Koopman Representation of Nonlinear Virtual Environments in Kinesthetic Haptic Systems

2026-08-11

AI Summary

This work addresses the challenge of accurately modeling nonlinear virtual environments in force-feedback systems, where traditional passivity-based approaches are overly conservative and sensitive to modeling errors. For the first time, Koopman operator theory is introduced into haptic rendering, enabling the lifting of nonlinear dynamics into a linear framework for efficient modeling and closed-loop stability analysis. The proposed method yields less restrictive stability conditions compared to conventional techniques and significantly improves robustness against device modeling uncertainties. Simulations based on the Duffing oscillator model and multi-user experiments demonstrate that the approach accurately captures the dynamics of nonlinear virtual environments, achieving superior performance with a more practical and effective stability analysis.

Abstract

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.

Problem

• nonlinear virtual environments
• haptic systems
• Koopman operator
• stability analysis
• force feedback

Innovation

• Koopman operator
• nonlinear virtual environments
• haptic feedback
• closed-loop stability
• robust modeling