The Research of Equal Probability Chaos Sequence Newton Iterative Methods and its Application to Mechanism Synthesis

Abstract

The discovery of dynamical chaos is one of the main achievements in the modern science and how to expand its application has important significance to the development of modern science. Many questions in natural science and engineering are transformed into nonlinear equations to be found. Newton iterative method is an important technique to one dimensional and multidimensional variables and iterative process exhibits sensitive dependence on initial guess point. To improve solving efficiency, the demand to chaos sequences is uniform distribution in every interval. The probability characteristics of three kinds of chaos were investigated. The simulations were work out with Matlab software. For the first time, a new method to find all solutions based on utilizing equal probability chaos sequences to obtain initial points to find all solutions of the nonlinear questions was proposed and it has higher solving efficiency compared with unequal probability chaos sequences to find all solutions. The numerical example in linkage synthesis shows that the method is correct and effective. And, different equal probability chaos sequences has different solving efficiency, so, for the same kind of question to be solved we can find the best equal probability chaos sequences to be used. This provides a simple realization method for mechanics design.

Inverse Displacement Analysis of a General 6R Manipulator Based on the Hyper-Chaotic Least Square Method

“…In 1991, C. W. Wampler and A. P. Morgan [9] completed the 6R inverse position problem using a generic-case solution methodology with 11 polynomial equations. In 1993, D. Kohli and M. Osvatic [10] used the product of the powers and the elimination method to obtain a set of equations containing 16 16  matrix coefficients, and to directly have 16th degree polynomials of no extraneous roots,since the solution to the problem can be simplified to the eigenvector problem. In the same year, D. Manocha and J. F. Canny [11] made use of the symbol processing and matrix computation to transform this problem into the decomposition problem of the matrix to be calculated, and solved the matrix eigenvalues and eigenvectors to obtain the robot joint variables.…”

“…However, this supposition has not been proven, and the process for solving the multivariable Jacobian matrix is very complex. The chaotic sequence method is new, in which the initial point of the Newton iteration is generated using the chaotic and hyper-chaotic system and all the real solutions on the mechanism synthesis can be effectively solved [15][16][17]. When the solutions are not in convergence using Newton method or a quasi-Newton method, the mathematical programming method can be adopted [18].…”

Mathematical Programming Method Based on Chaos Anti-Control for the Solution of Forward Displacement of Parallel Robot Mechanisms

“…So solving the matrix is quite complex. The chaotic sequence method is a new method, in which the initial point of the Newton iteration is generated using the chaotic and hyper-chaotic system and all the real solutions in the mechanism synthesis can be effectively solved [28][29][30][31][32]. But the Hénon hyper-chaotic Newton iteration method cannot solve the mechanism synthesis problem of 6-SPS.…”

LMF Algorithm Based on Hyper-Chaos for the Solving of Forward Displacement in a Parallel Robot Mechanism

“…So the matrix is quite complex to solve. The chaotic sequence method is a new method, in which the initial point of the Newton iteration is generated using the chaotic and hyper-chaotic system and all the real solutions of the mechanism synthesis can be effectively solved [22][23][24][25][26][27]. But the Hénon hyper-chaotic Newton iteration method cannot solve the mechanism synthesis problem of 6-SPS.…”