Solvability theory for a class of hemivariational inequalities involving copositive plus matrices applications in robotics

The study of the equilibrium of an object-robotic hand system including nonmonotone adhesive effects and nonclassical friction effects leads to new inequality methods in robotics. The aim of this paper is to describe these inequality methods and provide a corresponding suitable mathematical theory.     

Arc-Length Method for Frictional Contact Problems Using Mathematical Programming with Complementarity Constraints

A new formulation as well as a new solution technique is proposed for an equilibrium path-following method in two-dimensional quasistatic frictional contact problems. We consider the Coulomb friction law as well as a geometrical nonlinearity explicitly. Based on a criterion of maximum dissipation of energy, we propose a formulation as a mathematical program with complementarity constraints (MPEC) in order to avoid unloading solutions in which most contact candidate nodes become stuck. A regularization scheme for the MPEC is proposed, which can be solved by using a conventional nonlinear programming approach. The equilibrium paths of various structures are computed in cases such that there exist some limit points and/or infinite number of successive bifurcation points.     

On the variational formulation of the dynamics of systems with friction

We discuss the basic problem of the dynamics of mechanical systems with constraints, namely, the problem of finding accelerations as a function of the phase variables. It is shown that in the case of Coulomb friction, this problem is equivalent to solving a variational inequality. The general conditions for the existence and uniqueness of solutions are obtained. A number of examples are considered. For systems with ideal constraints the problem under discussion was solved by Lagrange in his “Analytical Dynamics” (1788), which became a turning point in the mathematization of mechanics. In 1829, Gauss gave his principle, which allows one to obtain the solution as the minimum of a quadratic function of acceleration, called the constraint. In 1872 Jellett gave examples of non-uniqueness of solutions in systems with static friction, and in 1895 Painlevé showed that in the presence of friction, the absence of solutions is possible along with the nonuniqueness. Such situations were a serious obstacle to the development of theories, mathematical models and the practical use of systems with dry friction. An elegant, and unexpected, advance can be found in the work [1] by Pozharitskii, where the author extended the Gauss principle to the special case where the normal reaction can be determined from the dynamic equations regardless of the values of the coefficients of friction. However, for systems with Coulomb friction, where the normal reaction is a priori unknown, there are still only partial results on the existence and uniqueness of solutions [2–4]. The approach proposed here is based on a combination of the Gauss principle in the form of reactions with the representation of the nonlinear algebraic system of equations for the normal reactions in the form of a variational inequality. The theory of such inequalities [5] includes results on the existence and uniqueness, as well as the developed methods of solution.     

Practical acceleration for computing the HITS ExpertRank vectors

A meaningful rank as well as efficient methods for computing such a rank are necessary in many areas of applications. Major methodologies for ranking often exploit principal eigenvectors. Kleinberg’s HITS model is one of such methodologies. The standard approach for computing the HITS rank is the power method. Unlike the PageRank calculations where many acceleration schemes have been proposed, relatively few works on accelerating HITS rank calculation exist. This is mainly because the power method often works quite well in the HITS setting. However, there are cases where the power method is ineffective, moreover, a systematic acceleration over the power method is desirable even when the power method works well. We propose a practical acceleration scheme for HITS rank calculations based on the filtered power method by adaptive Chebyshev polynomials. For cases where the gap-ratio is below 0.85 for which the power method works well, our scheme is about twice faster than the power method. For cases where gap-ratio is unfavorable for the power method, our scheme can provide significant speedup. When the ranking problems are of very large scale, even a single matrix–vector product can be expensive, for which accelerations are highly necessary. The scheme we propose is desirable in that it provides consistent reduction in number of matrix–vector products as well as CPU time over the power method, with little memory overhead.     

On the influence of design parameters and nonlinearities in vehicle suspensions on road deformation

Among others, two main objectives of modern vehicle design are road friendliness and ride comfort. Both aspects are strongly related since the dynamical tire forces depend on the vertical acceleration of the vehicle. In order to investigate the influence of design and operation parameters, different car models are considered which move with constant velocity on a rippled road. First, a linear half car model is examined and the influence of different design parameters is discussed. Second, nonlinear suspensions with Coulomb friction due to sealings as well as with bilinear shock absorbers are taken into account. The vertical dynamics of the vehicle model and the dynamic tire forces between vehicle and road are calculated using analytical methods. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

半导体与超晶格的数学模型及其分析Ⅱ:分析<英>

本文先介绍等熵可压缩Euler方程的相关补偿列紧框架.然后,我们综述基于补偿列紧方法的关于半导体流体动力模型的诸如整体弱解,松弛极限和拟中性.松弛极限的一些新近数学结果.     

