths N = 51, 101, and 151.Participants held the devices with their dominant hands and explored them by palpating with their index fingers.Source: D. Apparatus · p. 9, Figure 6 · p. 10
1) Warm-Up:
- A roughly five-minute warm-up familiarized participants with the devices and task before comparison trials.Participants explored the reference and virtual environments until comfortable with the task.Source: 1) Warm-Up · p. 9
2) Cross-Comparison:
- All three model pairs were tested, with randomized presentation order and 12 repetitions per pair, yielding 36 comparative trials per participant.The cross-comparison session lasted about 20 minutes; the full experiment took about 25 minutes.Source: 2) Cross-Comparison · p. 10
F. Results and Discussion
The results show that FO-SLS renderings with memory lengths N = 101 and N = 151 were judged close to the physical reference more often than the N = 51 rendering. The paper also presents passivity and rendering-characterization results, and concludes that N = 101 is a suitable practical window under the tested conditions.
F. Results and Discussion
- N = 101 and N = 151 were classified as close more often than the 50% reference, while N = 51 fell below it; both larger windows were preferred over N = 51, with no significant preference between them.The close-classification estimates were β = −0.849 for N = 51, β = 0.916 for N = 101, and β = 1.036 for N = 151 (all p < 0.001); pairwise preference estimates were β = 1.207 and β = 1.347 versus N = 51 (both p < 0.001), and β = 0.095 for N = 151 versus N = 101 (p = 0.503).Source: Figure 7 · p. 10, F. Results and Discussion · p. 10, Table IV · p. 10
- The classification analysis tested each window's “close” rate against 50%, while pairwise preference models tested whether the larger window was preferred; both used cluster-robust standard errors.Ordinal responses were recoded as “close” versus “similar” or “different.”Source: F. Results and Discussion · p. 10
VIII. CONCLUSIONS
- The paper derives closed-form passivity conditions for FO-SLS rendering under finite-memory discretization and reports experimental verification of the bounds.The condition provides a unified formulation from which passivity results for commonly used viscoelastic models can be derived as special cases.Source: §VIII · p. 11
- Closed-form effective stiffness and damping expressions characterize FO-SLS rendering under finite-memory discretization and support analysis of its frequency-dependent behavior.The formulations recover classical viscoelastic equations under appropriate parameter substitutions.Source: §VIII · p. 11
- The human-subject results indicate that the perceptual benefit of increasing memory length saturates beyond approximately N = 101 under the tested conditions.Both N = 101 and N = 151 achieved high close-classification rates.Source: §VIII · p. 11
- N = 101 is a suitable rendering window in this experiment, maintaining high perceived realism while avoiding the additional computational cost of a longer window.The conclusion is bounded to the present experimental conditions.Source: §VIII · p. 11