arXiv subjects

Leonel Rozo

Publications and source records attributed to Leonel Rozo.

At least 19 recordsLinked to original sources

arXiv2026

Let the Dynamics Flow: Stable Flow Matching Dynamical Systems

Flow matching has recently emerged as a powerful approach for imitation learning, enabling scalable, expressive, and multimodal motion policies. However, when modeling these policies as dynamical systems, incorporating formal stability guarantees into these generative models is a prerequisite to ensure safe and generalizable robot behaviors, which remains a significant challenge. This paper introduces Stable Flow Matching Dynamical Systems (SFMDS), a novel framework that bridges the gap between highly expressive generative modeling and formal stability guarantees. SFMDS parametrizes dynamical systems via flow matching while constraining the model to satisfy positive invariance and/or Lyapunov stability conditions. We propose two variants: a soft constraint based on a penalty term, and a hard structural constraint embedded directly into the model architecture. We further extend both formulations to Lie groups to robustly handle orientation trajectories. Experiments on benchmark datasets, in simulation, and on a humanoid robot show that SFMDS learns stable, scalable, and multimodal dynamical systems in low- and high-dimensional state spaces, enabling safe and expressive robot motion generation. SFMDS matches or outperforms state-of-the-art methods on unimodal datasets, while substantially improving performance on multimodal datasets, where competing approaches fail to capture multi-modal behaviors. Accompanying source code and video are available at: https://let-the-dynamics-flow.github.io/SFMDS/.

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arXiv2025

Contact Wasserstein Geodesics for Non-Conservative Schr\"odinger Bridges

The Schr\"odinger Bridge provides a principled framework for modeling stochastic processes between distributions; however, existing methods are limited by energy-conservation assumptions, which constrains the bridge's shape preventing it from model varying-energy phenomena. To overcome this, we introduce the non-conservative generalized Schr\"odinger bridge (NCGSB), a novel, energy-varying reformulation based on contact Hamiltonian mechanics. By allowing energy to change over time, the NCGSB provides a broader class of real-world stochastic processes, capturing richer and more faithful intermediate dynamics. By parameterizing the Wasserstein manifold, we lift the bridge problem to a tractable geodesic computation in a finite-dimensional space. Unlike computationally expensive iterative solutions, our contact Wasserstein geodesic (CWG) is naturally implemented via a ResNet architecture and relies on a non-iterative solver with near-linear complexity. Furthermore, CWG supports guided generation by modulating a task-specific distance metric. We validate our framework on tasks including manifold navigation, molecular dynamics predictions, and image generation, demonstrating its practical benefits and versatility.

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arXiv2025

Taxonomy-aware Dynamic Motion Generation on Hyperbolic Manifolds

Human-like motion generation for robots often draws inspiration from biomechanical studies, which often categorize complex human motions into hierarchical taxonomies. While these taxonomies provide rich structural information about how movements relate to one another, this information is frequently overlooked in motion generation models, leading to a disconnect between the generated motions and their underlying hierarchical structure. This paper introduces the \ac{gphdm}, a novel approach that learns latent representations preserving both the hierarchical structure of motions and their temporal dynamics to ensure physical consistency. Our model achieves this by extending the dynamics prior of the Gaussian Process Dynamical Model (GPDM) to the hyperbolic manifold and integrating it with taxonomy-aware inductive biases. Building on this geometry- and taxonomy-aware frameworks, we propose three novel mechanisms for generating motions that are both taxonomically-structured and physically-consistent: two probabilistic recursive approaches and a method based on pullback-metric geodesics. Experiments on generating realistic motion sequences on the hand grasping taxonomy show that the proposed GPHDM faithfully encodes the underlying taxonomy and temporal dynamics, and it generates novel physically-consistent trajectories.

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arXiv2025

Towards Safe Imitation Learning via Potential Field-Guided Flow Matching

Deep generative models, particularly diffusion and flow matching models, have recently shown remarkable potential in learning complex policies through imitation learning. However, the safety of generated motions remains overlooked, particularly in complex environments with inherent obstacles. In this work, we address this critical gap by proposing Potential Field-Guided Flow Matching Policy (PF2MP), a novel approach that simultaneously learns task policies and extracts obstacle-related information, represented as a potential field, from the same set of successful demonstrations. During inference, PF2MP modulates the flow matching vector field via the learned potential field, enabling safe motion generation. By leveraging these complementary fields, our approach achieves improved safety without compromising task success across diverse environments, such as navigation tasks and robotic manipulation scenarios. We evaluate PF2MP in both simulation and real-world settings, demonstrating its effectiveness in task space and joint space control. Experimental results demonstrate that PF2MP enhances safety, achieving a significant reduction of collisions compared to baseline policies. This work paves the way for safer motion generation in unstructured and obstaclerich environments.

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arXiv2025

Geometric Contact Flows: Contactomorphisms for Dynamics and Control

Accurately modeling and predicting complex dynamical systems, particularly those involving force exchange and dissipation, is crucial for applications ranging from fluid dynamics to robotics, but presents significant challenges due to the intricate interplay of geometric constraints and energy transfer. This paper introduces Geometric Contact Flows (GFC), a novel framework leveraging Riemannian and Contact geometry as inductive biases to learn such systems. GCF constructs a latent contact Hamiltonian model encoding desirable properties like stability or energy conservation. An ensemble of contactomorphisms then adapts this model to the target dynamics while preserving these properties. This ensemble allows for uncertainty-aware geodesics that attract the system's behavior toward the data support, enabling robust generalization and adaptation to unseen scenarios. Experiments on learning dynamics for physical systems and for controlling robots on interaction tasks demonstrate the effectiveness of our approach.

