arXiv subjects

Gregory S. Chirikjian

Publications and source records attributed to Gregory S. Chirikjian.

At least 19 recordsLinked to original sources

arXiv2026

Rotational Trajectory Synopsis Groups

We construct finite-dimensional Lie-group summaries of based rotational paths that retain the exact terminal attitude and selected iterated angular-velocity integrals. These summaries, called rotational trajectory synopses, compose exactly under path concatenation. Building on the classical works of Magnus and Chen, we establish endpoint-retaining body, spatial, and hybrid synopsis groups at every finite order and give a uniform matrix realization for the evaluated Magnus hierarchy. We distinguish this hierarchy from the full Chen signature from order three onward and prove \chg{that the exact endpoint is independent of every finite truncation of the body, spatial, or combined angular-velocity signature: the synopsis maps are surjective at every finite order}. An underactuated satellite example connects the retained coordinates to geometric phase in spacecraft attitude dynamics.

math.GR↗

arXiv2026

Hierarchical Object Representation for Spatial Robot Perception: Points, Meshes, and Superquadrics

Hierarchical 3D Scene Graphs (3DSG) have emerged as an actionable and scalable representation for long-term autonomy incorporating metric, semantic, and topological information in the scene. However, the question of geometric representation of objects in 3DSG has been overlooked as most methods use simplified geometric models such as partial point clouds or 3D bounding boxes. In this work, we introduce a hierarchical object representation that can be leveraged for high-fidelity object-level reconstruction, object-based robust re-localization or map alignment, and efficient and analytical collision checking for safe robot navigation planning in dense and cluttered environments. The representation is structurally organized into four distinct layers, progressively abstracting the scene from raw sensor data to dense 3D meshes to analytical primitives such as superquadrics, which provide a sparse and analytical representation for object geometry. We develop a pipeline that builds the hierarchical object representation from RGB-D image stream captured by a robot, and demonstrate its working in real-world open-set object scenes in both indoor and outdoor environments. Extensive experiments across diverse datasets including HOPE, ReplicaCAD, Kimera-Multi, and NUS Campus Dataset collected using Unitree B2 Robot validate our pipeline in both indoor and outdoor environments. We show that our superquadric-based map alignment method outperforms the current state-of-the-art object based map alignment method ROMAN. Our code can be found at https://github.com/perceptica-robotics/Hickory.

cs.RO↗

arXiv2026

Global Tensor Field Formulation of the Fokker-Planck Equation on Riemannian Manifolds

This paper presents a global, coordinate-free formulation of the Fokker-Planck equation on Riemannian manifolds. In the Stratonovich formulation, the infinitesimal generator is expressed intrinsically through Lie derivatives, and its adjoint is derived via the divergence theorem, yielding a concise geometric form of the Fokker-Planck equation. In the Ito formulation, a diffusion tensor field is introduced to generalize the Euclidean diffusion matrix, and a tensor-field-based analysis establishes an intrinsic double-divergence representation of the Fokker-Planck equation. The proposed framework provides a globally valid and geometrically consistent interpretation of diffusion and probability transport on Riemannian manifolds, supported by compact and intuitive proofs.

math.PR↗

arXiv2025

Geometric Interpretation of Brownian Motion on Riemannian Manifolds

This paper presents a unified geometric framework for Brownian motion on manifolds, encompassing intrinsic Riemannian manifolds, embedded submanifolds, and Lie groups. The approach constructs the stochastic differential equation by injecting noise along each axis of an orthonormal frame and designing the drift term so that the resulting generator coincides with the Laplace--Beltrami operator. Both Stratonovich and Itô formulations are derived explicitly, revealing the geometric origin of curvature-induced drift. The drift is shown to correspond to the covariant derivatives of the frame fields for intrinsic manifolds, the mean curvature vector for embedded manifolds, and the adjoint-trace term for Lie groups, which vanishes for unimodular cases. The proposed formulation provides a geometrically transparent and mathematically consistent foundation for diffusion processes on nonlinear configuration spaces.

