Internal Singularities Research Articles

From singularity analysis to singularity avoidance: Novel metric and convex allocation for spacecraft attitude control with control moment gyros

Singularities in robotics lead to a reduction in the number of degrees of freedom. When it comes to spacecraft employing control moment gyros (CMGs), singularities may occur due to the internal alignment of the gimbals, which inhibit the creation of torque in at least one direction. This translates into a loss of control authority that has a direct impact on the spacecraft’s pointing performance. In this work, a convex optimization-based allocation framework for singularity avoidance is put forward. The presented solution aims to provide a singularity-robust allocation scheme that can be used as an add-on to a conventional attitude controller. To this end, the proposed algorithm employs a novel, computationally efficient, and numerically robust singularity metric to assess the proximity to the singularities of a cluster of CMGs. With this information, a model predictive controller determines control actions that guide the system towards singularity-free configurations. A particularity about this allocation algorithm is that its formulation boils down to a simple set of convex equalities and inequalities, given that the new singularity metric can be written in a linear form, unlike most of the literature solutions, such as the condition number, which are highly complex and nonlinear. Lastly, the proposed approach is applied to a two-dimensional CMG cluster in a realistic simulation environment. The results confirm that the system effectively avoids all the internal singularities of the cluster at a relatively low computational expense.

Inverse-Kinematics Steering Laws for Double-Gimbal Scissored-Pair Control Moment Gyros

The generalized singularity robust inverse steering law (GSRISL) is widely used to avoid singularity problems in control moment gyroscopes (CMGs). However, GSRISL tends to generate perturbation torques near singularities, which hinder highly accurate attitude tracking control. Model predictive steering control can eliminate the necessity of considering singularities, but its implementation in satellites remains challenging because of its high computational cost. Therefore, the development of a steering law that avoids explicit consideration of internal singularities and costly calculations is critical. The inverse-kinematics steering law (IKSL) is one such law removing the need for the calculation of the inverse matrix. The double-gimbal scissored-pair CMG (DGSPCMG) addresses the singularity problem as it does not have internal singularities except along the outer gimbal axis. To enhance the attitude control by removing the need for the calculation of the inverse matrix, this study proposes the combining of IKSL with DGSPCMG. Additionally, to mitigate the risk of gimbal mechanical failure, the IKSL for inner or outer gimbal axes failure was designed by relaxing the identical steering constraint of the scissored-pair gimbal angles. Numerical simulations were performed to evaluate the effectiveness of the combination of IKSL and DGSPCMG in terms of settling time and maneuvering axis accuracy.

Thermocapillary dynamics of a surfactant-laden droplet with internal thermal singularity

Thermocapillary droplets with internal thermal singularities have potential applications in drug delivery and cell analysis. Inspired by the work of Pak et al. (J. Fluid Mech., vol. 753, 2014, pp. 535–552), which was investigated for a surfactant-laden non-deformable droplet in an isothermal Poiseuille flow, we have explored the droplet dynamics by taking account of additional internal thermal singularities, namely monopole and dipole. A generalized mathematical model is developed, which is solved by using the solenoidal decomposition to describe the flow field in any arbitrary Stokes flow, and results are shown extensively for the case of a non-isothermal Poiseuille flow. Under small Péclet number ( $Pe_s$ ) limit, the droplet with an off-centred monopole or a dipole oriented along the flow direction shows cross-stream migration at $O(Pe_s^2)$ . However, a dipole oriented perpendicular to the flow direction results in an $O(1)$ effect due to thermocapillarity, and from $O(Pe_s)$ onwards, we observe the combined impact of thermocapillary and surfactant-induced Marangoni stresses. As a surprise, we see cross-stream migration of the droplet from the Poiseuille flow centreline in a non-isothermal field, in contrast to existing findings which rule out any cross-stream migration. We show the trade-off between thermal Marangoni number ( $Ma_T$ ) and surfactant Marangoni number ( $Ma_\varGamma$ ). Our findings on droplet dynamics inspire new possibilities for microfluidics-based design.

Formula omitted]-smooth planar parameterization of complex domains for isogeometric analysis

The construction of high-quality parameterizations for complex domains remains a significant challenge in isogeometric analysis. To address this issue, we propose a G1-smooth parameterization method for planar domains with arbitrary topology. Firstly, we generate a coarse decomposition of the given complex shape without internal singularities utilizing the extracted skeleton of the domain. We then perform a finer domain partition to obtain more accurate boundary representations. Both rectangular and triangular Bézier patches are employed to represent the geometry. Secondly, by imposing G1 continuity constraints on the adjacent patches, we construct an initial parameterization that is globally G1 continuous. Finally, we enhance the parameterization quality of each patch separately using the quasi-conformal mapping technique. Various examples are presented to justify the capability of the proposed method and to demonstrate its superiority over existing multi-patch parameterizations.

