The GeometricKernels Package: Heat and Matérn Kernels for Geometric Learning on Manifolds, Meshes, and Graphs

Kernels are a fundamental technical primitive in machine learning. In recent years, kernel-based methods such as Gaussian processes are becoming increasingly important in applications where quantifying uncertainty is of key interest. In settings that involve structured data defined on graphs, meshes, manifolds, or other related spaces, defining kernels with good uncertainty-quantification behavior, and computing their value numerically, is less straightforward than in the Euclidean setting. To address this difficulty, we present GeometricKernels, a software package which implements the geometric analogs of classical Euclidean squared exponential - also known as heat - and Mat\'ern kernels, which are widely-used in settings where uncertainty is of key interest. As a byproduct, we obtain the ability to compute Fourier-feature-type expansions, which are widely used in their own right, on a wide set of geometric spaces. Our implementation supports automatic differentiation in every major current framework simultaneously via a backend-agnostic design. In this companion paper to the package and its documentation, we outline the capabilities of the package and present an illustrated example of its interface. We also include a brief overview of the theory the package is built upon and provide some historic context in the appendix.

Riemannian Flow Matching Policy for Robot Motion Learning

We introduce Riemannian Flow Matching Policies (RFMP), a novel model for learning and synthesizing robot visuomotor policies. RFMP leverages the efficient training and inference capabilities of flow matching methods. By design, RFMP inherits the strengths of flow matching: the ability to encode high-dimensional multimodal distributions, commonly encountered in robotic tasks, and a very simple and fast inference process. We demonstrate the applicability of RFMP to both state-based and vision-conditioned robot motion policies. Notably, as the robot state resides on a Riemannian manifold, RFMP inherently incorporates geometric awareness, which is crucial for realistic robotic tasks. To evaluate RFMP, we conduct two proof-of-concept experiments, comparing its performance against Diffusion Policies. Although both approaches successfully learn the considered tasks, our results show that RFMP provides smoother action trajectories with significantly lower inference times.

Bi-KVIL: Keypoints-based Visual Imitation Learning of Bimanual Manipulation Tasks

Visual imitation learning has achieved impressive progress in learning unimanual manipulation tasks from a small set of visual observations, thanks to the latest advances in computer vision. However, learning bimanual coordination strategies and complex object relations from bimanual visual demonstrations, as well as generalizing them to categorical objects in novel cluttered scenes remain unsolved challenges. In this paper, we extend our previous work on keypoints-based visual imitation learning (\mbox{K-VIL})~\cite{gao_kvil_2023} to bimanual manipulation tasks. The proposed Bi-KVIL jointly extracts so-called \emph{Hybrid Master-Slave Relationships} (HMSR) among objects and hands, bimanual coordination strategies, and sub-symbolic task representations. Our bimanual task representation is object-centric, embodiment-independent, and viewpoint-invariant, thus generalizing well to categorical objects in novel scenes. We evaluate our approach in various real-world applications, showcasing its ability to learn fine-grained bimanual manipulation tasks from a small number of human demonstration videos. Videos and source code are available at https://sites.google.com/view/bi-kvil.

Incremental Learning of Full-Pose Via-Point Movement Primitives on Riemannian Manifolds

Movement primitives (MPs) are compact representations of robot skills that can be learned from demonstrations and combined into complex behaviors. However, merely equipping robots with a fixed set of innate MPs is insufficient to deploy them in dynamic and unpredictable environments. Instead, the full potential of MPs remains to be attained via adaptable, large-scale MP libraries. In this paper, we propose a set of seven fundamental operations to incrementally learn, improve, and re-organize MP libraries. To showcase their applicability, we provide explicit formulations of the spatial operations for libraries composed of Via-Point Movement Primitives (VMPs). By building on Riemannian manifold theory, our approach enables the incremental learning of all parameters of position and orientation VMPs within a library. Moreover, our approach stores a fixed number of parameters, thus complying with the essential principles of incremental learning. We evaluate our approach to incrementally learn a VMP library from motion capture data provided sequentially.

Transfer Learning in Robotics: An Upcoming Breakthrough? A Review of Promises and Challenges

Transfer learning is a conceptually-enticing paradigm in pursuit of truly intelligent embodied agents. The core concept -- reusing prior knowledge to learn in and from novel situations -- is successfully leveraged by humans to handle novel situations. In recent years, transfer learning has received renewed interest from the community from different perspectives, including imitation learning, domain adaptation, and transfer of experience from simulation to the real world, among others. In this paper, we unify the concept of transfer learning in robotics and provide the first taxonomy of its kind considering the key concepts of robot, task, and environment. Through a review of the promises and challenges in the field, we identify the need of transferring at different abstraction levels, the need of quantifying the transfer gap and the quality of transfer, as well as the dangers of negative transfer. Via this position paper, we hope to channel the effort of the community towards the most significant roadblocks to realize the full potential of transfer learning in robotics.

