Many daily tasks exhibit a periodic nature, necessitating that robots possess the ability to execute them either alone or in collaboration with humans. A widely used approach to encode and learn such periodic patterns from human demonstrations is through periodic Dynamic Movement Primitives (DMPs). Periodic DMPs encode cyclic data independently across multiple dimensions of multi-degree of freedom systems. This method is effective for simple data, like Cartesian or joint position trajectories. However, it cannot account for various geometric constraints imposed by more complex data, such as orientation and stiffness. To bridge this gap, we propose a novel periodic DMP formulation that enables the encoding of periodic orientation trajectories and varying stiffness matrices while considering their geometric constraints. Our geometry-aware approach exploits the properties of the Riemannian manifold and Lie group to directly encode such periodic data while respecting its inherent geometric constraints. We initially employed simulation to validate the technical aspects of the proposed method thoroughly. Subsequently, we conducted experiments with two different real-world robots performing daily tasks involving periodic changes in orientation and/or stiffness, i.e., operating a drilling machine using a rotary handle and facilitating collaborative human–robot sawing.

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Biological systems, including human beings, have the innate ability to perform complex tasks in a versatile and agile manner. Researchers in sensorimotor control have aimed to comprehend and formally define this innate characteristic. The idea, supported by several experimental findings, that biological systems are able to combine and adapt basic units of motion into complex tasks finally leads to the formulation of the motor primitives’ theory. In this respect, Dynamic Movement Primitives (DMPs) represent an elegant mathematical formulation of the motor primitives as stable dynamical systems and are well suited to generate motor commands for artificial systems like robots. In the last decades, DMPs have inspired researchers in different robotic fields including imitation and reinforcement learning, optimal control, physical interaction, and human–robot co-working, resulting in a considerable amount of published papers. The goal of this tutorial survey is two-fold. On one side, we present the existing DMP formulations in rigorous mathematical terms and discuss the advantages and limitations of each approach as well as practical implementation details. In the tutorial vein, we also search for existing implementations of presented approaches and release several others. On the other side, we provide a systematic and comprehensive review of existing literature and categorize state-of-the-art work on DMP. The paper concludes with a discussion on the limitations of DMPs and an outline of possible research directions.

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Imitation learning techniques have been used as a way to transfer skills to robots. Among them, dynamic movement primitives (DMPs) have been widely exploited as an effective and an efficient technique to learn and reproduce complex discrete and periodic skills. While DMPs have been properly formulated for learning point-to-point movements for both translation and orientation, periodic ones are missing a formulation to learn the orientation. To address this gap, we propose a novel DMP formulation that enables encoding of periodic orientation trajectories. Within this formulation we develop two approaches: Riemannian metric-based projection approach and unit quaternion based periodic DMP. Both formulations exploit unit quaternions to represent the orientation. However, the first exploits the properties of Riemannian manifolds to work in the tangent space of the unit sphere. The second encodes directly the unit quaternion trajectory while guaranteeing the unitary norm of the generated quaternions. We validated the technical aspects of the proposed methods in simulation. Then we performed experiments on a real robot to execute daily tasks that involve periodic orientation changes (i.e., surface polishing/wiping and liquid mixing by shaking).@en