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.

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.

Patterned Bootstrap: A New Method Which Gives Efficiency for Precision Position Synthesis of Planar Linkages

2005 ◽  
Author(s):  
Zhenjun Luo ◽  
Jian S. Dai
Keyword(s):  
New Method ◽  
Continuation Methods ◽  
Interval Methods ◽  
Homotopy Continuation ◽  
Root Finding ◽  
Starting Point ◽  
Homotopy Continuation Methods ◽  
Upper Level ◽  
Planar Linkages ◽  
Position Synthesis

This paper presents a new method, termed as patterned bootstrap (PB), which is suitable for precision position synthesis of planar linkages. The method solves a determined system of equations using a new bootstrapping strategy. In principle, a randomly generated starting point is advanced to a final solution through solving a number of intermediate systems. The structure and the associated parameters of each intermediate system is defined as a pattern. In practice, a PB procedure generally consists of two levels: an upper level which controls the transition of patterns, and a lower level which solves intermediate systems using globally convergent root-finding algorithms. Besides introducing the new method, tunnelling functions have been added to several systems of polynomials derived by formal researchers in order to exclude degenerated solutions. Our numerical experiments demonstrate that many precision position synthesis problems can be solved efficiently without resorting to time-consuming polynomial homotopy continuation methods or interval methods. Finding over 95 percentages of the complete solutions of the 11 precision position function generation problem of a Stephenson-III linkage has been achieved for the first time.

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.

Kinematic Mapping Based Evaluation of Assembly Modes for Planar Four-Bar Synthesis

2005 ◽  
Author(s):  
Hans-Peter Schro¨cker ◽  
Manfred L. Husty ◽  
J. Michael McCarthy
Keyword(s):  
Image Space ◽  
Synthesis Problem ◽  
New Method ◽  
Kinematic Mapping ◽  
Position Synthesis ◽  
Four Bar Linkage

This paper presents a new method to determine if two task positions used to design a four-bar linkage lie on separate circuits of a coupler curve, known as a “branch defect.” The approach uses the image space of a kinematic mapping to provide a geometric environment for both the synthesis and analysis of four-bar linkages. In contrast to current methods of solution rectification, this approach guides the modification of the specified task positions, which means it can be used for the complete five position synthesis problem.

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