Singularity Analysis Research Articles

Prolongation structure and integrability of the modified Sawada-Kotera equation

The modified Sawada-Kotera equation is investigated by prolongation technique and Painlevé singularity analysis. As a result, the Lax pair and conservation laws of the modified Sawada-Kotera equation are formulated. It is proved that this equation pasts the Painlevé test in sense of having enough arbitrary functions at its resonant points. The auto-Bäcklund transformation and exact solutions of the modified Sawada-Kotera equation are obtained explicitly.

Design of a deployable mechanism based on 7R-6R-double-loop units for mars decelerators

The increasing interest in decelerators utilized in Mars exploration has heightened the need for a decelerator with the function of deploying and a low ballistic coefficient. This paper designed a novel deployable mechanism based on 7R-6R double-loop units for Mars decelerators. First, this paper designed a new double-loop deployable unit coupled by a 7R loop and 6R loop through type synthesis based on line geometry. Then, the mobility of the mechanism was proven to be single by using screw theory. In addition, the singularity of the mechanism was investigated, which contains 3 singular configurations. Geometric conditions to avoid singular configurations were obtained in the singularity analysis. Finally, a scaled model utilized to verify the configuration and assemblability of the deployable unit was developed.

The distributions under two species-tree models of the number of root ancestral configurations for matching gene trees and species trees

For a pair consisting of a gene tree and a species tree, the ancestral configurations at a species-tree internal node are the distinct sets of gene lineages that can be present at that node. The enumeration of root ancestral configurations—ancestral configurations at the species-tree root—assists in describing the complexity of gene-tree probability calculations in evolutionary biology. Assuming that the gene tree and species tree match in topology, we study the distribution of the number of root ancestral configurations of a random labeled tree topology under the uniform and Yule–Harding models. We employ analytic combinatorics, considering ancestral configurations in the context of additive tree parameters and using singularity analysis to evaluate asymptotic growth of the coefficients of generating functions. For both models, we obtain asymptotic lognormal distributions for the number of root ancestral configurations. For Yule–Harding random trees, we also obtain the asymptotic mean (∼1.425n) and variance (∼2.045n) of the number of root ancestral configurations, paralleling previous results for the uniform model (mean (4/3)n, variance ∼1.822n). A methodological innovation is that to obtain the Yule–Harding asymptotic variance, singularity analysis is conducted from the Riccati differential equation that the generating function satisfies—without possessing the generating function itself.

Singularity Analysis and Complete Methods to Compute the Inverse Kinematics for a 6-DOF UR/TM-Type Robot

Improving the strategies employed to control robotic arms is of great importance because of the increase in their use in advanced supervisory control strategies, such as digital twins. The inverse kinematic (IK) control of manipulators requires an IK solution and an awareness of the singular configurations. This work presents a complete IK calculation system with singularity analysis for the UR5 robotic arm created by Universal Robots. For a specific robot pose, different angle solution sets are obtained, and one of these solution sets has to be selected to achieve movement continuity and avoid singularities. Two methods for this double purpose are proposed: one calculates all the solution possibilities, and the other obtains only one solution set by following a sequence of decisions and calculations clearly stated by a finite state machine (FSM). Both methods are effective in managing singularities. The FSM-based method complements the IK solution procedure with advantages in the number of computations and performance by producing results that would not lead the joints to move abruptly. The results prove that the presented methods select an IK solution that does not result in a singular configuration, and that most of the time, they lead to the same valid IK solution.

Analysis and Design of a Novel Compact Three-Degree-of-Freedom Parallel Robot

Abstract This paper introduces a novel compact three-degree-of-freedom (DOF) parallel robot that will be used as a leg of a 9-DOF kinematically redundant parallel robot. First, the kinematic model of the robot is established based on geometric constraint conditions. Then, the inverse and forward kinematic problems are solved. The inverse problem is straightforward, while the forward problem can be solved analytically by three different approaches. Afterward, a singularity analysis is presented based on the Jacobian matrices derived from the kinematic model. The mathematical conditions for singularities are obtained and their geometric interpretation is given. Finally, the workspace of the robot is analyzed and is shown to correspond to a portion of a torus. The analysis reveals that the robot can have a singularity-free workspace of significant size relative to its footprint provided that some simple limitations are introduced at the design stage.

