# Periodic DMP formulation for Quaternion Trajectories

arXiv: https://arxiv.org/abs/2110.10510 · HTML: https://mlsanity.com/paper/2110.10510/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Robot learning from demonstration, one of 13 directions in Offline RL and imitation learning from data.
- Builds on 4 of its 25 references that are in the map.
- First posted on arXiv 20 Oct 2021.

## Placed in

- Offline RL and imitation learning from data / [Robot learning from demonstration](https://mlsanity.com/offlinerl/robot-learning-from-demonstration/)

## Builds on (most cited first, 4 of 4)

- [Dynamic Movement Primitives in Robotics: A Tutorial Survey](https://mlsanity.com/paper/2102.03861/) (2021)
- [Towards Orientation Learning and Adaptation in Cartesian Space](https://mlsanity.com/paper/1907.03918/) (2019)
- [Merging Position and Orientation Motion Primitives](https://mlsanity.com/paper/2003.11507/) (2020)
- [An Energy-based Approach to Ensure the Stability of Learned Dynamical Systems](https://mlsanity.com/paper/2003.11290/) (2020)

## Built on by (most cited first, 2 of 2)

- [Learning to Extrapolate: A Transductive Approach](https://mlsanity.com/paper/2304.14329/) (2023)
- [Fusion Dynamical Systems with Machine Learning in Imitation Learning: A Comprehensive Overview](https://mlsanity.com/paper/2403.19916/) (2024)

## Abstract (arXiv)

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).