arXiv subjects

Gregory S. Chirikjian

Publications and source records attributed to Gregory S. Chirikjian.

At least 37 records · Page 2Linked to original sources

arXiv2023

Non-Abelian Fourier Series on $\mathbb{Z}^2\backslash SE(2)$

This paper discusses computational structure of coefficients of non-Abelian Fourier series on the right coset space $\mathbb{Z}^2\backslash SE(2)$ expressed in the trigonometric basis, where $SE(2)$ is the group of handedness preserving Euclidean isometries of the plane and $\mathbb{Z}^2$ denotes the discrete subgroup of translations of the orthogonal (square) lattice in $\mathbb{R}^2$. Assume that $μ$ is the finite $SE(2)$-invariant measure on the right coset space $\mathbb{Z}^2\backslash SE(2)$, normalized with respect to Weil's formula. We present a constructive computational characterization including discrete sampling of non-Abelian Fourier matrix elements on $SE(2)$ for coefficients of $μ$-square integrable functions on $\mathbb{Z}^2\backslash SE(2)$ with respect to the concrete trigonometric basis. The paper is concluded with discussion of the method for non-Abelian Fourier coefficients of convolutions on $\mathbb{Z}^2\backslash SE(2)$.

math.FA↗

arXiv2023

Asymptotically Steerable Finite Fourier-Bessel Transforms and Closure under Convolution

This paper develops a constructive numerical scheme for Fourier-Bessel approximations on disks compatible with convolutions supported on disks. We address accurate finite Fourier-Bessel transforms (FFBT) and inverse finite Fourier-Bessel transforms (iFFBT) of functions on disks using the discrete Fourier Transform (DFT) on Cartesian grids. Whereas the DFT and its fast implementation (FFT) are ubiquitous and are powerful for computing convolutions, they are not exactly steerable under rotations. In contrast, Fourier-Bessel expansions are steerable, but lose both this property and the preservation of band limits under convolution. This work captures the best features of both as the band limit is allowed to increase. The convergence/error analysis and asymptotic steerability of FFBT/ iFFBT are investigated. Conditions are established for the FFBT to converge to the Fourier-Bessel coefficient and for the iFFBT to uniformly approximate the Fourier-Bessel partial sums. The matrix form of the finite transforms is discussed. The implementation of the discrete method to compute numerical approximation of convolutions of compactly supported functions on disks is considered as well.

math.NA↗

arXiv2023

A Lie-Theoretic Approach to Propagating Uncertainty Jointly in Attitude and Angular Momentum

Dynamic state estimation, as opposed to kinematic state estimation, seeks to estimate not only the orientation of a rigid body but also its angular velocity, through Euler's equations of rotational motion. This paper demonstrates that the dynamic state estimation problem can be reformulated as estimating a probability distribution on a Lie group defined on phase space (the product space of rotation and angular momentum). The propagation equations are derived non-parametrically for the mean and covariance of the distribution. It is also shown that the equations can be approximately solved by ignoring the third and higher moments of the probability distribution. Numerical experiments show that the distribution constructed from the propagated mean and covariance fits the sample data better than an extended Kalman filter.

eess.SY↗

arXiv2023

Robust and Accurate Superquadric Recovery: a Probabilistic Approach

Interpreting objects with basic geometric primitives has long been studied in computer vision. Among geometric primitives, superquadrics are well known for their ability to represent a wide range of shapes with few parameters. However, as the first and foremost step, recovering superquadrics accurately and robustly from 3D data still remains challenging. The existing methods are subject to local optima and sensitive to noise and outliers in real-world scenarios, resulting in frequent failure in capturing geometric shapes. In this paper, we propose the first probabilistic method to recover superquadrics from point clouds. Our method builds a Gaussian-uniform mixture model (GUM) on the parametric surface of a superquadric, which explicitly models the generation of outliers and noise. The superquadric recovery is formulated as a Maximum Likelihood Estimation (MLE) problem. We propose an algorithm, Expectation, Maximization, and Switching (EMS), to solve this problem, where: (1) outliers are predicted from the posterior perspective; (2) the superquadric parameter is optimized by the trust-region reflective algorithm; and (3) local optima are avoided by globally searching and switching among parameters encoding similar superquadrics. We show that our method can be extended to the multi-superquadrics recovery for complex objects. The proposed method outperforms the state-of-the-art in terms of accuracy, efficiency, and robustness on both synthetic and real-world datasets. The code is at http://github.com/bmlklwx/EMS-superquadric_fitting.git.

