A Study on Finding Finite Roots for Kinematic Synthesis

2017 ◽  
Author(s):  
Mark M. Plecnik ◽  
Ronald S. Fearing
Keyword(s):  
Random Process ◽  
Synthesis Problem ◽  
Homotopy Continuation ◽  
Synthesis System ◽  
Expected Number ◽  
Kinematic Synthesis ◽  
Scaling Down ◽  
Path Synthesis ◽  
Number Of Iterations

This study presents new results on a method to solve large kinematic synthesis systems termed Finite Root Generation. The method reduces the number of startpoints used in homotopy continuation to find all the roots of a kinematic synthesis system. For a single execution, many start systems are generated with corresponding startpoints using a random process such that start-points only track to finite roots. Current methods are burdened by computations of roots to infinity. New results include a characterization of scaling for different problem sizes, a technique for scaling down problems using cognate symmetries, and an application for the design of a spined pinch gripper mechanism. We show that the expected number of iterations to perform increases approximately linearly with the quantity of finite roots for a given synthesis problem. An implementation that effectively scales the four-bar path synthesis problem by six using its cognate structure found 100% of roots in an average of 16,546 iterations over ten executions. This marks a roughly six-fold improvement over the basic implementation of the algorithm.

scholarly journals Finding Only Finite Roots to Large Kinematic Synthesis Systems

2017 ◽  
Vol 9 (2) ◽  
Author(s):  
Mark M. Plecnik ◽  
Ronald S. Fearing
Keyword(s):  
Polynomial Systems ◽  
Synthesis Problem ◽  
New Method ◽  
Homotopy Continuation ◽  
Kinematic Synthesis ◽  
Root Finding ◽  
Large Polynomial Systems ◽  
Path Synthesis

In this work, a new method is introduced for solving large polynomial systems for the kinematic synthesis of linkages. The method is designed for solving systems with degrees beyond 100,000, which often are found to possess quantities of finite roots that are orders of magnitude smaller. Current root-finding methods for large polynomial systems discover both finite and infinite roots, although only finite roots have meaning for engineering purposes. Our method demonstrates how all infinite roots can be precluded in order to obtain substantial computational savings. Infinite roots are avoided by generating random linkage dimensions to construct startpoints and start systems for homotopy continuation paths. The method is benchmarked with a four-bar path synthesis problem.

Finding Only Finite Roots to Large Kinematic Synthesis Systems

2016 ◽  
Author(s):  
Mark M. Plecnik ◽  
Ronald S. Fearing
Keyword(s):  
Polynomial Systems ◽  
Synthesis Problem ◽  
New Method ◽  
Homotopy Continuation ◽  
Kinematic Synthesis ◽  
Root Finding ◽  
Large Polynomial Systems ◽  
Path Synthesis

In this work, a new method is introduced for solving large polynomial systems for the kinematic synthesis of linkages. The method is designed for solving systems with degrees beyond 100,000, which often are found to possess a number of finite roots that is orders of magnitude smaller. Current root-finding methods for large polynomial systems discover both finite and infinite roots, although only finite roots have meaning for engineering purposes. Our method demonstrates how all infinite roots can be avoided in order to obtain substantial computational savings. Infinite roots are avoided by generating random linkage dimensions to construct start-points and start-systems for homotopy continuation paths. The method is benchmarked with a four-bar path synthesis problem.

scholarly journals Comparison of some evolutionary algorithms for optimization of the path synthesis problem

2018 ◽  
Author(s):  
Jakub Krzysztof Grabski ◽  
Tomasz Walczak ◽  
Jacek Buśkiewicz ◽  
Martyna Michałowska
Keyword(s):  
Evolutionary Algorithms ◽  
Synthesis Problem ◽  
Path Synthesis

Geometric Design of Symmetric 3-RRS Constrained Parallel Platforms

2004 ◽  
Author(s):  
Eric Wolbrecht ◽  
Hai-Jun Su ◽  
Alba Perez ◽  
J. Michael McCarthy
Keyword(s):  
Geometric Design ◽  
Homotopy Continuation ◽  
Total Degree ◽  
Polynomial Equations ◽  
Dimensional Synthesis ◽  
Kinematic Synthesis ◽  
Real Solutions ◽  
Parallel Platform ◽  
Three Degree Of Freedom ◽  
Serial Chains

The paper presents the kinematic synthesis of a symmetric parallel platform supported by three RRS serial chains. The dimensional synthesis of this three degree-of-freedom system is obtained using design equations for each of three RRS chains obtained by requiring that they reach a specified set of task positions. The result is 10 polynomial equations in 10 unknowns, which is solved using polynomial homotopy continuation. An example is provided in which the direction of the first revolute joint (2 parameters) and the z component of the base and platform are specified as well as the two task positions. The system of polynomials has a total degree of 4096 which means that in theory it can have as many solutions. Our example has 70 real solutions that define 70 different symmetric platforms that can reach the specified positions.

scholarly journals Lower Bounds for the Size of Random Maximal $H$-Free Graphs

10.37236/93 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Guy Wolfovitz
Keyword(s):  
Lower Bound ◽  
Random Process ◽  
Lower Bounds ◽  
Random Permutation ◽  
Bipartite Graphs ◽  
Expected Number ◽  
Free Graph ◽  
Vertex Graph

