A Primer on SO(3) Action Representations in Deep Reinforcement Learning

In real-world reinforcement learning (RL), the choice of action space can play a key role in shaping motion smoothness, safety, and overall task performance. In this study, we evaluate pose increment, pose velocity, joint position increment, and joint velocity across two vision-based manipulation tasks: object picking and pushing. We train policies in simulation and deploy them to the real world using sim-to-real transfer. We find that action-space representation indeed significantly affects sim-to-real performance. In particular, we find that the joint velocity action space is best for the vision-based picking and pushing tasks in terms of smoothness and final task performance. We also provide practical guidance for RL practitioners in choosing action spaces for both simulation and real-world experiments.

Explicit 3D representations are attractive for manipulation because they expose object shape, workspace geometry, and robot-object relations in metric coordinates. However, sparse 3D encoders are often learned through downstream task objectives, tying the representation to a particular data distribution, policy architecture, and action parameterization. We introduce Sparse2Act, an observation-action alignment framework for pretraining sparse point-cloud encoders. The key idea is to use task-space end-effector actions as geometric supervision: masked sparse 3D tokens are trained to organize scene features around the workspace motion paired with the observation. After pretraining, only the encoder initialization is reused by downstream policies, allowing them to retain their own architectures and action spaces, including joint-space commands. On the LIBERO-10 benchmark, our method achieves 86.9% average success after 500 fine-tuning steps. The same pretrained encoder supports LIBERO-to-Meta-World cross-domain transfer, achieving 73.4% average success on the Meta-World-5 benchmark. Ablations on the objective and decoder capacity show that the gains come from the masked action-alignment signal and remain useful across downstream action decoders. In real-world experiments, simulation pretraining followed by limited real-data fine-tuning achieves an average success rate of 72.5% across four tasks, demonstrating effective sim-to-real transfer. These results suggest that robot actions can provide compact geometric supervision for reusable sparse 3D representations.

Diffusion-based Vision-Language-Action policies achieve remarkable success in robotic manipulation, yet commit a fundamental geometric error we term the $\textbf{Euclidean Fallacy}$: representing SE(3) poses as flat $\mathbb{R}^{12}$ vectors. This approximation induces (1) manifold drift violating SO(3) constraints, (2) broken equivariance under coordinate transformations, and (3) non-geodesic trajectories with excessive kinematic cost. We introduce $\textbf{Lie Diffuser Actor (LDA)}$, a diffusion framework operating intrinsically on SE(3). Our method injects noise through left-invariant SDEs, predicts scores in the tangent space, and retracts samples via the exponential map. This formulation eliminates manifold drift by construction while guaranteeing coordinate-frame equivariance and geodesic optimality. On CALVIN ABC$\rightarrow$D, LDA improves average task length from $3.27$ to $3.51$ ($+7.3\%$). We further validate our method on real robot and the results show that our methodology outperforms the baseline on majority tasks.