math.PR↗

arXiv2025

Continuous body 3-D reconstruction of limbless animals

Limbless animals such as snakes, limbless lizards, worms, eels, and lampreys move their slender, long bodies in three dimensions to traverse diverse environments. Accurately quantifying their continuous body's 3-D shape and motion is important for understanding body-environment interactions in complex terrain, but this is difficult to achieve (especially for local orientation and rotation). Here, we describe an interpolation method to quantify continuous body 3-D position and orientation. We simplify the body as an elastic rod and apply a backbone optimization method to interpolate continuous body shape between end constraints imposed by tracked markers. Despite over-simplifying the biomechanics, our method achieves a higher interpolation accuracy (~50% error) in both 3-D position and orientation compared with the widely-used cubic B-spline interpolation method. Beyond snakes traversing large obstacles as demonstrated, our method applies to other long, slender, limbless animals and continuum robots. We provide codes and demo files for easy application of our method.

physics.bio-ph↗

arXiv2025

INGRID: Intelligent Generative Robotic Design Using Large Language Models

The integration of large language models (LLMs) into robotic systems has accelerated progress in embodied artificial intelligence, yet current approaches remain constrained by existing robotic architectures, particularly serial mechanisms. This hardware dependency fundamentally limits the scope of robotic intelligence. Here, we present INGRID (Intelligent Generative Robotic Design), a framework that enables the automated design of parallel robotic mechanisms through deep integration with reciprocal screw theory and kinematic synthesis methods. We decompose the design challenge into four progressive tasks: constraint analysis, kinematic joint generation, chain construction, and complete mechanism design. INGRID demonstrates the ability to generate novel parallel mechanisms with both fixed and variable mobility, discovering kinematic configurations not previously documented in the literature. We validate our approach through three case studies demonstrating how INGRID assists users in designing task-specific parallel robots based on desired mobility requirements. By bridging the gap between mechanism theory and machine learning, INGRID enables researchers without specialized robotics training to create custom parallel mechanisms, thereby decoupling advances in robotic intelligence from hardware constraints. This work establishes a foundation for mechanism intelligence, where AI systems actively design robotic hardware, potentially transforming the development of embodied AI systems.

cs.RO↗

arXiv2025

Lateral oscillation and body compliance help snakes and snake robots stably traverse large, smooth obstacles

Snakes can move through almost any terrain. Similarly, snake robots hold the promise as a versatile platform to traverse complex environments like earthquake rubble. Unlike snake locomotion on flat surfaces which is inherently stable, when snakes traverse complex terrain by deforming their body out of plane, it becomes challenging to maintain stability. Here, we review our recent progress in understanding how snakes and snake robots traverse large, smooth obstacles that lack anchor points for gripping or bracing. First, we discovered that the generalist variable kingsnake combines lateral oscillation and cantilevering. Regardless of step height and surface friction, the overall gait is preserved. Next, to quantify static stability of the snake, we developed a method to interpolate continuous body in three dimensions (both position and orientation) between discrete tracked markers. By analyzing the base of support using the interpolated continuous body 3-D kinematics, we discovered that the snake maintained perfect stability during traversal, even on the most challenging low friction, high step. Finally, we applied this gait to a snake robot and systematically tested its performance traversing large steps with variable heights to further understand stability principles. The robot rapidly and stably traversed steps nearly as high as a third of its body length. As step height increased, the robot rolled more frequently to the extent of flipping over, reducing traversal probability. The absence of such failure in the snake with a compliant body inspired us to add body compliance to the robot. With better surface contact, the compliant body robot suffered less roll instability and traversed high steps at higher probability, without sacrificing traversal speed. Our robot traversed large step-like obstacles more rapidly than most previous snake robots, approaching that of the animal.

physics.bio-ph↗

arXiv2025

Means of Random Variables in Lie Groups

The concepts of mean (i.e., average) and covariance of a random variable are fundamental in statistics, and are used to solve real-world problems such as those that arise in robotics, computer vision, and medical imaging. On matrix Lie groups, multiple competing definitions of the mean arise, including the Euclidean, projected, distance-based (i.e., Fréchet and Karcher), group-theoretic, and parametric means. This article provides a comprehensive review of these definitions, investigates their relationships to each other, and determines the conditions under which the group-theoretic means minimize a least-squares type cost function. We also highlight the dependence of these definitions on the choice of inner product on the Lie algebra. The goal of this article is to guide practitioners in selecting an appropriate notion of the mean in applications involving matrix Lie groups.

math.ST↗

arXiv2025

Reasoning and Learning a Perceptual Metric for Self-Training of Reflective Objects in Bin-Picking with a Low-cost Camera