半导体与超晶格的数学模型及其分析Ⅱ:分析

本文先介绍等熵可压缩Euler方程的相关补偿列紧框架.然后,我们综述基于补偿列紧方法的关于半导体流体动力模型的诸如整体弱解,松弛极限和拟中性-松弛极限的一些新近数学结果.     

A Contribution to Jerk Detection

In technical applications the jerk information will be used for i.a. motion control, comfort evaluation or prediction of acceleration. In the majority of cases the jerk will be determined indirectly by differentiation of measured acceleration signals and depends consequently on the acceleration signals' quality and the differentiation's algorithms. An accurate jerk determination can only be realized by using an appropriate filter with respect to the relevant frequency range. In order to avoid such limitation a method of direct jerk detection will be required. Based on a 2 DOF mass system with viscous coupling a possible method was presented in [1] to detect directly the jerk. Thereby, in certain conditions the jerk is proportional to the displacement of one mass. Another method of direct jerk detection is given implicitly in [2]. With this method the jerk information will be generated by an additional inner variable of a damped single mass system based on the fundamental idea of the Mesoscopic Particle Method [3]. Both presented methods will be investigated and evaluated regarding its limits of applicability as a result of arbitrary periodical and impact-like excitations. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

计算代数方程组孤立奇异解的符号数值方法

求解代数方程组是计算代数几何的最基本问题之一,孤立奇异解的计算则是其中最具挑战性的课题之一,在科学与工程计算中有着广泛的应用,如机器人、计算机视觉、机器学习、人工智能、运筹学、密码学和控制论等.本文结合作者的研究成果,综述了符号数值方法在计算代数系统孤立奇异解、特别是近似奇异解精化与验证方面的研究进展,并对未来的研究方向提出了展望.     

Variational Based Stability Analysis in Coupled Electro-Mechanics at Finite Strains

Smart or functional materials are in focus in broad fields of recent research. In particular, electroactive polymers (EAPs) with their large actuator strains are used for innovative design in robotics. These materials become practical as soon as the material is robust, reliable and long lived. However, EAPs suffer from different types of instabilities, such as unstable thinning of actuators, local buckling induced by coexistent states and electrical breakdown. One distinguishes between structural instability, i.e. instabilities in the global structural response like buckling or wrinkling and local or material instability, e.g. the formation of microstructures and phase decompositions. These effects are well known and analyzed for purely mechanical phenomena, but not fully developed in the context of electro-mechanically coupled problems. To this end, we develop new variational-based stability criteria in finite electro-elastostatics, by postulating an electro-mechanical quasi-convexity condition for weakly convex energy and weakly convex-concave energy-enthalpy formulations. Starting from distinct variational principles, we develop criteria for local and global stability and construct stability checks, which accompany equilibrium paths for the detection of critical points in finite element models. Numerical examples validate the developed concepts for boundary value problems. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Calculation of frictional heat resistance characteristics

The search for self-lubricating materials, the selection of suitable ingredients and processing techniques for such materials, and quality control require the development of methods of testing for friction and wear. The frictional heat resistance can be used to evaluate the frictional properties of self-lubricating materials and to determine their limits of applicability. It is shown with reference to AF-3am self-lubricating material that it is possible to calculate the temperature dependence of the wear rate if the physicomechanical and geometric characteristics of the contact materials are known.Mekhanika Polimerov, Vol. 3, No. 3, pp. 524–532, 1967     

Solutions of quasi-geostrophic turbulence in multi-layered configurations

We consider quasi-geostrophic (QG) models in two- and three-layers that are useful in theoretical studies of planetary atmospheres and oceans. In these models, the streamfunctions are given by (1+2) partial differential systems of evolution equations. A two-layer QG model, in a simplified version, is dependent exclusively on the Rossby radius of deformation. However, the f-plane QG point vortex model contains factors such as the density, thickness of each layer, the Coriolis parameter, and the constant of gravitational acceleration, and this two-layered model admits a lesser number of Lie point symmetries, as compared to the simplified model. Finally, we study a three-layer oceanography QG model of special interest, which includes asymmetric wind curl forcing or Ekman pumping, that drives double-gyre ocean circulation. In three-layers, we obtain solutions pertaining to the wind-driven doublegyre ocean flow for a range of physically relevant features, such as lateral friction and the analogue parameters of the f-plane QG model. Zero-order invariants are used to reduce the partial differential systems to ordinary differential systems. We determine conservation laws for these QG systems via multiplier methods.     

