On Probabilistic Pullback Metrics on Latent Hyperbolic Manifolds

Gaussian Process Latent Variable Models (GPLVMs) have proven effective in capturing complex, high-dimensional data through lower-dimensional representations. Recent advances show that using Riemannian manifolds as latent spaces provides more flexibility to learn higher quality embeddings. This paper focuses on the hyperbolic manifold, a particularly suitable choice for modeling hierarchical relationships. While previous approaches relied on hyperbolic geodesics for interpolating the latent space, this often results in paths crossing low-data regions, leading to highly uncertain predictions. Instead, we propose augmenting the hyperbolic metric with a pullback metric to account for distortions introduced by the GPLVM's nonlinear mapping. Through various experiments, we demonstrate that geodesics on the pullback metric not only respect the geometry of the hyperbolic latent space but also align with the underlying data distribution, significantly reducing uncertainty in predictions.

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.

Neural Contractive Dynamical Systems

Stability guarantees are crucial when ensuring a fully autonomous robot does not take undesirable or potentially harmful actions. Unfortunately, global stability guarantees are hard to provide in dynamical systems learned from data, especially when the learned dynamics are governed by neural networks. We propose a novel methodology to learn neural contractive dynamical systems, where our neural architecture ensures contraction, and hence, global stability. To efficiently scale the method to high-dimensional dynamical systems, we develop a variant of the variational autoencoder that learns dynamics in a low-dimensional latent representation space while retaining contractive stability after decoding. We further extend our approach to learning contractive systems on the Lie group of rotations to account for full-pose end-effector dynamic motions. The result is the first highly flexible learning architecture that provides contractive stability guarantees with capability to perform obstacle avoidance. Empirically, we demonstrate that our approach encodes the desired dynamics more accurately than the current state-of-the-art, which provides less strong stability guarantees.

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.

Wasserstein Gradient Flows for Optimizing Gaussian Mixture Policies

Robots often rely on a repertoire of previously-learned motion policies for performing tasks of diverse complexities. When facing unseen task conditions or when new task requirements arise, robots must adapt their motion policies accordingly. In this context, policy optimization is the \emph{de facto} paradigm to adapt robot policies as a function of task-specific objectives. Most commonly-used motion policies carry particular structures that are often overlooked in policy optimization algorithms. We instead propose to leverage the structure of probabilistic policies by casting the policy optimization as an optimal transport problem. Specifically, we focus on robot motion policies that build on Gaussian mixture models (GMMs) and formulate the policy optimization as a Wassertein gradient flow over the GMMs space. This naturally allows us to constrain the policy updates via the $L^2$-Wasserstein distance between GMMs to enhance the stability of the policy optimization process. Furthermore, we leverage the geometry of the Bures-Wasserstein manifold to optimize the Gaussian distributions of the GMM policy via Riemannian optimization. We evaluate our approach on common robotic settings: Reaching motions, collision-avoidance behaviors, and multi-goal tasks. Our results show that our method outperforms common policy optimization baselines in terms of task success rate and low-variance solutions.

The e-Bike Motor Assembly: Towards Advanced Robotic Manipulation for Flexible Manufacturing

Robotic manipulation is currently undergoing a profound paradigm shift due to the increasing needs for flexible manufacturing systems, and at the same time, because of the advances in enabling technologies such as sensing, learning, optimization, and hardware. This demands for robots that can observe and reason about their workspace, and that are skillfull enough to complete various assembly processes in weakly-structured settings. Moreover, it remains a great challenge to enable operators for teaching robots on-site, while managing the inherent complexity of perception, control, motion planning and reaction to unexpected situations. Motivated by real-world industrial applications, this paper demonstrates the potential of such a paradigm shift in robotics on the industrial case of an e-Bike motor assembly. The paper presents a concept for teaching and programming adaptive robots on-site and demonstrates their potential for the named applications. The framework includes: (i) a method to teach perception systems onsite in a self-supervised manner, (ii) a general representation of object-centric motion skills and force-sensitive assembly skills, both learned from demonstration, (iii) a sequencing approach that exploits a human-designed plan to perform complex tasks, and (iv) a system solution for adapting and optimizing skills online. The aforementioned components are interfaced through a four-layer software architecture that makes our framework a tangible industrial technology. To demonstrate the generality of the proposed framework, we provide, in addition to the motivating e-Bike motor assembly, a further case study on dense box packing for logistics automation.

Learning Riemannian Stable Dynamical Systems via Diffeomorphisms

Dexterous and autonomous robots should be capable of executing elaborated dynamical motions skillfully. Learning techniques may be leveraged to build models of such dynamic skills. To accomplish this, the learning model needs to encode a stable vector field that resembles the desired motion dynamics. This is challenging as the robot state does not evolve on a Euclidean space, and therefore the stability guarantees and vector field encoding need to account for the geometry arising from, for example, the orientation representation. To tackle this problem, we propose learning Riemannian stable dynamical systems (RSDS) from demonstrations, allowing us to account for different geometric constraints resulting from the dynamical system state representation. Our approach provides Lyapunov-stability guarantees on Riemannian manifolds that are enforced on the desired motion dynamics via diffeomorphisms built on neural manifold ODEs. We show that our Riemannian approach makes it possible to learn stable dynamical systems displaying complicated vector fields on both illustrative examples and real-world manipulation tasks, where Euclidean approximations fail.

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.

Optimizing Demonstrated Robot Manipulation Skills for Temporal Logic Constraints

For performing robotic manipulation tasks, the core problem is determining suitable trajectories that fulfill the task requirements. Various approaches to compute such trajectories exist, being learning and optimization the main driving techniques. Our work builds on the learning-from-demonstration (LfD) paradigm, where an expert demonstrates motions, and the robot learns to imitate them. However, expert demonstrations are not sufficient to capture all sorts of task specifications, such as the timing to grasp an object. In this paper, we propose a new method that considers formal task specifications within LfD skills. Precisely, we leverage Signal Temporal Logic (STL), an expressive form of temporal properties of systems, to formulate task specifications and use black-box optimization (BBO) to adapt an LfD skill accordingly. We demonstrate our approach in simulation and on a real industrial setting using several tasks that showcase how our approach addresses the LfD limitations using STL and BBO.