This session aims at providing a summary of the main concepts of Riemannian geometry of particular relevance to robotics. Our goal is not to give an in-depth introduction to Riemannian geometry, but rather to provide the theoretical background required to make use of Riemannian geometry in robotics. Some of the topics that we will cover are: The notion of curvature, smooth manifolds, tangent spaces, Riemannian metrics, geodesics, exponential and logarithmic maps, and parallel transport operations.

This session will guide you through first-order Riemannian optimization and Bayesian optimization on Riemannian domains. Some of the optimization algorithms that we will cover include: Riemannian gradient descent and conjugate gradient descent, Riemannian SGD, Riemannian Adam.

Robots move and interact with a physical environment and energy is the Esperanto of physics which governs dynamics of any classical physical systems including robots. And yet, in robotics, it is not common to use the real structures of physical systems  to model or control robotic interaction. In this tutorial lecture the geometrical structure of interaction will be introduced giving some examples about why this is meaningful. The concepts will be presented in a general way giving focus to intuition by showing their generality and applicability to highly complicated systems. This concept of interaction can be just an arm touching something, a robotic hand or even a wind of a bird interacting with the air around it.

This session will introduce two types of learning methodologies on Riemannian manifolds, namely, learning on Riemannian spaces and learning a Riemannian manifold from data. 

One of the primary challenges in employing data-driven methods for generating robot movements is the high dimensionality of trajectories. This challenge is often further amplified by the small size of demonstration datasets. In this tutorial talk, we adopt the motion manifold hypothesis, which suggests that the high-dimensional trajectories approximately lie on a simpler, lower-dimensional space, referred to as the motion manifold. We then introduce a framework that leverages autoencoders to learn this motion manifold. This enables us to simultaneously learn the manifold and its coordinate chart, effectively addressing the issues of high dimensionality and small dataset sizes. Given this framework, several important questions naturally arise: (i) How can we deal with complex manifold structures, such as those with multiple connected components or holes; (ii) What constitutes good representations of trajectories; and (iii) How do we identify the most suitable latent coordinates for the manifold? Our tutorial will address these questions, providing both theoretical insights and practical strategies. Additionally, we will present experimental results from our research involving the 7-DoF Franka Panda robot arm, demonstrating the effectiveness of our approach in practical applications. This discussion is designed to offer a comprehensive overview of the challenges and solutions in learning the robot motion manifold from data, making it relevant for both researchers and practitioners in the field.

The recent adoption of Riemannian geometry in robotic applications has been largely characterized by a mathematically-flawed simplification, referred to as the “single tangent space fallacy”. This approach involves merely projecting the data of interest onto a single tangent (Euclidean) space, over which an off the-shelf algorithm is applied. This session will provide a theoretical elucidation of various misconceptions surrounding this approach and offers experimental evidence of its shortcomings. Moreover, it will present valuable insights to promote best practices when employing Riemannian geometry within robotic domains.