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

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