Extreme dynamic symmetry enables omnidirectional and multifunctional robots
Authors: Jiaxun Liu, Boxi Xia, Boyuan Chen
Organizations: Department of Mechanical Engineering and Materials Science, Duke University · Department of Electrical and Computer Engineering, Duke University · Department of Computer Science, Duke University
Symmetry is a central organizing principle in natural systems, yet its use as a unifying design strategy in robotics has largely remained limited to geometric form. We show that symmetry can instead be leveraged at the level of dynamic actuation capability. We introduce dynamic symmetry, the uniformity of a robot's attainable center-of-mass accelerations, and formalize it through a measure coined as dynamic isotropy. Across more than 1000 simulated morphologies, we found that higher dynamic symmetry consistently improved trajectory tracking, task success, robustness, resiliency, and energy efficiency, with the benefits becoming most pronounced as dynamic isotropy approached its theoretical limit. To study this regime systematically, we developed Argus, a family of spherical robots designed to explore the effects of increasing dynamic symmetry. Members of the Argus family vary in their actuation geometry and dynamic symmetry level while sharing a common architectural principle: radially oriented linear actuators that directly shape the robot's center-of-mass dynamics. Among them, we built a physical 20-leg Argus variant that achieved near-extreme dynamic isotropy and demonstrated orientation-invariant locomotion, agile traversal of cluttered and deformable terrain, rapid self-stabilization, and resilience to partial actuator failures. Its distributed sensing further enabled omnidirectional perception and object interaction during continuous motion. These results show that designing robots for symmetry not only in morphology but also in their attainable dynamics provides a powerful and general pathway toward agility, robustness, and multifunctionality in uncertain terrestrial and extraterrestrial environments.
Robots exhibit a rich variety of symmetries arising from their mechanical structure and the properties of their tasks. Although many robotics problems exhibit several symmetries simultaneously, existing approaches typically treat them in isolation, failing to exploit their combined potential. This paper introduces cross-space symmetry compositions, a framework for learning robot policies that are jointly equivariant to multiple symmetries across configuration and task spaces. Leveraging the differential-geometric structure of the forward kinematics map, we both descend symmetries from configuration to task space and lift symmetries from task to configuration space, enabling their composition within a unified representation space. We validate our framework on simulated and real-world experiments on a dual-arm robot, demonstrating that jointly leveraging multiple symmetries yields improved generalization.
While latent world models enable the proactive predictions required for extreme parkour, their purely data-driven nature forces them to redundantly encode left-right symmetric interactions as independent patterns. This inflates the learning burden and hinders the capture of geometric regularities, restricting the latent space's efficiency for downstream policies. To address this, we propose SWAP, an end-to-end equivariant symmetric world model. This framework embeds symmetry directly into both the world model and the actor-critic networks. In real-world tests, the robot leaps across a 2.13 m gap and climbs a 1.63 m platform, breaking records for quadruped parkour. Furthermore, the framework exhibits robust geometric generalization to unseen mirrored terrains and exceptional zero-shot transferability across diverse outdoor environments. These results demonstrate that symmetry equivariance is an effective structural prior for pushing the physical boundaries of learned legged locomotion.
Deep reinforcement policy learning directly in physical robots (on-robot learning) remains bottlenecked by slow wall-clock training times. We present SymmGrid, a trajectory level augmentation framework inspired by parallelized symmetries that super-scales group transformations to significantly accelerate on-robot learning in both egocentric and exocentric visual setups. We model a Markov Decision Process (MDP) under a symmetry tree, in which state-action pairs have admissible parallelized invariant transformations that yield a geometric grid structure. The state is modelled with ego- or exocentric images and proprioception information. The latter require special treatment, in the form of homographies, to warp visual scenes in line with their corresponding spatial transformations. These parallelized transformations produce a large set of unique symmetric equivalences that populate the replay buffer with diverse and consistent experiences that speed up learning and improve performance. We present extensive training and evaluations performed directly on real robot manipulation contact tasks including peg-insertions, cable routing, and object relocations. Relative to SOTA, SymmGrid achieved wall-clock training convergence speed-ups of 1.37-2.17x, evaluation success rate improvements of 1.09x-1.27x, fastest training convergence times of 16.6, 10.9, and 79.3 minutes respectively. For trajectory wide assessments, we used normalized area under the curve (nAUC) ratios. SymmGrid achieved improvements of up to 2.59x. These results confirm that simple branch symmetries can have an outsized result due to super-scaling and bring us closer to sub-10 minute on-robot learning training in manipulation tasks suitable for arms and humanoids. The project page is available at symmgrid-robot.github.io