Neighborhood Of Singular Point Research Articles

Kinematic and Static Characteristics of In-parallel Actuated Manipulators at Singular Points and in their Neighborhoods.

In the process of mechanism design and trajectory planning of in-parallel actuated manipulators, it is necessary to grasp the locations of all of singular points in the working space. This paper proposes a new method for numerically searching for the singular points of in-parallel actuated manipulators based on the pressure angle. A singular-point curve is defined as a set of singular points. Since the working space is divided into multiple regions by the singular-point curves, the end-effector cannot move freely from one region to any other regions. Kinematic and static characteristics at singular points are analyzed by pressure angles and instantaneous screw axes. Singular points of the Stewart-Platform are investigated based on the proposed method. Characteristics in the neighborhoods of singular points are analyzed from the points of view of output error, isotropy of output force and driving force, and the practical working space is defined.

Flow near the critical line in the gas-dynamic model of stellar wind with thermal conductivity

A one-dimensional, time-independent, spherically symmetric model of stellar wind is studied within the framework of gas dynamics. It is assumed that the energy transfer in the external stellar atmosphere occurs by electron thermal conduction. The form of the critical line on which the transition from subsonic to supersonic flow occurs and the types of singular points and their connection with the directions of separatrices is studied in detail. The stability of the solutions with respect to perturbations in the neighbourhood of singular points is considered, using the method suggested in [1].

The steady-state wave process in a piezoelectric layer and half-layer weakened by tunnel cuts (antiplane deformation)

The antiplane dynamic problem of electro-elasticity for a piezoelectric layer and half-layer containing curvilinear tunnel cuts along their bases is investigated. Integral representations of the solutions are constructed, by means of which the corresponding boundary value problems are reduced to a singular integro-differential equation in the jump in the amplitude of the displacement on the cuts. The asymptotic form of the combined mechanical and electric fields in the neighbourhood of singular points is investigated. The results of a numerical realization of the algorithm which enables the effect of the excitation frequency, the curvature of the cut (the crack), the type of boundary conditions and the effect of connectedness of the fields on the stress intensity factor K III are presented.

On the fast simulation of Direct and Inverse Jacobians for robot manipulators

The rapid development in VLSI technology makes it possible to implement highly complicated and time-consuming algorithms to accommodate real-time applications, Parallel processing techniques can now be used to reduce the computational time for models of high mathematical nature such as the computation of robot kinematics. This work proposes several computational methods for the evaluation of Direct and Inverse Jacobians for manipulators. Special attention is paid to the case of the robot operating in the neighbourhood of singular points. The algorithms are implemented using a Transputer-based development system. Detailed analysis is performed and successful results are obtained for a typical industrial robot manipulators (PUMA 560). Execution time comparisons between sequential and parallel processing architectures are also included to show the superiority of the latter approaches.

Robot control in the neighborhood of singular points

In this paper, we propose an alternative method for designing a robot controller in Cartasian coordinates using a time scale transformation. We will show how a suitable choice of the time scale allows us to control the robot in the neighborhood of singular points without reducing the performance outside the neighborhoods of these points. The resultant controlled system has slow poles in the area of poor manipulative ability and fast poles in that of good manipulative ability.

On a class of partial differential equations with singular coefficients

We study the equation div(fΔu)=0 for functions f which may become infinite at some subset of the domain of definition. After defining adequate Sobolev-like spaces of functions, we make a careful study of the behaviour of admissible solutions in the neighbourhood of singular points. This enables us to show that the Dirichlet problem possesses a space of solutions whose dimension is determined by the number and characteristics of the connected components of the set f=∞, a similar conclusion holds for the Neumann problem.

