Riemannian Motion Policies

TL;DR

The paper introduces Riemannian Motion Policies (RMPs), a modular framework that pairs motion policies with local Riemannian metrics to capture geometry-critical directions. Through well-defined pullback/pushforward and accumulation operators, RMPs enable geometrically optimal fusion and transformation of multiple policies across task spaces, while remaining compatible with both reactive controllers and horizon-based optimizers like MPC. The authors demonstrate that these operators are associative, commutative, and covariant, yielding path-independent, globally optimal behavior; they also show how to handle joint limits and integrate computationally intensive behaviors in a scalable, parallelizable way. Empirically, RMPs enable robust collision avoidance and long-range navigation on both simulated and real robotic platforms, including multi-arm systems, and reveal significant advantages over naive policy superposition. The framework unifies disparate motion-generation techniques under a single, modular, and efficient paradigm with practical applicability to real-world robotics.

Abstract

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.

Figures (2)

• Figure 1: A motivating application for this work is operating collaboratively in human environments, which are unstructured and highly dynamic. Here a two-armed ABB YuMi robot opens a drawer and inserts a cup, while avoiding unintended collisions. To solve such problems, fast reactive local control is critical. In this work, we present a framework for incorporating agile, reactive, highly adaptive behaviors in a modular way. The resulting motion generation system aggregates hundreds of controllers, is straightforward to implement, and performs well in practice.
• Figure 2: The left three images shows three of the four obstacle environments and representative start configurations of the retraction experiment. The retracted configuration is depicted by the left arm's configuration in these images; the second retraction heuristic described in Section \ref{['sec:longRange']} successfully retracts from all 20 trial configurations in these environments. The simpler heuristic successfully solves all trials except two from the rightmost world which generally has obstacles closer to the robot. The remaining right two images depict two of the three reaching environments and solutions found by the RMP motion generation system.

Theorems & Definitions (10)

• Lemma B.1: Commutativity of summation
• proof
• Lemma B.2: Associativity of summation
• proof
• Lemma B.3: Linearity of pullback
• proof
• Corollary B.1: Independence of computational path
• proof
• Corollary B.2: Optimality
• proof