# Riemannian Motion Policy Fusion through Learnable Lyapunov Function Reshaping

arXiv: https://arxiv.org/abs/1910.02646 · HTML: https://mlsanity.com/paper/1910.02646/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Oct 2019.

## Built on by (most cited first, 6 of 6)

- [Motion Planning by Learning the Solution Manifold in Trajectory Optimization](https://mlsanity.com/paper/2107.05842/) (2021)
- [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)
- [Learning Dynamical Systems Encoding Non-Linearity within Space Curvature](https://mlsanity.com/paper/2403.11948/) (2024)
- [A Constrained-Optimization Approach to the Execution of Prioritized Stacks of Learned Multi-Robot Tasks](https://mlsanity.com/paper/2301.05346/) (2023)
- [Stable Robot Motions on Manifolds: Learning Lyapunov-Constrained Neural Manifold ODEs](https://mlsanity.com/paper/2510.05707/) (2025)

## Abstract (arXiv)

RMPflow is a recently proposed policy-fusion framework based on differential geometry. While RMPflow has demonstrated promising performance, it requires the user to provide sensible subtask policies as Riemannian motion policies (RMPs: a motion policy and an importance matrix function), which can be a difficult design problem in its own right. We propose RMPfusion, a variation of RMPflow, to address this issue. RMPfusion supplements RMPflow with weight functions that can hierarchically reshape the Lyapunov functions of the subtask RMPs according to the current configuration of the robot and environment. This extra flexibility can remedy imperfect subtask RMPs provided by the user, improving the combined policy's performance. These weight functions can be learned by back-propagation. Moreover, we prove that, under mild restrictions on the weight functions, RMPfusion always yields a globally Lyapunov-stable motion policy. This implies that we can treat RMPfusion as a structured policy class in policy optimization that is guaranteed to generate stable policies, even during the immature phase of learning. We demonstrate these properties of RMPfusion in imitation learning experiments both in simulation and on a real-world robot.