Wavelet Movement Primitives: A Unified Framework for Learning Discrete and Rhythmic Movements

Real-world tasks often require combinations of both discrete and rhythmic movements. However, most of current methods can only address one of them. This letter proposes a unified framework, Wavelet Movement Primitives (WMPs), which are built on Probabilistic Movement Primitives (ProMPs) integrated with Discrete Wavelet Transform (DWT), to model and learn both discrete and rhythmic trajectories from demonstrations. The key advantage of WMPs lies in its ability to naturally identify and facilitate a smooth transition between discrete and rhythmic motions using wavelet transforms. Additionally, we propose local frame WMPs (LF-WMPs) for discrete tasks, enabling the learned movements to generalize to new environments. For rhythmic tasks, a phase-adaptive weight adjustment algorithm is proposed, allowing the system to capture time-frequency features of the demonstrations and safely guide the trajectory back to the desired region. Finally, the method is validated through several simulations and a real-world robotic stirring task, demonstrating its good extrapolation capabilities.

I. Introduction

With the rapid advancement of automation and robotics, the diversity and complexity of task execution have steadily increased. Among various task types, discrete and rhythmic tasks have garnered significant attention due to their distinct motion patterns [1]. Discrete tasks involve a series of well-defined action sequences, each with clear start and end points, such as grasping [2], moving [3], and placing objects [4]. In contrast, rhythmic tasks require repetitive motion over extended periods, as seen in actions like stirring [5], polishing [6], or welding [7]. Both types of tasks are widely needed in practical applications. However, many real-world tasks often comprise a combination of discrete and rhythmic motions. For example, in a cooking scenario, stirring involves rhythmic movements, while picking and placing ingredients are discrete actions [8]. In such compound tasks, robotic systems must not only learn and execute both types of movements but also ensure natural¹

The term ‘natural’ does not have a strict mathematical definition but instead refers to a continuous and graceful movement that appears smooth and natural. Similar meanings apply to the term ‘smooth’ throughout the letter.

transitions between these modes [9], [10]. This adaptability to multiple motion patterns is critical for improving the efficiency and precision of task execution, especially in unknown environments.