McGill's Koopman haptic method offers less conservative stability than passivity

Kinesthetic haptic systems, which render touch-based force feedback, often struggle with nonlinear virtual environments (VEs) because traditional stability guarantees rely on passivity theory—an overly conservative condition that restricts the range of rendered forces. Researchers at McGill University have published a new arXiv paper (2608.11461) that applies the Koopman operator to represent nonlinear VEs as a linear combination of lifted states. This shift from nonlinear to linear representation allows closed-loop (CL) stability analysis to be performed directly on the Koopman model, yielding stability bounds that are significantly less conservative than passivity-based methods.

The team validated their approach using a Duffing oscillator—a classic nonlinear system with chaotic behavior—and demonstrated effective representation of its dynamics. A multi-user study confirmed that haptic rendering with the Koopman representation maintains stability and improves force fidelity during interaction. Additionally, because the Koopman representation uses a linear combination of all lifted states, it is inherently more robust to uncertainties in modeling the haptic device itself. This means engineers can design haptic interfaces for nonlinear VEs (like soft tissue, deformable objects, or complex textures) without the performance penalties imposed by passivity constraints. The work opens the door to more expressive force feedback in surgical simulators, teleoperation, and virtual prototyping, where accuracy and stability are critical.