# RieMO

**Riemannian Motion Optimization (RieMO)**

*Nathan Ratliff, Marc Toussaint, Stefan Schaal*

[**ICRA 2015 paper**][**longer tech report**]

**Understanding the Geometry of Workspace Obstacles in Motion Optimization**

*Nathan Ratliff, Marc Toussaint, Stefan Schaal *[**longer tech report**]

*Proc. ICRA 2015, May 2015.*

What is it that makes movement around obstacles hard? The answer seems clear: obstacles contort the geometry of the workspace and make it difficult to leverage what we consider easy and intuitive straight-line Cartesian geometry. But is Cartesian motion actually easy? It's certainly well-understood and has numerous applications. But beneath the details of linear algebra and pseudoinverses, lies a non-trivial Riemannian metric driving the solution. Cartesian motion is easy only because the pseudoinverse, our powerhouse tool, correctly represents how Euclidean workspace geometry pulls back into the configuration space. In light of that observation, it reasons that motion through a field of obstacles could be just as easy as long as we correctly account for how those obstacles warp the geometry of the space.

This paper explores extending our geometric model of the robot beyond the notion of a Cartesian workspace space to fully model and leverage how geometry changes in the presence of obstacles. Intuitively, impenetrable obstacles form topological holes and geodesics curve accordingly around them. We formalize this intuition and develop a general motion optimization framework called Riemannian Motion Optimization (RieMO) to efficiently find motions using our geometric models. We also prove that Gauss-Newton approximations for Hessian calculations across the entire trajectory are natural representations of the problem's first-order Riemannian Geometry and capture all of the relevant curvature information encoded in the Hessian. Our experiments demonstrate that, for many problems, obstacle avoidance can be much more natural when placed within the right geometric context.