Singularity‐Free Frame Fields for Line Drawing Vectorization

AbstractState‐of‐the‐art methods for line drawing vectorization rely on generated frame fields for robust direction disambiguation, with each of the two axes aligning to different intersecting curve tangents around junctions. However, a common source of topological error for such methods are frame field singularities. To remedy this, we introduce the first frame field optimization framework guaranteed to produce singularity‐free fields aligned to a line drawing. We first perform a convex solve for a roughly‐aligned orthogonal frame field (cross field), and then comb away its internal singularities with an optimal transport–based matching. The resulting topology of the field is strictly maintained with the machinery of discrete trivial connections in a final, non‐convex optimization that allows non‐orthogonality of the field, improving smoothness and tangent alignment. Our frame fields can serve as a drop‐in replacement for frame field optimizations used in previous work, improving the quality of the final vectorizations.

Singularity Parametrization With a Novel Kinematic Decoupled Model for Non-Spherical Wrist Robots

Abstract Most of the current commercial collaborative robots present a non-spherical wrist, so they cannot utilize singularity handling techniques efficiently to avoid excessive safety stops while dynamically avoiding collisions. These robots usually require heavier algorithms due to their kinematics or online methods that shift the original singularities. Therefore, to enable more efficient computations on singularity handling and collision avoidance controllers, this paper proposes a novel method to characterize singular configurations of non-spherical wrist collaborative robots (6 and 7 degrees-of-freedom). This method is based on a new decoupled kinematic model that allows lighter kinematic computations and enables the joint-dependant characterization of the robot singularities to avoid shifting the singular configurations. Finally, the proposed kinematic model is particularized for a UR10e, where its kinematic behavior has been tested against two different literature models in simulation. In this manner, a novel singularity category (belonging to the internal singularities) is proposed, and a new closed set of characterized singular solutions is obtained.

On the mother bodies of steady polygonal uniform vortices. Part I: numerical experiments

ABSTRACT The existence of an integral relation between self-induced velocity of a uniform, planar vortex and Schwarz function of its boundary opens the way to understand the kinematics of the vortex by analysing the internal singularities of that function. In general, they are branch cuts and form the so-called “mother body” of the vortex, because they generate the same external velocities of the vortex, by means of a relation identical to the Biot–Savart law for a vortex sheet. The jump of the Schwarz function across the cuts plays the role of the (complex) density of circulation. This paper investigates the singularities of polygonal vortices, which are highly nontrivial steady vortices widely present in Nature, and having fascinating properties, some of them still not well understood. By means of the equation of the dynamics of the Schwarz function specialised for steady vortices, a numerical tool based on elementary properties of the holomorphic functions is used for detecting the internal singularities and evaluating their strengths. In this way, it is shown that an nagonal vortex possesses n internal branch cuts. In a reference system having origin on the centre of vorticity of the vortex and real axis crossing one of its vertices, these cuts start from the origin and are directed along the n roots of the unity, so that they are aligned with the vertices. The positions of the branch points and the values assumed by the Schwarz function in these points are calculated by evaluating this function just outside the vortex boundary. Once the conditions on the branch points are defined, a power series representation of the Schwarz function is proposed, that is able to explain the behaviour of its real and imaginary parts in neighbourhoods of these points. Some conjectures about the external singularities are also discussed.

Evocube: A Genetic Labelling Framework for Polycube‐Maps

AbstractPolycube‐maps are used as base‐complexes in various fields of computational geometry, including the generation of regular all‐hexahedral meshes free of internal singularities. However, the strict alignment constraints behind polycube‐based methods make their computation challenging for CAD models used in numerical simulation via finite element method (FEM). We propose a novel approach based on an evolutionary algorithm to robustly compute polycube‐maps in this context.We address the labelling problem, which aims to precompute polycube alignment by assigning one of the base axes to each boundary face on the input. Previous research has described ways to initialize and improve a labelling via greedy local fixes. However, such algorithms lack robustness and often converge to inaccurate solutions for complex geometries. Our proposed framework alleviates this issue by embedding labelling operations in an evolutionary heuristic, defining fitness, crossover, and mutations in the context of labelling optimization. We evaluate our method on a thousand smooth and CAD meshes, showing Evocube converges to accurate labellings on a wide range of shapes. The limitations of our method are also discussed thoroughly.

