Superquadric Functions Research Articles

Discrete element method simulation of high-speed vehicle collisions with road barrier systems

AbstractThe behavior of road or perimeter protection barriers under vehicle impact are usually investigated based on crash tests and finite element (FE) numerical approaches, which are ether expensive or time-consuming. Several studies have proposed to reduce the computation time of the numerical analysis by substituting the complex FE models of vehicles using simplified mass–spring–damper system models. However, these models have drawbacks since consideration of different vehicle impact angles is difficult and they are unable to correctly simulate the risk of high-speed vehicle collision running over the barrier. In this paper, a new approach is proposed to simulate the collision of vehicles with barriers based on the discrete element method (DEM). Here, to save computation time only a handful of 3D non-spherical particles are used to represent the barrier and vehicle. These particles are generated based on the super-quadric function, which is capable of generating a variety of shapes needed for the model. The contact detection and evaluation are carried out based on discrete function representation of the particles with uniform sampling. The bond between two discrete elements is defined using a nonlinear cohesive beam model since the distance between the elements is relatively large. The simulation results obtained based on this approach are more accurate and complete than the simplified mass–spring models and computationally more efficient than the FE model.

Angle of repose for superquadric particles: Investigating the effects of shape parameters

This study develops a model for the angle of repose (AOR) of granular materials composed of non-spherical particles represented by a superquadric function featuring five distinct shape parameters. Our investigation integrates experimental and numerical analyses based on the lifting cylinder approach, using a total of 30 different particle shapes. Initially, we validate a superquadric-based discrete element method using spherical particles and two spheroidal particles, emulating m&ms and Pinto Beans. Subsequently, we examine the impact of each shape parameter on the AOR and heap geometry to identify the most influential parameters. We then construct the AOR model as a function of the particle aspect ratio defined using a combination of the shape parameters. To assess the accuracy and predictive capacity of the proposed AOR model, we conduct six blind tests involving particles with varying shapes and numbers. The results demonstrate a remarkable accuracy, with an average R2 value of 98.6%. Additionally, to showcase the practicality of the proposed AOR model, we apply it to estimate the internal friction angle and the soaked California Bearing Ratio (CBR) of dry sand samples. The results show estimation errors below one percent, underscoring the predictive accuracy and reliability of the model.

Development of resolved CFD–DEM coupling model for three-phase flows with non-spherical particles

In this work, a resolved CFD–DEM coupling model for gas–liquid-solid three-phase flows with non-spherical particles is developed where the shape of the particle is implicitly captured by a superquadric function. Both gas–liquid and fluid–solid interfaces are smoothly represented with a specified thickness so that the interface thicknesses and CFD cell size are independent of each other. Several sensitivity studies are carried out and criteria for mesh- and thickness-independent results are proposed. It is confirmed that the proposed model can properly predict the hydrodynamic and capillary forces acting on particles with a wide range of shapes and contact angles. Finally, the model proposed is applied to perform several virtual experiments of typical chemical engineering processes: a liquid–solid fluidised bed and bubbly flow with various particle shapes. It is found that particle shape can have a significant impact on the fluid-particle interactions and resultant particle movement, leading to segregation and preferential alignment.

Wave propagation in granular material: What is the role of particle shape?

Wave propagation in granular materials is one of the most fundamental problems in mechanics and physics, with both scientific fascination and practical importance. Whether elastic waves in granular media are affected by particle morphology and, if yes, how they are affected remain open questions. Here we present a novel grain-scale model to address these fundamental questions. Efficient techniques are incorporated in the model to cope with a huge number of non-spherical particles which are randomly packed to propagate elastic waves. The variability of particle shape is mathematically described using the superquadric function and a family of geometric shapes is produced so that a systematic investigation of the effect of particle shape becomes viable. The difficulty with the detection of particle contacts that are represented by nonlinear Hertzian contact law is tackled using an efficient algorithm. A marked finding from the multiple series of simulations using this new model is that: an increase in the aspect ratio of particles (i.e. particles changing from spherical to ellipsoidal) leads to a notable rise in the elastic wave velocity, for both compression and shear waves, whereas for non-spherical particles with a given aspect ratio, an increased particle blockiness causes a moderate reduction in the wave velocity. Moreover, it is found that an assembly of particles with higher aspect ratio is associated with a broader range of transmitted frequencies while an assembly of particles with magnified blockiness holds back the conduction of higher frequencies. Based on statistical analyses, we further show that the transition from spherical to non-spherical particles is associated with a broader range of void ratios and increased coordination numbers whereas inflated blockiness brings an opposite impact, and these changes are linked with the observed effect of particle shape on the characteristics of elastic waves, in both time and frequency domains.