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arXiv2025

Diffeomorphic Obstacle Avoidance for Contractive Dynamical Systems via Implicit Representations

Ensuring safety and robustness of robot skills is becoming crucial as robots are required to perform increasingly complex and dynamic tasks. The former is essential when performing tasks in cluttered environments, while the latter is relevant to overcome unseen task situations. This paper addresses the challenge of ensuring both safety and robustness in dynamic robot skills learned from demonstrations. Specifically, we build on neural contractive dynamical systems to provide robust extrapolation of the learned skills, while designing a full-body obstacle avoidance strategy that preserves contraction stability via diffeomorphic transforms. This is particularly crucial in complex environments where implicit scene representations, such as Signed Distance Fields (SDFs), are necessary. To this end, our framework called Signed Distance Field Diffeomorphic Transform, leverages SDFs and flow-based diffeomorphisms to achieve contraction-preserving obstacle avoidance. We thoroughly evaluate our framework on synthetic datasets and several real-world robotic tasks in a kitchen environment. Our results show that our approach locally adapts the learned contractive vector field while staying close to the learned dynamics and without introducing highly-curved motion paths, thus outperforming several state-of-the-art methods.

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arXiv2025

Riemann$^2$: Learning Riemannian Submanifolds from Riemannian Data

Latent variable models are powerful tools for learning low-dimensional manifolds from high-dimensional data. However, when dealing with constrained data such as unit-norm vectors or symmetric positive-definite matrices, existing approaches ignore the underlying geometric constraints or fail to provide meaningful metrics in the latent space. To address these limitations, we propose to learn Riemannian latent representations of such geometric data. To do so, we estimate the pullback metric induced by a Wrapped Gaussian Process Latent Variable Model, which explicitly accounts for the data geometry. This enables us to define geometry-aware notions of distance and shortest paths in the latent space, while ensuring that our model only assigns probability mass to the data manifold. This generalizes previous work and allows us to handle complex tasks in various domains, including robot motion synthesis and analysis of brain connectomes.

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arXiv2024

Fast and Robust Visuomotor Riemannian Flow Matching Policy

Diffusion-based visuomotor policies excel at learning complex robotic tasks by effectively combining visual data with high-dimensional, multi-modal action distributions. However, diffusion models often suffer from slow inference due to costly denoising processes or require complex sequential training arising from recent distilling approaches. This paper introduces Riemannian Flow Matching Policy (RFMP), a model that inherits the easy training and fast inference capabilities of flow matching (FM). Moreover, RFMP inherently incorporates geometric constraints commonly found in realistic robotic applications, as the robot state resides on a Riemannian manifold. To enhance the robustness of RFMP, we propose Stable RFMP (SRFMP), which leverages LaSalle's invariance principle to equip the dynamics of FM with stability to the support of a target Riemannian distribution. Rigorous evaluation on ten simulated and real-world tasks show that RFMP successfully learns and synthesizes complex sensorimotor policies on Euclidean and Riemannian spaces with efficient training and inference phases, outperforming Diffusion Policies and Consistency Policies.

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arXiv2024

Extended Neural Contractive Dynamical Systems: On Multiple Tasks and Riemannian Safety Regions

Stability guarantees are crucial when ensuring that a fully autonomous robot does not take undesirable or potentially harmful actions. We recently proposed the Neural Contractive Dynamical Systems (NCDS), which is a neural network architecture that guarantees contractive stability. With this, learning-from-demonstrations approaches can trivially provide stability guarantees. However, our early work left several unanswered questions, which we here address. Beyond providing an in-depth explanation of NCDS, this paper extends the framework with more careful regularization, a conditional variant of the framework for handling multiple tasks, and an uncertainty-driven approach to latent obstacle avoidance. Experiments verify that the developed system has the flexibility of ordinary neural networks while providing the stability guarantees needed for autonomous robotics.

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arXiv2024

On Probabilistic Pullback Metrics for Latent Hyperbolic Manifolds

Probabilistic Latent Variable Models (LVMs) excel at modeling complex, high-dimensional data through lower-dimensional representations. Recent advances show that equipping these latent representations with a Riemannian metric unlocks geometry-aware distances and shortest paths that comply with the underlying data structure. This paper focuses on hyperbolic embeddings, a particularly suitable choice for modeling hierarchical relationships. Previous approaches relying on hyperbolic geodesics for interpolating the latent space often generate paths crossing low-data regions, leading to highly uncertain predictions. Instead, we propose augmenting the hyperbolic manifold with a pullback metric to account for distortions introduced by the LVM's nonlinear mapping and provide a complete development for pullback metrics of Gaussian Process LVMs (GPLVMs). Our experiments 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.

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arXiv2024

The GeometricKernels Package: Heat and Mat\'ern 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 Python 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.

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arXiv2024

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.

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arXiv2024

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.

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arXiv2023

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.

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arXiv2023

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.

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arXiv2023

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.

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arXiv2022

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.

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arXiv2022

Bringing motion taxonomies to continuous domains via GPLVM on hyperbolic manifolds

Human motion taxonomies serve 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 substantial efforts devoted to design their hierarchy and underlying categories, their use remains limited. 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. We achieve this by formulating a novel Gaussian process hyperbolic latent variable model that incorporates the taxonomy structure through graph-based priors on the latent space and distance-preserving back constraints. We validate our model on three different human motion taxonomies to learn hyperbolic embeddings that faithfully preserve the original graph structure. We show that our model properly encodes unseen data from existing or new taxonomy categories, and outperforms its Euclidean and VAE-based counterparts. Finally, through proof-of-concept experiments, we show that our model may be used to generate realistic trajectories between the learned embeddings.

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