  In robotic tasks, changes in reference frames typically do not influence the
underlying physical properties of the system, which has been known as
invariance of physical laws.These changes, which preserve distance, encompass
isometric transformations such as translations, rotations, and reflections,
collectively known as the Euclidean group. In this work, we delve into the
design of improved learning algorithms for reinforcement learning and planning
tasks that possess Euclidean group symmetry. We put forth a theory on that
unify prior work on discrete and continuous symmetry in reinforcement learning,
planning, and optimal control. Algorithm side, we further extend the 2D path
planning with value-based planning to continuous MDPs and propose a pipeline
for constructing equivariant sampling-based planning algorithms. Our work is
substantiated with empirical evidence and illustrated through examples that
explain the benefits of equivariance to Euclidean symmetry in tackling natural
control problems.
