  We present a comprehensive framework for studying and leveraging
morphological symmetries in robotic systems. These are intrinsic properties of
the robot's morphology, frequently observed in animal biology and robotics,
which stem from the replication of kinematic structures and the symmetrical
distribution of mass. We illustrate how these symmetries extend to the robot's
state space and both proprioceptive and exteroceptive sensor measurements,
resulting in the equivariance of the robot's equations of motion and optimal
control policies. Thus, we recognize morphological symmetries as a relevant and
previously unexplored physics-informed geometric prior, with significant
implications for both data-driven and analytical methods used in modeling,
control, estimation and design in robotics. For data-driven methods, we
demonstrate that morphological symmetries can enhance the sample efficiency and
generalization of machine learning models through data augmentation, or by
applying equivariant/invariant constraints on the model's architecture. In the
context of analytical methods, we employ abstract harmonic analysis to
decompose the robot's dynamics into a superposition of lower-dimensional,
independent dynamics. We substantiate our claims with both synthetic and
real-world experiments conducted on bipedal and quadrupedal robots. Lastly, we
introduce the repository MorphoSymm to facilitate the practical use of the
theory and applications outlined in this work.
