Type-II Singularities Research Articles

CURVATURE-TORSION ENTROPY FOR TWISTED CURVES UNDER CURVE SHORTENING FLOW

AbstractWe study curve-shortening flow for twisted curves in $\mathbb {R}^3$ (that is, curves with nowhere vanishing curvature $\kappa $ and torsion $\tau $ ) and define a notion of torsion-curvature entropy. Using this functional, we show that either the curve develops an inflection point or the eventual singularity is highly irregular (and likely impossible). In particular, it must be a Type-II singularity which admits sequences along which ${\tau }/{\kappa ^2} \to \infty $ . This contrasts strongly with Altschuler’s planarity theorem, which shows that ${\tau }/{\kappa } \to 0$ along any essential blow-up sequence.

A Numerical Stability Analysis of Mean Curvature Flow of Noncompact Hypersurfaces with Type-II Curvature Blowup: II

In previous work, we have presented evidence from numerical simulations that the Type-II singularities of mean curvature flow (MCF) of rotationally symmetric, complete, noncompact embedded hypersurfaces, constructed by the second and the fourth authors of this paper, are stable. In this work, we again use numerical simulations to show that MCF subject to initial perturbations that are not rotationally symmetric behaves asymptotically like it does for rotationally symmetric perturbations. In particular, if we impose sinusoidal angular dependence on the initial embeddings, we find that for perturbations near the tip, evolutions by MCF asymptotically lose their angular dependence—becoming round—and develop Type-II bowl soliton singularities. As well, if we impose sinusoidal angular dependence on the initial embeddings for perturbations sufficiently far from the tip, the angular dependence again disappears as Type-I neckpinch singularities develop. The numerical analysis carried out in this paper is an adaptation of the “overlap” method introduced in our previous work and permits angular dependence.

Rotationally symmetric Ricci flow on [formula omitted

We study the Ricci flow on Rn+1, with n≥2, starting at some complete bounded curvature rotationally symmetric metric g0. We first focus on the case where (Rn+1,g0) does not contain minimal hyperspheres; we prove that if g0 is asymptotic to a cylinder, then the solution develops a Type-II singularity and converges to the Bryant soliton after scaling, while if the curvature of g0 decays at infinity, then the solution is immortal. As a corollary, we prove a conjecture by Chow and Tian about Perelman's standard solutions. We then consider a class of asymptotically flat initial data (Rn+1,g0) containing a neck and we prove that if the neck is sufficiently pinched, in a precise way, the Ricci flow encounters a Type-I singularity.

Ricci flow of warped Berger metrics on {mathbb {R}}^{4}

We study the Ricci flow on {mathbb {R}}^{4} starting at an SU(2)-cohomogeneity 1 metric g_{0} whose restriction to any hypersphere is a Berger metric. We prove that if g_{0} has no necks and is bounded by a cylinder, then the solution develops a global Type-II singularity and converges to the Bryant soliton when suitably dilated at the origin. This is the first example in dimension n > 3 of a non-rotationally symmetric Type-II flow converging to a rotationally symmetric singularity model. Next, we show that if instead g_{0} has no necks, its curvature decays and the Hopf fibres are not collapsed, then the solution is immortal. Finally, we prove that if the flow is Type-I, then there exist minimal 3-spheres for times close to the maximal time.

Reconfiguration of a parallel kinematic manipulator with 2T2R motions for avoiding singularities through minimizing actuator forces

This paper aims to develop an approach for the reconfiguration of a parallel kinematic manipulator (PKM) with four degrees of freedom (DoF) designed to tackle tasks of diagnosis and rehabilitation in an injured knee. The original layout of the 4-DoF manipulator presents Type-II singular configurations within its workspace. Thus, we proposed to reconfigure the manipulator for avoiding such singularities (owing to the Forward Jacobian of the PKM) during typical rehabilitation trajectories. We achieve the reconfiguration of the PKM through a minimization problem where the design variables correspond to the anchoring points of the robot limbs on fixed and mobile platforms. The objective function relies on the minimization of the forces exert by the actuators for a specific trajectory. The minimization problem considers constraints equations to avoid Type-II singularities, which guarantee the feasibility of the active generalized coordinates for a particular path. To evaluate the proposed conceptual strategy, we build a prototype where reconfiguration occurs by moving the position of the anchoring points to holes bored in the fixed and mobile platforms. Simulations and experiments of several study cases enable testing the strategy performance. The results show that the reconfiguration strategy allows obtaining trajectories having minimum actuation forces without Type-II singularities.

Efficient Singularity-Free Workspace Approximations Using Sum-of-Squares Programming

Abstract In this paper, we provide a general framework to determine inner and outer approximations to the singularity-free workspace of fully actuated robotic manipulators, subject to Type-I and Type-II singularities. This framework utilizes the sum-of-squares optimization technique, which is numerically implemented by semidefinite programming. In order to apply the sum-of-squares optimization technique, we convert the trigonometric functions in the kinematics of the manipulator to polynomial functions with an additional constraint. We define two quadratic forms, describing two ellipsoids, whose volumes are optimized to yield inner and outer approximations of the singularity-free workspace.

A robust geometric method of singularity avoidance for kinematically redundant planar parallel robot manipulators

Jacobian-based methods of singularity analysis are known to be unreliable when applied to kinematically redundant parallel robot manipulators, due to their potential to miss certain singularities and incorrectly identify others in the manipulator’s workspace. In this paper, a geometric method of singularity avoidance for kinematically redundant planar parallel robot manipulators is presented, which firstly determines the manipulator’s proximity to a singularity and then computes how the kinematically redundant degree(s) of freedom should be optimised for the given pose of the end-effector. The singularity analysis is conducted by examining the mechanism in terms of the instantaneous centres of rotation of its corresponding mobility one sub-mechanisms when all but one of the actuators are locked, where the manipulator is in a type-II singularity when these points either are indeterminable or coincide with one another, and an index, rmin, is introduced which describes the minimum normalised distance from such conditions being met. A predictor-corrector method is employed to compute the configuration for which rmin is optimised, and is reachable without crossing a singularity. Finally, the advantages of the geometric method of singularity analysis are shown in comparison to traditional Jacobian-based methods when applied to kinematically redundant parallel robot manipulators.

