Kinematic Geometry Research Articles

Kinematic geometry of hyperbolic dual spherical motions and Euler–Savary’s equation

In this paper, differential properties of the one-parameter hyperbolic dual spherical kinematics are developed with explicit expressions independent of coordinates systems. We calculate Euler–Savary equations of spherical kinematics in the dual Lorentzian 3-space [Formula: see text]. Then from E. Study’s map new proofs are directly attained for the Disteli’s formulae and their spatial equivalents are examined in detail. Lastly for spherical and planar motions, the point trajectories theoretical expressions of the point trajectories are investigated with a certain value of acceleration and velocity, which are regarded as different forms of Euler–Savary equation form.

Optimal Design of a Novel 3-DOF Orientational Parallel Mechanism for Pelvic Assistance on a Wheelchair: An Approach Based on Kinematic Geometry and Screw Theory

Pelvis mobility is essential to the daily seated activities of wheelchair users, however it is not yet fully addressed in the field of active wearable devices. This letter presents a novel design and optimization methodology of an in-parallel actuated robotic brace for assisting the human pelvis during seated maneuvers on wheelchair. This design uses human data captured by cameras in conjunction with the knowledge of kinematic geometry and screw theory. The mechanism has full rotational three degrees-of-freedom (DOFs) and also accommodates coupled translation of the human pelvic segment. This type of motion was realized by employing three kinematic limbs that impose non-intersecting zero-pitch constraint wrenches on the platform. Our multi-objective optimization (MOO) routine consists of two stages: (I) platform constraint synthesis, where the geometric parameters of the limb constraints were determined to minimize the pelvis-platform errors of trajectories and instantaneous screw axes (ISAs); and (II) limb structural synthesis, where limb types and dimensions, workspace, transmission performances, singularities, and actuated joint displacements were considered. The optimized mechanism has an asymmetrical [RRR]U-2[RR]S architecture. This mechanism can also be integrated into our previously developed Wheelchair Robot for Active Postural Support (WRAPS).

Reflecting on the life of Kenneth H. Hunt and his contributions to Mechanism

Kenneth Henderson Hunt, see Fig. 1, was the Foundation Professor of Engineering, and the inaugural Dean of the Faculty of Engineering at Monash University, Victoria, Australia. He was the Dean of Engineering from 1961 until 1975 and then served as the Chair of Mechanism from 1976 until 1985. He is remembered as the scientist who rediscovered Ball's theory of screws and who made significant contributions to screw system theory which he applied to the kinematics of spatial mechanisms, shaft couplings, and robotics. Kenneth authored two classical books Mechanisms and Motion and Kinematic Geometry of Mechanisms, and co-authored the more modern text Robots and Screw Theory: Applications of Kinematics and Statics to Robotics. He was also one of the founding fathers, contributing to the foundation act with the initial member organizations, of the International Federation for the Theory of Machines and Mechanisms (in 1998 the official name was changed to IFToMM, acronym for the International Federation for the Promotion of Mechanism and Machine Science). The federation was officially born in 1969 and Kenneth served on the first Executive Council and was the founding chairman of the Australian Committee.

Kinematic Geometry Description of a Line With Four Positions and Its Application in Dimension Synthesis of Spatial Linkage

Abstract In the dimension synthesis of the spatial linkages, the geometric characteristics of floating links in mechanisms reveal the geometric relationship between the motion task and the dimensions of the mechanisms. In order to establish the kinematic geometry rules corresponding to the motion of the floating link, this paper transforms the kinematic problems of geometric elements on the floating link into geometric problems and uses the geometric procedure to solve the spatial linkages synthesis problem with four given positions. In a previous work, we have extracted the kinematic geometry rules of a line with two and three positions. However, as the number of task positions increases, the kinematic characteristics representing the position transformation become more complicated. The method proposed in this paper extends the previous work to four given positions and builds up the geometric relationships among the kinematic rules for two, three, and four positions. The establishment of this geometric relationship is helpful to unify the synthesis procedure of synthesis problems with different number of positions. After that, the two-plane projection system and the transformation of projection are introduced to establish a procedural graphical synthesis method.

Kinematic Geometry Principles and Invariants Method for Rotational Accuracy Measurements and Evaluations

Kinematic Geometry Principles and Invariants Method for Rotational Accuracy Measurements and Evaluations

An Application of Jacobian Matrix in Kinematic of Multi-loop Spatial Mechanism

Mechanism is an intended to turn input motion into a required set of output motion and forces. Now a days the spatial mechanism is widely used in various industrial applications like parallel kinematic machines as a robot manipulator. The main aim of the review paper is to use the Jacobian matrix into the kinematic analysis. From the review papers analysis, it has been concluded that the Kinematic analysiswith help of Jacobian matrix performed various Multi-loop Spatial Mechanism settings for Kinematic geometry, its topology and to carried out displacement modelling. It also used to Kinematic Design, synthesis and optimization. Finally, Comparison of existing methods for Kinematic Singularities with Jacobian in Parallel Manipulator.

