Geometric Iteration Research Articles

Shape-controllable inverse kinematics of hyper-redundant robots based on the improved FABRIK method

AbstractHyper-redundant robots have good prospects for applications in confined space due to their high flexibility and slim body size. However, the super-redundant structure brings great challenges for its inverse kinematics with shape constraints. Unfortunately, traditional Jacobian pseudo-inverse-based inverse kinematics method and forward and backward reaching inverse kinematics (FABRIK) method are difficult to constrain the arm shape and realize trajectory tracking in confined spaces. To solve this problem, we propose a shape-controllable FABRIK method to satisfy the given path and shape constraints. Firstly, the kinematic model of the hyper-redundant robot is established, and the canonical FABRIK method is introduced. Based on the preliminary works, the single-layer improved FABRIK method is developed to solve the position and pointing inverse kinematics considering path environment and joint angle constraints instead of two-layer geometric iterations. For tracking the desired end roll angles, the polygonal virtual arm is designed. The real arm roll angle is achieved by controlling its winding on the virtual arm. In this way, the shape can be controlled. Finally, we compare the proposed method with other three approaches by simulations. Results show that the proposed method is more efficient and the arm shape is controllable.

A novel thermal management system design based on variable contact area to maintain uniform temperature in Li-ion battery module

A battery thermal management system (BTMS) aims to control battery temperature and maintain thermal uniformity in a battery module to avoid thermal degradation of the battery and thermal runaway. Hence, it is indispensable for an electric vehicle's safe and efficient operation. The present study proposes a novel variable contact area relationship for designing a liquid-cooled BTMS. The contact area between the battery and the coolant is increased along the flow direction to ensure almost identical heat removal from each battery to maintain a uniform temperature across the battery module at extreme operating conditions. A battery module containing twenty-four 18650 Lithium-ion batteries is considered. The heat generated in the batteries is removed using the coolant (water). A three-dimensional computational fluid dynamics (CFD) model of the battery module is developed and validated. Multiple geometric iterations are performed using the proposed relationship to obtain the contact area between each battery and aluminum block. The ensuing variable contact area geometry achieves significant reductions of 73.9 % and 72.5 % in the maximum temperature difference ∆Tmax of the battery module compared to the constant contact area geometries. Furthermore, a significant reduction in the maximum temperature (Tmax) within the battery module is also observed. The variable contact area geometry is 57.2 % lighter than the maximum constant contact area geometry and performs significantly better. The influence of the coolant flow rate, the contact angle between the battery and aluminum block, and the heat generation rate of the battery on Tmax and ∆Tmax are also explored.

Stochastic Geometric Iterative Method for Loop Subdivision Surface Fitting

In this paper, we propose a stochastic geometric iterative method (S-GIM) to approximate the high-resolution 3D models by finite loop subdivision surfaces. Given an input mesh as the fitting target, the initial control mesh is generated using the mesh simplification algorithm. Then, our method adjusts the control mesh iteratively to make its finite loop subdivision surface approximate the input mesh. In each geometric iteration, we randomly select part of points on the subdivision surface to calculate the difference vectors and distribute the vectors to the control points. Finally, the control points are updated by adding the weighted average of these difference vectors. We prove the convergence of S-GIM and verify it by demonstrating error curves in the experiment. In addition, compared with existing geometric iterative methods, S-GIM has a shorter running time under the same number of iteration steps.

A cable-driven highly compact single port laparoscopic surgical robot with sequentially inserted arms.

The single port surgical robot causes only one incision and brings many benefits to patients. It is very challenging to design a single port surgical robot that causes a smaller incision than current products. This paper presents a highly compact single port laparoscopy surgical robot, which makes full use of the space of the port and only needs a 15 mm-diameter port. The robot is composed of a camera manipulator and two operating manipulators. The non-fully cylindrical manipulators enter the port sequentially, and the equivalent diameter of each operating manipulator is 12mm. An additional 9 mm-diameter channel is left for other surgical tools to pass through after all manipulators entering the port. The kinematics model of the robot is established, including detailed forward kinematics model and inverse kinematics solution based on geometric iteration method. The teleoperation experiment shows that the manipulator can complete the object-grasping, object-transfer and weight-lifting tasks. The proposed single port surgical robot design concept can also be extended to the field of natural orifice transluminal endoscopic surgical robots.

