Local Kinematic Analysis of Closed-Loop Linkages—Mobility, Singularities, and Shakiness

2016 ◽  
Vol 8 (4) ◽  
Author(s):  
Andreas Müller
Keyword(s):  
Kinematic Analysis ◽  
Regular Point ◽  
Geometric Constraints ◽  
Local Analysis ◽  
Kinematic Singularity ◽  
Space Geometry ◽  
Geometric Entity ◽  
Kinematic Singularities ◽  
Space Singularity

The mobility of a linkage is determined by the constraints imposed on its members. The geometric constraints define the configuration space (c-space) variety as the geometric entity in which the finite mobility of a linkage is encoded. The aim of a local kinematic analysis of a linkage is to deduce its finite mobility, in a given configuration, from the local c-space geometry. In this paper, a method for the local analysis is presented adopting the concept of the tangent cone to a variety. The latter is an algebraic variety approximating the c-space. It allows for investigating the mobility in regular as well as singular configurations. The instantaneous mobility is determined by the constraints, rather than by the c-space geometry. Shaky and underconstrained linkages are prominent examples that exhibit a permanently higher instantaneous than finite DOF even in regular configurations. Kinematic singularities, on the other hand, are reflected in a change of the instantaneous DOF. A c-space singularity as a kinematic singularity, but a kinematic singularity may be a regular point of the c-space. The presented method allows to identify c-space singularities. It also reveals the ith-order mobility and allows for a classification of linkages as overconstrained and underconstrained. The method is applicable to general multiloop linkages with lower pairs. It is computationally simple and only involves Lie brackets (screw products) of instantaneous joint screws. The paper also summarizes the relevant kinematic phenomena of linkages.

Local Analysis of Closed-Loop Linkages: Mobility, Singularities, and Shakiness

2015 ◽  
Author(s):  
Andreas Müller
Keyword(s):  
Tangent Cone ◽  
Local Approximation ◽  
Approximation Order ◽  
Local Analysis ◽  
Space Geometry ◽  
Geometric Entity ◽  
Best Local Approximation ◽  
Kinematic Singularities ◽  
General Configuration

The mobility of a linkage is determined by the constraints imposed on its members. The constraints define the configuration space (c-space) variety as the geometric entity in which the finite mobility of a linkage is encoded. The instantaneous motions are determined by the constraints, rather than by the c-space geometry. Shaky linkages are prominent examples that exhibit a higher instantaneous than finite DOF even in regular configurations. Inextricably connected to the mobility are kinematic singularities that are reflected in a change of the instantaneous DOF. The local analysis of a linkage, aiming at determining the instantaneous and finite mobility in a given configuration, hence needs to consider the c-space geometry as well as the constraint system. A method for the local analysis is presented based on a higher-order local approximation of the c-space adopting the concept of the tangent cone to a variety. The latter is the best local approximation of the c-space in a general configuration. It thus allows for investigating the mobility in regular as well as singular configurations. Therewith the c-space is locally represented as an algebraic variety whose degree is the necessary approximation order. In regular configurations the tangent cone is the tangent space. The method is generally applicable and computationally simple. It allows for a classification of linkages as overconstrained and underconstrained, and to identify singularities.

scholarly journals Parabolic limits of renormalization

2000 ◽  
Vol 20 (1) ◽  
pp. 173-229 ◽  
Author(s):  
BENJAMIN HINKLE
Keyword(s):  
Fixed Point ◽  
Period Doubling ◽  
Unit Interval ◽  
Local Analysis ◽  
Unimodal Maps ◽  
Unimodal Map ◽  
Combinatorial Description ◽  
Combinatorial Rigidity

A unimodal map $f:[0,1] \to [0,1]$ is renormalizable if there is a sub-interval $I \subset [0,1]$ and an $n > 1$ such that $f^n|_I$ is unimodal. The renormalization of $f$ is $f^n|_I$ rescaled to the unit interval.We extend the well-known classification of limits of renormalization of unimodal maps with bounded combinatorics to a classification of the limits of renormalization of unimodal maps with essentially bounded combinatorics. Together with results of Lyubich on the limits of renormalization with essentially unbounded combinatorics, this completes the combinatorial description of limits of renormalization. The techniques are based on the towers of McMullen and on the local analysis around perturbed parabolic points. We define a parabolic tower to be a sequence of unimodal maps related by renormalization or parabolic renormalization. We state and prove the combinatorial rigidity of bi-infinite parabolic towers with complex bounds and essentially bounded combinatorics, which implies the main theorem.As an example we construct a natural unbounded analogue of the period-doubling fixed point of renormalization, called the essentially period-tripling fixed point.