Fundamental equation of mechanism kinematic geometry: Mapping curve in se(3) to counterpart in SE(3)

The most general continuous rigid motion can be described by a curve in SE(3) of whose tangent vector the left or right invariant representation is a curve in se(3). The integral operation between them has not been achieved because there is no general solution for a system of the first-order linear differential equations with variable coefficients in mathematics. This paper develops a matrix equation (termed as the fundamental equation) based on the geometric properties of a pair of conjugate axodes equivalent to a curve in se(3) in algebras to achieve this integral operation. Moreover, the first-order and second-order derivations of the fundamental equation are derived. Furthermore, an algebraic method representing the body and spatial velocity twists as vector functions of dimensions and input parameters of mechanisms is founded on the theory of reciprocal screws. After that, this method and the fundamental equation are validated by the numerical examples of Bennett, spherical and planar four-bar linkages. Finally, this work presents a notion of generalized-involute which is a useful tool in the study of gear tooth profile and cam profile.

On the one-parameter Lorentzian spatial motions

In this paper, differential properties of the one-parameter Lorentzian spatial motions are developed with explicit expressions independent of coordinates systems. In term of this, we calculate the Disteli formulae of a spacelike line trajectory and derive the connections with kinematic geometry of the axodes. Lastly, a theoretical expression of a spacelike inflection line congruence are investigated in detail.

Compensation method for profile deviations caused by the complex shape of electrodes in orbital electrical discharge machining

In the orbital mode of electrical discharge machining (EDM), an electrode with a complex profile causes deviation in the cavity profile from the design objective. In this paper, electrode compensation techniques according to the envelope theory are presented to eliminate deviation errors. The succession of orbits traced out a family of electrode curves based on kinematic geometry integrated with the motion path of tool bodies and constraints. The boundary of the family of electrode curves is referred to as the orbiting profile. The deviation error between the orbiting and objective profiles can be eliminated by using a compensation electrode in the orbiting process. Thus, the generated orbiting profile conforms to the objective profile. Experimental results of the tested example for the machining profile of the compensation electrode by using the orbital process revealed that profile deviation can be eliminated to ensure that the EDM profile closely conforms to the objective profile. This compensation method for orbiting EDM provided an exact process to engineers for easy control of the machining precision of EDM.

Stability Analysis of Stiffened FG Sandwich Plates Based on FEA

Abstract This paper presents the modal and buckling analysis, and static analysis (i.e., flexural and stress analysis) of simple supported stiffened FG sandwich plates (SFGS) using finite elements. The geometry of SFGS mid-plane kinematics has been modelled by using ABAQUS, which is based on first order shear deformation theory (FSDT). The whole SFGS has been formulated using an eight node serendipity S8R element. Firstly, the natural frequency values of unstiffened FG sandwich plate are validated with available benchmark results. Then, several numerical examples have been illuminated to study the influence of plate thickness, core thickness, and depth and width of stiffener on the natural frequencies, buckling load, central deflection, and Mises stress of the SFGS and discussed in detail.

Prediction of Eigen-frequency of Stiffened Sandwich Composite Plates Using Finite Elements

Abstract This paper presents the natural vibration analysis of stiffened sandwich composite plates (SSCP) using finite elements. The geometry of SSCP mid-plane kinematics has been modelled by using ANSYS parametric design language (APDL) code, which is based on first order shear deformation theory (FSDT). The whole SSCP has been formulated using an eight node serendipity shell 281 element. Firstly, the eigen-frequency values of unstiffened sandwich plate and stiffened composite plates are validated with available benchmark results. Then, the natural frequencies of SSCP by using different combination of lamina orientation as well as no of stiffeners are examined by solving suitable numerical examples and findings are debated in detail.

An efficient method for minimum zone cylindricity error evaluation using kinematic geometry optimization algorithm

Cylindricity error evaluation is necessary to verify the cylindrical product function and to control the manufacturing process. This paper proposes a novel method for minimum zone cylindricity error evaluation based on kinematic geometry optimization algorithm (KGOA). The proposed method shows excellent performance in the cylindricity error evaluation process. First, the theoretical basis and representation for the minimum zone programming evaluation model are presented. Next, a new approach for extracting feature points is proposed using projective transformation and convex set construction. Instead of all measured points, the extracted feature points are used for the cylindricity error evaluation. On comparing the proposed method with other existing methods based on some literature data sets, agreement or improved solutions are obtained. The simulation and experiment results show that the proposed method can achieve optimum solutions accurately and improve the implementation efficiency. Compared with the commercial software of Coordinate Measuring Machine, the proposed method can reduce the average calculating time by half while retaining the accuracy of evaluation results. The basis of this method can also be applied for other geometrical error evaluation in engineering metrology.

