  This paper presents a new data-driven framework for analyzing periodic
physical human-robot interaction (pHRI) in latent state space. To elaborate
human understanding and/or robot control during pHRI, the model representing
pHRI is critical. Recent developments of deep learning technologies would
enable us to learn such a model from a dataset collected from the actual pHRI.
Our framework is developed based on variational recurrent neural network
(VRNN), which can inherently handle time-series data like one pHRI generates.
This paper modifies VRNN in order to include the latent dynamics from robot to
human explicitly. In addition, to analyze periodic motions like walking, we
integrate a new recurrent network based on reservoir computing (RC), which has
random and fixed connections between numerous neurons, with VRNN. By augmenting
RC into complex domain, periodic behavior can be represented as the phase
rotation in complex domain without decaying the amplitude. For verification of
the proposed framework, a rope-rotation/swinging experiment was analyzed. The
proposed framework, trained on the dataset collected from the experiment,
achieved the latent state space where the differences in periodic motions can
be distinguished. Such a well-distinguished space yielded the best prediction
accuracy of the human observations and the robot actions. The attached video
can be seen in youtube: https://youtu.be/umn0MVcIpsY