Progressive Iterative Approximation of Non-Uniform Cubic B-Spline Curves and Surfaces via Successive Over-Relaxation Iteration

Geometric iteration (GI) is one of the most efficient curve- or surface-fitting techniques in recent years, which is famous for its remarkable geometric significance. In essence, GI can be thought of as the sum of iterative methods for solving systems of linear equations, such as progressive iterative approximation (PIA) which relies on the theory of Richardson iteration. Thus, when the curve- or surface-fitting error is at a desired level, we want to have as few iterations as possible to improve efficiency when dealing with large data sets. Based on the idea of successive over-relaxation (SOR) iteration, we formulate a faster PIA curve and surface interpolation scheme using classical non-uniform cubic B-splines, named SOR-PIA. The genetic algorithm is utilized to estimate the best approximate relaxation factor of SOR-PIA. Similar to standard PIA, SOR-PIA can also be regarded as a process in which the control points move in one direction, but it can greatly reduce the number of iterations in the iterative process with the same fitting accuracy. By comparing with the standard PIA and WPIA algorithms, the effectiveness of the SOR-PIA iterative interpolation algorithm can be verified.

Efficient Inverse Kinematics and Planning of a Hybrid Active and Passive Cable-Driven Segmented Manipulator

A cable-driven segmented manipulator (CDSM) has superior dexterity for operations in confined space due to its light-slender body and redundant degree of freedoms (DOFs). However, its inverse kinematics resolving and configuration planning are very challenging due to the complex structure and strict constraints. In this article, we propose a two-layer geometric iteration (TLGI) method for inverse kinematics resolving and configuration-constrained Cartesian path planning. The computation efficiency is largely improved and singularities are avoided. First, the end-effector attitude is decomposed into a direction vector and a rotation angle. The former and the end-effector position are combined into state variables of the inner layer, and the latter is treated separately as the state variable of the outer layer. Then, the TLGI method enables to rapidly reach the desired 6-DOF pose by two-layer iterations, i.e., the inner and outer loop iteration. Second, during the inner loop iteration, the CDSM is modeled as an equivalent articulated arm whose end-effector position and direction is the same as that of CDSM, but its links length and joint angles depend on the current configuration of CDSM. Then, the efficient forward and backward reaching inverse kinematics (FABRIKs) method is extended to apply on CDSM so that it can fast reach the inner state variables. During the outer loop iteration, three different rotation cases, i.e., the rotating around the end, root, and both end and root, are designed to switch automatically to reach the outer state variable iteratively. Moreover, by parameterizing geometric constraints of the environment, a TLGI-based configuration-pose simultaneous planning method is also put forward to efficiently achieve additional configuration constraints for operations of CDSM in confined space. Finally, the proposed method is verified by both the simulations and experiments.

Thermal and hydraulic analysis of slotted plate fins heat sinks using numerical and experimental techniques

A heatsink's thermal and hydraulic performance can be optimized by making several modifications in the geometry. Modifying the heatsinks usually increases the weight by adding material. This study focuses on enhancing heat transfer by the removal of material in the form of slots. Fabrication of slots is one of the advantageous geometric modifications in the plate fins. This study also focuses on optimizing the slots for heat transfer augmentation without compromising the pressure drop. In this study, the authors performed Computational Fluid Dynamics (CFD) analysis of various geometric iterations of the plate fins. The heat sinks with comparatively better thermal performance are pointed out (S8 and S9). On average, the Nusselt number of the heat sink S9 is 28.4% higher than the plain heatsink P at various Reynold's number values. The introduction of slots drops the pressure across the heat sinks with an additional average of 9.3% pressure drop across the heatsink S9 compared to the heatsink P at various Re number values. The authors performed an experimental analysis to validate the numerical analysis. The maximum difference between the observed and simulated bast temperature values is 2.6%, 2.9%, and 3% for the heat sinks P, S8, and S9, respectively. • Performance analysis of slotted plate-fin heat sinks. • Hydraulic and thermal analysis of the heat sinks. • Mass reduction of the plate fins. • Analytical model validation with experiments. • Utilization of 3D printing techniques for the development of various elements.

Sharp regularity for the inhomogeneous porous medium equation

We show that locally bounded solutions of the inhomogeneous porous medium equation $$u_{t} - {\rm div} \left( m |u|^{m-1} \nabla u \right) = f \in L^{q,r}, \quad m >1 ,$$ are locally H\"older continuous, with exponent $$\gamma =\min \left\{ \frac{\alpha_{0}^-}{m}, \frac{[(2q - n)r -2q]}{q[(mr - (m-1)]} \right\},$$ where $\alpha_{0}$ denotes the optimal H\"older exponent for solutions of the homogeneous case. The proof relies on an approximation lemma and geometric iteration in the appropriate intrinsic scaling.