Direct Kinematic Singularities of 3-3 Stewart-Gough Platforms

2005 ◽  
Vol 219 (4) ◽  
pp. 311-324 ◽  
Author(s):  
C H Liu ◽  
J Chiu
Keyword(s):  
Parallel Manipulator ◽  
Degrees Of Freedom ◽  
Polynomial Equation ◽  
Numerical Results ◽  
Polynomial Equations ◽  
Kinematic Singularity ◽  
Cubic Polynomial ◽  
Singular Configuration ◽  
Kinematic Singularities ◽  
Stewart Gough Platform

In this article, a method to locate direct kinematic singularities of a 3-3 Stewart-Gough parallel manipulator (called a Stewart manipulator henceforth) is proposed. The Stewart manipulator is first replaced by an analogous manipulator, the 3PRPS parallel manipulator, and as the first three active joints of this manipulator remain fixed, this manipulator reduces to an asymmetric 3RPS parallel manipulator. With all moving platform's degrees of freedom, except its height, properly specified, there exists at least one height that gives rise to direct kinematic singularity of the asymmetric 3RPS manipulator and this height is a root of a cubic polynomial equation. The procedure to locate direct kinematic singularities thus reduces to solving cubic polynomial equations. Numerical results show that every singular configuration of the asymmetric 3RPS manipulator thus-determined is also a singular configuration of the 3-3 Stewart-Gough platform.

Kinematics Analysis of the Greenhouse Robot

2011 ◽  
Vol 305 ◽  
pp. 112-116
Author(s):  
Jun Liu ◽  
Sheng Wang ◽  
Xiao Lei Wang ◽  
Wen Kui Ma
Keyword(s):  
Dynamic Analysis ◽  
Kinematic Analysis ◽  
Kinematics Analysis ◽  
Kinematic Equation ◽  
Analysis Of Motion

The greenhouse robots are regard as the research object, the essential conception and classification of the robots as well as the developing state of the research in foreign and in domestic are presented. The kinematic equation and the calculation mold of locus are established according to the kinematic analysis, motion designation, dynamic analysis and the analysis of motion mold, which make the function of automatically avoiding obstacles work.

Synthesis and Analysis of 4-DOF Parallel Manipulators of Computer Aided Geometry Approach

2014 ◽  
Vol 912-914 ◽  
pp. 1010-1016
Author(s):  
Yan Hua Zhang ◽  
Xiu Ju Du ◽  
Bai Yong Zhang
Keyword(s):  
Computer Simulation ◽  
Kinematic Analysis ◽  
Geometric Approach ◽  
Parallel Manipulators ◽  
Geometric Constraints ◽  
Type Synthesis ◽  
Moving Platforms ◽  
Kinematic Characteristics ◽  
Computer Aided

A novel computer aided geometry approach for type synthesis and analysis of new spatial 4-DOF parallel manipulators is put forward, and create the computer simulation mechanisms of parallel manipulators using the geometric constraints and dimension driving techniques in CAD software, Based on the computer simulation mechanisms of parallel manipulators, several new spatial 4-DOF parallel manipulators are synthesized, the kinematic characteristics of the moving platforms are analyzed by computer simulation. The results of computer simulation prove that the computer aided geometric approach for solving type synthesis and kinematic analysis is not only fairly quick and straightforward, but also has the advantages of accuracy.

scholarly journals Kinematic Analysis Of Tricept Parallel Manipulator

2012 ◽  
Vol 12 (5) ◽  
Author(s):  
Mir Amin Hosseini ◽  
Hamid-Reza Mohammadi Daniali
Keyword(s):  
Kinematic Analysis ◽  
Parallel Manipulator ◽  
Parallel Manipulators ◽  
Degree Of Freedom ◽  
Geometric Constraints ◽  
Design Parameters ◽  
Forward Kinematics ◽  
Jacobian Matrices ◽  
Moving Platforms ◽  
Workspace Optimization

Parallel manipulators consist of fixed and moving platforms connected to each other with some actuated links. They have some significant advantages over their serial counterparts. While, they suffer from relatively small workspaces, complex kinematics relations and highly singular points within their workspaces. In this paper, forward kinematics of Tricept parallel manipulator is solved analytically and its workspace optimization is performed. This parallel manipulator has a complex degree of freedom, therefore leads to dimensional in-homogeneous Jacobian matrices. Thus, we divide some entries of the Jacobian by units of length, thereby producing a new Jacobian that is dimensionally homogeneous. Moreover, its workspace is parameterized using some design parameters. Then, using GA method, the workspace is optimized subjects to some geometric constraints. Finally, dexterity of the design is evaluated. Keywords- Kinematic, Workspace, Singularity, TriceptABSTRAK - Manipulator selari terdiri daripada platform tetap dan bergerak yang bersambung antara satu sama lain dengan beberapa pautan bergerak. Manipulator selari mempunyai beberapa kebaikan tertentu dibandingkan dengan yang bersamaan dengannya. Walaupun ia mempunyai ruang kerja yang sempit, hubungan kinematik kompleks dan titik tunggal tinggi dalam linkungan ruang kerjanya. Dalam kajian ini, kinematik ke hadapan manipulator selari Tricept diselesaikan secara analisa dan pengoptimuman ruang kerja dijalankan. Manipulator selari ini mempunyai darjah kebebasan yang kompleks, yang menyebabkan ia mendorong kepada kehomogenan dimensi matriks Jacobian. Catatan Jacobian dibahagikan kepada unit panjang, dimana ia menghasilkan Jacobian baru yang homogen dimensinya. Tambahan, ruang kerjanya diparameterkan dengan menggunakan beberapa parameter reka bentuk. Kemudian, dengan kaedah GA, ruang kerja mengoptimakan subjek kepada beberapa kekangan geometrik. Akhirnya, kecakatan reka bentuk dinilaikan.Keywords- Kinematic, Workspace, Singularity, Tricept