Bin-picking of metal objects using low-cost RGB-D cameras often suffers from sparse depth information and reflective surface textures, leading to errors and the need for manual labeling. To reduce human intervention, we propose a two-stage framework consisting of a metric learning stage and a self-training stage. Specifically, to automatically process data captured by a low-cost camera (LC), we introduce a Multi-object Pose Reasoning (MoPR) algorithm that optimizes pose hypotheses under depth, collision, and boundary constraints. To further refine pose candidates, we adopt a Symmetry-aware Lie-group based Bayesian Gaussian Mixture Model (SaL-BGMM), integrated with the Expectation-Maximization (EM) algorithm, for symmetry-aware filtering. Additionally, we propose a Weighted Ranking Information Noise Contrastive Estimation (WR-InfoNCE) loss to enable the LC to learn a perceptual metric from reconstructed data, supporting self-training on untrained or even unseen objects. Experimental results show that our approach outperforms several state-of-the-art methods on both the ROBI dataset and our newly introduced Self-ROBI dataset.

cs.CV↗

arXiv2025

Fast Convolutions on $\mathbb{Z}^2\backslash SE(2)$ via Radial Translational Dependence and Classical FFT

Let $\mathbb{Z}^2\backslash SE(2)$ denote the right coset space of the subgroup consisting of translational isometries of the orthogonal lattice $\mathbb{Z}^2$ in the non-Abelian group of planar motions $SE(2)$. This paper develops a fast and accurate numerical scheme for approximation of functions on $\mathbb{Z}^2\backslash SE(2)$. We address finite Fourier series of functions on the right coset space $\mathbb{Z}^2\backslash SE(2)$ using finite Fourier coefficients. The convergence/error analysis of finite Fourier coefficients are investigated. Conditions are established for the finite Fourier coefficients to converge to the Fourier coefficients. The matrix forms of the finite transforms are discussed. The implementation of the discrete method to compute numerical approximation of $SE(2)$-convolutions with functions which are radial in translations are considered. The paper is concluded by discussing capability of the numerical scheme to develop fast algorithms for approximating multiple convolutions with functions which are radial in translations.

math.NA↗

arXiv2025

Uncertainty Propagation and Bayesian Fusion on Unimodular Lie Groups from a Parametric Perspective

We address the problem of uncertainty propagation and Bayesian fusion on unimodular Lie groups. Starting from a stochastic differential equation (SDE) defined on Lie groups via Mckean-Gangolli injection, we first convert it to a parametric SDE in exponential coordinates. The coefficient transform method for the conversion is stated for both Ito's and Stratonovich's interpretation of the SDE. Then we derive a mean and covariance fitting formula for probability distributions on Lie groups defined by a concentrated distribution on the exponential coordinate. It is used to derive the mean and covariance propagation equations for the SDE defined by injection, which coincides with the result derived from a Fokker-Planck equation in previous work. We also propose a simple modification to the update step of Kalman filters using the fitting formula, which improves the fusion accuracy with moderate computation time.

eess.SY↗

arXiv2024

Parameter Estimation on Homogeneous Spaces

The Fisher Information Metric (FIM) and the associated Cramér-Rao Bound (CRB) are fundamental tools in statistical signal processing, which inform the efficient design of experiments and algorithms for estimating the underlying parameters. In this article, we investigate these concepts for the case where the parameters lie on a homogeneous space. Unlike the existing Fisher-Rao theory for general Riemannian manifolds, our focus is to leverage the group-theoretic structure of homogeneous spaces, which is often much easier to work with than their Riemannian structure. The FIM is characterized by identifying the homogeneous space with a coset space, the group-theoretic CRB and its corollaries are presented, and its relationship to the Riemannian CRB is clarified. The application of our theory is illustrated using two examples from engineering: (i) estimation of the pose of a robot and (ii) sensor network localization. In particular, these examples demonstrate that homogeneous spaces provide a natural framework for studying statistical models that are invariant with respect to a group of symmetries.

eess.SP↗

arXiv2024

RaggeDi: Diffusion-based State Estimation of Disordered Rags, Sheets, Towels and Blankets

Cloth state estimation is an important problem in robotics. It is essential for the robot to know the accurate state to manipulate cloth and execute tasks such as robotic dressing, stitching, and covering/uncovering human beings. However, estimating cloth state accurately remains challenging due to its high flexibility and self-occlusion. This paper proposes a diffusion model-based pipeline that formulates the cloth state estimation as an image generation problem by representing the cloth state as an RGB image that describes the point-wise translation (translation map) between a pre-defined flattened mesh and the deformed mesh in a canonical space. Then we train a conditional diffusion-based image generation model to predict the translation map based on an observation. Experiments are conducted in both simulation and the real world to validate the performance of our method. Results indicate that our method outperforms two recent methods in both accuracy and speed.

