Internal Singularities Research Articles

Steering law of control moment gyros using artificial potential function approach

A Control Moment Gyro (CMG) cluster for an agile spacecraft usually suffers from a singularity problem where it does not meet a desired torque. To solve the problem, a proposed steering law consists of SR (singularity robustness) inverse and SR null motion to compensate each other. The SR null motion solves the singularity problem that the SR inversestays in the singularities. Especially, an APF (artificial potential function) is included in the SR null motion to avoid internal singularities and to reach external singularity simultaneously. The APF newly considers the distance between the current gimbal angle and the gimbal angle corresponding to all singularities. All the combinations of singular gimbal angles are regarded as a target and obstacles of the potential function. After a spacecraft maneuver, an initial gimbal angle is regarded as the target to secure the repeatability of next maneuver. When avoiding or escaping the obstacles corresponding to the unwanted singular gimbal angles, the proposed steering law rapidly changes the gimbal angle by taking torque error. To overcome the local minima problem which is generally a serious drawback in using the potential function, an algorithm for changing parameters of the potential function are proposed in the steering law. To avoid the internal singularity, moreover, the present study introduces a configuration change of the CMG cluster. The proposed steering law is mathematicallyanalyzed and verified through numerical simulations. It shows that the CMG cluster can avoid an internal singularity rapidly, resulting in faster maneuver time of the spacecraft.

Gimbal-Angle Vectors of the Nonredundant CMG Cluster

This paper deals with the method using the preferred gimbal angles of a control moment gyro (CMG) cluster for controlling spacecraft attitude. To apply the method to the nonredundant CMG cluster, analytical gimbal-angle solutions for the zero angular momentum state are derived, and the gimbal-angle vectors for the nonzero angular momentum states are studied by a numerical method. It will be shown that the number of the gimbal-angle vectors is determined from the given skew angle and the angular momentum state of the CMG cluster. Through numerical examples, it is shown that the method using the preferred gimbal-angle is an efficient approach to avoid internal singularities for the nonredundant CMG cluster.

Algorithm 988

The algorithm AMGKQ for adaptive multivariate Gauss-Kronrod quadrature over hyperrectangular regions of arbitrary dimensionality is proposed and implemented in Octave/MATLAB. It can approximate numerically any number of integrals over a common domain simultaneously. Improper integrals are addressed through singularity weakening coordinate transformations. Internal singularities are addressed through the use of breakpoints. Its accuracy performance is investigated thoroughly, and its running time is compared to other commonly available routines in two and three dimensions. Its running time can be several orders of magnitude faster than recursively called quadrature routines. Its performance is limited only by the memory structure of its operating environment. Included with the software are numerous examples of its invocation.

Performance evaluation of robust attitude control system using control moment gyros for rapid multi-target observation

Since observation tasks of micro-satellite require rapid multi-target acquisition and high pointing accuracy, the single-gimbal control moment gyro (CMG) is considered as an ideal torque-generating actuator due to its practical performance for the satellite. This paper compares two useful robust attitude control algorithms (sliding mode control method and high order sliding mode control method). These control methods have improved the system's robustness and accuracy. The pseudo-inverse steering logic based on mixed two-norm and least-squares minimisation is chosen in this paper which can deal with both the internal singularity problem and momentum saturation singularities. This attitude control system is verified through numerical simulation. The results demonstrate the validity and feasibility of the proposed approach.

All-hex meshing using closed-form induced polycube

The polycube-based hexahedralization methods are robust to generate all-hex meshes without internal singularities. They avoid the difficulty to control the global singularity structure for a valid hexahedralization in frame-field based methods. To thoroughly utilize this advantage, we propose to use a frame field without internal singularities to guide the polycube construction. Theoretically, our method extends the vector fields associated with the polycube from exact forms to closed forms, which are curl free everywhere but may be not globally integrable. The closed forms give additional degrees of freedom to deal with the topological structure of high-genus models, and also provide better initial axis alignment for subsequent polycube generation. We demonstrate the advantages of our method on various models, ranging from genus-zero models to high-genus ones, and from single-boundary models to multiple-boundary ones.