Motion singularity analysis of the thick-panel kirigami

Thick-panel kirigami, constructed by cutting open creases on origami structures, can be applied to deployable structures with different Gaussian curvatures. By converting creases to rotational hinges, thick-panel kirigami can be treated as the assembly of spatial linkage, which provides accurate and precise deployment motion. Compared with origami, kirigami structures are able to optimize jagged deployed surfaces by geometry parameter adjustment. However, replacing several creases with slits in structures may lead to shortcomings in kinematics, i.e., motion singularity. In this study, motion singularity of thick-panel kirigami structure is discussed. Starting with kirigami units, bifurcation paths are detected and followed by the predictor-corrector numerical method. We then check the corresponding extension kirigami arrays and explore their complex bifurcation behaviors. The result indicates that the bifurcation motion paths within kirigami structures may lead to incomplete deployment. Physical prototypes are carried out, which successfully validate the singularity analysis. Also, motor- and spring-based hinges are arranged for driving, and the redundant actuator method is considered to avoid bifurcations and accomplish the full deployment. The findings are applicable to the design of kirigami-inspired deployable structures for engineering applications in the future.

Analysis of cone-like singularities in twisted elastic ribbons

Twisting a thin elastic ribbon is known to produce a localised deformation pattern resembling a cone whose tip is located on the edge of the ribbon. Using the theory of inextensional ribbons, we present a matched asymptotic analysis of these singularities for ribbons whose width-to-length ratio w/ℓ≪1 is small. An inner layer solution is derived from the finite-w Wunderlich model and captures the fast, local variations of the bending and twisting strains in the neighbourhood of the cone-like region; it is universal up to a load intensity factor. The outer solution is given by the zero-w Sadowsky model. Based on this analysis, we propose a new standalone ribbon model that combines the Sadowsky equations with jump conditions providing a coarse-grained description of cone-like singularities, and give a self-contained variational derivation of this model. Applications to the Möbius band and to an end-loaded open ribbon are presented. Overall, the new model delivers highly accurate approximations to the solutions of the Wunderlich model in the limit w≪ℓ while avoiding the numerical difficulties associated with cone-like singularities.

Modeling and Analysis of Tendon-Driven Parallel Continuum Robots Under Constant Curvature and Pseudo-Rigid-Body Assumptions

Abstract This article unifies the approaches of kinematic and static modeling, and singularity analysis for tendon-driven parallel continuum robots under constant curvature as well as pseudo-rigid-body assumptions with those implemented in conventional rigid parallel robots. Constraint conditions are determined for the legs of this type of parallel continuum robots based on which the velocity equations and Jacobian matrices are derived. These are further exploited for inverse kinematic and singularity analysis. Static models for the robot as well as for each of the continuum links under pseudo-rigid-body assumption are derived. Finally, a simulation example is given to validate the kinematic models. It is shown that singularities can be determined using Grassmann line geometry or by detecting the numerical values of three performance indices.

Kinematic Analysis and Workspace Investigation of Novel 3-RPS/(H) Metamorphic Parallel Mechanism

Abstract 3-RPS/(H) Metamorphic parallel mechanism based on conventional mechanisms is proposed in this study. In contrast to the 3-RPS mechanism, the 3-RPS/(H) mechanism comprises a novel intermediate metamorphic limb. The different constraints provided by the intermediate metamorphic limb and locking device enhance the dexterity of the metamorphic parallel mechanism. The configurations of the 3-RPS/(H) metamorphic parallel mechanism are analyzed and their degrees-of-freedom are obtained. Additionally, an inverse kinematic analysis of the mechanism is performed, and its workspace is compared with that of a conventional 3-RPS parallel mechanism with similar parameters. The results demonstrate that the 3-RPS/(H) metamorphic parallel mechanism exhibits better dexterity than that of the 3-RPS mechanism. Moreover, the effects of different structural parameters of the metamorphic parallel mechanism on the workspace are analyzed. Finally, the singularity analysis of the mechanism is performed. The results provide a theoretical basis for the design and optimization of mechanical legs.

