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

Many robotic control tasks require policies to act on orientations, yet the geometry of SO(3) makes this nontrivial. Because SO(3) admits no global, smooth, minimal parameterization, common representations such as Euler angles, quaternions, rotation matrices, and Lie algebra coordinates introduce distinct constraints and failure modes. While these trade-offs are well studied for supervised learning, their implications for actions in reinforcement learning remain unclear. We systematically evaluate SO(3) action representations across three standard continuous control algorithms, PPO, SAC, and TD3, under dense and sparse rewards. We compare how representations shape exploration, interact with entropy regularization, and affect training stability through empirical studies and analyze the implications of different projections for obtaining valid rotations from Euclidean network outputs. Across a suite of robotics benchmarks, we quantify the practical impact of these choices and distill simple, implementation-ready guidelines for selecting and using rotation actions. Our results highlight that representation-induced geometry strongly influences exploration and optimization and show that representing actions as tangent vectors in the local frame yields the most reliable results across algorithms. Accurate reasoning over 3D rotations is a core requirement for machine learning algorithms applied in computer graphics, state estimation and control. In robotics and embodied intelligence, the problem extends to controlling physical orientations through learned actions, e.g., in manipulation policies that command full task-space poses or aerial vehicles that regulate attitude. These tasks rely on trained policies with action spaces including rotations in SO(3). This restriction has led to multiple parameterizations, each with its own tradeoffs (Macdonald, 2011; Barfoot, 2017). Euler angles are minimal and intuitive but suffer from order dependence, angle wrapping, and gimbal-lock singularities. Quaternions are smooth and numerically robust with a simple unit-norm constraint, but double-cover SO(3). Rotation matrices are a smooth and unique mapping, but are heavily over-parameterized and require orthonormalization. Viewing SO(3) as a Lie group, one can use tangent spaces, i.e., the Lie algebra m of skew-symmetric matrices, together with the exponential and logarithm maps to represent orientations. Tangent spaces are locally smooth, but globally exhibit singularities at large angles (Sol ` a et al., 2018). Irrespective of the choice of parameterization, any minimal 3-parameter chart must incur singularities, and global parameterizations that avoid singularities are necessarily redundant and constrained. Applications in deep learning that require reasoning over rotations and orientations have renewed interest in this topic by adding another perspective: irrespective of any mathematical properties, what is the best representation to learn from data in SO(3)?