A tolerance constrained G2 continuous path smoothing and interpolation method for industrial SCARA robots

Abstract Piecewise-linear robot paths lack G1 and G2 continuity. The discontinuity results in feedrate reduction and fluctuation during real-time execution. State-of-art linear path smoothing methods have issues such as lack of data or chord tolerance constraints, numerical instability, and lack of synchronized position and orientation parameterization. This study proposes a robot path smoothing method for SCARA (Selective Compliance Assembly Robot Arm) robots. The smoothing method simultaneously satisfies requirements of G2 continuity, tolerance constraints (target point interpolation and chord tolerance constrained), shape-preserving (free of self-intersection), unified parameterization, and real-time performance. A global geometric iteration method combined with local corner transition scheme is used to generate smooth path. A convex combination presentation is proposed to simplify the formulation, which also makes it possible to unify various smoothing schemes into one extensible framework. Simulations with data analysis and actual experiments are performed for the verification of quality and efficiency. Comparisons with other solutions are included to evaluate the advantages of the proposed robot path smoothing method.

A Single-Image Super-Resolution Method Based on Progressive-Iterative Approximation

In this paper, a novel single image super-resolution (SR) method based on progressive-iterative approximation is proposed. To preserve textures and clear edges, the image SR reconstruction is treated as an image progressive-iterative fitting procedure and achieved by iterative interpolation. Due to different features in different regions, we first employ the nonsubsampled contourlet transform (NSCT) to divide the image into smooth regions, texture regions, and edges. Then, a hybrid interpolation scheme based on curves and surfaces is proposed, which differs from the traditional surface interpolation methods. Specifically, smooth regions are interpolated by the non-uniform rational basis spline (NURBS) surface geometric iteration. To retain textures, control points are increased, and the progressive-iterative approximation of the NURBS surface is employed to interpolate the texture regions. By considering edges in an image as curve segments that are connected by pixels with dramatic changes, we use NURBS curve progressive-iterative approximation to interpolate the edges, which sharpens the edges and can maintain the image edge structure without jaggy and block artifacts. The experimental results demonstrate that the proposed method significantly outperforms the state-of-the-art methods in terms of both subjective and objective measures.

Discontinuous deformation analysis for SH-body

Based on micro X-ray computed tomography data, the spherical harmonic (SH) function can be employed to reconstruct the nature sand particle, referred to as SH-body. Thus, the sand particle micromorphology can be described by analytic functions. Thanks to these analytic functions, the closest points between SH-bodies for contact detection can be identified by using a second-order geometric iteration algorithm. Meanwhile, the equivalent variational inequality is formulated to determine the inter-particle contact forces for the three-dimensional cases. To capture the kinematics and kinetics of SH-bodies, this study deduces discontinuous deformation analysis (DDA) formula based on the principle of minimum potential energy. The study establishes a framework for simulating the dynamic behavior of SH-bodies, and the proposed methodology is validated through numerical examples.

Survey on geometric iterative methods and their applications

Geometric iterative methods (GIM), including the progressive–iterative approximation (PIA) and the geometric interpolation/approximation method, are a class of iterative methods for fitting curves and surfaces with clear geometric meanings. In this paper, we provide an overview of the interpolatory and approximate geometric iteration methods, present the local properties and accelerating techniques, and show their convergence. Moreover, because it is easy to integrate geometric constraints in the iterative procedure, GIM has been widely applied in geometric design and related areas. We survey the successful applications of geometric iterative methods, including applications in geometric design, data fitting, reverse engineering, mesh and NURBS solid generation.

Modeling circular inductors coupled to a semi-infinite magnetic medium considering the proximity effect

This paper presents a simple 2D electromagnetic model that solves the high-frequency current distribution, considering the proximity effect, in a planar spiral coil above a linear conductive or non-conductive semi-infinite magnetic medium. The conductive regions are divided into axisymmetric elements in which the current is assumed constant. The current flowing in each element depends on its complex impedance and is computed by Kirchhoff’s circuit law. To take the effect of the magnetic medium into account, a new set of mutual inductance formulas are presented. Those formulas are expressed in terms of elliptic integrals and the fast-converging arithmetic–geometric mean iteration of Gauss. The geometric mean distance method is used to deal with elements of arbitrarily shaped cross-section. Elliptic integrals are also used to express the magnetic flux density. The current distribution, the magnetic field and the equivalent impedance computed with the multifilament model agree well with the results obtained using commercial finite element software.

Solid Mass Fractal Model for Pore Structure in Cement Paste

Though discussed a lot, it remains a practical challenge to modeling pore structure in cement paste. The fractal approach shows a great advantage since it allows to generate complex pore structure via simple geometric iterations and to incorporate the wide scope of pores in a self-similar manner. In this paper, the solid mass fractal model is proposed for pore structure in cement paste. The parametric analysis is performed in conjunction with the porosimetric test. It is shown that the proposed solid mass fractal well describes pore structure in cement paste.