Application of Singularity Theory to the Distribution of Heavy Metals in Surface Sediments of the Zhongsha Islands

This research aimed to use nonlinear theory and technology to describe the spatial distribution of heavy metals in the surface sediments of the Zhongsha Islands Sea region. The goal of this study is to explore the spatial distribution characteristics of heavy metals in the surface sediments of the Zhongsha Islands. The singularity theory and method were used to delineate heavy metal geochemical anomalies and the generalized self-similarity analysis method was used to decompose heavy metal geochemical anomalies and background concentrations. The results showed that there were abnormally high concentrations of heavy metals in the deep-sea plain area and in the western central sand trough area. The results of this study can inform priority areas for environmental monitoring. The element anomalies extracted by the singularity analysis method in this paper can be a guide to the next step in the investigation and provide the basis for the regional environmental assessment.

Nonlinear structural stability and linear dynamic instability of transonic steady-states to hydrodynamic model for semiconductors

For unipolar hydrodynamic model of semiconductor device represented by Euler-Poisson equations, when the doping profile is supersonic, and the boundary data are in subsonic region and supersonic region separately, the system possesses the shock transonic steady-states and the smooth transonic steady-states. In this paper we study the nonlinear structural stability and the linear dynamic instability of these steady transonic solutions. For any relaxation time: 0<τ≤+∞, by means of elaborate singularity analysis, we first investigate the structural stability of the C1-smooth transonic steady-states, once the perturbations of the initial data and the doping profiles are small enough. We note that, when the C1-smooth transonic steady-states pass through the sonic line, they produce singularities for the system, and cause some essential difficulty in the proof of structural stability. Moreover, when the relaxation time is large enough τ≫1, under the condition that the electric field is positive at the shock location, we prove that the transonic shock steady-states are structurally stable with respect to small perturbations of the supersonic doping profile. Furthermore, we show the linearly dynamic instability for these transonic shock steady-states provided that the electric field is suitable negative. The proofs for the structural stability results are based on singularity analysis, a monotonicity argument on the shock position and the downstream density, and the stability analysis of supersonic and subsonic solutions. The linear dynamic instability of the steady transonic shock for Euler-Poisson equations can be transformed to the ill-posedness of a free boundary problem for the Klein-Gordon equation. By using a nontrivial transformation and the shooting method, we prove that the linearized problem has a transonic shock solution with exponential growths. These results enrich and develop the existing studies.

Nonlinear dynamic control of a photovoltaic pumping PMSM at MPP: Low-Frequency instability induced by Hopf bifurcation

ABSTRACT In a photovoltaic permanent magnet synchronous motor pumping system (PVPMSM), both photovoltaic modules and switching-mode converters present nonlinear and time-variant characteristics, which result in a difficult control problem. This paper presents an in-depth analysis of the operating point dynamics. A potential trade-off exists between maximum power and stability of the system operating point. Analysis of parametric singularities based on bifurcation theory permits to optimize the maximum power-stability trade-off. Tracing the bifurcation structures related to the parameter set (T, G), we can determine the maximum power point(MPP) in midway and far enough from the bifurcation points from these structures. An algorithm aiming to determine the MPP within the range of stability through power bifurcation analysis is proposed in this paper. Our methodology was validated with the simulation that was carried out on MATCONT/MATLAB, and the results are demonstrated.

Design and experiment of a new translational parallel manipulator with large payload and high repeatability

Parallel manipulators are known to have higher structural rigidity and end-effector speed than serial manipulators. However, the size, payload, rigidity, and repeatability of parallel manipulators highly depend on the types of active and passive joints used. Compared with other joint types, revolute joints are structurally simpler and have an unlimited rotation range. They can be easily preloaded to increase the stiffness and eliminate clearance. Hence, revolute joints are preferable for parallel manipulators that need to have a smaller size, larger payload, and higher reparability. This paper presents a new translational parallel manipulator (TPM) that uses only revolute joints as the active and passive joints. The kinematics, workspace analysis, and singularity analysis are provided to facilitate the design of the new TPM. Finite element analyses are given to evaluate the compliance contribution of the revolute joints and links of the new TPM. Compliance, payload, and repeatability experiments are provided to verify the merits of the proposed design. We expect that the proposed TPM can provide an alternative for selecting parallel manipulators to meet different industrial applications.

