# ProDMPs: A Unified Perspective on Dynamic and Probabilistic Movement Primitives

arXiv: https://arxiv.org/abs/2210.01531 · HTML: https://mlsanity.com/paper/2210.01531/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 35 references that are in the map.
- First posted on arXiv 4 Oct 2022.

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

- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Conditional Neural Processes](https://mlsanity.com/paper/1807.01613/) (2018)
- [Neural Processes](https://mlsanity.com/paper/1807.01622/) (2018)
- [Neural Dynamic Policies for End-to-End Sensorimotor Learning](https://mlsanity.com/paper/2012.02788/) (2020)
- [Dynamic Movement Primitives in Robotics: A Tutorial Survey](https://mlsanity.com/paper/2102.03861/) (2021)
- [ImitationFlow: Learning Deep Stable Stochastic Dynamic Systems by Normalizing Flows](https://mlsanity.com/paper/2010.13129/) (2020)
- [Specializing Versatile Skill Libraries using Local Mixture of Experts](https://mlsanity.com/paper/2112.04216/) (2021)
- [A Probabilistic Representation for Dynamic Movement Primitives](https://mlsanity.com/paper/1612.05932/) (2016)

## Built on by (most cited first, 25 of 25)

- [Movement Primitive Diffusion: Learning Gentle Robotic Manipulation of Deformable Objects](https://mlsanity.com/paper/2312.10008/) (2023)
- [Motion Planning Diffusion: Learning and Adapting Robot Motion Planning with Diffusion Models](https://mlsanity.com/paper/2412.19948/) (2024)
- [BEAST: Efficient Tokenization of B-Splines Encoded Action Sequences for Imitation Learning](https://mlsanity.com/paper/2506.06072/) (2025)
- [Enhancing the LLM-Based Robot Manipulation Through Human-Robot Collaboration](https://mlsanity.com/paper/2406.14097/) (2024)
- [TOP-ERL: Transformer-based Off-Policy Episodic Reinforcement Learning](https://mlsanity.com/paper/2410.09536/) (2024)
- [Chunking the Critic: A Transformer-based Soft Actor-Critic with N-Step Returns](https://mlsanity.com/paper/2503.03660/) (2025)
- [Acquiring Diverse Skills using Curriculum Reinforcement Learning with Mixture of Experts](https://mlsanity.com/paper/2403.06966/) (2024)
- [A Survey on Imitation Learning for Contact-Rich Tasks in Robotics](https://mlsanity.com/paper/2506.13498/) (2025)
- [Open the Black Box: Step-based Policy Updates for Temporally-Correlated Episodic Reinforcement Learning](https://mlsanity.com/paper/2401.11437/) (2024)
- [MP3: Movement Primitive-Based (Re-)Planning Policy](https://mlsanity.com/paper/2306.12729/) (2023)
- [Bridging the gap between Learning-to-plan, Motion Primitives and Safe Reinforcement Learning](https://mlsanity.com/paper/2408.14063/) (2024)
- [Using Implicit Behavior Cloning and Dynamic Movement Primitive to Facilitate Reinforcement Learning for Robot Motion Planning](https://mlsanity.com/paper/2307.16062/) (2023)
- [PUMA: Deep Metric Imitation Learning for Stable Motion Primitives](https://mlsanity.com/paper/2310.12831/) (2023)
- [Fusion Dynamical Systems with Machine Learning in Imitation Learning: A Comprehensive Overview](https://mlsanity.com/paper/2403.19916/) (2024)
- [Enhancing Exploration with Diffusion Policies in Hybrid Off-Policy RL: Application to Non-Prehensile Manipulation](https://mlsanity.com/paper/2411.14913/) (2024)
- [MuTT: A Multimodal Trajectory Transformer for Robot Skills](https://mlsanity.com/paper/2407.15660/) (2024)
- [Use the Force, Bot! -- Force-Aware ProDMP with Event-Based Replanning](https://mlsanity.com/paper/2409.11144/) (2024)
- [FRMD: Fast Robot Motion Diffusion with Consistency-Distilled Movement Primitives for Smooth Action Generation](https://mlsanity.com/paper/2503.02048/) (2025)
- [Context-aware Learned Mesh-based Simulation via Trajectory-Level Meta-Learning](https://mlsanity.com/paper/2511.05234/) (2025)
- [Frequency-Aware Flow Matching for Continuous and Consistent Robotic Action Generation](https://mlsanity.com/paper/2606.20135/) (2026)
- [Latent Task-Specific Graph Network Simulators](https://mlsanity.com/paper/2311.05256/) (2023)
- [TamedPUMA: safe and stable imitation learning with geometric fabrics](https://mlsanity.com/paper/2503.17432/) (2025)
- [MoRe-ERL: Learning Motion Residuals using Episodic Reinforcement Learning](https://mlsanity.com/paper/2508.01409/) (2025)
- [FODMP: Fast One-Step Diffusion of Movement Primitives Generation for Time-Dependent Robot Actions](https://mlsanity.com/paper/2603.24806/) (2026)
- [Sparse Compositional Flow Matching by geometric assembly from motion primitives](https://mlsanity.com/paper/2605.23341/) (2026)

## Abstract (arXiv)

Movement Primitives (MPs) are a well-known concept to represent and generate modular trajectories. MPs can be broadly categorized into two types: (a) dynamics-based approaches that generate smooth trajectories from any initial state, e. g., Dynamic Movement Primitives (DMPs), and (b) probabilistic approaches that capture higher-order statistics of the motion, e. g., Probabilistic Movement Primitives (ProMPs). To date, however, there is no method that unifies both, i. e. that can generate smooth trajectories from an arbitrary initial state while capturing higher-order statistics. In this paper, we introduce a unified perspective of both approaches by solving the ODE underlying the DMPs. We convert expensive online numerical integration of DMPs into basis functions that can be computed offline. These basis functions can be used to represent trajectories or trajectory distributions similar to ProMPs while maintaining all the properties of dynamical systems. Since we inherit the properties of both methodologies, we call our proposed model Probabilistic Dynamic Movement Primitives (ProDMPs). Additionally, we embed ProDMPs in deep neural network architecture and propose a new cost function for efficient end-to-end learning of higher-order trajectory statistics. To this end, we leverage Bayesian Aggregation for non-linear iterative conditioning on sensory inputs. Our proposed model achieves smooth trajectory generation, goal-attractor convergence, correlation analysis, non-linear conditioning, and online re-planing in one framework.