We address the challenge of effectively controlling the locomotion of legged robots by incorporating precise frequency and phase characteristics, which is often ignored in locomotion policies that do not account for the periodic nature of walking. We propose a hierarchical architecture that integrates a low-level phase tracker, oscillators, and a high-level phase modulator. This controller allows quadruped robots to walk in a natural manner that is synchronized with external musical rhythms. Our method generates diverse gaits across different frequencies and achieves real-time synchronization with music in the physical world. This research establishes a foundational framework for enabling real-time execution of accurate rhythmic motions in legged robots. Video is available at website: https://music-walker.github.io/.

Hierarchical Imitation Learning (HIL) is an effective way for robots to learn sub-skills from long-horizon unsegmented demonstrations. However, the learned hierarchical structure lacks the mechanism to transfer across multi-tasks or to new tasks, which makes them have to learn from scratch when facing a new situation. Transferring and reorganizing modular sub-skills require fast adaptation ability of the whole hierarchical structure. In this work, we propose Dual Meta Imitation Learning (DMIL), a hierarchical meta imitation learning method where the high-level network and sub-skills are iteratively meta-learned with model-agnostic meta-learning. DMIL uses the likelihood of state-action pairs from each sub-skill as the supervision for the high-level network adaptation, and use the adapted high-level network to determine different data set for each sub-skill adaptation. We theoretically prove the convergence of the iterative training process of DMIL and establish the connection between DMIL and Expectation-Maximization algorithm. Empirically, we achieve state-of-the-art few-shot imitation learning performance on the Meta-world \cite{metaworld} benchmark and competitive results on long-horizon tasks of Kitchen environments.