SWAP: Symmetric Equivariant World-Model for Agile Robot Parkour

A latent world model built from an equivariant encoder and predictor inherits a provable symmetry of its training loss: when the dynamics carries a group $G$ acting on latents by an orthogonal representation $ρ(g)$, the one-step prediction relMSE is exactly invariant across the whole group, so fitting a restricted slice of orientations mathematically determines it on the entire orbit. The symmetry survives a real Muon/AdamW+EMA+VICReg run -- composed residual $\sim 10^{-6}$ after training, under any optimiser (intrinsic Vector-Neuron/e3nn parametrisation) -- and one-step error is flat across the group (5-seed medians: equivariant $\times 1.00$ vs a higher-capacity non-equivariant baseline $\times 12.7$ in 2D, $\times 17.2$ in 3D), while that baseline fits the slice but breaks out-of-distribution. The flatness is not a synthetic artefact: on real-robot DROID end-effector trajectories the equivariant model stays flat across the orbit ($\times 1.000$, rotation residual $1.5\times 10^{-16}$) while a $4.5\times$-larger baseline degrades $\times 11$. One caution is load-bearing: flatness is necessary, not sufficient -- the theorem transports the in-distribution error level unchanged but does not lower it (3D relMSE $\approx 0.43$): across-group error is constant, not low. The same isometry lifts to a closed-loop corollary: under a matching equivariant planner the control error is invariant across the group -- float-floor-exact in 2D/SO(2), statistically flat in 3D/SE(3). Stress-tested against Sutton's Bitter Lesson (augmentation, scale, soft-equivariance), each closes at most the across-group task metric, never the float-floor exactness. This is the generalisation-side foundation of a certified-world-models programme (arXiv:2606.13092, 2606.24945, 2606.24946): flatness transports competence, and the trust bounds built on it are downstream products.

Reinforcement learning (RL) algorithms classically suffer from poor sample efficiency. In robotics, a recent line of work has emerged addressing this problem by encoding physics priors in the learning process. However, most of these approaches are validated on well-defined, low-dimensional benchmark systems rather than high-dimensional robots with complex nonlinear dynamics. In this paper, we introduce \textit{SKooP (Symmetric Koopman Predictions)}, an approach combining the advantages of morphological symmetries with those of a Koopman model learned via autoencoder to enhance policy learning. SKooP learns a Koopman model of the system dynamics alongside the policy. The resulting Koopman predictions are used as privileged observations for the critic, allowing the agent to learn based on smoother, more informative features. We also incorporate group symmetries into the actor, critic, encoder and decoder networks to produce a highly equivariant policy. The SKooP approach is validated via in-depth analysis of the learned Koopman models and symmetric policies to showcase how each of these influences the agent's performance. We also show that the learned policies are transferable to different simulation environments. Our results show that SKooP consistently reduces convergence time and increases the learned reward for multiple challenging bipedal locomotion tasks on a quadruped robot. Project page: https://evelyd.github.io/SymmetricKoopmanPredictions

Parkour tasks for quadrupeds have emerged as a promising benchmark for agile locomotion. While human athletes can effectively perceive environmental characteristics to select appropriate footholds for obstacle traversal, endowing legged robots with similar perceptual reasoning remains a significant challenge. Existing methods often rely on hierarchical controllers that follow pre-computed footholds, thereby constraining the robot's real-time adaptability and the exploratory potential of reinforcement learning. To overcome these challenges, we present PUMA, an end-to-end learning framework that integrates visual perception and foothold priors into a single-stage training process. This approach leverages terrain features to estimate egocentric polar foothold priors, composed of relative distance and heading, guiding the robot in active posture adaptation for parkour tasks. Extensive experiments conducted in simulation and real-world environments across various discrete complex terrains, demonstrate PUMA's exceptional agility and robustness in challenging scenarios.