Linear-sized independent sets in random cographs and increasing subsequences in separable permutations

This paper is interested in independent sets (or equivalently, cliques) in uniform random cographs. We also study their permutation analogs, namely, increasing subsequences in uniform random separable permutations. First, we prove that, with high probability as $n$ gets large, the largest independent set in a uniform random cograph with $n$ vertices has size $o(n)$. This answers a question of Kang, McDiarmid, Reed and Scott. Using the connection between graphs and permutations via inversion graphs, we also give a similar result for the longest increasing subsequence in separable permutations. These results are proved using the self-similarity of the Brownian limits of random cographs and random separable permutations, and actually apply more generally to all families of graphs and permutations with the same limit. Second, and unexpectedly given the above results, we show that for $\beta >0$ sufficiently small, the expected number of independent sets of size $\beta n$ in a uniform random cograph with $n$ vertices grows exponentially fast with $n$. We also prove a permutation analog of this result. This time the proofs rely on singularity analysis of the associated bivariate generating functions.

Machine Learning and Singularity Analysis Reveal Zircon Fertility and Magmatic Intensity: Implications for Porphyry Copper Potential

Machine Learning and Singularity Analysis Reveal Zircon Fertility and Magmatic Intensity: Implications for Porphyry Copper Potential

An enhanced artificial hummingbird algorithm and its application in truss topology engineering optimization

In view of the shortcomings such as slow search speed, low optimization precision and premature convergence of artificial hummingbird algorithm, an enhanced artificial hummingbird algorithm based on golden sine factor named DGSAHA is proposed. Firstly, chaos mapping is used to generate the initial candidate solution to increase the diversity of the population, which lays the foundation for the global search. Then, perturb the individuals by means of the differential variation between individuals on the group, thereby enhancing the diversity of the population, preserving the excellent individuals, eliminating the inferior individuals, and guiding the search process to approach the global optimal solution, avoiding the phenomenon of premature convergence. Finally, the golden sine factor were introduced in the guided foraging stage is conducive to the full exploration of the global optimal solution, reducing the search space for individuals to approach the optimal solution. And, it facilitates the balance between “exploration” and “exploitation” of algorithm. Thereby, the accuracy and speed of the DGSAHA can be improved to a certain extent. 25 classic functions, the CEC2014 and CEC2019 benchmark functions were tested, and several representative meta-heuristic algorithms and its improved algorithm are compared for evaluate the validity of DGSAHA. Meanwhile, the dimensional scalability of the variable-dimensional test function is discussed. The results of non-parametric statistical analysis and performance index show that DGSAHA in this paper has better comprehensive optimization performance, significantly improves the search speed and convergence precision, and has strong ability to get rid of the local optimal solution. Finally, the performance of DGSAHA and the practicability of truss structure are answered by three engineering examples of plane and space truss topology optimization problem. This optimization problem considers not only the static constraints such as stress, displacement and buckling, but also the dynamic constraints of frequency and motion stability. In order to avoid singularity and unnecessary analysis, the stiffness, mass and load matrices are reconstructed in finite element analysis. A lighter truss structure than the existing solution is obtained. The validity, extensibility and practicability of the algorithm are further illustrated.

Enumeration of chordal planar graphs and maps

We determine the number of labelled chordal planar graphs with n vertices, which is asymptotically g⋅n−5/2γnn! for a constant g>0 and γ≈11.89235. We also determine the number of rooted simple chordal planar maps with n edges, which is asymptotically s⋅n−3/2δn, where s>0, δ=1/σ≈6.40375, and σ is an algebraic number of degree 12. The proofs are based on combinatorial decompositions and singularity analysis. Chordal planar graphs (or maps) are a natural example of a subcritical class of graphs in which the class of 3-connected graphs is relatively rich. The 3-connected members are precisely chordal triangulations, those obtained starting from K4 by repeatedly adding vertices adjacent to an existing triangular face.

