Haptic Rendering of Fractional-Order Viscoelasticity: Passivity and Rendering Fidelity

Rendering haptic feedback with nonlinear virtual environments (VEs) is important in many applications that require highly accurate force feedback. This paper considers the use of the Koopman operator to represent a nonlinear VE interacting with a haptic system. Simulation and experimental results demonstrated that the proposed method provides an effective representation of the nonlinear dynamics of a Duffing-oscillator VE. A multi-user study further confirmed this conclusion. In addition, a closed-loop (CL) stability analysis is performed leveraging the Koopman representation of the nonlinear VE to access stability of the overall haptic system. This alternative way of representing nonlinear VEs enables a convenient CL stability analysis that is less conservative than traditional passivity-based methods. Since a linear combination of all lifted states is used to represent the nonlinearity, such representation is also more robust to uncertainties in the modeling of the haptic device than a traditional nonlinear model.

Fast prediction of the response of adhesive soft viscoelastic contacts represents a current challenge in soft robotics and for gripping and manipulation tasks. Determining the complete time-resolved force trajectory requires full numerical simulations, whose computational cost is strongly parameter-dependent, making them impractical for real-time application or design-optimization loops. In this work, we overcome this limitation by training a scalar-conditioned, stateful, sequence-to-sequence deep learning model to predict the full force evolution from a prescribed displacement history for both short- and long-range adhesion regimes. The data set spans four orders of magnitude in loading and unloading rates and includes varied dwell times, with the Tabor parameter ranging from $0.2$ to $3.2$. To enable learning across these heterogeneous time scales, we introduce a fixed-measurement-step (FMS) representation that converts variable-length trajectories into fixed-length sequences while preserving their physical-time information. Different architectures were trained, including long short-term memory (LSTM) networks, temporal convolutional neural (TCN) networks, and time-distributed dense layers with three different Tabor-conditioning mechanisms. The models were compared using global waveform and error metrics. We found that the best-performing model has an LSTM architecture with concatenated conditioning, which achieves a held-out mean-squared error of $5.0\times10^{-4}$, a median pull-off-force error of $\approx2.2\%$, and a median hysteresis error of $\approx1.1\%$. For the held-out protocols, the model predicts a complete force trajectory with a median inference time of $0.16$ s. The model is tested across unseen parameter combinations and against analytical limiting cases, providing a rapid surrogate for repeated numerical evaluations with potential use in control-oriented applications.

Standard segmentation metrics such as Dice and Hausdorff distance measure geometric overlap but say nothing about whether a segmented surface is suitable for haptic rendering in surgical simulation. We propose SC-MFJ (Surface-Constrained Mean Force Jerk), a simple, inexpensive metric that samples a segmented organ surface with many short virtual stylus walks and measures how jerky the resulting contact forces are. The metric is computed from existing segmentation outputs and uses roughly one minute of CPU time per case. We evaluate three pancreas CT segmentation approaches-binary nnU-Net output, Gaussian-smoothed output, and learned signed distance function (SDF) regression-across 80 cases in five-fold cross-validation. SC-MFJ reveals a 147x gap in haptic quality between the raw binary baseline and simple Gaussian post-processing, a difference entirely invisible to Dice and HD95. It also shows that learned SDF regression, despite requiring full model retraining, produces more variable haptic quality than Gaussian smoothing, with a case-level standard deviation of 168 N/s2 compared with 22 N/s2 for Gaussian. A second evaluation on the LiTS liver dataset (131 cases) confirms the generality of these findings: the binary-to-Gaussian gap widens to 189x, and Gaussian smoothing again produces consistently low force jerk across all folds. Our results suggest that for haptic simulation applications, a one-line post-processing step may be sufficient, and that a cheap metric like SC-MFJ can flag problems that geometric metrics miss.