Trajectories and singular points in steady-state models of two-phase flows

The purpose of this paper is to adapt the methodology of dynamical systems to the study of a wide class of mathematical models currently used to solve problems of steady 1-D two-phase flows. A detailed study of the geometrical features of the ensemble of solutions is then used for two purposes. First, it makes it possible to understand the physical characteristics of such flows without the need to produce complete solutions. Secondly, the methodology gives valuable indications as to how to supplement computer codes which become inadequate in the neighborhood of singular points. This leads to avoidable numerical difficulties and incorrect interpretations. This methodology is particularly useful in the study of the phenomena of cooking. The important contribution of this general analysis is to show that, regardless of the number of equations in the model, the generic solutions involve only three types of singular points: saddle points, spirals or nodes. By way of example, the method is applied to the mathematically simplest case, the homogeneous flow model with adiabatic boundary conditions. The channel consists of a convergent portion with transition to a divergent portion through a smooth throat. The divergent portion possesses an inflection point after which the rate of area divergence decreases. The fluid is a mixture of water and steam in thermodynamic equilibrium. The expected saddle point, located just downstream from the throat, is followed by an unexpected spiral. The phase space consisting of pressure P, enthalpy h and spatial coordinate z divides itself into four distinct areas. In area A of figure 13 all solutions are single-valued, possess a pressure minimum and correspond to physically acceptable subcritical flows. In area B the solutions of the differential equations (all possessing a turning point) are irrelevant for the physical problem at hand. Through the saddle point there passes a trajectory which is unique in the subcritical portion and continues with two branches, one subcritical ending at pressure P′, the other supercritical, ending at pressure P″. Area C between those branches contains states which are not described by the postulated model. It is conjectured that flows against a back-pressure, P″ < P a < P′, evidently observed in nature, must presumably be described by an extension to the model involving enhanced entropy production. The remaining area D contains supercritical state points and is of no present interest.

Inverse Kinematic Solutions With Singularity Robustness for Robot Manipulator Control

The singularity problem is an inherent problem in controlling robot manipulators with articulated configuration. In this paper, we propose to determine the joint motion for the requested motion of the endeffector by evaluating the feasibility of the joint motion. The determined joint motion is called an inverse kinematic solution with singularity robustness, because it denotes feasible solution even at or in the neighborhood of singular points. The singularity robust inverse (SR-inverse) is introduced as an alternative to the pseudoinverse of the Jacobian matrix. The SR-inverse of the Jacobian matrix provides us with an approximating motion close to the desired Cartesian trajectory of the endeffector, even when the inverse kinematic solution by the inverse or the pseudoinverse of the Jacobian matrix is not feasible at or in the neighborhood of singular points. The properties of the SR-inverse are clarified by comparing it with the inverse and the pseudoinverse. The computational complexity of the SR-inverse is considered to discuss its implementability. Several simulation results are also shown to illustrate the singularity problem and the effectiveness of the inverse kinematic solution with singularity robustness.

Singular cauchy problems for systems of ordinary differential equations

Cauchy problems are investigated for systems of nonlinear ordinary differential equations with initial conditions specified at a singular point. Existence and uniqueness theorems for the solution of singular Cauchy problems, and theorems on the asymptotic representation of the solutions in the neighbourhood of singular points are presented.

Adiabatic One-Dimensional Flow of a Perfect Gas through a Rotating Tube of Uniform Cross Section

SummaryThe paper contains an analysis of the flow of a perfect gas with constant specific heats through a rotating channel of constant cross-sectional area, as used in certain helicopter propulsion systems and wind-driven gas turbines. The analysis is restricted to the adiabatic one-dimensional treatment, the Coriolis accelerations acting across a section being disregarded.The equations of motion and energy are deduced and, together with the equation of continuity, reduced to an ordinary non-linear differential equation of the first order, involving a dimensionless form of velocity and distance.The patterns of the integral curves of the differential equation are discussed and sketched by examining their asymptotic behaviour as well as that in the neighbourhood of singular points. It is shown that there exists a critical value of the angular velocity below which the flow remains subsonic in the pipe, if the entrance velocity is subsonic; it may, however, become sonic at the exit of the pipe. For supercritical angular velocities the flow may become sonic or supersonic in the pipe if the entrance velocity attains a given “ correct” but subsonic value. A method of examining for the possibility of shock formation is indicated.The initial conditions for the differential equation are deduced for the design and for the performance problem, two new flow functions being introduced and tabulated to facilitate practical calculations. Formulae are also deduced for the calculation of the pressure and Mach number variation from the previously calculated velocity variation along the pipe.Finally an approximate solution in closed terms is given for the case of small entrance velocities.

Transformation of differential equations in the neighborhood of singular points

Transformation of differential equations in the neighborhood of singular points

Singular Point in Two-dimensional Elasticity

Singular point is defined as a point at which the direction of principal stress is indeterminate, and divided into 3 kinds, viz. zero-point, pole and point of discontinuity. It is usually so difficult in the well known experiment of photo-elasticity to determine the stress-curves in the immediate neighbourhood of singular points, that the features of the stress-curves near the singular points have not been ascertained up to date. In this paper they are studied analytically with some experimental verifications, paying special regard to how the direction of principal stress changes, when we go around a singular point along any closed curve. As the result, we obtained an effective resort in the experiment of photo-elasticity to determine, to what sort of singular points a point in question belongs, by observing only the surroundings apart enough from the point, without necessarily closing about it.