cs.RO↗

arXiv2024

Grasping by Hanging: a Learning-Free Grasping Detection Method for Previously Unseen Objects

This paper proposes a novel learning-free three-stage method that predicts grasping poses, enabling robots to pick up and transfer previously unseen objects. Our method first identifies potential structures that can afford the action of hanging by analyzing the hanging mechanics and geometric properties. Then 6D poses are detected for a parallel gripper retrofitted with an extending bar, which when closed forms loops to hook each hangable structure. Finally, an evaluation policy qualities and rank grasp candidates for execution attempts. Compared to the traditional physical model-based and deep learning-based methods, our approach is closer to the human natural action of grasping unknown objects. And it also eliminates the need for a vast amount of training data. To evaluate the effectiveness of the proposed method, we conducted experiments with a real robot. Experimental results indicate that the grasping accuracy and stability are significantly higher than the state-of-the-art learning-based method, especially for thin and flat objects.

cs.RO↗

arXiv2024

RAIL: Robot Affordance Imagination with Large Language Models

This paper introduces an automatic affordance reasoning paradigm tailored to minimal semantic inputs, addressing the critical challenges of classifying and manipulating unseen classes of objects in household settings. Inspired by human cognitive processes, our method integrates generative language models and physics-based simulators to foster analytical thinking and creative imagination of novel affordances. Structured with a tripartite framework consisting of analysis, imagination, and evaluation, our system "analyzes" the requested affordance names into interaction-based definitions, "imagines" the virtual scenarios, and "evaluates" the object affordance. If an object is recognized as possessing the requested affordance, our method also predicts the optimal pose for such functionality, and how a potential user can interact with it. Tuned on only a few synthetic examples across 3 affordance classes, our pipeline achieves a very high success rate on affordance classification and functional pose prediction of 8 classes of novel objects, outperforming learning-based baselines. Validation through real robot manipulating experiments demonstrates the practical applicability of the imagined user interaction, showcasing the system's ability to independently conceptualize unseen affordances and interact with new objects and scenarios in everyday settings.

cs.RO↗

arXiv2024

Design, Calibration, and Control of Compliant Force-sensing Gripping Pads for Humanoid Robots

This paper introduces a pair of low-cost, light-weight and compliant force-sensing gripping pads used for manipulating box-like objects with smaller-sized humanoid robots. These pads measure normal gripping forces and center of pressure (CoP). A calibration method is developed to improve the CoP measurement accuracy. A hybrid force-alignment-position control framework is proposed to regulate the gripping forces and to ensure the surface alignment between the grippers and the object. Limit surface theory is incorporated as a contact friction modeling approach to determine the magnitude of gripping forces for slippage avoidance. The integrated hardware and software system is demonstrated with a NAO humanoid robot. Experiments show the effectiveness of the overall approach.

cs.RO↗

arXiv2024

Non-Abelian Fourier Analysis on $\mathbfΓ\backslash SE(d)$

This paper presents a systematic study for the general theory of non-Abelian Fourier series of integrable functions on the homogeneous space $\mathbfΓ\backslash SE(d)$, where $SE(d)$ is the special Euclidean group in dimension $d$, and $\mathbf{\mathbfΓ}$ is a discrete and co-compact subgroup of $SE(d)$. Suppose that $μ$ is the finite $SE(d)$-invariant measure on the right coset space $\mathbfΓ\backslash SE(d)$, normalized with respect to Weil's formula. The analytic aspects of the proposed method works for any given orthonormal basis of the Hilbert function space $L^2(\mathbfΓ\backslash SE(d),μ)$. The paper is concluded with some convolution and Plancherel formulas.

math.FA↗

arXiv2023

Uncertainty Propagation on Unimodular Matrix Lie Groups

This paper addresses uncertainty propagation on unimodular matrix Lie groups that have a surjective exponential map. We derive the exact formula for the propagation of mean and covariance in a continuous-time setting from the governing Fokker-Planck equation. Two approximate propagation methods are discussed based on the exact formula. One uses numerical quadrature and another utilizes the expansion of moments. A closed-form second-order propagation formula is derived. We apply the general theory to the joint attitude and angular momentum uncertainty propagation problem and numerical experiments demonstrate two approximation methods. These results show that our new methods have high accuracy while being computationally efficient.

eess.SY↗