# Riemannian Motion Policies

arXiv: https://arxiv.org/abs/1801.02854 · HTML: https://mlsanity.com/paper/1801.02854/

## Key facts

- Cited by 46 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Jan 2018.

## Built on by (most cited first, 30 of 46)

- [VoxPoser: Composable 3D Value Maps for Robotic Manipulation with Language Models](https://mlsanity.com/paper/2307.05973/) (2023)
- [ReKep: Spatio-Temporal Reasoning of Relational Keypoint Constraints for Robotic Manipulation](https://mlsanity.com/paper/2409.01652/) (2024)
- [Affordances from Human Videos as a Versatile Representation for Robotics](https://mlsanity.com/paper/2304.08488/) (2023)
- [AnyTeleop: A General Vision-Based Dexterous Robot Arm-Hand Teleoperation System](https://mlsanity.com/paper/2307.04577/) (2023)
- [Motion Policy Networks](https://mlsanity.com/paper/2210.12209/) (2022)
- [Potential Based Diffusion Motion Planning](https://mlsanity.com/paper/2407.06169/) (2024)
- [DexWild: Dexterous Human Interactions for In-the-Wild Robot Policies](https://mlsanity.com/paper/2505.07813/) (2025)
- [FACTR: Force-Attending Curriculum Training for Contact-Rich Policy Learning](https://mlsanity.com/paper/2502.17432/) (2025)
- [Differentiable Spatial Planning using Transformers](https://mlsanity.com/paper/2112.01010/) (2021)
- [SAIL: Faster-than-Demonstration Execution of Imitation Learning Policies](https://mlsanity.com/paper/2506.11948/) (2025)
- [Hierarchical Neural Dynamic Policies](https://mlsanity.com/paper/2107.05627/) (2021)
- [Composable Energy Policies for Reactive Motion Generation and Reinforcement Learning](https://mlsanity.com/paper/2105.04962/) (2021)
- [Regularized Deep Signed Distance Fields for Reactive Motion Generation](https://mlsanity.com/paper/2203.04739/) (2022)
- [Learning Stable Vector Fields on Lie Groups](https://mlsanity.com/paper/2110.11774/) (2021)
- [RGB-Only Reconstruction of Tabletop Scenes for Collision-Free Manipulator Control](https://mlsanity.com/paper/2210.11668/) (2022)
- [Model Predictive Control for Fluid Human-to-Robot Handovers](https://mlsanity.com/paper/2204.00134/) (2022)
- [Deep Reactive Policy: Learning Reactive Manipulator Motion Planning for Dynamic Environments](https://mlsanity.com/paper/2509.06953/) (2025)
- [RMP2: A Structured Composable Policy Class for Robot Learning](https://mlsanity.com/paper/2103.05922/) (2021)
- [Hierarchical Policy Blending as Inference for Reactive Robot Control](https://mlsanity.com/paper/2210.07890/) (2022)
- [Reaching Through Latent Space: From Joint Statistics to Path Planning in Manipulation](https://mlsanity.com/paper/2210.11779/) (2022)
- [Sim-Suction: Learning a Suction Grasp Policy for Cluttered Environments Using a Synthetic Benchmark](https://mlsanity.com/paper/2305.16378/) (2023)
- [Reactive Motion Generation on Learned Riemannian Manifolds](https://mlsanity.com/paper/2203.07761/) (2022)
- [Sampling-free obstacle gradients and reactive planning in Neural Radiance Fields (NeRF)](https://mlsanity.com/paper/2205.01389/) (2022)
- [Hierarchical Policy Blending As Optimal Transport](https://mlsanity.com/paper/2212.01938/) (2022)
- [TempoVLA: Learning Speed-Controllable Vision-Language-Action Policies](https://mlsanity.com/paper/2606.06491/) (2026)
- [Learning Dynamical Systems Encoding Non-Linearity within Space Curvature](https://mlsanity.com/paper/2403.11948/) (2024)
- [From Single Images to Motion Policies via Video-Generation Environment Representations](https://mlsanity.com/paper/2505.19306/) (2025)
- [OptiWorld: Optimal Control for Video World Generation under Physical Constraints](https://mlsanity.com/paper/2606.00499/) (2026)
- [ADEPT: Accelerating Dexterity via Pre-Training and Post-Training using Reinforcement Learning](https://mlsanity.com/paper/2608.19182/) (2026)
- [Imitation Learning via Simultaneous Optimization of Policies and Auxiliary Trajectories](https://mlsanity.com/paper/2105.03019/) (2021)

## Abstract (arXiv)

We introduce the Riemannian Motion Policy (RMP), a new mathematical object for modular motion generation. An RMP is a second-order dynamical system (acceleration field or motion policy) coupled with a corresponding Riemannian metric. The motion policy maps positions and velocities to accelerations, while the metric captures the directions in the space important to the policy. We show that RMPs provide a straightforward and convenient method for combining multiple motion policies and transforming such policies from one space (such as the task space) to another (such as the configuration space) in geometrically consistent ways. The operators we derive for these combinations and transformations are provably optimal, have linearity properties making them agnostic to the order of application, and are strongly analogous to the covariant transformations of natural gradients popular in the machine learning literature. The RMP framework enables the fusion of motion policies from different motion generation paradigms, such as dynamical systems, dynamic movement primitives (DMPs), optimal control, operational space control, nonlinear reactive controllers, motion optimization, and model predictive control (MPC), thus unifying these disparate techniques from the literature. RMPs are easy to implement and manipulate, facilitate controller design, simplify handling of joint limits, and clarify a number of open questions regarding the proper fusion of motion generation methods (such as incorporating local reactive policies into long-horizon optimizers). We demonstrate the effectiveness of RMPs on both simulation and real robots, including their ability to naturally and efficiently solve complicated collision avoidance problems previously handled by more complex planners.