scholarly journals Kinematic analysis of scissor-like elements using dual quaternions

2020 ◽  
Author(s):  
Juan Gabriel Guerrero Grijalva ◽  
Edson Roberto De Pieri
Keyword(s):  
Kinematic Analysis ◽  
Standard Approach ◽  
Dual Quaternions ◽  
Kinematic Singularities

Abstract This paper presents a standard approach, based on dual quaternions (DQ), to deal with the kinematic analysis of scissor-like elements (SLE). Kinematic relations of homogeneous SLE are simplified by using DQ exponentiation. The steps to identify kinematic singularities are detailed by an algorithm. A second algorithm is introduced to deal with the computation and plotting of reachable and dexterous workspace. In this approach, not just points and vectors, but lines and planes are represented by DQ. The motion of points, lines and planes can be appreciated in practical examples based on domotics.

Structural mechanic material damping in fabric reinforced composites: a review

2017 ◽  
Vol 1 (88) ◽  
pp. 12-41
Author(s):  
M. Romano ◽  
I. Ehrlich ◽  
N. Gebbeken
Keyword(s):  
Material Properties ◽  
Geometric Constraints ◽  
Separation Procedure ◽  
Reinforced Composites ◽  
Material Damping ◽  
Structural Dynamic ◽  
Geometric Conditions ◽  
Dynamic Material Properties ◽  
Practical Implications

Purpose: A review regarding the acting mechanisms of structural dynamic material damping in fabric reinforced composites is presented. Design/methodology/approach: Mechanical acting principles identified by different investigations are considered. Aspects of the determination and calculation of structural mechanical material properties of fabric reinforced composites are described. Approaches intending the description and classification of ondulations in fabrics reinforced single layers are demonstrated. Findings: The mesomechanic geometry of fabrics is not considered sufficiently by relatively simple homogenization approaches. Yet, it significantly affects its structural dynamic material properties, especially the dynamic ones. Research limitations/implications: In each case the different damping mechanisms act coupled and occur at the same time. Therefore a separation procedure is required in any case. Practical implications: Against the background of the comparison and remarks of the presented papers a reasonable further procedure is recommended. Thereby, FE-calculations with a parametrical variation of the mesomechanic geometry in order to identify kinematic correlations due to geometric constraints are suggested. Originality/value: The idea of the representation of the geometric conditions in terms of a degree of ondulation is described. Such a non-dimensional specific value representing the intensity of the ondulation would enable the comparability of the results of different kinds of investigations.

scholarly journals Conceptual Design of Schoenflies Motion Generators Based on the Wrench Graph

2013 ◽  
Author(s):  
Semaan Amine ◽  
Latifah Nurahmi ◽  
Philippe Wenger ◽  
Stéphane Caro
Keyword(s):  
Conceptual Design ◽  
Parallel Manipulator ◽  
Screw Theory ◽  
Parallel Manipulators ◽  
Design Stage ◽  
The Subject ◽  
Grassmann Geometry ◽  
Kinematic Singularities ◽  
General Configuration

The subject of this paper is about the conceptual design of parallel Schoenflies motion generators based on the wrench graph. By using screw theory and Grassmann geometry, some conditions on both the constraint and the actuation wrench systems are generated for the assembly of limbs of parallel Schoenflies motion generators, i.e., 3T1R parallel manipulators. Those conditions are somehow related to the kinematic singularities of the manipulators. Indeed, the parallel manipulator should not be in a constraint singularity in the starting configuration for a valid architecture, otherwise it cannot perform the required motion pattern. After satisfying the latter condition, the parallel manipulator should not be in an actuation singularity in a general configuration, otherwise the obtained parallel manipulator is permanently singular. Based on the assembly conditions, six types of wrench graphs are identified and correspond to six typical classes of 3T1R parallel manipulators. The geometric properties of these six classes are highlighted. A simplified expression of the superbracket decomposition is obtained for each class, which allows the determination and the comparison of the singularities of 3T1R parallel manipulators at their conceptual design stage. The methodology also provides new architectures of parallel Schoenflies motion generators based on the classification of wrench graphs and on their singularity conditions.

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