A Geometric Approach to Actuator Failure in Robotic Manipulators

The use of robotic manipulators in remote and sensitive areas calls for more robust solutions when handling joint failure, and the industry demands mathematically robust approaches to handle even the worst case scenarios. Thus, a systematic analysis of the effects of external forces on manipulators with passive joints is presented. In parallel manipulators passive joints can appear as a design choice or as a result of torque failure. In both cases a good understanding of the effects that passive joints have on the mobility and motion of the parallel manipulator is crucial. We first look at the effect that passive joints have on the mobility of the mechanism. Then, if the mobility, considering passive joints only, is not zero we find a condition for which the parallel manipulator is conditionally equilibrated with respect to a specific external force. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling the angular capability of the ball joints in a complex mechanism with two degrees of mobility

The paper puts forward a complex linkage mechanism with two degrees of mobility and three kinematic loops, which is used for the guiding (suspension and steering) system of the vehicles. The geometric parameters and the coordinates frames that define the mechanical system are presented, as well as the specific kinematic functions. For this complex mechanical system, the angular capability of the ball (spherical) joints is defined by two angles. The equations for these angles have been determined by matrix algebra tools, considering the transformation matrices between the bodies reference frames. The diagrams of the angular capability of the ball joints, which are represented in angular coordinates, describe the form, orientation and size of the sockets from the spherical casings. Wears, shocks, functional locks or the compromising of the joint strength can occur if scarce sockets are implemented. The risk points, in which the angular parameters have maximum values, have been determined, the simulation being performed for a real system (vehicle).     

Kinematic analysis of multibody systems

We discuss how to use constructive methods in commutative algebra to study multibody systems. The main focus is in the kinematic analysis, i.e. the analysis of the geometry of the configuration space. We show how to define and compute the mobility of the system and study various singularities of the configuration space. We also discuss implications of this analysis for numerical computations. AMS subject classification (2000)  13P10, 65L05, 68W30, 70B10     

Low-energy theorems for the scattering of photons and gravitons by a target with arbitrary spin

Conclusions In the framework of the method developed here we have established the kinematic nature of the low-energy theorems. They are a consequence of Lorentz invariance and the general analytic properties of the amplitude.The structure of the kinematic singularities of the helicity amplitudes is completely determined by Lorentz and gauge invariance. The kinematic-singular structure of some of the helicity amplitudes readily yields low-energy theorems for the elastic scattering of photons and gravitons by a target with arbitrary spin. For the remaining helicity amplitudes we had to consider crossing relations, which are a consequence of the generalized (unphysical) Lorentz invariance and analyticity.Decisive importance in the derivation of the low-energy theorems attaches to the separation of the kinematic singularities and zeros from the helicity amplitudes in the manner proposed in [9].JINR, Dubna; Tbilisi State University. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 8, No. 1, pp. 16–22, July, 1971.     

On geometric dynamics of rigid multi-body systems

Standard methods to model multibody systems are aimed at systems with configuration spaces isomorphic to ℝⁿ . This limitation leads to singularities and other artifacts in case the configuration space has a different topology, for example in the case of ball joints or a free-floating mechanism. This paper discusses an extension of classical methods to allow for a very general class of joints, including all joints with a Lie group structure. The model equations are derived using the Boltzmann-Hamel equations and have very similar structure and complexity as obtained using classical methods, but they do not suffer from singularities. Furthermore, the equations are explicit differential equations that can be directly implemented in simulation software. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Motion equations of cooperative multi flexible mobile manipulator via recursive Gibbs–Appell formulation

