# Orientation Probabilistic Movement Primitives on Riemannian Manifolds

arXiv: https://arxiv.org/abs/2110.15036 · HTML: https://mlsanity.com/paper/2110.15036/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 32 references that are in the map.
- First posted on arXiv 28 Oct 2021.

## Builds on (most cited first, 8 of 8)

- [A micro Lie theory for state estimation in robotics](https://mlsanity.com/paper/1812.01537/) (2018)
- [Neural Dynamic Policies for End-to-End Sensorimotor Learning](https://mlsanity.com/paper/2012.02788/) (2020)
- [Kernelized Movement Primitives](https://mlsanity.com/paper/1708.08638/) (2017)
- [Bayesian Optimization Meets Riemannian Manifolds in Robot Learning](https://mlsanity.com/paper/1910.04998/) (2019)
- [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)
- [Learning and Sequencing of Object-Centric Manipulation Skills for Industrial Tasks](https://mlsanity.com/paper/2008.10471/) (2020)
- [Benchmark for Skill Learning from Demonstration: Impact of User Experience, Task Complexity, and Start Configuration on Performance](https://mlsanity.com/paper/1911.02725/) (2019)

## Built on by (most cited first, 10 of 10)

- [TOP-ERL: Transformer-based Off-Policy Episodic Reinforcement Learning](https://mlsanity.com/paper/2410.09536/) (2024)
- [Learning Riemannian Stable Dynamical Systems via Diffeomorphisms](https://mlsanity.com/paper/2211.03169/) (2022)
- [Open the Black Box: Step-based Policy Updates for Temporally-Correlated Episodic Reinforcement Learning](https://mlsanity.com/paper/2401.11437/) (2024)
- [Wasserstein Gradient Flows for Optimizing Gaussian Mixture Policies](https://mlsanity.com/paper/2305.10411/) (2023)
- [MP3: Movement Primitive-Based (Re-)Planning Policy](https://mlsanity.com/paper/2306.12729/) (2023)
- [Reactive Motion Generation on Learned Riemannian Manifolds](https://mlsanity.com/paper/2203.07761/) (2022)
- [Unraveling the Single Tangent Space Fallacy: An Analysis and Clarification for Applying Riemannian Geometry in Robot Learning](https://mlsanity.com/paper/2310.07902/) (2023)
- [Riemann2: Learning Riemannian Submanifolds from Riemannian Data](https://mlsanity.com/paper/2503.05540/) (2025)
- [The Unreasonable Effectiveness of Discrete-Time Gaussian Process Mixtures for Robot Policy Learning](https://mlsanity.com/paper/2505.03296/) (2025)
- [Non-parametric regression for robot learning on manifolds](https://mlsanity.com/paper/2310.19561/) (2023)

## Abstract (arXiv)

Learning complex robot motions necessarily demands to have models that are able to encode and retrieve full-pose trajectories when tasks are defined in operational spaces. Probabilistic movement primitives (ProMPs) stand out as a principled approach that models trajectory distributions learned from demonstrations. ProMPs allow for trajectory modulation and blending to achieve better generalization to novel situations. However, when ProMPs are employed in operational space, their original formulation does not directly apply to full-pose movements including rotational trajectories described by quaternions. This paper proposes a Riemannian formulation of ProMPs that enables encoding and retrieving of quaternion trajectories. Our method builds on Riemannian manifold theory, and exploits multilinear geodesic regression for estimating the ProMPs parameters. This novel approach makes ProMPs a suitable model for learning complex full-pose robot motion patterns. Riemannian ProMPs are tested on toy examples to illustrate their workflow, and on real learning-from-demonstration experiments.