  In this study, we propose the use of the phase-amplitude reduction method to
construct an imitation learning framework. Imitating human movement
trajectories is recognized as a promising strategy for generating a range of
human-like robot movements. Unlike previous dynamical system-based imitation
learning approaches, our proposed method allows the robot not only to imitate a
limit cycle trajectory but also to replicate the transient movement from the
initial or disturbed state to the limit cycle. Consequently, our method offers
a safer imitation learning approach that avoids generating unpredictable
motions immediately after disturbances or from a specified initial state. We
first validated our proposed method by reconstructing a simple limit-cycle
attractor. We then compared the proposed approach with a conventional method on
a lemniscate trajectory tracking task with a simulated robot arm. Our findings
confirm that our proposed method can more accurately generate transient
movements to converge on a target periodic attractor compared to the previous
standard approach. Subsequently, we applied our method to a real robot arm to
imitate periodic human movements.