cs.CV↗

arXiv2023

Prepare the Chair for the Bear! Robot Imagination of Sitting Affordance to Reorient Previously Unseen Chairs

In this letter, a paradigm for the classification and manipulation of previously unseen objects is established and demonstrated through a real example of chairs. We present a novel robot manipulation method, guided by the understanding of object stability, perceptibility, and affordance, which allows the robot to prepare previously unseen and randomly oriented chairs for a teddy bear to sit on. Specifically, the robot encounters an unknown object and first reconstructs a complete 3D model from perceptual data via active and autonomous manipulation. By inserting this model into a physical simulator (i.e., the robot's "imagination"), the robot assesses whether the object is a chair and determines how to reorient it properly to be used, i.e., how to reorient it to an upright and accessible pose. If the object is classified as a chair, the robot reorients the object to this pose and seats the teddy bear onto the chair. The teddy bear is a proxy for an elderly person, hospital patient, or child. Experiment results show that our method achieves a high success rate on the real robot task of chair preparation. Also, it outperforms several baseline methods on the task of upright pose prediction for chairs.

cs.RO↗

arXiv2023

Marching-Primitives: Shape Abstraction from Signed Distance Function

Representing complex objects with basic geometric primitives has long been a topic in computer vision. Primitive-based representations have the merits of compactness and computational efficiency in higher-level tasks such as physics simulation, collision checking, and robotic manipulation. Unlike previous works which extract polygonal meshes from a signed distance function (SDF), in this paper, we present a novel method, named Marching-Primitives, to obtain a primitive-based abstraction directly from an SDF. Our method grows geometric primitives (such as superquadrics) iteratively by analyzing the connectivity of voxels while marching at different levels of signed distance. For each valid connected volume of interest, we march on the scope of voxels from which a primitive is able to be extracted in a probabilistic sense and simultaneously solve for the parameters of the primitive to capture the underlying local geometry. We evaluate the performance of our method on both synthetic and real-world datasets. The results show that the proposed method outperforms the state-of-the-art in terms of accuracy, and is directly generalizable among different categories and scales. The code is open-sourced at https://github.com/ChirikjianLab/Marching-Primitives.git.

cs.CV↗

arXiv2023

PRIMP: PRobabilistically-Informed Motion Primitives for Efficient Affordance Learning from Demonstration

This paper proposes a learning-from-demonstration method using probability densities on the workspaces of robot manipulators. The method, named "PRobabilistically-Informed Motion Primitives (PRIMP)", learns the probability distribution of the end effector trajectories in the 6D workspace that includes both positions and orientations. It is able to adapt to new situations such as novel via poses with uncertainty and a change of viewing frame. The method itself is robot-agnostic, in which the learned distribution can be transferred to another robot with the adaptation to its workspace density. The learned trajectory distribution is then used to guide an optimization-based motion planning algorithm to further help the robot avoid novel obstacles that are unseen during the demonstration process. The proposed methods are evaluated by several sets of benchmark experiments. PRIMP runs more than 5 times faster while generalizing trajectories more than twice as close to both the demonstrations and novel desired poses. It is then combined with our robot imagination method that learns object affordances, illustrating the applicability of PRIMP to learn tool use through physical experiments.