Dynamic Movement Primitives: Volumetric Obstacle Avoidance Using Dynamic Potential Functions

Obstacle avoidance for Dynamic Movement Primitives (DMPs) is still a challenging problem. In our previous work, we proposed a framework for obstacle avoidance based on superquadric potential functions to represent volumes. In this work, we extend our previous work to include the velocity of the system in the definition of the potential. Our formulations guarantee smoother behavior with respect to state-of-the-art point-like methods. Moreover, our new formulation allows obtaining a smoother behavior in proximity of the obstacle than when using a static (i.e. velocity independent) potential. We validate our framework for obstacle avoidance in a simulated multi-robot scenario and with different real robots: a pick-and-place task for an industrial manipulator and a surgical robot to show scalability; and navigation with a mobile robot in a dynamic environment.

Learning to Predict Superquadric Parameters From Depth Images With Explicit and Implicit Supervision

Reconstruction of 3D space from visual data has always been a significant challenge in the field of computer vision. A popular approach to address this problem can be found in the form of bottom-up reconstruction techniques which try to model complex 3D scenes through a constellation of volumetric primitives. Such techniques are inspired by the current understanding of the human visual system and are, therefore, strongly related to the way humans process visual information, as suggested by recent visual neuroscience literature. While advances have been made in recent years in the area of 3D reconstruction, the problem remains challenging due to the many possible ways of representing 3D data, the ambiguity of determining the shape and general position in 3D space and the difficulty to train efficient models for the prediction of volumetric primitives. In this article, we address these challenges and present a novel solution for recovering volumetric primitives from depth images. Specifically, we focus on the recovery of superquadrics, a special type of parametric models able to describe a wide array of 3D shapes using only a few parameters. We present a new learning objective that relies on the superquadric (inside-outside) function and develop two learning strategies for training convolutional neural networks (CNN) capable of predicting superquadric parameters. The first uses explicit supervision and penalizes the difference between the predicted and reference superquadric parameters. The second strategy uses implicit supervision and penalizes differences between the input depth images and depth images rendered from the predicted parameters. CNN predictors for superquadric parameters are trained with both strategies and evaluated on a large dataset of synthetic and real-world depth images. Experimental results show that both strategies compare favourably to the existing state-of-the-art and result in high quality 3D reconstructions of the modelled scenes at a much shorter processing time.

Multi-Contact Heavy Object Pushing With a Centaur-Type Humanoid Robot: Planning and Control for a Real Demonstrator

Performing a demanding manipulation task with a multi-legged loco-manipulation platform may require the exploitation of multiple external contacts with different environment surfaces for counteracting the manipulation forces. This is the case of the pushing task of a heavy object, where the grip forces at ground may not be adequate and establishing leg contacts against a wall turns out to be an effective solution to the manipulation problem. In order to produce such behaviour, this letter presents a control architecture that is able to freely exploit the environment complexity to perform loco-manipulation actions, e.g. pushing a heavy object, while meeting the implementation requirements to achieve a real demonstrator. The proposed approach, conceived for torque-controlled platforms, combines the planning capabilities of nonlinear optimization over the robot centroidal statics, based on a continuous description of the environment through superquadric functions, with the instantaneous capabilities of hierarchical inverse kinematics and reactive contact force distribution. Experimental validation has been performed on the pushing task of a wooden cabinet loaded with bricks, using the CENTAURO robot developed at Istituto Italiano di Tecnologia (IIT).