Unraveling the Single Tangent Space Fallacy: An Analysis and Clarification for Applying Riemannian Geometry in Robot Learning

In the realm of robotics, numerous downstream robotics tasks leverage machine learning methods for processing, modeling, or synthesizing data. Often, this data comprises variables that inherently carry geometric constraints, such as the unit-norm condition of quaternions representing rigid-body orientations or the positive definiteness of stiffness and manipulability ellipsoids. Handling such geometric constraints effectively requires the incorporation of tools from differential geometry into the formulation of machine learning methods. In this context, Riemannian manifolds emerge as a powerful mathematical framework to handle such geometric constraints. Nevertheless, their recent adoption in robot learning has been largely characterized by a mathematically-flawed simplification, hereinafter 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 learning algorithm is applied. This paper provides a theoretical elucidation of various misconceptions surrounding this approach and offers experimental evidence of its shortcomings. Finally, it presents valuable insights to promote best practices when employing Riemannian geometry within robot learning applications.

On the Design of Region-Avoiding Metrics for Collision-Safe Motion Generation on Riemannian Manifolds

The generation of energy-efficient and dynamic-aware robot motions that satisfy constraints such as joint limits, self-collisions, and collisions with the environment remains a challenge. In this context, Riemannian geometry offers promising solutions by identifying robot motions with geodesics on the so-called configuration space manifold. While this manifold naturally considers the intrinsic robot dynamics, constraints such as joint limits, self-collisions, and collisions with the environment remain overlooked. In this paper, we propose a modification of the Riemannian metric of the configuration space manifold allowing for the generation of robot motions as geodesics that efficiently avoid given regions. We introduce a class of Riemannian metrics based on barrier functions that guarantee strict region avoidance by systematically generating accelerations away from no-go regions in joint and task space. We evaluate the proposed Riemannian metric to generate energy-efficient, dynamic-aware, and collision-free motions of a humanoid robot as geodesics and sequences thereof.

Bringing robotics taxonomies to continuous domains via GPLVM on hyperbolic manifolds

Robotic taxonomies have appeared as high-level hierarchical abstractions that classify how humans move and interact with their environment. They have proven useful to analyse grasps, manipulation skills, and whole-body support poses. Despite the efforts devoted to design their hierarchy and underlying categories, their use in application fields remains scarce. This may be attributed to the lack of computational models that fill the gap between the discrete hierarchical structure of the taxonomy and the high-dimensional heterogeneous data associated to its categories. To overcome this problem, we propose to model taxonomy data via hyperbolic embeddings that capture the associated hierarchical structure. To do so, we formulate a Gaussian process hyperbolic latent variable model and enforce the taxonomy structure through graph-based priors on the latent space and distance-preserving back constraints. We test our model on the whole-body support pose taxonomy to learn hyperbolic embeddings that comply with the original graph structure. We show that our model properly encodes unseen poses from existing or new taxonomy categories, it can be used to generate trajectories between the embeddings, and it outperforms its Euclidean counterparts.

Riemannian geometry as a unifying theory for robot motion learning and control

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.

K-VIL: Keypoints-based Visual Imitation Learning

Visual imitation learning provides efficient and intuitive solutions for robotic systems to acquire novel manipulation skills. However, simultaneously learning geometric task constraints and control policies from visual inputs alone remains a challenging problem. In this paper, we propose an approach for keypoint-based visual imitation (K-VIL) that automatically extracts sparse, object-centric, and embodiment-independent task representations from a small number of human demonstration videos. The task representation is composed of keypoint-based geometric constraints on principal manifolds, their associated local frames, and the movement primitives that are then needed for the task execution. Our approach is capable of extracting such task representations from a single demonstration video, and of incrementally updating them when new demonstrations become available. To reproduce manipulation skills using the learned set of prioritized geometric constraints in novel scenes, we introduce a novel keypoint-based admittance controller. We evaluate our approach in several real-world applications, showcasing its ability to deal with cluttered scenes, new instances of categorical objects, and large object pose and shape variations, as well as its efficiency and robustness in both one-shot and few-shot imitation learning settings. Videos and source code are available at https://sites.google.com/view/k-vil.