IGA-suitable planar parameterization with patch structure simplification of closed-form polysquare

A primary challenge for isogeometric analysis (IGA)-suitable parameterization is to efficiently decompose a complex computational domain into a small number of high-quality IGA-suitable patches without internal singularities. Aiming at domains with arbitrary topology, we propose integrating frame field and polysquare structure, resulting in a structure associated with a frame field that is locally integrable everywhere, i.e. a closed differential form or closed-form. Such a structure is more general than a common polysquare but has no internal singularity as well, which makes it suitable for IGA because it usually has fewer undesired boundary corners and low distortion. Since IGA prefers a small number of patches in the parameterization structure, we further propose a patch simplification method with distortion control. A key challenge in simplification is to prevent the resulting patches from degenerating. Previous methods rely on a mesh of the domain to characterize the non-degenerate condition. In sharp contrast, our approach simply uses the boundary of the domain, which obviously improves efficiency because the number of variables is much lower. The patch degeneration issue in such a boundary-only strategy is addressed by solving a constrained optimization problem with a set of automatically constructed inequality constraints. We prove that such constraints are sufficient to make the common polysquare structure homeomorphic to the input shape. We also demonstrate that it prevents the patches from degenerating for the structure of closed-form polysquares, and it can be extendedly applied to 3D closed-form polycubes. Numerical experiments using bi-quadratic B-spline surfaces with C1-constraints are also provided to justify the accuracy of applying the resulting planar parameterization for IGA application.

Robust gimbal reorientation singularity-avoidance steering strategy for regular pentagonal pyramid configuration with two control moment gyroscopes failed

The failure configuration which consists of four single-gimbal control moment gyroscopes (SGCMGs) of the widely-used regular pentagonal pyramid SGCMG system, i.e. failure 4-SGCMG configuration, is investigated in this paper. On account of the symmetry of regular pentagonal pyramid configuration, a conversion method which could transform any failure 4-SGCMG configuration to the chosen failure system is introduced. Then, the zero-momentum gimbal angle of the chosen failure system is derived and its distribution in gimbal angle space is analyzed. In order to avoid internal singularity, preferred initial gimbal angles are calculated basing on derivation of zero-momentum gimbal angles and the corresponding terminal singular surfaces are obtained through numerical simulations. Furthermore, a constraint about maneuver direction is introduced and a robust gimbal reorientation steering strategy is proposed. Finally, simulations on a non-chosen failure configuration with inertia matrix uncertainty demonstrate the effectiveness of the conversion method and robustness of the proposed steering law.

A Gimballed Control Moment Gyroscope Cluster Design for Spacecraft Attitude Control

This paper addresses the problem of singularity avoidance in a cluster of four Single-Gimbal Control Moment Gyroscopes (SGCMGs) in a pyramid configuration when used for the attitude control of a satellite by introducing a new gimballed control moment gyroscope (GCMG) cluster scheme. Four SGCMGs were used in a pyramid configuration, along with an additional small and simple stepper motor that was used to gimbal the full cluster around its vertical (z) axis. Contrary to the use of four variable-speed control moment gyroscopes (VSCMGs), where eight degrees of freedom are available for singularity avoidance, the proposed GCMG design uses only five degrees of freedom (DoFs), and a modified steering law was designed for the new setup. The proposed design offers the advantages of SGCMGs, such as a low weight, size, and reduced complexity, with the additional benefit of overcoming the internal elliptic singularities, which create a minor attitude error. A comparison with the four-VSCMG cluster was conducted through numerical simulations, and the results indicated that the GCMG design was considerably more efficient in terms of power while achieving a better gimbal configuration at the end of the simulation, which is essential when it is desired for different manoeuvres to be consecutively executed. Additionally, for a nano-satellite of a few kilograms, the results prove that it is feasible to manufacture the GCMG concept by using affordable and lightweight commercial off-the-shelf (COTS) stepper motors.

Two Mathematical Approaches to Inferring the Internal Structure of Proteins from their Shape

sing a simple mathematical model, we propose two approaches to externally infer how the amino-acid sequence is folded in a protein. One is the previously proposed differential geometric approach. The other is a new category theoretical approach proposed in this paper. As an example, we consider detecting the presence of internal singularities from the outside. Knowledge of Category theory is not required. Proteins are represented as a loop of triangles. In both approaches, the outer contour of the loop is examined to detect the presence of singular triangles (such as isolated triangles) inside. By considering the interaction between loops, the new approach allows us to detect more singular triangles than the previous approach. We hope that this research will provide a new perspective on protein structure analysis and promote further collaboration between mathematics and biology.