Photo-acoustic inversion using plasmonic contrast agents: The full Maxwell model

We analyze the inversion of the photo-acoustic imaging modality using electromagnetic plasmonic nano-particles as contrast agents. We prove that the generated pressure, before and after injecting the plasmonic nano-particles, measured at a single point, located away from the targeted inhomogeneity to image, and at a given band of incident frequencies is enough to reconstruct the (eventually complex valued) permittivity. Indeed, from these measurements, we define an indicator function which depends on the used incident frequency and the time of measurement. This indicator function has differentiating behaviors in terms of both time and frequency. First, from the behavior in terms of time, we can estimate the arrival time of the pressure from which we can localize the injected nano-particle. Second, we show that this indicator function has maximum picks at incident frequencies close to the plasmonic resonances. This allows us to estimate these resonances from which we construct the permittivity.To justify these properties, we derive the dominant electric field created by the injected nano-particle when the incident frequency is close to plasmonic resonances. This is done for the full Maxwell system. To this end, we use a natural spectral decomposition of the vector space (L2(D))3 based on the spectra of the Newtonian and the Magnetization operators. As another key argument, we provide the singularity analysis of the Green's tensor of the Maxwell problem with varying permittivity. Such singularity is unusual if compared to the ones of the elliptic case (as the acoustic or elastic models). In addition, we show how the derived approximation of the electric fields propagates, as a source, in the induced pressure with the time.

Singularity analysis of linear hardening V-notch by asymptotic expansion coupled BEM

In this paper, a general numerical method by the asymptotic expansion coupled the boundary element method (CBEM), considering the singular term and the higher order terms, is developed to analyze the complete singular stress field of linear hardening elastic–plastic material with a V-notch. The method firstly transforms the singularity analysis for the governing equation of the V-notch tip into an ordinary differential eigenvalue equation, which is then solved by the interpolating matrix method. Then, the complete singular stress field and the notch stress intensity factors (SIFs) can be obtained by coupling the asymptotic expansion of the elastic–plastic singular stresses at the V-notch tip with the conventional boundary elements method for the remaining region of the structure. Two types of examples are analyzed for the V-notch problem of linear hardening material. It is shown that the elastic–plastic singular stress field can be evaluated by the equivalent Young’s modulus and Poisson’s ratio. The proposed new approach is found to be accurate and effective in evaluating singular stress fields in a V-notch structure with linear hardening material.

On finite time singularities in scalar field dark energy models based in the RS-II Braneworld

The quest of deciphering the true nature of dark energy has proven to be one of the most exciting in recent times in cosmology. Various ideas have been put forward in this regard besides the usual cosmological constant approach, ranging from scalar field based models like Quintessence and Phantom dark energy to various modified gravity approaches as well. A very interesting idea then is to consider scalar field dark energy models in quantum gravitationally corrected cosmologies with the RS-II Braneworld being one of the most well known in this regard. So in this work, we consider RS-II Braneworld based scalar field dark energy models and try to look out for the existence of finite time singularities in these regimes both through a dynamical system perspective, for which we employ the Goriely–Hyde singularity analysis method, and a physical perspective. Our approach is general in the sense that it is not limited to any particular class of potentials or for any constrained parameter region for the brane tension and is valid for both Quintessence and phantom dark energy regimes. We firstly show through Goriely–Hyde procedure that finite time singularities can exist in these models for a limited set of initial conditions and that this result would hold irrespective of any consideration given to the swampland dS conjecture. We then discuss the physical nature of the singularities that can occur in this regime, where we use a well motivated ansatz for the Hubble parameter and show that these models of dark energy can allow for weak singularities like those of Type III and Type IV and can also allow for strong singularities like the Big Rip (Type I).