cs.RO↗

arXiv2023

Learning-Free Grasping of Unknown Objects Using Hidden Superquadrics

Robotic grasping is an essential and fundamental task and has been studied extensively over the past several decades. Traditional work analyzes physical models of the objects and computes force-closure grasps. Such methods require pre-knowledge of the complete 3D model of an object, which can be hard to obtain. Recently with significant progress in machine learning, data-driven methods have dominated the area. Although impressive improvements have been achieved, those methods require a vast amount of training data and suffer from limited generalizability. In this paper, we propose a novel two-stage approach to predicting and synthesizing grasping poses directly from the point cloud of an object without database knowledge or learning. Firstly, multiple superquadrics are recovered at different positions within the object, representing the local geometric features of the object surface. Subsequently, our algorithm exploits the tri-symmetry feature of superquadrics and synthesizes a list of antipodal grasps from each recovered superquadric. An evaluation model is designed to assess and quantify the quality of each grasp candidate. The grasp candidate with the highest score is then selected as the final grasping pose. We conduct experiments on isolated and packed scenes to corroborate the effectiveness of our method. The results indicate that our method demonstrates competitive performance compared with the state-of-the-art without the need for either a full model or prior training.

cs.RO↗

arXiv2023

On the Inertial Rotational Brownian Motion of Arbitrarily Shaped Particles

This article reports the modeling of inertial rotational Brownian motion as an Ornstein-Uhlenbeck process evolving on the cotangent bundle of the rotation group, SO(3). The benefit of this approach and the use of a different parameterization of rotations allows the handling of particles with arbitrary shapes, without requiring any simplifying assumptions on the shape or the structure of the viscosity tensors. The resultant Fokker-Planck equation for the joint orientation and angular momentum probability distribution can be solved approximately using an `ansatz' Gaussian distribution in exponential coordinates.

cond-mat.stat-mech↗

arXiv2023

A Learning-Based Approach for Estimating Inertial Properties of Unknown Objects from Encoder Discrepancies

Many robots utilize commercial force/torque sensors to identify inertial properties of unknown objects. However, such sensors can be difficult to apply to small-sized robots due to their weight, size, and cost. In this paper, we propose a learning-based approach for estimating the mass and center of mass (COM) of unknown objects without using force/torque sensors at the end-effector or on the joints. In our method, a robot arm carries an unknown object as it moves through multiple discrete configurations. Measurements are collected when the robot reaches each discrete configuration and stops. A neural network is designed to estimate joint torques from encoder discrepancies. Given multiple samples, we derive the closed-form relation between joint torques and the object's inertial properties. Based on the derivation, the mass and COM of object are identified by weighted least squares. In order to improve the accuracy of inferred inertial properties, an attention model is designed to generate weights of joints, which indicate the relative importance for each joint. Our framework requires only encoder measurements without using any force/torque sensors, but still maintains accurate estimation capability. The proposed approach has been demonstrated on a 4 degree of freedom (DOF) robot arm.

cs.RO↗

arXiv2023

Model-Free and Learning-Free Proprioceptive Humanoid Movement Control

This paper presents a novel model-free method for humanoid-robot quasi-static movement control. Traditional model-based methods often require precise robot model parameters. Additionally, existing learning-based frameworks often train the policy in simulation environments, thereby indirectly relying on a model. In contrast, we propose a proprioceptive framework based only on sensory outputs. It does not require prior knowledge of a robot's kinematic model or inertial parameters. Our method consists of three steps: 1. Planning different pairs of center of pressure (CoP) and foot position objectives within a single cycle. 2. Searching around the current configuration by slightly moving the robot's leg joints back and forth while recording the sensor measurements of its CoP and foot positions. 3. Updating the robot motion with an optimization algorithm until all objectives are achieved. We demonstrate our approach on a NAO humanoid robot platform. Experiment results show that it can successfully generate stable robot motions.

