Larisa Rybak

Approximating a solution set of nonlinear inequalities

In this paper we propose a method for solving systems of nonlinear inequalities with predefined accuracy based on nonuniform covering concept formerly adopted for global optimization. The method generates inner and outer approximations of the solution set. We describe the general concept and three ways of numerical implementation of the method. The first one is applicable only in a few cases when a minimum and a maximum of the constraints convolution function can be found analytically. The second implementation uses a global optimization method to find extrema of the constraints convolution function numerically. The third one is based on extrema approximation with Lipschitz under- and overestimations. We obtain theoretical bounds on the complexity and the accuracy of the generated approximations as well as compare proposed approaches theoretically and experimentally.

Synthesis of a Multi-Connected Digital Controller for a Robotized Vibration Isolation Platform Based on H∞-Optimization

We consider the problem of constructing multi-connected control of a robotic platform designed to protect technological objects and human operators from low-frequency influences on part of the moving base. The platform includes six drive mechanisms with stepper motors. The problem is solved by the methods of the modern theory of robust stabilization and optimal control based on H∞-optimization in the state space. We construct a mathematical model of the multidimensional system, taking into account the characteristics of electromechanical drives and using signals of feedback sensors as state variables. We give an example of synthesizing a multidimensional optimal stabilizing controller in the form of state feedback for a system with disturbances bounded in L2-norm. We define the feedback control structure and obtain the matrix of feedback coefficients. We also show the results of mathematical modeling.

On Approach Based on Lie Groups and Algebras to the Structural Synthesis of Parallel Robots

This paper proposes an approach to the structural synthesis of parallel robots based on the Lie algebras of the displacement groups. This approach allows the automation of the synthesis and eliminates the disadvantages of both methods based on the using of displacement subgroups, and methods based on the screw theory. Parallel robot is presented consisting of the moving platform and the stationary base connected via several serial kinematic chains. It is shown that movement of kinematic chain correspond to Lie subgroups of displacement, called displacement subgroups generators. These generators play an important role in determining the possible architectures of parallel mechanisms. Structural synthesis was performed for the case in which the mobile platform speed relative to the fixed basement forms a subalgebra of the Lie algebra of the displacement group of Euclidean space. In this case, we obtain three types of parallel robots. We also consider the case when the speed of mobile platform relative to the fixed basement forms a subspace, which is not a subalgebra of the displacement group of Euclidean space. It is divided into two types of robots. Is discussed case of parallel robot with serial chains generates Lie subalgebras in displacement space synthesis.

Finding sets of solutions to systems of nonlinear inequalities

The problem of approximating the set of all solutions to a system of nonlinear inequalities is studied. A method based on the concept of nonuniform coverings is proposed. It allows one to obtain an interior and exterior approximation of this set with a prescribed accuracy. The efficiency of the method is demonstrated by determining the workspace of a parallel robot.

Structural Synthesis of a Class of the Parallel Mechanisms Providing Plane-Parallel Displacement of a Mobile Platform

In this article the structural synthesis method of a class of the parallel mechanisms providing plane-parallel displacement of a mobile platform is considered. The considered method is based on application of the screw theory and the concept of virtual chains. The structures of all parallel mechanisms containing three legs are received.