Experimental and numerical analysis of self-excited friction oscillator

A self-excited friction oscillator has been designed and manufactured to carry out experimental analysis of dry friction phenomenon. A mathematical model of this oscillator has been formulated. The influence of the different types of classical friction characteristics on the dynamical behaviour of the model is investigated by way of numerical analysis. A comparison with dynamics of real oscillator is presented and some reasons of observed differences are explained. A particular analysis of experimental data leads to the confirmation of non-reversible friction characteristics and allows to formulate a hypothesis that a course of such characteristics also depends on value (not only on the sign) of acceleration.     

Curves of stationary acceleration in SE(3)

^** Email: seligjm{at}lsbu.ac.uk The concept of curves of minimal acceleration seems to havebeen introduced by efran and Kumar and independently by Noakes,Heinzinger and Paden. In part, the motivation was to extendthe notion of spline curves to curves in groups, specificallythe groups associated with robotics. A curve in the rigid-bodymotion group SE(3), e.g. can be thought of as a trajectory ofa rigid body. Hence, these ideas have applications to motionplanning and interpolation. In this work, the analysis is repeatedbut using bi-invariant metrics on the group. Since these metricsare not positive definite, the curves specified by the equationsderived are only stationary, not minimal. It is possible tosolve these non-linear coupled differential equations in somesimple cases. However, these simple cases turn out to be highlyrelevant to robotics and mechanism theory.     

2-Binary trees: Bijections and related issues

A 2-binary tree is a binary rooted tree whose root is colored black and the other vertices are either black or white. We present several bijections concerning different types of 2-binary trees as well as other combinatorial structures such as ternary trees, non-crossing trees, Schröder paths, Motzkin paths and Dyck paths. We also obtain a number of enumeration results with respect to certain statistics.     

Multibody dynamics with unilateral constraints and friction

In the view of robotics or granular media mechanics, we ask for the statement of the dynamical evolution problem for multibody systems with contacts and friction. The “event driven” approach does not allow to state an evolution problem in a systematic way: it gives a peculiar status to the instants of impact and to the contact points although they are unknowns of the problem. This matter of fact has given rise to a new approach in the frictionless case [4, 6] we generalize to the case including friction. Following to the point of view of Lagrange on the equation of the dynamics, we emphasize the systematic use of virtual powers and duality. This bias suggests to put emphasis on generalized reaction forces in the statement of the evolution problem and not on local reaction forces of the real world. This point of view allows to escape from pathologies e.g. Painlevé paradox [5] and Kane Paradox [3]. The proposed evolution problem accounts for impacts and friction in case of multiple unilateral constraints. We prove the existence and uniqueness of the solution to the associated rate problem. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Possibilities and Challenges in Kinematic Modeling of Highly Redundant,Non-Holonomic and Omnidirectional Mobile Robots

Industry as well as the private domain show an increasing interest to the field of mobile robotics. With a higher number of applications, the complexity of the required tasks rises, and therefore the community tends to use robots with many degrees of freedom (DOF). The present paper takes this topic up and focuses on challenges of the kinematical modeling of redundant, non-holonomic mobile robots by considering a 12 DOF platform. To reach an omnidirectional behaviour, the actuated wheels are diagonally mounted on the chassis. Well known problems resulting from this set-up are parametric singularities which unnecessarily restrict the motion of the mathematical model. As it turns out, this problem can be avoided by the use of a non-minimal parametrized model. An additional challenge results from the inverse kinematic (IK) problem on velocity level. The corresponding equations are highly underdetermined and, due to the non-holonomic wheels, not directly affected by the steering angle velocities. This problem is solved by performing a local optimization on acceleration level. Finally, simulation results are presented. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling of the robotic Powerball®: a nonholonomic,underactuated and variable structure-type system

The Powerball® is the commercial name for a gyroscopic device that is marketed as a wrist exerciser. The device has a rotor with two underactuated degrees of freedom, which can be actuated by the appropriate motion of human or robot wrist axes. After the initial spin, applying the appropriate motion and torques to the housing leads to a spin-up of the rotor. Finding these torques intuitively is an easy task for human operators, but a complex task for a technical consideration, for example, in robotics.

This article's main contribution is a novel dynamic model that considers friction effects. The presented model includes all three working principles of the device: free rotor mode and both modes of rotor rolling in the housing. The work introduces models with one and two degrees of freedom actuation, both of which are suitable for laboratory control experiments. An estimation of the friction is discussed, and both the simulation and the experimental results are presented to evaluate the models.