The canonical stewart platform as a six DOF pose sensor for automotive applications

The growing demand for prolonged fatigue life of automotive parts and components requires elaboration of their motion in Cartesian space having six degrees of freedom (DOF). Recently, the canonical Steward platform, consisting of six displacement sensors mounted on two parallel platforms, was introduced to comply with this request. In order to apply this pose sensor to automotive applications, the following two important matters are investigated in this study. First, update Jacobian is proposed as a faster and more stable numerical method to solve the forward kinematic problem without any iteration process. Second, the attachment position and initial configuration of the Stewart platform must be adjustable to avoid the interference with other components due to space constraints under the hood of automotive vehicle. In this case, however, the Jacobian matrix which converts six displacement components into a six DOF pose vector is prone to be ill-conditioned so that the converting accuracy becomes worse. The L1-norm of each row in the Jacobian matrix quantifies how much the error would be provoked according to the given kinematic geometry. Hence, it can be used here as a reliable error indicator. Furthermore, several numerical examples are discussed to demonstrate what to consider when designing a six DOF pose sensor for automotive applications.

Analytical prediction of the geometry of contact ellipses and kinematics in a roller screw versus experimental results

This paper brings together the geometry of threaded contact areas and roller screw kinematics in order to shine some light on the local phenomena which dissipate power in the mechanism. The subject has received little attention in current literature, often because only the global kinematics is studied. Moreover, principal directions of curvature are usually assumed from the start. In this paper, differential geometry is used to calculate the result, which proves to be different from published research. Next, the classic Hertzian theory is adapted for slightly conforming contacts, which can be encountered in common roller screw designs. The generalized equations are used to deduce the shape, size and orientation of the contact ellipses for both the roller-screw and the roller-nut contacts. The two appear to be very different in terms of local kinematics. A new stationary model is introduced, which can calculate the sliding velocity field at any point within the contact area. The model has only one degree of freedom in the form of a slip ratio, which depends on lubricant properties and dynamics. An experimental setup was designed to measure this ratio and thus allow comparison with analytical models available in the literature.

Electron momentum spectroscopy study of outer valence electronic structure of pyrrole

The binding energy spectrum and electron momentum distributions (EMDs) for the outer valence shells of pyrrole molecule are measured by a high-sensitivity (e, 2e) spectrometer at incident energy of 1200 eV plus the binding energy employing non-coplanar symmetric kinematic geometry. The obtained EMDs are compared with the theoretical calculations at equilibrium molecular geometry as well as considering vibrational effect by density functional theory method. The result shows that the molecular vibrational motion has obvious influence on EMDs for some of the outer valence orbitals. The controversy over the assignment of the ionization bands, especially the positions of satellite states concerning the ionization of 1b1 orbital, in the binding energy range of 11–16.5 eV has been clarified.

A new approach of kinematic geometry for error identification and compensation of industrial robots

A new approach for kinematic calibration of industrial robots, including the kinematic pair errors and the link errors, is developed in this paper based on the kinematic invariants. In most methods of kinematic calibration, the geometric errors of the robots are considered in forms of variations of the link parameters, while the kinematic pairs are assumed ideal. Due to the errors of mating surfaces in kinematic pairs, the fixed and moving axes of revolute pairs, or the fixed and moving guidelines of prismatic pairs, are separated, which can be concisely identified as the kinematic pair errors and the link errors by means of the kinematic pair errors model, including the self-adaption fitting of a ruled surface, or the spherical image curve fitting and the striction curve fitting. The approach is applied to the kinematic calibration of a SCARA robot. The discrete motion of each kinematic pair in the robot is completely measured by a coordinate measuring machine. Based on the global kinematic properties of the measured motion, the fixed and moving axes, or guidelines, of the kinematic pairs are identified, which are invariants unrelated to the positions of the measured reference points. The kinematic model of the robot is set up using the identified axes and guidelines. The results validate the approach developed has good efficiency and accuracy.

