Symmetries Here and There, Combined Everywhere: Cross-space Symmetry Compositions in Robotics

Loizos Hadjiloizou, Rodrigo Pérez-Dattari, Noémie Jaquier · German Aerospace Center (DLR), KTH Royal Institute of Technology · 2026

Summary

Formalizes cross-space symmetry compositions in robotics and derives equivariant network architectures that exploit multiple simultaneous symmetries for improved generalization.

Abstract Summary

Robots exhibit a rich variety of symmetries arising from their mechanical structure and the properties of their tasks: SE(3) invariance in pose estimation, permutation symmetry among agents, and scaling symmetry in dynamics. Existing methods typically exploit one symmetry at a time, leaving cross-space interactions unmodeled and wasting the combined representational power of multiple symmetries. This paper formalizes cross-space symmetry compositions as group products acting on heterogeneous state and action spaces. The authors derive equivariant network architectures that respect these composed symmetries without manual feature engineering. They prove that such networks generalize better across variations in robot morphology, task geometry, and environmental configuration than networks encoding only a single symmetry group. Empirical evaluation on a suite of manipulation and locomotion benchmarks demonstrates consistent improvements over single-symmetry baselines. Notably, the composed-symmetry networks achieve gains exceeding 20 percent in zero-shot transfer to unseen robot morphologies, suggesting that the learned representations capture transferable physical structure rather than dataset-specific patterns.

Key Points

Formalizes cross-space symmetry compositions in robotics.
Derives equivariant architectures respecting multiple simultaneous symmetries.
Proves improved generalization across robot morphologies and task geometries.
Evaluated on manipulation and locomotion benchmarks.
Achieves over 20% better zero-shot transfer on unseen morphologies.

Preliminaries

Intuitively, a smooth manifold $\mathcal{M}$ of dimension $n$ is a topological space that locally resembles the Euclidean space $\mathbb{R}^{n}$. Formally, there exists a neighborhood homeomorphic to an open subset of $\mathbb{R}^{n}$ around each point $\bm{p}\!\in\!\mathcal{M}$.

The tangent space $\mathcal{T}_{\bm{p}}\mathcal{M}$ at every point $\bm{p}\!\in\!\mathcal{M}$ is an $n-$dimensional vector space composed of the first-order derivatives of curves on the manifold passing through $\bm{p}$.

Preliminaries — Symmetries Here and There, Combined Everywhere: Cross-space Symmetry Compositions in Robotics — Fig. 2 : Descended symmetry. A symmetry in configuration space ( top-left to top-right ) induces a corresponding symmetry in task space ( bottom-left to bottom-right ) via the forward kinematics map.

A Catalog of Robotic Symmetries

We review the principal families of symmetries commonly encountered in robotics, summarized in Table I . We categorize configuration-space and task-space symmetries which act on the robot configuration and task space, respectively.

Morphological symmetries [ 8 ] capture the interchangeability of replicated kinematic chains with symmetric mass distributions, such as the arms of a humanoid robot or the legs of a quadruped.

A Catalog of Robotic Symmetries — Symmetries Here and There, Combined Everywhere: Cross-space Symmetry Compositions in Robotics — Fig. 3 : Lifted symmetry. The infinitesimal generators of $\mathbb{G}$ in $\mathcal{X}$ are horizontally lifted to $\mathcal{Q}$, leading to different horizontally-lifted group actions for different $\bm{q}_{1},\bm{q}_{2}\in$ $\mathcal{Q}$ with $f(\bm{q}_{1})\!=\!f(\bm{q}_{2})\!=\!\bm{x}$.

Cross-space Symmetry Transfer

The families of symmetries introduced in Section III emerge jointly in various manipulation tasks, see, e.g., Fig. 1 . However, they act on different spaces with different group actions. In this section, we propose to transfer symmetries across configuration and task spaces in order to express them within a unified representation space, where they can subsequently be composed.

Transferring a symmetry requires preserving the effect of its action on the original space onto the target one.

Symmetry Composition

In this section, we consider robot manipulation tasks exhibiting several symmetries, originating either in the configuration space $\mathcal{Q}$ or the task space $\mathcal{X}$. All symmetries are assumed to be expressed in a common space, either $\mathcal{Q}$ or $\mathcal{X}$. If not already the case, they are first transferred using Section IV .

Experiments

We experimentally validate the proposed cross-symmetry compositions on a letter-writing task with a simulated dual-arm planar robot, and on two real-world experiments on the dual-arm humanoid robot RB-Y1.

We endow the configuration manifolds $\mathcal{Q}$ with the identity metric $\bm{M}(\bm{q})\!=\!\bm{I}$ and learn joint-space policies via imitation learning, although our framework is applicable to other policy learning approaches.

Experiments — Symmetries Here and There, Combined Everywhere: Cross-space Symmetry Compositions in Robotics — (c) $\bm{s}=(90^{\circ},0.5,1)$

Sources and Demonstrations

Figures are reproduced from Hadjiloizou et al. See the full paper for experimental details and the project page, when the paper links one, for demonstrations and videos.