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Delay-Compensated Stiffness Estimation for Robot-Mediated Dyadic Interaction

Mingtian Du, Suhas Raghavendra Kulkarni, Bernardo Noronha, Domenico Campolo

TL;DR

The paper tackles the challenge of inaccurate stiffness perception in robot-mediated dyadic therapy caused by network-induced haptic delays. It introduces a delay-compensated stiffness estimator derived from quasi-static equilibrium and implements it with Normalised Weighted Least Squares to filter dynamic bias. Through H-MAN based experiments simulating téléoperation, the Naive estimator's performance degrades with delay, while the proposed NWLS method maintains accurate stiffness tracking up to 320 ms of delay. The approach promises improved fidelity of remote haptic perception, enabling reliable stiffness assessments in rehabilitative networks.

Abstract

Robot-mediated human-human (dyadic) interactions enable therapists to provide physical therapy remotely, yet an accurate perception of patient stiffness remains challenging due to network-induced haptic delays. Conventional stiffness estimation methods, which neglect delay, suffer from temporal misalignment between force and position signals, leading to significant estimation errors as delays increase. To address this, we propose a robust, delay-compensated stiffness estimation framework by deriving an algebraic estimator based on quasi-static equilibrium that explicitly accounts for temporally aligning the expert's input with the novice's response. A Normalised Weighted Least Squares (NWLS) implementation is then introduced to robustly filter dynamic bias resulting from the algebraic derivation. Experiments using commercial rehabilitation robots (H-MAN) as the platform demonstrate that the proposed method significantly outperforms the standard estimator, maintaining consistent tracking accuracy under multiple introduced delays. These findings offer a promising solution for achieving high-fidelity haptic perception in remote dyadic interaction, potentially facilitating reliable stiffness assessment in therapeutic settings across networks.

Delay-Compensated Stiffness Estimation for Robot-Mediated Dyadic Interaction

TL;DR

The paper tackles the challenge of inaccurate stiffness perception in robot-mediated dyadic therapy caused by network-induced haptic delays. It introduces a delay-compensated stiffness estimator derived from quasi-static equilibrium and implements it with Normalised Weighted Least Squares to filter dynamic bias. Through H-MAN based experiments simulating téléoperation, the Naive estimator's performance degrades with delay, while the proposed NWLS method maintains accurate stiffness tracking up to 320 ms of delay. The approach promises improved fidelity of remote haptic perception, enabling reliable stiffness assessments in rehabilitative networks.

Abstract

Robot-mediated human-human (dyadic) interactions enable therapists to provide physical therapy remotely, yet an accurate perception of patient stiffness remains challenging due to network-induced haptic delays. Conventional stiffness estimation methods, which neglect delay, suffer from temporal misalignment between force and position signals, leading to significant estimation errors as delays increase. To address this, we propose a robust, delay-compensated stiffness estimation framework by deriving an algebraic estimator based on quasi-static equilibrium that explicitly accounts for temporally aligning the expert's input with the novice's response. A Normalised Weighted Least Squares (NWLS) implementation is then introduced to robustly filter dynamic bias resulting from the algebraic derivation. Experiments using commercial rehabilitation robots (H-MAN) as the platform demonstrate that the proposed method significantly outperforms the standard estimator, maintaining consistent tracking accuracy under multiple introduced delays. These findings offer a promising solution for achieving high-fidelity haptic perception in remote dyadic interaction, potentially facilitating reliable stiffness assessment in therapeutic settings across networks.
Paper Structure (12 sections, 10 equations, 5 figures)

This paper contains 12 sections, 10 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Stiffness Estimation Framework
  3. The Naive Estimator
  4. Delay-Compensated Estimator: Ordinary Least Square
  5. Implementation: Normalised Weighted Least Squares
  6. Experimental Evaluation
  7. System Description
  8. Experimental Protocol
  9. Data Analysis and Performance Metrics
  10. Experimental Results
  11. Discussion
  12. Conclusion

Figures (5)

  • Figure 1: (a) Schematic representation of the stiffness estimation framework for robot-mediated dyadic interaction. The system couples an expert (therapist) and a novice (patient) via a virtual stiffness $k$, aiming to estimate the novice's intrinsic stiffness $k_0$ (representing muscle tone) in the presence of communication delay. (b) The physical experimental setup. Two planar H-MAN robots serve as the expert and novice interfaces, where human agents are replaced by HEBI actuator-based mechanisms to simulate programmable human joint impedance and ensure consistent ground truth generation.
  • Figure 2: Block diagram of the haptic-delayed dyadic interaction system. The expert and novice interfaces are connected via a virtual coupling stiffness $k$ across a communication channel with haptic delay $\delta$. The diagram illustrates the signal causality, where the local expert force $f_1(t)$ and the delayed novice position $\hat{x}_2(t) = x_2(t-\delta)$ act as the observable variables for the stiffness estimator. Additionally, an observer on the expert side retrieves the round-trip delayed position $\tilde{x}_1(t) = x_1(t-2\delta)$ to enable the proposed delay-compensated estimation.
  • Figure 3: Ordinary Least Squares (OLS) add weights bias denoted by $x_\mathrm{geom} = \tilde{x}_1(t) - \hat{x}_2(t)$, and $x_\mathrm{local} = x_1(t) - \hat{x}_2(t)$. When there is no delay, $x_\mathrm{geom} = x_\mathrm{local}$. Data points with a larger $x_\mathrm{geom}$ or $x_\mathrm{local}$ will be taken into regression with higher weights. Normalised Weighted Least Squares (NWLS) can remove this bias by placing a quadratic weight inversely proportional to the magnitude of the displacements on each data point, i.e. $(x^2_\mathrm{geom} + x^2_\mathrm{local})^{-1/2}$.
  • Figure 4: Comparative stiffness regression analysis across experimental conditions. Each subplot displays the force-displacement data (grey dots) collected from 10 independent trials for a specific combination of haptic delay ($\delta$), commanded novice stiffness ($\mathrm{K}_0$), and operational axis (X or Y-Axis). The coloured bands represent the range of stiffness estimates (minimum to maximum slope) generated by the Naive (blue), OLS (red), and Proposed NWLS (yellow) estimators, with the solid lines indicating the mean estimate. The alignment of these regression bands with the distribution of the raw force-displacement data provides a qualitative measure of estimation accuracy and robustness against dynamic transients. The displacement axes (X-Axis) limits are fixed to $\pm 0.02\,$m to maintain a consistent visual scale across stiffness conditions.
  • Figure 5: Statistical comparison of stiffness estimation accuracy. The boxplots illustrate the distribution of the Absolute Percentage Error ($E_{\mathrm{APE}}$) across 10 trials for each experimental condition, grouped by operational axis (X or Y-Axis) and novice stiffness ($\mathrm{K_0}$). The results compare the performance of the Naive (blue), OLS (red), and proposed NWLS (yellow) estimators across four delay conditions ($0$, $80$, $160$, and $320$ ms). An asterisk denotes a statistically significant difference ($p < 0.05$) between the OLS or NWLS estimator and the Naive method under identical conditions. Outliers are omitted for visual clarity.