Morphological Symmetries in Robot Learning: A Hitchhiker's Guide to Symmetry-aware Robot Modelling, Control, and Estimation

This thesis presents a comprehensive theoretical study of symmetries in robot dynamics, characterizing them as foundational, physics-informed geometric priors that can significantly enhance the efficiency, generalization, and optimality of learning methods for dynamics modeling, optimal control, and state estimation in robotics. Our main focus is on morphological symmetries—a previously underexplored type of symmetry arising from regularities in a robot's morphology, associated with the replication of kinematic structures and the balanced distribution of mass. In analytical methods, we characterize how these symmetries extend to the robot's equations of motion, generalized mass matrix, Jacobians, configuration space, and the observation space of both proprioceptive and exteroceptive sensor measurements. For data-driven methods, we characterize the conditions under which these symmetries transform the problems of dynamics modeling, optimal control, and state estimation into symmetry-constrained learning problems, where optimal models must satisfy strict invariance and equivariance properties. Thus providing a clear theoretical justification for exploiting symmetries in these applications. To substantiate our claims, we present extensive empirical evidence in locomotion and bimanual manipulation control, deterministic and probabilistic state estimation, and dynamics modeling via transfer/Koopman operators, demonstrating that leveraging symmetry in robot learning leads to significant improvements in sample efficiency, generalization, and robustness. Lastly, to facilitate the practical use of the theory and applications outlined in this work, we introduce two open-access repositories, `morpho_symm` and `symm_learning`, which provide a comprehensive collection of tools and resources for symmetry exploitation in robot learning.