On one approach to the approximation of solutions to the direct kinematics problem of parallel manipulators

Andrei Gorchakov,Vyacheslav Mozolenko

doi:10.1515/comp-2020-0007

Abstract

AbstractAny real continuous bounded function of many variables is representable as a superposition of functions of one variable and addition. Depending on the type of superposition, the requirements for the functions of one variable differ. The article investigated one of the options for the numerical implementation of such a superposition proposed by Sprecher. The superposition was presented as a three-layer Feedforward neural network, while the functions of the first’s layer were considered as a generator of space-filling curves (Peano curves). The resulting neural network was applied to the problems of direct kinematics of parallel manipulators.

Highlights

For the approximation of solutions of systems of nonlinear equations, machine learning methods are widely used, neural networks are mainly used
The superposition was presented as a three-layer Feedforward neural network, while the functions of the first’s layer were considered as a generator of space-filling curves (Peano curves)
The resulting neural network was applied to the problems of direct kinematics of parallel manipulators

Summary

Introduction

For the approximation of solutions of systems of nonlinear equations (many authors use the phrase “to solve systems of nonlinear equations”), machine learning methods are widely used, neural networks are mainly used. For any integer n ≥ 2 there exist continuous real functions ψpq (x) defined on the unit segment E1 =. It should be noted that, by their construction, functions ψpq are independent of the form of the function f , increase monotonically and are continuous This approach was further developed by Sprecher [5,6,7,8,9]. – Each continuous function of n variables on a unit 2 Construction space-filling curve cube En can be represented as f The first and second layers utilize a fixed processing element that is independent of n, and only the output layer depends on f It is worth mentioning a rather interesting opinion expressed in [15]: “Representation Properties of Networks: Kolmogorov’s Theorem Is Irrelevant”. N is both the number of nodes of the curve and the number of partitions of the piecewise constant function g

Parallel manipulators

Numerical experiment

Conclusions