We consider the next random process for generating a maximal $H$-free graph: Given a fixed graph $H$ and an integer $n$, start by taking a uniformly random permutation of the edges of the complete $n$-vertex graph $K_n$. Then, traverse the edges of $K_n$ according to the order imposed by the permutation and add each traversed edge to an (initially empty) evolving $n$-vertex graph - unless its addition creates a copy of $H$. The result of this process is a maximal $H$-free graph ${\Bbb M}_n(H)$. Our main result is a new lower bound on the expected number of edges in ${\Bbb M}_n(H)$, for $H$ that is regular, strictly $2$-balanced. As a corollary, we obtain new lower bounds for Turán numbers of complete, balanced bipartite graphs. Namely, for fixed $r \ge 5$, we show that ex$(n, K_{r,r}) = \Omega(n^{2-2/(r+1)}(\ln\ln n)^{1/(r^2-1)})$. This improves an old lower bound of Erdős and Spencer. Our result relies on giving a non-trivial lower bound on the probability that a given edge is included in ${\Bbb M}_n(H)$, conditioned on the event that the edge is traversed relatively (but not trivially) early during the process.

A Random Recolouring Method for Graphs and Hypergraphs

1993 ◽  
Vol 2 (3) ◽  
pp. 363-365 ◽  
Author(s):  
Colin McDiarmid
Keyword(s):  
Chromatic Number ◽  
Expected Number ◽  
Expected Time ◽  
Number Of Iterations

We consider a simple randomised algorithm that seeks a weak 2-colouring of a hypergraph H; that is, it tries to 2-colour the points of H so that no edge is monochromatic. If H has a particular well-behaved form of such a colouring, then the method is successful within expected number of iterations O(n3) when H has n points. In particular, when applied to a graph G with n nodes and chromatic number 3, the method yields a 2-colouring of the vertices such that no triangle is monochromatic in expected time O(n4).

Preparation, Characterization of P(AA-AM)/Organic Montmorillonite Super Absorbent Resin under Focused Signal-Mode Microwave Irradiation

2013 ◽  
Vol 746 ◽  
pp. 142-146
Author(s):  
Shui Li Lai ◽  
Ying Hua Gao
Keyword(s):  
Microwave Irradiation ◽  
Single Mode ◽  
X Ray Diffraction ◽  
Synthesis System ◽  
Organic Montmorillonite ◽  
Modified Montmorillonite ◽  
Aqueous Solution Polymerization ◽  
Signal Mode ◽  
Irradiation Technology

A super absorbent resin (SAR) of acrylic acid (AA)/acrylamide (AM) /organic montmorillonite (OMMT) composite was synthesized by aqueous solution polymerization. This process was carried out under precision microwave organic synthesis system and its single-mode focusing microwave irradiation technology was studied. The montmorillonite (MMT) was modified by treating with cetyltrimethylammonium bromide.The characteristics of the obtained SAR, natural and modified montmorillonite were characterized by X-ray diffraction (XRD), fourier transform infrared spectroscopy (FTIR) and scanning electron microscopy (SEM). The results show that the intercalation is successful, the CTAB as a organic modifier introuduced into lattice layers of montmorillonite by cation-exchange reaction.The SAR have higher liquid absorbency rate.

scholarly journals Rényi’s parking problem revisited

2016 ◽  
Vol 16 (02) ◽  
pp. 1660006 ◽  
Author(s):  
Matthew P. Clay ◽  
Nándor J. Simányi
Keyword(s):  
Random Process ◽  
Recent Work ◽  
Computer Simulations ◽  
Line Segment ◽  
Exclusion Process ◽  
Packing Problem ◽  
Expected Number ◽  
Parking Problem ◽  
Long Line ◽  
Recursion Formulas

Rényi’s parking problem (or 1D sequential interval packing problem) dates back to 1958, when Rényi studied the following random process: Consider an interval [Formula: see text] of length [Formula: see text], and sequentially and randomly pack disjoint unit intervals in [Formula: see text] until the remaining space prevents placing any new segment. The expected value of the measure of the covered part of [Formula: see text] is [Formula: see text], so that the ratio [Formula: see text] is the expected filling density of the random process. Following recent work by Gargano et al. [4], we studied the discretized version of the above process by considering the packing of the 1D discrete lattice interval [Formula: see text] with disjoint blocks of [Formula: see text] integers but, as opposed to the mentioned [4] result, our exclusion process is symmetric, hence more natural. Furthermore, we were able to obtain useful recursion formulas for the expected number of [Formula: see text]-gaps ([Formula: see text]) between neighboring blocks. We also provided very fast converging series and extensive computer simulations for these expected numbers, so that the limiting filling density of the long line segment (as [Formula: see text]) is Rényi’s famous parking constant, [Formula: see text]

Synthesis of Path-Generating Mechanisms by Numerical Methods

1963 ◽  
Vol 85 (3) ◽  
pp. 298-304 ◽  
Author(s):  
Bernard Roth ◽  
Ferdinand Freudenstein
Keyword(s):  
Numerical Methods ◽  
Numerical Solution ◽  
Initial Approximation ◽  
Algebraic Methods ◽  
Algebraic Equations ◽  
Kinematic Synthesis ◽  
Path Synthesis ◽  
Special Case

Algebraic methods in kinematic synthesis are extended to cases in which the development of iterative numerical procedures are required. Algorithms are developed for the numerical solution of nonlinear, simultaneous, algebraic equations. Convergence is obtained without the need for a “good” initial approximation. The theory is applied to the nine-point path synthesis of geared five-bar motion, in terms of which four-bar motion may be considered as a special case.

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