Self-excited vibration of a 3-PRR planar parallel robot

Self-excited vibration of parallel robots can seriously affect the motion performance and damage the mechanical structure. In order to study the self-excited vibration characteristics of a 3-PRR (where P and R represent the prismatic and revolute joints respectively and the underlined letter represents the actuated joint) planar parallel robot, dynamic model is firstly established and the singularity is analyzed theoretically. Then the dynamic characteristics at internal and boundary singularities are both explored experimentally. The motion form and generating mechanism of the self-excited vibration are researched. The influencing factors on vibration frequency are obtained and the self-excited vibration during trajectory tracking motion is analyzed. Finally, a singularity escaping strategy is proposed and tested. Theoretical analysis and experimental results show that the performance of parallel robot deteriorates dramatically at singularities. Nevertheless, the parallel robot can pass through the internal singularity successfully with optimized load and motion speed so that the workspace can be expanded. The 3-PRR planar parallel robot exhibits self-excited vibration at internal singularities, which is mainly caused by the singularity, self-regulation of motors and the closed-chain coupling effect. The vibration frequency is mainly determined by singular configuration and the structural parameters. The parallel robot can maintain self-excited vibration state while carrying out trajectory tracking under a singular attitude angle. Moreover, the proposed singularity escaping strategy is verified to be feasible so that the self-excited vibration can be eliminated effectively.

Design Optimization of 3-DOF Redundant Planar Parallel Kinematic Mechanism Based Finishing Cut Stage for Improving Surface Roughness of FDM 3D Printed Sculptures

This paper describes the optimal design of a 3-DOF redundant planar parallel kinematic mechanism (PKM) based finishing cut stage to improve the surface roughness of FDM 3D printed sculptures. First, to obtain task-optimized and singularity minimum workspace of the redundant PKM, a weighted grid map based design optimization was applied for a task-optimized workspace without considering the redundancy. For the singularity minimum workspace, the isotropy and manipulability of the end effector of the PKM were carefully modeled under the previously obtained redundancy for optimality. It was confirmed that the workspace size increased by 81.4%, and the internal singularity significantly decreased. To estimate the maximum rated torque and torsional stiffness of all active joints and prevent an undesired end effector displacement of more than 200 μμm, a kinematic stiffness model composed of active and passive kinematic stiffness was derived from the virtual work theorem, and the displacement characteristic at the end effector was examined by applying the reaction force for the PLA surface finishing as an external force acting at the end effector. It was confirmed that the displacement of the end effector of a 1-DOF redundant PKM was not only less than 200 μμm but also decreased from 40.9% to 67.4% compared to a nonredundant actuation.

Manifold path selection steering law for pyramid-type single-gimbal control moment gyros

The properties of singularities for pyramid-type single-gimbal control moment gyro (SGCMG) configuration are investigated in this paper. Manifold theory is applied to the SGCMG system and singular surfaces are classified by passability and directionality. A terminal domain is introduced as a small angular momentum space wrapped by internal elliptic singular surfaces and intersected hyperbolic singular surfaces. The hyperbolic singular surfaces in a terminal domain can provide the SGCMG system with a chance to select a manifold path containing no internal elliptic singularity. Thereafter, a manifold path selection (MPS) steering law based on a modified gradient method is proposed, which can achieve internal elliptic singularity avoidance without introducing torque errors. Compared with the other two steering laws, the effectiveness of the proposed method is demonstrated using attitude maneuver examples.

Chatter detection for milling using novel p-leader multifractal features

Chatter in machining results in poor workpiece surface quality and short tool life. An accurate and reliable chatter detection method is needed before its complete development. This paper applies a novel p-leader multifractal formalism for chatter detection in milling processes. This novel formalism can discover internal singularities rising on unstable signals due to chatter without prior knowledge of the natural frequencies of the machining system. The p-leader multifractal features are selected by using a multivariate filter method for feature selection, and verified by both numerical simulations and experimental studies with detailed parameter selection discussions when applying this formalism. The proposed method is assessed in terms of their dynamic monitoring abilities and classification accuracies under wide cutting conditions. The results show that the multifractal features can successfully detect chatter with high accuracies and short computation time. For further verification, the proposed method is compared with two commonly-used methods, which indicates that the proposed method gives better classification accuracies, especially when identifying unstable tests.

