SoftNash: Entropy-Regularized Nash Games for Non-Fighting Virtual Fixtures

TL;DR

Soft-Nash Virtual Fixtures address user–controller conflict in teleoperation by introducing an entropy-regularized two-player LQ game with a single softness parameter τ. This parameter smoothly interpolates between a hard, performance-optimized VF and a near-pass-through of human input while preserving Nash equilibrium and stability. Empirical results on a 6-DoF haptic device show that moderate softness (τ ≈ 2) maintains classic VF accuracy while substantially reducing conflict and workload and increasing perceived agency. The work provides a practical, theoretically grounded knob for personalized, non-fighting shared control in haptics and teleoperation.

Abstract

Virtual fixtures (VFs) improve precision in teleoperation but often ``fight'' the user, inflating mental workload and eroding the sense of agency. We propose Soft-Nash Virtual Fixtures, a game-theoretic shared-control policy that softens the classic two-player linear-quadratic (LQ) Nash solution by inflating the fixture's effort weight with a single, interpretable scalar parameter $τ$. This yields a continuous dial on controller assertiveness: $τ=0$ recovers a hard, performance-focused Nash / virtual fixture controller, while larger $τ$ reduce gains and pushback, yet preserve the equilibrium structure and continuity of closed-loop stability. We derive Soft-Nash from both a KL-regularized trust-region and a maximum-entropy viewpoint, obtaining a closed-form robot best response that shrinks authority and aligns the fixture with the operator's input as $τ$ grows. We implement Soft-Nash on a 6-DoF haptic device in 3D tracking task ($n=12$). Moderate softness ($τ\approx 1-3$, especially $τ=2$) maintains tracking error statistically indistinguishable from a tuned classic VF while sharply reducing controller-user conflict, lowering NASA-TLX workload, and increasing Sense of Agency (SoAS). A composite BalancedScore that combines normalized accuracy and non-fighting behavior peaks near $τ=2-3$. These results show that a one-parameter Soft-Nash policy can preserve accuracy while improving comfort and perceived agency, providing a practical and interpretable pathway to personalized shared control in haptics and teleoperation.

Figures (7)

• Figure 1: Experiment Setup. Each participant performed a 60 second 3D tracking task with a haptic device followed by a questionnaire.
• Figure 2: RMS tracking error by assistance mode (participant means $\pm$95% CI). Classic VF and SoftNash with $\tau \le 3$ achieve similar error, while very soft controllers ($\tau \ge 5$) drift toward the no-assistance baseline.
• Figure 3: Conflict energy by assistance mode. Classic VF and hard-Nash ($\tau=0$) generate the most controller work directly opposing the user, whereas SoftNash with $\tau \in [1,3]$ reduces conflict by roughly 60–80% and very soft modes approach the NONE floor.
• Figure 4: Non-Fighting Index (NFI) by assistance mode. Lower NFI indicates less opposition per unit of delivered assistance; SoftNash with $\tau \in [1,3]$ roughly halves NFI relative to classic VF and hard-Nash, while very soft modes and NONE provide little assistance at all.
• Figure 5: Sense of Agency Scale (SoAS; 1–7) by assistance mode. Perceived agency is lowest for classic VF and hard-Nash, rises sharply for SoftNash around $\tau=2$, and then plateaus, with NONE yielding the highest agency as expected.