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Human joint dynamic stiffness plays an important role in the stability of performance augmentation exoskeletons. In this paper, we consider a new frequency domain model of the human joint dynamics which features a complex value stiffness. This complex stiffness consists of a real stiffness and a hysteretic damping. We use it to explain the dynamic behaviors of the human connected to the exoskeleton, in particular the observed non-zero low frequency phase shift and the near constant damping ratio of the resonance as stiffness and inertia vary. We validate this concept with an elbow-joint exoskeleton testbed (attached to a subject) by experimentally varying joint stiffness behavior, exoskeleton inertia, and the strength augmentation gain. We compare three different models of elbow-joint dynamic stiffness: a model with real stiffness, viscous damping and inertia; a model with complex stiffness and inertia; and a model combining the previous two models. Our results show that the hysteretic damping term improves modeling accuracy (via a statistical F-test). Moreover, this term contributes more to model accuracy than the viscous damping term. In addition, we experimentally observe a linear relationship between the hysteretic damping and the real part of the stiffness which allows us to simplify the complex stiffness model down to a 1-parameter system. Ultimately, we design a fractional order controller to demonstrate how human hysteretic damping behavior can be exploited to improve strength amplification performance while maintaining stability.

Complex Stiffness Model of Physical Human-Robot Interaction: Implications for Control of Performance Augmentation Exoskeletons