Steering control law for double-gimbal scissored-pair CMG

Because control moment gyroscopes (CMGs) can generate a large torque compared to reaction wheels, they are used as actuators for attitude control of large spacecraft. However, when the number of Single-Gimbal CMG (SGCMG) units is five or less, there can be internal singularities that cannot generate torque around the desired direction. To construct a system that has no internal singularities, six or more SGCMGs are required, because an orthogonally arrayed, three scissored-pair CMG system has no internal singularities. Because CMG singularities are disruptive for attitude control, a great deal of effort has been devoted to overcoming the CMG singularity problem; the various designs include Variable-Speed CMGs (VSCMGs), Double-Gimbal CMGs (DGCMGs), and Double-Gimbal Variable-Speed CMGs (DGVSCMGs). However, these designs still have problems, such as a slow response to torque generation commands about the wheel axis in VSCMGs and DGVSCMGs, and difficulty in precise attitude tracking when perturbation torque is generated to avoid singularities. To overcome the problems of the traditional CMG configurations, this paper proposes a new CMG system configuration that we call the Double-Gimbal Scissored-pair CMG (DGSPCMG) system. Because the DGSPCMG system is a hybrid system combining a Scissored-Pair CMG and a DGCMG, the DGSPCMG system does not have internal singularities except at the origin and along the x-axis. Moreover, this system can recover from an internal singularity by null motion only, and from outer singularities (saturation singularities) by steering the scissored-pair gimbals only. Thus, generation of perturbation torque is unnecessary for recovering from singularities, and a precise attitude tracking maneuver can be more easily achieved. This paper presents a conceptual design of a DGSPCMG system and describes a steering control law for the proposed system. Furthermore, the validity of the proposed steering control law is demonstrated through numerical simulations and results of comparison experiments are shown to demonstrate the advantage of the DGSPCMG over a VSCMG.

A singularity handling algorithm based on operational space control for six-degree-of-freedom anthropomorphic manipulators

In this article, we analyze the singularities of six-degree-of-freedom anthropomorphic manipulators and design a singularity handling algorithm that can smoothly go through singular regions. We show that the boundary singularity and the internal singularity points of six-degree-of-freedom anthropomorphic manipulators can be identified through a singularity analysis, although they do not possess the nice kinematic decoupling property as six-degree-of-freedom industrial manipulators. Based on this discovery, our algorithm adopts a switching strategy to handle these two cases. For boundary singularities, the algorithm modifies the control input to fold the manipulator back from the singular straight posture. For internal singularities, the algorithm controls the manipulator with null space motion. We show that this strategy allows a manipulator to move within singular regions and back to non-singular regions, so the usable workspace is increased compared with conventional approaches. The proposed algorithm is validated in simulations and real-time control experiments.

How information crosses Schwarzschild’s central singularity

We study an extension of spacetime across Schwarzschild’s central singularity and the behavior of the geodesics crossing it. Locality implies that this extension is independent from the future fate of black holes. We argue that this extension could be the limit of the effective quantum geometry inside a black hole, and show that the central region contains causal diamonds with area satisfying Bousso’s bound for an entropy that can be as large as Hawking’s radiation entropy. This result sheds light on the possibility that Hawking radiation is purified by information crossing the internal singularity and supports the black hole to white hole transition scenario.

Plotting the map projection graticule involving discontinuities based on combined sampling

This article presents new algorithm for interval plotting the projection graticule on the interval $\varOmega=\varOmega_{\varphi}\times\varOmega_{\lambda}$ based on the combined sampling technique. The proposed method synthesizes uniform and adaptive sampling approaches and treats discontinuities of the coordinate functions $F,G$. A full set of the projection constant values represented by the projection pole $K=[\varphi_{k},\lambda_{k}]$, two standard parallels $\varphi_{1},\varphi_{2}$ and the central meridian shift $\lambda_{0}^{\prime}$ are supported. In accordance with the discontinuity direction it utilizes a subdivision of the given latitude/longitude intervals $\varOmega_{\varphi}=[\underline{\varphi},\overline{\varphi}]$, $\varOmega_{\lambda}=[\underline{\lambda},\overline{\lambda}]$ to the set of disjoint subintervals $\varOmega_{k,\varphi}^{g},$$\varOmega_{k,\lambda}^{g}$ forming tiles without internal singularities, containing only "good" data; their parameters can be easily adjusted. Each graticule tile borders generated over $\varOmega_{k}^{g}=\varOmega_{k,\varphi}^{g}\times\varOmega_{k,\lambda}^{g}$ run along singularities. For combined sampling with the given threshold $\overline{\alpha}$ between adjacent segments of the polygonal approximation the recursive approach has been used; meridian/parallel offsets are $\Delta\varphi,\Delta\lambda$. Finally, several tests of the proposed algorithms are involved.