In this paper, the developed model of an N-flexible-link mobile manipulator with revolute-prismatic joints is presented for the cooperative flexible multi mobile manipulator. In this model, the deformation of flexible links is calculated by using the assumed modes method. In additions, non-holonomic constraints of the robots’ mobile platforms that bound its locomotion are considered. This limitation is alleviated through the concurrent motion of revolute and prismatic joints, although it results in computational complexity and changes the final motion equations to time-varying form. Not only is the proposed dynamic model implemented for the multi-mobile manipulators with arms having independent motion, but also for multi-mobile manipulators in cooperation after defining gripper's kinematic constraints. These constraints are imported to the dynamic equations by defining Lagrange multipliers. The recursive Gibbs–Appell formulation is preferred over other similar approaches owing to the capability of solving the equations without the need to use Lagrange multipliers for eliminating non-holonomic constraints in addition to the novel optimized process of obtaining system equations. Hence, cumbersome simultaneous computations for eliminating the constraints of platform and arms are circumvented. Therefore, this formulation is improved for the first time by importing Lagrange multipliers for solving kinematic constrained systems. In the simulation section, the results of forward dynamics solution for two flexible single-arm manipulators with revolute-prismatic joints while carrying a rigid object are presented. Inverse dynamics equations of the system are also presented to obtain the maximum dynamic load-carrying capacity of the two-rigid-link mobile manipulators on a predefined path. Two constraints, namely the capacity of joint motors torque and robot motion stability are considered as the limitation criteria. The concluded motion equations are used to accurately control the movement of sensitive bodies, which is not achievable through the use of one platform.     

Entropies and weak solutions of the compressible isentropic Euler equations

In this Note, we study the system of isentropic Euler equations for compressible fluids, with a general equation of state. We establish the existence of the fundamental kernel that generates the family of weak entropies, and study its singularities. The kernel is the solution of an equation of Euler-Poisson-Darboux type, and its partial derivative with respect to the density variable tends to a Dirac measure as the density approaches zero. We prove a new reduction theorem for the Young measures associated with the compressible Euler system. From these results, we deduce the existence, compactness, and asymptotic decay of measurable and bounded entropy solutions.     

Stokes flow inside a porous spherical shell: Stress jump boundary condition

An arbitrary Stokes flow of a viscous, incompressible fluid inside a sphere with internal singularities, enclosed by a porous spherical shell, using Brinkmans equation for the flow in the porous region is discussed. At the interface of the clear fluid and porous region stress jump boundary condition for tangential stresses is used. The drag and torque are found by deriving the corresponding Faxens laws. It is found that drag and torque not only change with the varying permeability, but also change for different values of stress jump coefficient. Critical permeability is found for which drag and torque change their behavior. As a limiting case the corresponding Faxens laws for the rigid spherical shell with internal singularities has been obtained.Received: December 17, 2002; revised: February 3, 2004     

Stability in systems with aftereffect when there are singularities in the integral kernels

Systems with aftereffect are considered. The state of these systems is described by integrodifferential equations of the Volterra type, which depend on functionals in integral form and, in particular, on analytic functionals which are represented by Frechet series. The integral kernels can allow of singularities of Abel kernel singularities. The total stability (i.e. stability under persistent disturbances) is investigated, and the structure of the general solution is investigated in the neighbourhood of zero for an equation with a holomorphic non-linearity assuming asymptotic stability of the trivial solution of the linearized unperturbed equation. The conditions for instability are given in the critical case of a single zero root, which generalise results obtained previously.     

Development and experimental validation of a control-oriented Diesel engine model for fuel consumption and brake torque predictions

This article describes the development and experimental validation of a control-oriented, real-time capable, Diesel engine instantaneous fuel consumption and brake torque model under warmed-up conditions with only two inputs: torque request and the engine speed and no other measurements. Such a model, with the capability of reliably and computationally efficiently estimating the aforementioned variables at both steady-state and transient engine-operating conditions, can be utilized in the context of real-time control and optimization of hybrid power train systems. Although Diesel engine dynamics are highly non-linear and very complex, by considering the Diesel engine and its control system, that is, engine control unit together as an entity, it becomes possible to predict the engine instantaneous fuel consumption and torque based on only those two inputs. A synergy between different modelling methodologies including physically based grey-box and data-driven black-box approaches were integrated in the Diesel engine model. The fuelling and torque predictions have been validated by means of experimental data from a medium-duty Diesel engine at both steady-state and transient operations, including engine start-ups and shutdowns.     

Refined Numerical Schemes for a Stressed-Strained State of Structural Joints of Layered Elements

A numerical approach based on block-variational modeling of a stressed state of structural joints of layered elements is developed. A refined theory based on a kinematic hypothesis for every layer and joints and numerical results are presented.     

