Morphological Symmetries in Robotics

TL;DR

This work recognizes morphological symmetries as a relevant and previously unexplored physics-informed geometric prior, with significant implications for both data-driven and analytical methods used in modeling, control, estimation and design in robotics.

Abstract

We present a comprehensive framework for studying and leveraging morphological symmetries in robotic systems. These are intrinsic properties of the robot's morphology, frequently observed in animal biology and robotics, which stem from the replication of kinematic structures and the symmetrical distribution of mass. We illustrate how these symmetries extend to the robot's state space and both proprioceptive and exteroceptive sensor measurements, resulting in the equivariance of the robot's equations of motion and optimal control policies. Thus, we recognize morphological symmetries as a relevant and previously unexplored physics-informed geometric prior, with significant implications for both data-driven and analytical methods used in modeling, control, estimation and design in robotics. For data-driven methods, we demonstrate that morphological symmetries can enhance the sample efficiency and generalization of machine learning models through data augmentation, or by applying equivariant/invariant constraints on the model's architecture. In the context of analytical methods, we employ abstract harmonic analysis to decompose the robot's dynamics into a superposition of lower-dimensional, independent dynamics. We substantiate our claims with both synthetic and real-world experiments conducted on bipedal and quadrupedal robots. Lastly, we introduce the repository MorphoSymm to facilitate the practical use of the theory and applications outlined in this work.

Figures (2)

• Figure 1: (a) https://en.wikipedia.org/wiki/Cayley_graph of the morphological symmetries of the Mini Cheetah quadruped robot katz2019mini (see https://danfoa.github.io/MorphoSymm/static/animations/mini_cheetah-C2xC2xC2-symmetries_anim_static.gif?utm_source=\source Morphological symmetry group of the Mini Cheetah robot $\mathbb{G}_{\hbox{$$}}= \mathbb{C}_{2} \times \mathbb{C}_{2} \times \mathbb{C}_{2}$: See animation at https://danfoa.github.io/MorphoSymm/static/animations/mini_cheetah-C2xC2xC2-symmetries_anim_static.gif?utm_source=\source ). The robot's symmetries relate any system state with eight distinct symmetric states, each exhibiting identical or equivalent dynamics. These symmetries stem from the robot's ability to reconfigure its state, emulating three orthogonal reflections in $3D$ space: reflection with respect to the $yz$-plane ($g_s$), $xz$-plane ($g_t$), and $xy$-plane ($g_f$). The robot's symmetry group is composed of the three reflections and any composition of these transformations (e.g., $g_s \mathrel{\hbox{$\circ$}} g_t$), resulting in the group $\mathbb{G}_{\hbox{$$}}= \mathbb{K}_{4} \times \mathbb{C}_{2} = \mathbb{C}_{2} \times \mathbb{C}_{2} \times \mathbb{C}_{2} = \{e, g_s\} \times \{e, g_t\} \times \{e, g_f\}$ (see \ref{['fig:G_C2xC2xC2']}). (b) Morphological symmetries manifest in the robot's dynamics, control policies, and both proprioceptive and exteroceptive sensor readings (e.g., contact points/forces, RGBD images). Consequently, any controlled trajectory of motion $[({\bm{q}}_{t},\dot{{\bm{q}}}_{t})]_{t=0}^{T}$, along with the trajectory of sensor readings, can be augmented to feasible controlled trajectories and sensor readings for each of the $8$ symmetric states $[(g \mathrel{\hbox{\stackon[0.5pt]{$\triangleright$}{$\diamond$}}} {\bm{q}}_{t},g \mathrel{\hbox{\stackon[0.5pt]{$\triangleright$}{$\diamond$}}} \dot{{\bm{q}}}_{t})]_{t=0}^{T}$ for all $g \in \mathbb{G}_{\hbox{$$}}$ (see https://danfoa.github.io/MorphoSymm/static/dynamic_animations/mini-cheetah_animation_C2xC2xC2.gif Equivariant temporal evolution of symmetric states of the Mini Cheetah robot See animation at https://danfoa.github.io/MorphoSymm/static/dynamic_animations/mini-cheetah_animation_C2xC2xC2.gif ). (c) These trajectories, conceptualized as point trajectories evolving within the robot's configuration space $\mathcal{Q} := \mathbb{SE}_{3} \times \mathcal{M}$, are decomposed into the $6$-dimensional manifold of the robot's base configurations (the special Euclidean group $\mathbb{SE}_{3}$) and the $12$-dimensional manifold of joint space configurations $\mathcal{M}$. Morphological symmetries, imposing geometric constraints on $\mathbb{SE}_{3}$ and $\mathcal{M}$, serve as a physics-informed geometric prior useful in robotics methods (refer to \ref{['sec:applications']}).
• Figure 2: Structure of the group $\mathbb{G}_{\hbox{$$}}= \mathbb{C}_{2} \times \mathbb{C}_{2} \times \mathbb{C}_{2}$