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.

Dynamic motions of humans and robots are widely driven by posture-dependent nonlinear interactions between their degrees of freedom. However, these dynamical effects remain mostly overlooked when studying the mechanisms of human movement generation. Inspired by recent works, we hypothesize that human motions are planned as sequences of geodesic synergies, and thus correspond to coordinated joint movements achieved with piecewise minimum energy. The underlying computational model is built on Riemannian geometry to account for the inertial characteristics of the body. Through the analysis of various human arm motions, we find that our model segments motions into geodesic synergies, and successfully predicts observed arm postures, hand trajectories, as well as their respective velocity profiles. Moreover, we show that our analysis can further be exploited to transfer arm motions to robots by reproducing individual human synergies as geodesic paths in the robot configuration space.