Riemannian geometry as a unifying theory for robot motion learning and control

Riemannian geometry is a mathematical field which has been the cornerstone of revolutionary scientific discoveries such as the theory of general relativity. Despite early uses in robot design and recent applications for exploiting data with specific geometries, it mostly remains overlooked in robotics. With this blue sky paper, we argue that Riemannian geometry provides the most suitable tools to analyze and generate well-coordinated, energy-efficient motions of robots with many degrees of freedom. Via preliminary solutions and novel research directions, we discuss how Riemannian geometry may be leveraged to design and combine physically-meaningful synergies for robotics, and how this theory also opens the door to coupling motion synergies with perceptual inputs.