What Is a Riemannian Manifold?

From school, we are familiar with Euclidean spaces: flat geometry, the Pythagorean theorem, and straight lines. This works perfectly well as long as the world is actually flat. However, when navigating on the surface of a sphere or describing robot joints, this geometry reaches its limits.

For example, the set of all possible orientations of a gripper in three-dimensional space forms a curved space of its own. Orientations cannot simply be added together like numbers. A Riemannian manifold generalizes this concept. It creates a space that appears flat locally but may be curved globally, while still allowing distances, angles, and shortest paths to be defined in a meaningful way.

Why Is This Relevant to Robotics?

A robot’s configuration space is geometrically complex. Using Euclidean calculations can lead to systematic errors. Orientations in space, meaning the way a gripper or tool is aligned, also move within a curved geometry rather than a flat space.

The controllers responsible for individual tasks must take this into account. Otherwise, errors can occur when calculating orientations and avoidance movements. When motion planning is based on manifolds, it becomes possible to determine physically natural and energy-efficient trajectories.

How Is the Mathematics Implemented in the Robot?

Once the mathematical framework is clear and the intended implementation has been defined, the most important step in the coding process has already been completed. The code can then run directly on the robot’s control computer.

Further Information

Riemannian Manifolds in Robot Learning, Optimization, and Control by Noémie Jaquier and Leonel Rozo, with contributions from Hans-Peter Schröcker, Arne Sachtler, Soren Hauberg, Andra Kupcsik, Suvrit Sra, and Alin Albu-Schäffer.

Cambridge University Press, publication planned for 2027.

Information and pre-publication PDF: https://github.com/Riemannian-Robotics/Book

Text: Andreas Schmitz