Optimal SUAS Path Planning in Three-Dimensional Constrained Environments

Small Unmanned Aircraft Systems have grown in autonomy and capability and continue to complement Department of Defense mission objectives. Teaming unmanned aircraft with manned vehicles can expand mission profiles and reduce risk to human life. To fully leverage unmanned systems, vehicles must be efficient and autonomous in path planning development. The work herein explores direct orthogonal collocation optimal control techniques combined with fast geometric path planning algorithms to reduce computation time and increase solution accuracy for small unmanned aircraft systems path planning missions. Previous work in the two-dimensional plane demonstrated a methodology to provide optimal flight paths through defined simplex corridors and simplified the optimal control parameter bounds by formulating the problem in the barycentric coordinate system. These methodologies are extended in this paper for three-dimensional flight and are solved with two different formulations for flight in an urban environment. The first formulation solves the constrained optimal control problem using a single phase while modeling the building constraints with superquadric functions. The second formulation implements the simplex methodology, eliminating polygonal constraints from the search domain, and solving the optimal path in a multiple phase approach. Results illustrate the benefits gained in computation time and accuracy when implementing simplex methods into the optimal control design and provide a foundation for closing the gap to real-time, onboard operations for unmanned vehicle path planning.

A Hybrid Obstacle-Avoidance Method of Spatial Hyper-Redundant Manipulators for Servicing in Confined Space

SummaryDue to a large number of redundant degrees of freedom (DOFs), the hyper-redundant manipulator shows outstanding dexterity and adaptability in avoiding the obstacles in confined space. In this paper, a hybrid obstacle-avoidance method of spatial hyper-redundant manipulators is proposed, with both efficiency and accuracy considered. The space around an obstacle is classified into safe, warning, and dangerous zones. A two-level protection strategy is then addressed to handle the obstacle-avoidance problem from qualitative (i.e., pseudo-distance based on super-quadric function) and quantitative (i.e., Euclidean distance based on practical geometry function) perspectives, respectively. The only condition for switching between the two-level protections is the value of pseudo-distance. Then, a modified modal method, which is a trajectory planning method, is presented to plan the collision-free trajectory of the manipulator by maximizing the minimum pseudo-distance or Euclidean distance in different zones. Some parameters, including the arm-angle parameters and the equivalent link length parameters, are defined to represent the manipulator configuration. They are adjusted to avoid the obstacle, singularity, and joint limit. The simulations of 12-DOF manipulator and an experiment of 18-DOF manipulator verify the proposed methods.

Real-time Pipeline for Object Modeling and Grasping Pose Selection via Superquadric Functions

This work provides a novel real-time pipeline for modeling and grasping of unknown objects with a humanoid robot. Such a problem is of great interest for the robotic community, since conventional approaches fail when the shape, dimension or pose of the objects are missing. Our approach reconstructs in real-time a model for the object under consideration and represents the robot hand with proper and mathematically usable models, i.e. superquadric functions. The volume graspable by the hand is represented by an ellipsoid and is defined a-priori, because the shape of the hand is known in advance. The superquadric representing the object is obtained in real-time from partial vision information instead, e.g. one stereo view of the object under consideration, and provides an approximated 3D full model. The optimization problem we formulate for the grasping pose computation is solved online by using the Ipopt software package and, thus, does not require off-line computation or learning. Even though our approach is for a generic humanoid robot, we developed a complete software architecture for executing this approach on the iCub humanoid robot. Together with that, we also provide a tutorial on how to use this framework. We believe that our work, together with the available code, is of a strong utility for the iCub community for three main reasons: object modeling and grasping are relevant problems for the robotic community, our code can be easily applied on every iCub and the modular structure of our framework easily allows extensions and communications with external code.

Visualization of tensor fields using superquadric glyphs

The spatially varying tensor fields that arise in magnetic resonance imaging are difficult to visualize due to the multivariate nature of the data. To improve the understanding of myocardial structure and function a family of objects called glyphs, derived from superquadric parametric functions, are used to create informative and intuitive visualizations of the tensor fields. The superquadric glyphs are used to visualize both diffusion and strain tensors obtained in canine myocardium. The eigensystem of each tensor defines the glyph shape and orientation. Superquadric functions provide a continuum of shapes across four distinct eigensystems (lambda(i), sorted eigenvalues), lambda(1) = lambda(2) = lambda(3) (spherical), lambda(1) < lambda(2) = lambda(3) (oblate), lambda(1) > lambda(2) = lambda(3) (prolate), and lambda(1) > lambda(2) > lambda(3) (cuboid). The superquadric glyphs are especially useful for identifying regions of anisotropic structure and function. Diffusion tensor renderings exhibit fiber angle trends and orthotropy (three distinct eigenvalues). Visualization of strain tensors with superquadric glyphs compactly exhibits radial thickening gradients, circumferential and longitudinal shortening, and torsion combined. The orthotropic nature of many biologic tissues and their DTMRI and strain data require visualization strategies that clearly exhibit the anisotropy of the data if it is to be interpreted properly. Superquadric glyphs improve the ability to distinguish fiber orientation and tissue orthotropy compared to ellipsoids.