cs.RO↗

arXiv2022

Bootstrapping Globally Optimal Variational Calculus Solutions

Whereas in a coordinate-dependent setting the Euler-Lagrange equations establish necessary conditions for solving variational problems in which both the integrands of functionals and the resulting paths are assumed to be sufficiently smooth, uniqueness and global optimality are generally hard to prove in the absence of convexity conditions, and often times they may not even exist. This is also true for variational problems on Lie groups, with the Euler-Poincaré equation establishing necessary conditions. The difficulties compound when integrands and/or the optimal paths are not sufficiently regular, since in this case the classical necessary conditions no longer apply. This article therefore reviews several nonstandard cases where unique globally optimal solutions can be guaranteed, and establishes a ``bootstrapping'' process to build new globally optimal variational solutions on larger spaces from existing ones on smaller spaces. Surprisingly, it is possible to prove global optimality in some nonconvex cases where even the regularity conditions required for classical necessary conditions do not hold. This general theory is then applied to several topics such as optimal framing of curves in three-dimensional Euclidean space, optimal motion interpolation, and optimal reparametrization of video sequences to compare salient actions.

math.OC↗

arXiv2022

Watch Me Calibrate My Force-Sensing Shoes!

This paper presents a novel method for smaller-sized humanoid robots to self-calibrate their foot force sensors. The method consists of two steps: 1. The robot is commanded to move along planned whole-body trajectories in different double support configurations. 2. The sensor parameters are determined by minimizing the error between the measured and modeled center of pressure (CoP) and ground reaction force (GRF) during the robot's movement using optimization. This is the first proposed autonomous calibration method for foot force-sensing devices in smaller humanoid robots. Furthermore, we introduce a high-accuracy manual calibration method to establish CoP ground truth, which is used to validate the measured CoP using self-calibration. The results show that the self-calibration can accurately estimate CoP and GRF without any manual intervention. Our method is demonstrated using a NAO humanoid platform and our previously presented force-sensing shoes.

cs.RO↗

arXiv2022

Primitive-based Shape Abstraction via Nonparametric Bayesian Inference

3D shape abstraction has drawn great interest over the years. Apart from low-level representations such as meshes and voxels, researchers also seek to semantically abstract complex objects with basic geometric primitives. Recent deep learning methods rely heavily on datasets, with limited generality to unseen categories. Furthermore, abstracting an object accurately yet with a small number of primitives still remains a challenge. In this paper, we propose a novel non-parametric Bayesian statistical method to infer an abstraction, consisting of an unknown number of geometric primitives, from a point cloud. We model the generation of points as observations sampled from an infinite mixture of Gaussian Superquadric Taper Models (GSTM). Our approach formulates the abstraction as a clustering problem, in which: 1) each point is assigned to a cluster via the Chinese Restaurant Process (CRP); 2) a primitive representation is optimized for each cluster, and 3) a merging post-process is incorporated to provide a concise representation. We conduct extensive experiments on two datasets. The results indicate that our method outperforms the state-of-the-art in terms of accuracy and is generalizable to various types of objects.

cs.CV↗

arXiv2022

Efficient Path Planning in Narrow Passages for Robots with Ellipsoidal Components

Path planning has long been one of the major research areas in robotics, with PRM and RRT being two of the most effective classes of planners. Though generally very efficient, these sampling-based planners can become computationally expensive in the important case of "narrow passages". This paper develops a path planning paradigm specifically formulated for narrow passage problems. The core is based on planning for rigid-body robots encapsulated by unions of ellipsoids. Each environmental feature is represented geometrically using a strictly convex body with a $\mathcal{C}^1$ boundary (e.g., superquadric). The main benefit of doing this is that configuration-space obstacles can be parameterized explicitly in closed form, thereby allowing prior knowledge to be used to avoid sampling infeasible configurations. Then, by characterizing a tight volume bound for multiple ellipsoids, robot transitions involving rotations are guaranteed to be collision-free without needing to perform traditional collision detection. Furthermore, by combining with a stochastic sampling strategy, the proposed planning framework can be extended to solving higher dimensional problems in which the robot has a moving base and articulated appendages. Benchmark results show that the proposed framework often outperforms the sampling-based planners in terms of computational time and success rate in finding a path through narrow corridors for both single-body robots and those with higher dimensional configuration spaces. Physical experiments using the proposed framework are further demonstrated on a humanoid robot that walks in several cluttered environments with narrow passages.

