Riemann and Gauss meet Asimov
2nd Tutorial on Geometric Methods in Robot Learning, Optimization and Control
May 17th, 2024 - Yokohama, Japan (hybrid)
Conference Center Room 502
Thank you all for attending!
The recordings of the talks are now available in our dedicated Youtube playlist!
Outline and objectives
Incorporating robots into our daily lives demands their proficiency in navigating and operating in unstructured and dynamic environments, to effectively react to unforeseen situations, and to integrate these experiences into their knowledge base. This mandate extends beyond the requirement for reliable and steadfast controllers; it also hinges on their remarkable adaptability, ensuring that robot actions consistently yield successful outcomes. To achieve this, a pivotal facet of data-driven learning and adaptation in robotics leverages explicit (e.g., domain knowledge) or implicit (e.g., learned) structures inherent in collected data.
Domain knowledge and data structures in robotics can be viewed through the lens of geometry. Various robotic variables exhibit distinct geometric characteristics, collected data often resides in curved spaces, and many problems naturally lend themselves to geometric interpretations. In this context, the realms of differential geometry, specifically encompassing Lie groups and Riemannian manifold theories, provide invaluable methodologies to grapple with the intricate geometry of non-Euclidean spaces. Although geometric methods have been successfully applied to robotics from early on, their recent resurgence within robot learning, control, and optimization has garnered significant attention. Recent research underscores the transformative potential of exploiting geometry-awareness into these robotic challenges, yielding improved performance, data-efficient learning, and robust stability guarantees.
This tutorial serves a dual purpose: to kindle the interest of the robotics community in geometric methods, often overlooked in robot learning, control, and optimization, and to underscore the critical role of differential geometry across various branches of robotics. By providing a comprehensive introduction to geometric methods, an overview of recent relevant works in robotics, and guidance on best practices for incorporating Riemannian geometry, this tutorial equips researchers with essential tools to seamlessly integrate geometry into their work, fostering innovation and advancements in the field.
Program
Time (Yokohama)
Session
9:00 - 9:05
Welcome from the organizers
9:05 - 10:00
Tutorial session: Basics of differential geometry for robotics
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.
10:00 - 10:30
Coffee break
10:30 - 11:15
Tutorial session: Optimization on manifolds
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.
11:15 - 12:00
Practical session: Optimization on manifolds
12:00 - 12:45
Invited talk: Geometry of interaction: port-based and energy-aware robotics (Stefano Stramigioli)
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.
12:45 - 14:00
Lunch break
14:00 - 14:45
Tutorial session: Learning
This session will introduce two types of learning methodologies on Riemannian manifolds, namely, learning on Riemannian spaces and learning a Riemannian manifold from data.
14:45 - 15:15
Practical session: Learning
15:15 - 15:45
Coffee break
15:45 - 16:30
Invited talk: A Geometric Take on Motion Manifold Learning from Demonstration (Yonghyeon Lee)
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.
16:30 - 17:00
Tutorial session: Good practices to use geometric methods
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.
17:00 - 17:05
Closing words
Invited tutorial speakers
Organizers
Karlsruhe Institute of Technology
Bosch Center for Artificial Intelligence
Technical University of Denmark
Universität Innsbruck
Bosch Center for Artificial Intelligence
Support
This tutorial is supported by the following committees: