Measuring the closeness to singularities of a planar parallel manipulator using geometric algebra

A new index for measuring the closeness to the singularities of parallel manipulators using geometric algebra is proposed in this paper. Constraint wrenches acting on the moving platform of a parallel manipulator are derived using the outer product and dual operations. Removing the redundant constraint wrenches, a singularity polynomial is obtained when the coefficient of the outer product of all the non-redundant constraint wrenches equals zero. A singularity surface can be drawn using the singularity polynomial. Similarly, an approximate singularity polynomial and approximate singularity surface can be obtained by imposing a threshold to the singular polynomial. Then the singularity volume is calculated as the space between singularity surface and approximate singularity surface. The new index is derived by calculating the ratio of the non-singularity workspace volume (the workspace volume minus the singularity volume) to the workspace volume. The proposed index is coordinate-free and has a clear geometrical and physical interpretation. This index can be a basis for selecting structural parameters, path planning and mechanism design.     

Optimal Trajectory Tracking of Underwater Vehicle-Manipulator Systems Through the Clifford Algebras and of the Davies Method

The manipulation in singular regions generates an instantaneous reduction in the mechanism mobility which can result in some disturbances in the trajectory tracking. In proximity of the singularities, small velocities in the end-effectors generates high speeds in the joints due to the gradual reduction of the mobility. The phenomenon of kinematic singularity generates a instantaneous instability in torque profile of the redundant robotic systems by the transformation of secondary joints to primary joints. The disturbances of the underwater environment intensifies the effects of the kinematic singularities because the hydrodynamic strongly oppose to torque variations. This work presents a methodology for using dual quaternions in the posture feedback of a Underwater Vehicle-Manipulator System (UVMS) using the Davies method which avoids kinematic singularities and ensures the optimal torque profiles.     

Algorithm for the Semigroup of a Space Curve Singularity

The semigroup of values of irreducible space curve singularities is the set of intersection multiplicities among hypersurfaces and the given curve. It is an invariant of the singularity, and for plane curves it characterizes the equisingularity type considered by Zariski. For space curve singularities the semigroup of values is a numerical semigroup and it can not be computed by means of the exponents of any Puiseux parametrization, as in the plane case. We obtain an algorithm for calculating the semigroup of values of a space curve singularity, which determines the generators of the semigroup and the valuation ideals associated with the semigroup. We give a Maple version of the algorithm.     

Phase Transformation and Singularity Theory

Phase transformation plays an important role in thermodynamics and materials science. Based on the theory of singularities, a new method to construct phase diagrams is presented. Analysing singularities on base of root sequences, see Tamaschke [16], will help to develop singularity graphs, where workings by H. Whitney, R. Thom, and V. I. Arnold provide fundamentals. The generated singularity graphs build the starting point for singularity phase diagrams. A powerful characteristic of such singularity graphs is that higher-dimensional surfaces can be transformed to a two-dimensional diagram. The attained singularity diagram can be used in materials science for analytical models of temperature-concentrated diagrams. As tools from algebra and analysis build a sound basis for singularity diagrams, it is possible to evolve computer software generating these phase diagrams. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convex- and Monotone-Transformable Mathematical Programming Problems and a Proximal-Like Point Method

The problem of finding the singularities of monotone vectors fields on Hadamard manifolds will be considered and solved by extending the well-known proximal point algorithm. For monotone vector fields the algorithm will generate a well defined sequence, and for monotone vector fields with singularities it will converge to a singularity. It will also be shown how tools of convex analysis on Riemannian manifolds can solve non-convex constrained problems in Euclidean spaces. To illustrate this remarkable fact examples will be given.     

Computation of the attractive force of an ellipsoid

A numerical method for computing the attractive force of an ellipsoid is proposed that does not involve separating subdomains with singularities. The sought function is represented as a triple integral such as the inner integral of the kernel can be evaluated analytically with the kernel treated as a weight function. The inner integral is approximated by a quadrature for the product of functions, of which one has an integrable singularity. As a result, the integrand obtained before the second integration has only a weak logarithmic singularity. The subsequent change of variables yields an integrand without singularities. Based on this approach, at each stage of integral evaluation with respect to a single variable, quadrature formulas are derived that do not have singularities at integration nodes and do not take large values at these nodes. For numerical experiments, a rather complicated test function is constructed that is the exact attractive force of an ellipsoid of revolution with an elliptic density distribution.     

Singularities of the Laplacian at corners and edges of three-dimensional domains and their treatment with finite element methods

Three-dimensional Poisson problems containing boundary singularities are treated. The forms of the solutions for certain problems of this type are derived, where the domains of the problems can be represented in terms either of spherical- or of cylindrical-polar co-ordinates. These singular forms are used to augment the basis of a standard piecewise polynomial Galerkin space, thus producing an augmented Galerkin technique which is suited to the context of a problem involving a singularity. Error estimates are derived.     

Foveal detection and approximation for singularities

Projections in a foveal space at u approximate functions with a resolution that decreases proportionally to the distance from u. Such spaces are defined by dilating a finite family of foveal wavelets, which are not translated. Their general properties are studied and illustrated with spline functions. Orthogonal bases are constructed with foveal wavelets of compact support and high regularity. Foveal wavelet coefficients give pointwise characterization of nonoscillatory singularities. An algorithm to detect singularities and choose foveal points is derived. Precise approximations of piecewise regular functions are obtained with foveal approximations centered at singularity locations.     

Boundary singularities with a simple decomposition

We define the decomposition of a boundary singularity as a pair (a singularity in the ambient space together with a singularity of the restriction to the boundary). We prove that the Lagrange transform is an involution on the set of boundary singularities that interchanges the singularities that occur in the decomposition of a boundary singularity. We classify the boundary singularities for which both of these singularities are simple. Bibliography: 8 titles.Translated fromTrudy Seminara imeni I. G. Petrovskogo, No. 15, pp. 55–69, 1991.     

Geometry of singularities for the steady Boussinesq equations

Analysis and computations are presented for singularities in the solution of the steady Boussinesq equations for two-dimensional, stratified flow. The results show that for codimension 1 singularities, there are two generic singularity types for general solutions, and only one generic singularity type if there is a certain symmetry present. The analysis depends on a special choice of coordinates, which greatly simplifies the equations, showing that the type is exactly that of one dimensional Legendrian singularities, generalized so that the velocity can be infinite at the singularity. The solution is viewed as a surface in an appropriate compactified jet space. Smoothness of the solution surface is proved using the Cauchy-Kowalewski Theorem, which also shows that these singularity types are realizable. Numerical results from a special, highly accurate numerical method demonstrate the validity of this geometric analysis. A new analysis of general Legendrian singularities with blowup, i.e., at which the derivative may be infinite, is also presented, using projective coordinates.Research supported in part by the ARPA under URI grant number #N00014092-J-1890.Research supported in part by the NSF under grant number #DMS93-02013.Research supported in part by the NSF under grant #DMS-9306488.     

Triangulated categories of Gorenstein cyclic quotient singularities

We prove there is an equivalence of derived categories between Orlov's triangulated category of singularities for a Gorenstein cyclic quotient singularity and the derived category of representations of a quiver with relations, which is obtained from a McKay quiver by removing one vertex and half of the arrows. This result produces examples of distinct quivers with relations which have equivalent derived categories of representations.

     

On corner singularities in Reissner-Mindlin's theory of elastic plates

In the framework of linear elasticity, singularities occur in domains with non-smooth boundaries. Particularly in Fracture Mechanics, the local stress field near stress concentrations is of interest. In this work, singularities at re-entrant corners or sharp notches in Reissner-Mindlin plates are studied. Therefore, an asymptotic solution of the governing system of partial differential equations is obtained by using a complex potential approach which allows for an efficient calculation of the singularity exponent λ. The effect of the notch opening angle and the boundary conditions on the singularity exponent is discussed. The results show, that it can be distinguished between singularities for symmetric and antisymmetric loading and between singularities of the bending moments and the transverse shear forces. Also, stronger singularities than the classical crack tip singularity with free crack faces are observed. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cobordism of algebraic knots

Summary The isolated singularities of complex hypersurfaces are studied by considering the topology of the highly connected submanifolds of spheres determined by the singularity. By introducing the notion of the link of a perturbation of the singularity and using techniques of surgery theory, we are able to describe which invariants associated to a singularity can be used to determine the cobordism type of the singularity.It is shown that the cobordism type is determined by the set of weakly distinguished bases. This result is used to draw a distinction between the classical case of two variables and the higher dimensional problem. That is, we show that the result of Le which states that the cobordism and topological classifications of singularities coincide in the classical dimension does not hold for singularities of functions of more than three variables. Examples of topologically distinct but cobordant singularities are obtained using results of Ebeling.     

Higher-Order Differential Equations Having a Singularity in an Interior Point

Ordinary differential equations of an arbitrary order having a non-integrable singularity inside the interval are considered under additional matching conditions for solutions at the singular point. We construct special fundamental systems of solutions for this class of differential equations, study their asymptotical, analytical and structural properties and the behavior of the corresponding Stokes multipliers. These fundamental systems of solutions are used in spectral analysis of differential operators with singularities.     

The solution of infinitely ill-conditioned weakly-singular problems

It is demonstrated on examples that a weak singularity (i.e., with converging improper integral) may produce in computations (depending on the algorithm employed) an infinitely ill-conditioned situation when arbitrarily small imprécisions introduced by the algorithm or by a software create divergent approximations for mathematically convergent integrals. The possibility of hidden singularities is shown, and the double error phenomenon is identified and demonstrated in a simple example. Construction of test problems is proposed to check the applicability of existing software prior to its use for the solution of real life problems with weakly-singular equations. It is shown that the application of the integration by parts formula to weakly-singular integrals may create strong singularities (i.e., unbounded terms or divergent improper integrals). Methods of removal of singularities with and without compensation are studied for the numerical solution of infinitely ill-conditioned weakly-singular problems.     

Scaled boundary finite element analysis of stress singularities in piezoelectric multi-material systems

The scaled boundary finite element method (SBFEM) in an extension for piezoelectric materials is used to analyze twoand three-dimensional stress singularities in piezoelectric multi-material systems. It is found to be an efficient tool for the analysis of singularity orders of such situations, that turn out to be rather complex. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Closed-form solutions for stress singularities at bi-material wedges in Reissner-Mindlin plates

In this work, stress singularities in isotropic bi-material junctions are investigated using Reissner-Mindlin plate theory by means of a complex potential formalism. The governing system of partial differential equations is solved employing methods of asymptotic analysis. The resulting asymptotic near-fields including the singularity exponent λ are obtained in a closed-form analytical manner as solutions of a corresponding eigenvalue problem. The singular solution character is discussed for different geometrical configurations. In particular, the present study investigates the influence of the material constants on the singularity exponent. It is shown, that the Reissner-Mindlin theory allows for distinguishing between singularities of the bending moments and the transverse shear forces. Further, stronger singularities than the classical crack-tip singularity are observed. The results allow for further application such as a combination with numerical methods. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Singularities of solutions to linear,second order analytic elliptic equations in two independent variables I. The completely regular boundary

Two methods are described for the a priori location of singularities of solutions to exterior boundary value problems. One uses an expansion for the solution in a circle centered on a regular exterior point P. A singularity lies on the circle of convergence. The envelope of these circles, generated as P makes a circuit about the closed boundary, circumscribes the singularities. The radius of convergence depends on singularities of the solution u(s) and its normal derivative v(s) on the boundary. The second method employs complex characteristics to relate singularities of the boundary data to real singularities of the solution. Integral equations connecting (y), v(s) and the analytic boundary condition are used to continue the data into the complex s-plane and to locate their singularities. Explicit solution of the integral equations is unnecessary; some nonlinear boundary conditions can be handled.     

Detection of the singularities of a complex function by numerical approximations of its Laurent coefficients

For several applications, it is important to know the location of the singularities of a complex function: just for example, the rightmost singularity of a Laplace Transform is related to the exponential order of its inverse function. We discuss a numerical method to approximate, within an input accuracy tolerance, a finite sequence of Laurent coefficients of a function by means of the Discrete Fourier Transform (DFT) of its samples along an input circle. The circle may also enclose some singularities, since the method works with the Laurent expansion. The DFT is computed by the FFT algorithm so that, from a computational point of view, the efficiency is guaranteed. The function samples may be obtained by solving a numerical problem such as, for example, a differential problem. We derive, as consequences of the method, some new outcomes able to detect those singularities which are close to the circle and to discover if the singularities are all external or internal to the circle so that the Laurent expansion reduces to its regular or singular part, respectively. Other singularities may be located by means of a repeated application of the method, as well as an analytic continuation. Some examples and results, obtained by a first implementation, are reported.     

Multi-valued solutions and branch point singularities for nonlinear hyperbolic or elliptic systems

Multi-valued solutions are constructed for 2 × 2 first-order systems using a generalization of the hodograph transformation. The solution is found as a complex analytic function on a complex Riemann surface for which the branch points move as part of the solution. The branch point singularities are envelopes for the characteristics and thus move at the characteristic speeds. We perform an analysis of stability of these singularities with respect to perturbations of the initial data. The generic singularity types are folds, cusps, and nondegenerate umbilic points with non-zero 3-jet. An isolated singularity is generically a square root branch point corresponding to a fold. Two types of collisions between singularities are generic: At a “tangential” collision between two singularities moving at the same characteristic speed, a cube root branch point is formed, corresponding to a cusp. A “non-tangential” collision, between two square root branch points moving at different characteristic speeds, remains a square root branch point at the collision and corresponds to a nondegenerate umbilic point. These results are also valid for a diagonalizable n-th order system for which there are exactly two speeds. © 1993 John Wiley & Sons, Inc.