cs.RO↗

arXiv2022

When Kinematics Dominates Mechanics: Locally Volume-Preserving Primitives for Model Reduction in Finite Elasticity

A new, and extremely fast, computational modeling paradigm is introduced here for specific finite elasticity problems that arise in the context of soft robotics. Whereas continuum mechanics is a very classical area of study, and significant effort has been devoted to the development of intricate constitutive models for finite elasticity, we show that in the kinds of large-strain mechanics problems arising in soft robotics, many of the parameters in constitutive models are irrelevant. For the most part, the isochoric (locally volume-preserving) constraint dominates behavior, and this can be built into closed-form kinematic deformation fields before even considering other aspects of constitutive modeling. We therefore focus on developing and applying primitive deformations that each observe this constraint. It is shown that by composing a wide enough variety of such deformations that the most common behaviors observed in soft robots can be replicated. Case studies include an inflatable rubber chamber, a slender rubber rod, and a rubber block subjected to different boundary conditions. We show that this method is at least 50 times faster than the ABAQUS implementation of the finite element method (FEM). Physical experiments and measurements show that both our method and ABAQUS have approximately 10% error relative to experimentally measured displacements, as well as to each other. Our method provides a real-time alternative to FEM, and captures essential degrees of freedom for use in feedback control systems.

cs.RO↗

arXiv2022

Entropy, Symmetry, and the Difficulty of Self-Replication

The defining property of an artificial physical self-replicating system, such as a self-replicating robot, is that it has the ability to make copies of itself from basic parts. Three questions that immediately arises in the study of such systems are: 1) How complex is the whole robot in comparison to each basic part ? 2) How disordered can the parts be while having the robot successfully replicate ? 3) What design principles can enable complex self-replicating systems to function in disordered environments generation after generation ? Consequently, much of this article focuses on exploring different concepts of entropy as a measure of disorder, and how symmetries can help in reliable self replication, both at the level of assembly (by reducing the number of wrong ways that parts could be assembled), and also as a parity check when replicas manufacture parts generation after generation. The mathematics underpinning these principles that quantify artificial physical self-replicating systems are articulated here by integrating ideas from information theory, statistical mechanics, ergodic theory, group theory, and integral geometry.

math.GM↗

arXiv2021

Rate of Entropy Production in Stochastic Mechanical Systems

Entropy production in stochastic mechanical systems is examined here with strict bounds on its rate. Stochastic mechanical systems include pure diffusions in Euclidean space or on Lie groups, as well as systems evolving on phase space for which the fluctuation-dissipation theorem applies, i.e., return-to-equilibrium processes. Two separate ways for ensembles of such mechanical systems forced by noise to reach equilibrium are examined here. First, a restorative potential and damping can be applied, leading to a classical return-to-equilibrium process wherein energy taken out by damping can balance the energy going in from the noise. Second, the process evolves on a compact configuration space (such as random walks on spheres, torsion angles in chain molecules, and rotational Brownian motion) lead to long-time solutions that are constant over the configuration space, regardless of whether or not damping and random forcing balance. This is a kind of potential-free equilibrium distribution resulting from topological constraints. Inertial and noninertial (kinematic) systems are considered. These systems can consist of unconstrained particles or more complex systems with constraints, such as rigid-bodies or linkages. These more complicated systems evolve on Lie groups and model phenomena such as rotational Brownian motion and nonholonomic robotic systems. In all cases, it is shown that the rate of entropy production is closely related to the appropriate concept of Fisher information matrix of the probability density defined by the Fokker-Planck equation. Classical results from information theory are then repurposed to provide computable bounds on the rate of entropy production in stochastic mechanical systems.

math-ph↗