Kinematical synthesis of 1-dof mechanisms using finite elements and genetic algorithms

The use of finite elements in both kinematic analysis and synthesis of mechanisms has shown good performance. Its only drawback is the need for an initial reasonable quality solution.

To address this topic, the use of genetic algorithms with a finite-element-based error function is proposed in this paper. This approach has not only shown good behaviour with simple mechanisms, but also with complex kinematic chains.

The energy-based error function, which has demonstrated such good behaviour in the optimization via second-order Newton–Raphson methods, presented some limitations. To solve them, a new distance-based error function has been developed. Simultaneously, new finite elements have been created to represent some joints and kinematic elements. The need to address some specific configuration problems has led to the development of a correction algorithm.     

Fundamentals of Quaternionic Kinematics in Euclidean 4-Space

Based on a kinematic mapping for the group SE(4) of displacements of Euclidean 4-space, we show that the mapping of basic elements (points, oriented lines, oriented planes, oriented hyperplanes, instantaneous screws) can be written compactly in terms of 2 × 2 quaternionic matrices. Moreover we discuss the kinematics on the velocity level by investigating instantaneous screws and their geometric parameters.     

Structure of a Linear Array of Uniform Vortices

The shape and properties of an infinite steady linear array of uniform vortices are calculated. A nonlinear singular integrodifferential equation is obtained for the shapes, which is solved numerically by Newton's method and Euler continuation to give a one parameter family of shapes as size over separation is varied. The kinematic properties and energy of the array are obtained. It is found that there exists an array of maximum area, for given separation, which also possesses minimum energy in accordance with a general argument of Kelvin. A simple model based on elliptical vortices is constructed, which reproduces the qualitative kinematic properties and is quantitatively quite accurate. Continuation of the numerical solution past the array of maximum area leads to a limit of finite, lens shaped, touching vortices. This array is also shown to be limit of a finite amplitude bifurcation of a vortex sheet of finite thickness. The stability of the array to two dimensional subharmonic and superharmonic disturbances is considered. General arguments, based on ideas of Kelvin, are given to show that the array is stable to superharmonic disturbances if the area is less than the maximum and otherwise unstable, and that it is always unstable to subharmonic disturbances, of which the pairing instability is a special case. It is verified by direct calculation in an Appendix that hollow vortices, whose shapes can be determined analytically in closed form, are unstable to the pairing instability whatever their size. Some speculations are made about the possible relevance of the results to the observed properties of organized structures in the turbulent mixing layer.     

Modeling of Friction-Induced Vibrations during Tightening of Bolted Joints

During the automated tightening process of bolted joints, the preload force can only be determined with greater effort. This is usually realized by using indirect methods, one of which is the torque-controlled procedure. The dependency between the tightening torque and the preload force is deeply connected to the frictional conditions under the screw head and in the screw thread. Several methods exist to support the tightening process. Nonetheless, errors occur, which are, besides other reasons, due to friction-induced vibrations. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the exceptional case of the characteristic singular equation with Cauchy kernel

We study the exceptional case of the characteristic singular integral equation with Cauchy kernel in which its coefficients admit zeros or singularities of complex orders at finitely many points of the contour. By reduction to a linear conjugation problem, we obtain an explicit solution formula and solvability conditions for this equation in weighted Hölder classes.     

Singularities of the boundaries of stability domains

The singularities of the general position which arise on the boundaries of stability domains of a linear, autonomous system of differential equations in a two- or three-dimensional parameter space are investigated. A constructive approach is proposed which enables one to determine the geometry of the singularities (the orientation in space, the magnitude of the angles, etc.) by constructing cones which are tangential to the stability domain using the first derivatives of the matrix operator of the system with respect to the parameters and its eigenvectors and associated vectors at the singular points of the boundary. Examples are presented.     

The use of infinite grid refinements at singularities in the solution of Laplace's equation

Summary The approximate solution of Laplace's equation using the finite element method is considered. Particular emphasis is given to problems in which there are boundary singularities and the use of infinite refinements in the grid of triangles in the neighbourhood of these singularities is analysed. A particular type of infinite grid refinement is proposed and some examples are given.     

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.