Parameter Identification of Lorenz System with Incomplete Information: Case of One Unknown Function

Inverse problem of the Lorenz system parametric identification is considered in the case of incomplete information about solutions of the system. In the present paper, it is assumed that only two solutions of the system from three are known in different combinations. The problem of the parameter identification of the system is solved by means of elimination of unknown functions from the original system. The obtained system of equations has the same order as the original one, but contains the unknown original parameters in new combinations. Sometimes, the number of new unknown parameters is higher than number of the original unknowns. In this case, the method of the constrained least squares minimization is used in the special formulation, developed by the authors. This novel formulation exploits linearity of the system with respect to the new unknown parameters, by means of which the number of nonlinear equations becomes equal to the number of the constraints between the new parameters. Two methods of the constraint minimization are considered: the classical method of Lagrange’s multipliers and a novel method of the auxiliary parameters. Numerical simulations demonstrate effectiveness of the algorithms.

Measuring the Pollutants in a System of Three Interconnecting Lakes by the Semianalytical Method

Pollution has become an intense danger to our environment. The lake pollution model is formulated into the three-dimensional system of differential equations with three instances of input. In the present study, the new iterative method (NIM) was applied to the lake pollution model with three cases called impulse input, step input, and sinusoidal input for a longer time span. The main feature of the NIM is that the procedure is very simple, and it does not need to calculate any special type of polynomial or multipliers such as Adomian polynomials and Lagrange’s multipliers. Comparisons with the Adomian decomposition method (ADM) and the well-known purely numerical fourth-order Runge-Kutta method (RK4) suggest that the NIM is a powerful alternative for differential equations providing more realistic series solutions that converge very rapidly in real physical problems.

Maggi’s Equations Used in the Finite Element Analysis of the Multibody Systems with Elastic Elements

The main method used to determine the equations of motion of a multibody system (MBS) with elastic elements is the method of Lagrange’s multipliers. The assembly of equations for the whole system represents an important step in the elastodynamic analysis of such a system. This paper presents a new method of approaching this stage, by applying Maggi’s equations. In this way, the links that exist between the finite elements and the connections that exist between different bodies of the MBS system are conveniently taken into account, each body having a distinct velocity and acceleration field. Although Maggi’s equations have been used, sporadically, in some applications so far, we are not aware that they have been used in the study of elastic systems using the finite element method. Finally, an algorithm is presented that uses the Maggi formalism to obtain the equations of motion for an MBS system.

Dynamic Answer of a Human Ankle Joint Implant

This paper addresses to a research of a dynamic answer obtained through numerical simulations of a human ankle joint implant with finite element method. The research background consists of an inverse dynamic analysis based on Newton-Euler formalism completed with Lagrange’s multipliers method. Thus, a parameterized virtual model of a human ankle joint was elaborated and simulated together with the implant, in dynamic conditions similar with real ones in reality. A results numerical processing was obtained with the aid of MSC Nastran and important results were obtained for orthopedic implants design.

A new method for determining the reactions of mechanical constraints

In the present paper it is introduced the method for determining the reactions of mechanical constraints (holonomic and nonholonomic constraints).As is known, for studying dynamical characters of a mechanical system it is necessary to determine the constraint reactions acting on the system. Up to now, the reactions are calculated through Lagrange's multipliers. By such a way the reactions are determined only indirectly. In the [3, 4], two methods of determining directly the reactions are discussed. However, for applying these methods, it is necessary to compute the inverse matrix of the matrix of inertia. This thing in general is not convenient, specially when the matrix of inertial is of large size and dense.In the present paper it is represented the method for determining the constraint reactions, by which it is possible to avoid inertia the computation of the inverse matrix of the matrix of inertia is avoided. For this in the paper it is used the middle variables by which we obtain a closed set of algebraic equations for directly determining reactions.

Principles of superposition for controlling pinch motions by means of robot fingers with soft tips

This paper analyzes the dynamics and control of pinch motions generated by a pair of two multi-degrees-of-freedom robot fingers with soft and deformable tips pinching a rigid object. It is shown firstly that passivity analysis leads to an effective design of a feedback control signal that realizes dynamic stable pinching (grasping), even if extra terms of Lagrange's multipliers arise from holonomic constraints of tight area-contacts between soft finger-tips and surfaces of the rigid object and exert torques and forces on the dynamics. It is shown secondly that a principle of superposition is applicable to the design of additional feedback signals for controlling both the posture (rotational angle) and position (some of task coordinates of the mass center) of the object provided that the number of degrees of freedom of each finger is specified for satisfying a condition of stationary resolution of controlled position state variables. The details of feedback signals are presented in the case of a special setup consisting of two robot fingers with two degrees of freedom.