Scattering forms and the positive geometry of kinematics, color and the worldsheet

The search for a theory of the S-Matrix over the past five decades has revealed surprising geometric structures underlying scattering amplitudes ranging from the string worldsheet to the amplituhedron, but these are all geometries in auxiliary spaces as opposed to the kinematical space where amplitudes actually live. Motivated by recent advances providing a reformulation of the amplituhedron and planar mathcal{N} = 4 SYM amplitudes directly in kinematic space, we propose a novel geometric understanding of amplitudes in more general theories. The key idea is to think of amplitudes not as functions, but rather as differential forms on kinematic space. We explore the resulting picture for a wide range of massless theories in general spacetime dimensions. For the bi-adjoint ϕ3 scalar theory, we establish a direct connection between its “scattering form” and a classic polytope — the associahedron — known to mathematicians since the 1960’s. We find an associahedron living naturally in kinematic space, and the tree level amplitude is simply the “canonical form” associated with this “positive geometry”. Fundamental physical properties such as locality and unitarity, as well as novel “soft” limits, are fully determined by the combinatorial geometry of this polytope. Furthermore, the moduli space for the open string worldsheet has also long been recognized as an associahedron. We show that the scattering equations act as a diffeomorphism between the interior of this old “worldsheet associahedron” and the new “kinematic associahedron”, providing a geometric interpretation and simple conceptual derivation of the bi-adjoint CHY formula. We also find “scattering forms” on kinematic space for Yang-Mills theory and the Non-linear Sigma Model, which are dual to the fully color-dressed amplitudes despite having no explicit color factors. This is possible due to a remarkable fact—“Color is Kinematics”— whereby kinematic wedge products in the scattering forms satisfy the same Jacobi relations as color factors. Finally, all our scattering forms are well-defined on the projectivized kinematic space, a property which can be seen to provide a geometric origin for color-kinematics duality.

Modeling of the kinematic geometry of spur gears using matlab

The analytical method of gear design is calculation-intensive and it is usually difficult to achieve optimum backlash and interference-free involute profile that are required to generate geometrical compatibility in a pair of meshing gears when design procedure is entirely based on this method. Some amount of backlash is often required in the assembly of gears but excess backlash can lead to increase vibration and wear of the gear assembly. Also, interference-risk profile can result in undercutting of gear tooth. This paper optimized a spur external involute-profile gear by developing an application for the modeling of its geometrical compatibility using Matlab®. The application uses existing models to test for interference and a proposed model to determine effective backlash in a gear. The backlash values resulting from the application are more confined and the model is applicable to a wider range of modules suggested by American Gear Manufacturers Association. Simulation of the gear-set in Solidworks® for kinematic geometry presents an interference-free tooth contour and an effective backlash.

The General Mathematical Model for the Production Engineering (ProMAT)

The objective of this publication is the generalization of geometrically correct production of worm surfaces (e.g. turning, milling, grinding) based on the results of kinematical geometry and toothing theory, the production geometry analysis of tools, the mathematical defining of the geometric and connection relation. Our purpose is to be able to define every thread surface in one common system so that they could be produced in a modern manufacturing system.We managed to develop such a mathematical model (ProMAT) in which the contacting and machining geometry of all surfaces can be discussed and summarized in a single system in order that they could be manufactured in modern manufacturing systems (CAD, CAM, CAQ, CIM).The contain of the article is about:investigation of geometric problems when manufacturing cylindrical helicoid surfaces using general mathematical-kinematic models,possible uses of the model described,mathematical generation of helicoid thread,realization of the general mathematical model,general mathematical model for the mathematical modelling of the methods of manufacturing technology.

Кинематическая геометрия кривой линии и ее приложение к геометрическому моделированию плоского зубчатого зацепления

The paper presents the results of investigations in the field of kinematic geometry of the spatial curve of a line. The basis of the research is the method of the movable trihedron of the curve. The components of the trihedron motion along the spatial curve are considered and it is shown that its resultant instantaneous motion is screw motion. This result differs from the representation of the motion of a trihedron known in geometry as a rotation described by the Darboux vector. An analytical description of the set of axes of instantaneous helical motions of a trihedron in a moving and fixed system of assigning a spatial curve is given. The possibility of applying the obtained general results to the investigation of a plane curve is shown. The paper proposes a flat tooth gearing model based on the geometric interpretation of the motions of a trihedron of a plane curve and known in the geometric theory of plane mechanisms of the construction of Bobillier. The geometric scheme of this construction is expanded due to the introduction of evolutes simulating instantaneous motions of trihedron of the corresponding construction curves. As a result, a geometric model is obtained, which is more complete in comparison with the known models of flat gearing. It allows to perform both direct and inverse tasks of profiling the teeth of the wheels while simultaneously obtaining the curvature of the desired profiles in the absence of such. The proposed model can be used as the basis for the development of gears with a planar gearing scheme by the condition of achieving the necessary transmission performance due to the geometric shape of the teeth of the wheels.