Analysis of metallic waveguides of a large class of cross sections using polynomial approximation and superquadric functions

By using the polynomial approximation and superquadric functions in the Rayleigh-Ritz procedure, a unified method has been proposed to analyze conducting hollow waveguides of a large class of cross sections in our previous paper. Some useful and complicated cross-sectional waveguides in the microwave system, namely, eccentric annular, pentagonal, L-shaped, single-ridged, and double-ridged waveguides are analyzed in this paper. Compared with other numerical methods, this method has the advantages of being straightforward, accurate, and computational effective.

Analysis of hollow conducting waveguides using superquadric functions-A unified representation

Waveguides of various geometries have found many applications. The analysis of the wave modes inside these waveguides are usually subject to cross sections of the waveguides and specific, but convenient, coordinate systems have to be chosen in the analysis. In this paper, the boundary geometries of waveguides (which include rectangular, circular, elliptical, triangular, coaxial, etc.) are represented in a unified manner by a superquadric function. In this paper, with the Rayleigh-Ritz method, the wave propagation characteristics in a hollow conducting waveguide of the superquadric cross section are analyzed in a unified manner. From the analysis, it is realized that the superquadric function can be utilized to accurately model the boundary of various waveguide structures through variation of the shape parameters. The comparisons between the analytical and computational results show this method is accurate and efficient.

Fast numerical algorithms for fitting multiresolution hybrid shape models to brain MRI

In this paper, we present new and fast numerical algorithms for shape recovery from brain MRI using multiresolution hybrid shape models. In this modeling framework, shapes are represented by a core rigid shape characterized by a superquadric function and a superimposed displacement function which is characterized by a membrane spline discretized using the finite-element method. Fitting the model to brain MRI data is cast as an energy minimization problem which is solved numerically. We present three new computational methods for model fitting to data. These methods involve novel mathematical derivations that lead to efficient numerical solutions of the model fitting problem. The first method involves using the nonlinear conjugate gradient technique with a diagonal Hessian preconditioner. The second method involves the nonlinear conjugate gradient in the outer loop for solving global parameters of the model and a preconditioned conjugate gradient scheme for solving the local parameters of the model. The third method involves the nonlinear conjugate gradient in the outer loop for solving the global parameters and a combination of the Schur complement formula and the alternating direction-implicit method for solving the local parameters of the model. We demonstrate the efficiency of our model fitting methods via experiments on several MR brain scans.

Universal anisotropic yield criterion based on superquadric functional representation: Part 1. Algorithmic issues and accuracy analysis

A robust and efficient stress update algorithm for the anisotropic yield criterion represented in the form of superquadric function introduced by Barlat et al. is presented. It relies on a discrete variational formulation of the Lagrangian functional which results from the principle of maximum plastic dissipation. Numerical solution of the discretized equations of evolution is based on the operator split methodology and the Newton-Raphson method. A line search algorithm is implemented to overcome numerical instabilities and to expand the convergence region of the basic Newton-Raphson solution procedure. The consistent tangent modulus is expressed in a closed form as a result of the exact linearization of the discrete Kuhn-Tucker condition. Numerical testing of the integration algorithm based on iso-error maps is provided for several variants of the yield function.

Reflector antennas with superquadric aperture boundaries

A unified treatment for reflector antennas with elliptical (circular) or rounded-rectangular (rounded-square) aperture boundaries utilizing superquadric functions is presented. Reflector antennas with superquadric aperture boundaries possess unique properties. To reveal these properties and properly implement them in diffraction analysis formulation, important geometrical features of the superquadric aperture, including a closed-form expression for the area, are presented. Parametric representations of the superquadric aperture are investigated, and a superior parameterization is introduced for evaluating the diffraction integrals. Representative results of physical optics (PO) analysis for superquadric reflector antennas are presented, and key features of the radiation pattern characteristics are summarized. Although the focus of this work is on reflector antennas, the main feature of superquadric aperture characteristics and its parametric representations are generally applicable to any aperture type antennas.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>