Benchmark analytical solutions from beams with shared eigenpair

Structures with governing equations having identical inertial terms but somewhat differing stiffness terms can be termed flexurally analogous. An example of such a structure includes an axially loaded non-uniform beam and an unloaded uniform beam, for which an exact solution exists. We find that there exist shared eigenpairs (frequency and mode shapes) for a particular mode between such structures. Non-uniform beams with uniform axial loads, gravity loaded beams and rotating beams are considered and shared eigenpairs with uniform beams are found. In general, the derived flexural stiffness functions (FSF׳s) for the non-uniform beams required for the existence of shared eigenpair have internal singularities, but some of the singularities can be removed by an appropriate selection of integration constants using the theory of limits. The derived functions yield an insight into the relationship between the axial load and flexural stiffness of axially loaded beam structures. The derived functions can serve as benchmark solutions for numerical methods.

Dynamic singularity avoidance for parallel manipulators using a task-priority based control scheme

This paper proposes a control scheme that allows parallel manipulators to overcome a major inherent challenge, namely, avoiding singularities occurring inside their workspaces. The scheme lets a parallel manipulator follow a path that originally contains such singularities. By means of a suitably chosen artificial potential function arising out of the proximity to the singularities, the behaviour of the manipulator is modified dynamically, enabling it to push away the singularities in its path, without deviating from the path itself. Such behaviour is obtained, however, at the cost of one of the DoF of the manipulator. Nevertheless, various application scenarios involving axi-symmetric tools, e.g., industrial operations such as welding and gas cutting, can afford such a trade-off, and thereby benefit by eliminating the requirement of checking each path for singularities a priori. An additional benefit is the increase in the effective size of the workspaces due to the avoidance of internal singularities. The theory developed is applicable to any parallel manipulator, in principle. In this paper, it has been applied to a three- DoF planar 3-RRR manipulator. The experimental results clearly demonstrate the efficacy of the proposed scheme.

Spherically symmetric vacuum solutions arising from trace dynamics modifications to gravitation

We derive the equations governing static, spherically symmetric vacuum solutions to the Einstein equations, as modified by the frame-dependent effective action (derived from trace dynamics) that gives an alternative explanation of the origin of "dark energy". We give analytic and numerical results for the solutions of these equations, first in polar coordinates, and then in isotropic coordinates. General features of the static case are that: (i) there is no horizon, since g00 is nonvanishing for finite values of the polar radius, and only vanishes (in isotropic coordinates) at the internal singularity, (ii) the Ricci scalar R vanishes identically, and (iii) there is a physical singularity at cosmological distances. The large distance singularity may be an artifact of the static restriction, since we find that the behavior at large distances is altered in a time-dependent solution using the McVittie Ansatz.

Approximate controllability for a 2D Grushin equation with potential having an internal singularity

This paper is dedicated to approximate controllability for Grushin equation on the rectangle (x, y) ∈ (−1, 1) × (0, 1) with an inverse square potential. This model corresponds to the heat equation for the Laplace-Beltrami operator associated to the Grushin metric on R^2 , studied by Boscain and Laurent. The operator is both degenerate and singular on the line {x = 0}. The approximate controllability is studied through unique continuation of the adjoint system. For the range of singularity under study, approximate controllability is proved to hold whatever the degeneracy is. Due to the internal inverse square singularity, a key point in this work is the study of well-posedness. An extension of the singular operator is designed imposing suitable transmission conditions through the singular-ity. Then, unique continuation relies on the Fourier decomposition of the 2d solution in one variable and Carleman estimates for the 1d heat equa-tion solved by the Fourier components. The Carleman estimate uses a suitable Hardy inequality.

Imagining Counter-Dispositif: The Case of Money Through Servitude, Slavery, Abolition and Free Labour

The paper explains how money as a master dispositif had guided history through institutions of servitude, slavery, its abolition and free labour through networks of human relations. The first section of the paper deals with the idea of money as dispositif. Second section explains how the motive of accumulation of wealth had driven the early modern Europeans into slave traders. Third section explains the abolition of slavery is in continuity with the motive of accumulation. It is explained, the free market for labour is a further extension of the logic of wealth accumulation through control and domination. The fourth section opens the discussion towards imagining counter-dispositif.Dispositif is naturalised power, the second nature that works from within as our mentality organising our thought patterns, attitudes, behavior and actions. It is an internal singularity, habitus, functioning as strange attractor in enabling us to abstract sense from the chaos of experiences. The internal singularity is an affect of historical and biographical engagement with the network of the socius that empowers us to be culturally conditioned sense making beings. Master dispositif is the deep singularity that draws other cleavages of mentalities towards it ‘naturally’ figuratively comparable to a deep valley drawing streams towards it. In this paper, I argue, money beyond its multiple appearances and usages such as, means of exchange, currency, cash apparatus, or financial statement is a master dispositive, primarily an internal psychic apparatus, drawn from our collective historical ontology, that rules our subjective perceptions.The central argument here is that, qua the master dispositif, money has an agential role in guiding history. For instance, though the institutions of servitude, slavery, slavery-abolition, and free-labour appear to us a progress from servitude to freedom I argue, they are just various manifestations in historical junctures since money as a master dispositif has possessed our internal cartography of pathways and ‘progress’. Said in Heideggerian terminology thrown into money’s dispositif is the state of das-man. Human subject moments, as truth moments and authentic freedom, if have to have liberative expressions, that should be through overcoming the eclipse of the Self and not in that being subjected to the dispositif that overwhelms.

Frame Field Singularity Correction for Automatic Hexahedralization.

We present an automatic hexahedralization tool, based on a systematic treatment that removes some of the singularities that would lead to degenerate volumetric parameterization. Such singularities could be abundant in automatically generated frame fields guiding the interior and boundary layouts of the hexahedra in an all hexahedral mesh. We first give the mathematical definitions of the inadmissible singularities prevalent in frame fields, including newly introduced surface singularity types. We then give a practical framework for adjusting singularity graphs by automatically modifying the rotational transition of frames between charts (cells of a tetrahedral mesh for the volume) to resolve the issues detected in the internal and boundary singularity graph. After applying an additional re-smoothing of the frame field with the modified transition conditions, we cut the volume into a topologically trivial domain, with the original topology encoded by the self-intersections of the boundary of the domain, and solve a mixed integer problem on this domain for a global parameterization. Finally, a properly connected hexahedral mesh is constructed from the integer isosurfaces of (u,v,w) in the parameterization. We demonstrate the applicability of the method on complex shapes, and discuss its limitations.

Inverse free steering law for small satellite attitude control and power tracking with VSCMGs

Recent developments in integrated power and attitude control systems (IPACSs) for small satellite, has opened a new dimension to more complex and demanding space missions. This paper presents a new inverse free steering approach for integrated power and attitude control systems using variable-speed single gimbal control moment gyroscope. The proposed inverse free steering law computes the VSCMG steering commands (gimbal rates and wheel accelerations) such that error signal (difference in command and output) in feedback loop is driven to zero. H∞ norm optimization approach is employed to synthesize the static matrix elements of steering law for a static state of VSCMG. Later these matrix elements are suitably made dynamic in order for the adaptation. In order to improve the performance of proposed steering law while passing through a singular state of CMG cluster (no torque output), the matrix element of steering law is suitably modified. Therefore, this steering law is capable of escaping internal singularities and using the full momentum capacity of CMG cluster. Finally, two numerical examples for a satellite in a low earth orbit are simulated to test the proposed steering law.

PERFORMANCE MEASUREMENT AND DYNAMIC ANALYSIS OF TWO DOF ROBOTIC ARM MANIPULATOR

Forward and inverse kinematic analysis of 2DOF robo t is presented to predict singular configurations. Cosine function is used for servo motor simulation of kinematics and dynamics u sing Pro/Engineer. The significance of joint-2 for reducing internal singularities is highlighted. Performance analysis in terms of co ndition number, local conditioning index and mobili ty index is carried out for the manipulator. Dynamic analysis using Lagrangian’s an d Newton’s Euler approach is worked out analyticall y using MATLAB and results are plotted for their comparison.

Rotating beams and non-rotating beams with shared eigenpair for pinned-free boundary condition

In this paper we look for a rotating beam, with pinned-free boundary conditions, whose eigenpair (frequency and mode-shape) is same as that of a uniform non-rotating beam for a particular mode. It is seen that for any given mode, there exists a flexural stiffness function (FSF) for which the ith mode eigenpair of a rotating beam with uniform mass distribution, is identical to that of a corresponding non-rotating beam with same length and mass distribution. Inserting these derived FSF's in a finite element code for a rotating pinned-free beam, the frequencies and mode shapes of a non-rotating pinned-free beam are obtained. For the first mode, a physically realistic equivalent rotating beam is possible, but for higher modes, the FSF has internal singularities. Strategies for addressing these singularities in the FSF for finite element analysis are provided. The proposed functions can be used as test functions for rotating beam codes and also for targeted destiffening of rotating beams.

Closed-form solutions for stepped Timoshenko beams with internal singularities and along-axis external supports

The Timoshenko beam model in presence of internal singularities causing deflection and rotation discontinuities and resting on external concentrated supports along the span is studied in a static context. The internal singularities are modelled as concentrated reductions in the flexural and the shear stiffness by making use of the distribution theory. Along-axis supports are treated as unknown concentrated loads and moments. An exact integration procedure of the proposed model, not requiring continuity conditions at all, is presented. Closed-form solutions are provided for both cases of homogeneous and stepped Timoshenko beams. The so-called static Green’s functions are also obtained by the proposed procedure and their explicit expressions are provided.

Generalized Framework for Linearly Constrained Control Moment Gyro Steering

Constrained steering laws for control-moment gyroscope (CMG) arrays can avoid internal singularities without inducing torque error or requiring precomputed momentum paths, making such steering laws worthy of further study. This paper proposes a generalized framework for constrained steering laws that are linear with respect to the gimbal rates by augmenting the system’s Jacobian matrix with constraint equations. The derivation of this generalized framework is followed by a discussion of constraints used in laws of this type and general principles for designing a singularity-free constraint. A CMGarray consisting of three scissored pairs provides an example of how aknownsteering lawfits into this framework.Anew steering lawbased onorthogonal triplets of single-gimbalCMGs is developed with the help of the proposed methodology. Simulations of the Violet nanosatellite’s attitude-control performance verify the results of this new steering law.

Nonlinear Attitude Control of Flexible Spacecraft With Scissored Pairs of Control Moment Gyros

Dynamic equations describing the attitude motion of flexible spacecraft with scissored pairs of control moment gyroscopes are established. A nonlinear controller is designed to drive the flexible spacecraft to implement three-axis large-angle attitude maneuvers with the vibration suppression. Singularity analysis for three orthogonally mounted scissored pairs of control moment gyros shows that there exists no internal singularity in this configuration. A new pseudo-inverse steering law is designed based on the synchronization of gimbal angles of the twin gyros in each pair. To improve the synchronization performance, an adaptive nonlinear feedback controller is designed for each pairs of control moment gyros by using the stability theory of Lyapunov. Simulation results are provided to show the validity of the controllers and the steering law.

On the Calculation of Two-Dimensional Added Mass Coefficients by the Taylor Theorem and the Method of Fundamental Solutions

ABSTRACTThis work integrates the Taylor theorem and the method of fundamental solutions to develop a numerical tool for estimating the added mass coefficient tensor for a solid object of any convex shape moving in potential flow. In potential flow theory, the Taylor theorem calculates the added mass coefficient tensor for a Rankine body with algebraic manipulations of the properties of the internal singularities employed to generate the corresponding flow. To apply this theorem for objects in other shapes, the singularity strength and locations are required information which is facilitated numerically in this work by the method of fundamental solutions (MFS). The developed scheme is tested on a circle, an ellipse, a square, and a rhombus and the numerical results are in good agreement with the corresponding analytical values. A final example of a Cassini oval is also considered to show the potential applications on bio-engineering problems.

A generalised Milne-Thomson theorem for the case of an elliptical inclusion

An ℝ-linear conjugation problem modelling the process of power fields forming in a heterogeneous infinite planar structure with an elliptical inclusion is considered. Exact analytical solutions are derived in the class of piece-wise meromorphic functions with their principal parts fixed. Cases with internal singularities and with singularities of the given principal parts at the interface are investigated.

Revisiting the inverse dynamics of the Gough–Stewart platform manipulator with special emphasis on universal–prismatic–spherical leg and internal singularity

This article discusses the dynamic modeling for control of Gough–Stewart platform manipulator with special emphasis on universal–prismatic–spherical leg kinematics. Inverse dynamic model of these six degrees of freedom parallel manipulator robots is reviewed, while complete dynamics with true kinematics of universal–prismatic–spherical legs is compared with several models found in the literature. Most existing models have not taken into account some of the legs kinematical effects, namely the legs angular velocity around their axes and the internal singularities due to passive joints; some other used a simplified parameterization to describe the leg kinematics. Furthermore, some kinetic assumption can be used to reduce the computational burden. This article shows the effect of all these simplifications on the driving forces by simulating the different dynamic models for a commercial manipulator and for different sets of geometric and dynamic parameters of manipulator.