  Daily manipulation tasks are characterized by geometric primitives related to
actions and object shapes. Such geometric descriptors are poorly represented by
only using Cartesian coordinate systems. In this paper, we propose a learning
approach to extract the optimal representation from a dictionary of coordinate
systems to encode an observed movement/behavior. This is achieved by using an
extension of Gaussian distributions on Riemannian manifolds, which is used to
analyse a set of user demonstrations statistically, by considering multiple
geometries as candidate representations of the task. We formulate the
reproduction problem as a general optimal control problem based on an iterative
linear quadratic regulator (iLQR), where the Gaussian distribution in the
extracted coordinate systems are used to define the cost function. We apply our
approach to object grasping and box opening tasks in simulation and on a 7-axis
Franka Emika robot. The results show that the robot can exploit several
geometries to execute the manipulation task and generalize it to new
situations, by maintaining the invariant characteristics of the task in the
coordinate system(s) of interest.