Lagrangian mean curvature flow of Whitney spheres

It is shown that an equivariant Lagrangian sphere with a positivity condition on its Ricci curvature develops a type-II singularity under the Lagrangian mean curvature flow that rescales to the product of a grim reaper with a flat Lagrangian subspace. In particular this result applies to the Whitney spheres.

Dimensional synthesis of the single-loop translational parallel manipulator PRRR-PRPU

The dimensional synthesis of translational parallel manipulators (TPMs) of type PRRR-PRPU is addressed by using an overall novel method. Addressing this design step on such TPMs is interesting for the scientific community since, in a previous paper, one of the authors showed that it has the following promising features: a single-loop not-overconstrained architecture with all the actuators on or near to the base, a simple position analysis, easy-to-find workspace boundaries, no constraint singularity, a type-II singularity locus that is a plane easy to keep far from the useful workspace, and a double infinity of isotropic configurations. The presented synthesis includes the analysis of isotropy and dexterity by local and global conditioning indexes, the useful workspace optimization, and the accuracy and stiffness analyses. The result is the identification of a normalized TPM of type PRRR-PRPU with performances that are comparable with those of commercial TPMs. The identified normalized TPM yields a set of actual TPMs with the same performances by changing the value of a reference geometric length. Also, the chosen shape of the useful workspace (i.e., a cuboid) matches the needs of many industrial applications.

Type-II singularities of two-convex immersed mean curvature flow

AbstractWe show that any strictly mean convex translator of dimension n ≥ 3 which admits a cylindrical estimate and a corresponding gradient estimate is rotationally symmetric. As a consequence, we deduce that any translating solution of the mean curvature flow which arises as a blow-up limit of a two-convex mean curvature flow of compact immersed hypersurfaces of dimension n ≥ 3 is rotationally symmetric. The proof is rather robust, and applies to a more general class of translator equations. As a particular application, we prove an analogous result for a class of flows of embedded hypersurfaces which includes the flow of twoconvex hypersurfaces by the two-harmonic mean curvature.

Ricci Flow Emerging from Rotationally Symmetric Degenerate Neckpinches

In [3], the authors created Type-II Ricci flow neckpinch singularities. In this paper, we construct solutions to Ricci flow whose initial data are the singular metric resulting from these singularities. We show in particular that the curvature decreases at the same rate at which it blew up. This is the first example of Ricci flow starting from a Type-II singularity.

On Type-II Singularities in Ricci Flow on ℝ N

In each dimension N ≥ 3 and for each real number λ ≥1, we construct a family of complete rotationally symmetric solutions to Ricci flow on ℝ N , which encounter a global singularity at a finite time T. The singularity forms arbitrarily slowly with the curvature blowing up arbitrarily fast at the rate (T − t)−(λ+1). Near the origin, blow-ups of such a solution converge uniformly to the Bryant soliton. Near spatial infinity, blow-ups of such a solution converge uniformly to the shrinking cylinder soliton. As an application of this result, we prove that there exist standard solutions of Ricci flow on ℝ N whose blow-ups near the origin converge uniformly to the Bryant soliton.

A novel method to avoid turning point singularities for the semi-regular hexagons Gough-Stewart parallel manipulator

This paper presents a novel disturbance function method to avoid turning point singularities for the semi-regular hexagons 6-SPS Gough-Stewart manipulator. Through analysis of the configuration bifurcation characteristics of the manipulator at the type-II singular points, it is found that the type-II singularities under signal input parameter belong to turning point bifurcation. The configuration patterns for the manipulator to pass through the turning points are divided into three types: persistent, non-persistent and path configuration. Utilizing the universal unfolding approach, the configuration bifurcation characteristics under the perturbation parameters applied to the extendable legs are analyzed. The investigation reveals that all configuration branches converged in the same singular point in the unperturbed system will be separated in the disturbed system. Based on this discovery, a novel method for the parallel manipulator to pass through the singular points with the desired configuration is presented. Then the disturbance functions for the manipulator to pass through the turning points with the persistent configuration and the non-persistent configuration are constructed. The method presented in this paper can be applied to avoiding the singularities in such cases where the path and orientation of the manipulator are strictly programmed.

Study on the method for 6-SPS Gough—Stewart platforms to pass through type-II singular points with its original configuration

It is well known that there exist many more singularities in parallel manipulators. At the singular point, the motion of the parallel manipulator is uncertain. In order to make parallel manipulators produce concrete output at singular points, and go away from singular points with the desired configuration, the method for the manipulator to pass through type-II singularities with its original configuration has been investigated in the paper. First, the semi-regular hexagon Gough—Stewart manipulator is taken as an example to analyse the configuration bifurcation characteristics at the vicinities of the type-II singularities going with the input parameters. The studies show that on different configuration branches, the singularity-free moving region is different, and there are two or four assembly configurations in the space above the base for the same group of input parameters. By researching transition behaviours of the configuration bifurcation curves under the disturbances applied to the other input parameters, it is found that under a suitable disturbance the perturbed persistent configuration will go away from the non-persistent configuration. Based on this kind of character, a novel method for the manipulator to pass through the turning point with its original configuration has been presented.