A Geometric Orthogonal Projection Strategy for Computing the Minimum Distance Between a Point and a Spatial Parametric Curve

A new orthogonal projection method for computing the minimum distance between a point and a spatial parametric curve is presented. It consists of a geometric iteration which converges faster than the existing Newton’s method, and it is insensitive to the choice of initial values. We prove that projecting a point onto a spatial parametric curve under the method is globally second-order convergence.

Projecting points onto planar parametric curves by local biarc approximation

This paper proposes a geometric iteration algorithm for computing point projection and inversion on planar parametric curves based on local biarc approximation. The iteration begins with initial estimation of the projection of the prescribed test point. For each iteration, we construct a biarc that locally approximates a segment on the original curve starting from the current projective point. Then we compute the projective point for the next iteration, as well as the parameter corresponding to it, by projecting the test point onto this biarc. The iterative process terminates when the projective point satisfies the required precision. Examples demonstrate that our algorithm converges faster and is less dependent on the choice of the initial value compared to the traditional geometric iteration algorithms based on single-point approximation.

Education

Take a random polygon, find the the midpoints of its sides, and construct a new polygon from them. Repeat ad infinitum. What happens? This simple geometric iteration is at the heart of this issue's Education section offering, “From Random Polygon to Ellipse: An Eigenanalysis” by Adam N. Elmachtoub and Charles F. Van Loan. A cursory numerical experiment (although the traditionalist may wish to try this with a compass and straightedge) shows that the polygon shrinks to a point which the astute observer will realize is the centroid of the original vertices. If we now zoom in on the sequence of shrinking polygons, most easily accomplished by rescaling at each iteration, we see even more order. The rescaled vertices approach a binary oscillation between two polygons, both of which can be inscribed in the same ellipse. While students posed with these questions may think they are pursuing a geometry problem, the not-too-secret agenda is to introduce them to some fundamental ideas of numerical linear algebra, namely iteration, rescaling, and convergence. The authors guide us to write the problem as an iterative algorithm whose properties can be analyzed via eigenvalues and eigenvectors and whose convergence is akin to the power method for finding the eigenvalue of largest magnitude of a matrix. One might consider threading this example through a course in numerical analysis or numerical linear algebra, having the students explore the behavior at the beginning of the course (a great programming exercise), and eventually leading them through the analysis after introducing iterative methods for computing eigenvalues and eigenvectors. The problem opens the door to many generalizations, including exploring what happens for different geometric constructions and asking for geometric interpretations for other iterative linear algebra algorithms. The midpoint and rescaling construction is also simple enough to show to freshmen or even high school students; who knows, perhaps a few of them may be drawn to the study of mathematics by the allure of binary blinking ellipses?

A torus patch approximation approach for point projection on surfaces

This paper proposes a second order geometric iteration algorithm for point projection and inversion on parametric surfaces. The iteration starts from an initial projection estimation. In each iteration, we construct a second order osculating torus patch to the parametric surface at the previous projection. Then we project the test point onto the torus patch to compute the next projection and its parameter. This iterative process is terminated when the parameter satisfies the required precision. Experiments demonstrate the convergence speed of our algorithm.

FLOW OF NEWTONIAN FLUIDS THROUGH SINUSOID ALLY CONSTRICTED TUBES. NUMERICAL AND EXPERIMENTAL RESULTS

Numerical and experimental predictions of pressure drops in the flow of Newtonian fluids through sinusoidally constricted tubes (SCT) are carried out. The numerical evaluations analyzed in this work are obtained from the following methods: Geometric Iteration (GJM), Geometric Iteration with First and Second Upwinds (GIM1, GIM2), Successive Over Relaxation by Line (SORBLM), Global Galerkin Spectral Method (GGSM), Collocation (CM) and Dufort-Frankel (DFM). The GIM1, GIM2 and SORBLM are applied to SCT and explained in this work. The other methods have been previously reported in the literature with the same purpose. Experiments are accomplished for constrictions of approximately 40,60 and 80% of the average tube diameter and results compare well with numerical predictions of the steady flow. It is concluded that special attention should be given to evaluations of the friction factor ƒ for Reynolds numbers Re between the onsets of flow separation and turbulence due to flow instability. Finally, the SCT as a model for porous media (PM) is discussed.

On the geometrization of the Peano curve and the arithmetization of the Hilbert curve

Peano, who presented the first space‐filling curve, did so by means of an arithmetic‐analytic definition. Hilbert recognized that such a curve can be generated by a geometric iteration process and defined his own space‐filling curve at the outset by such a process. It appears that an arithmetic‐analytic definition of the Hilbert curve was never attempted. The four similarity transformations that are needed to generate Hilbert's curve are used to obtain such an arithmetic‐analytic representation in closed form. It so happens that this new representation lends itself readily to evaluation by computer.