Application of Hyper-Dual Numbers to Multibody Kinematics

2015 ◽  
Vol 8 (1) ◽  
Author(s):  
Avraham Cohen ◽  
Moshe Shoham

Hyper-dual numbers (HDNs) are applied in this paper to multibody kinematics. First, the hyper-dual angle that encompasses a body's position, orientation, as well as its velocity, is defined as an element of the hyper-dual transformation matrix. Then, the “automatic differentiation” feature of the dual numbers is used to obtain the second derivative of a body pose. The body's velocity and acceleration are obtained from the elements of the hyper-dual transformation matrix by algebraic manipulations only, with no need for further time derivatives of the body pose. A robot manipulator is presented as an exemplary application of HDNs to multibody kinematics.

Author(s):  
László Szirmay-Kalos

In engineering applications, we often need the derivatives of functions defined by a program. The approach chosen for derivative computation must be algebraic to allow computer implementation. A particular solution to obtain first derivatives is the application of dual numbers. This paper proposes simple and compact generalizations of this idea to obtain derivatives of arbitrary order for single or multi-variate functions and the automatic handling of 0/0 ambiguities in the calculations. We also provide the C++ code that takes advantage of operator overloading and recursion. The method is demonstrated by path animation, Gaussian curvature computation, and curve fairing.


2017 ◽  
Vol 919 (1) ◽  
pp. 7-12
Author(s):  
N.A Sorokin

The method of the geopotential parameters determination with the use of the gradiometry data is considered. The second derivative of the gravitational potential in the correction equation on the rectangular coordinates x, y, z is used as a measured variable. For the calculated value of the measured quantity required for the formation of a free member of the correction equation, the the Cunningham polynomials were used. We give algorithms for computing the second derivatives of the Cunningham polynomials on rectangular coordinates x, y, z, which allow to calculate the second derivatives of the geopotential at the rectangular coordinates x, y, z.Then we convert derivatives obtained from the Cartesian coordinate system in the coordinate system of the gradiometer, which allow to calculate the free term of the correction equation. Afterwards the correction equation coefficients are calculated by differentiating the formula for calculating the second derivative of the gravitational potential on the rectangular coordinates x, y, z. The result is a coefficient matrix of the correction equations and corrections vector of the free members of equations for each component of the tensor of the geopotential. As the number of conditional equations is much more than the number of the specified parameters, we go to the drawing up of the system of normal equations, from which solutions we determine the required corrections to the harmonic coefficients.


Geophysics ◽  
1952 ◽  
Vol 17 (2) ◽  
pp. 344-364 ◽  
Author(s):  
Fraser S. Grant

A method is developed for determining the approximate size and shape of the three‐dimensional mass distribution that is required to produce a given gravitational field. The first few reduced multipole moments of the distribution are calculated from the derivatives of the surface field, and the approximative structure is determined from the values of these moments and a knowledge of the density contrast between the body and its surroundings. A system of classification of problems by symmetry is introduced and its practical usage discussed. A relaxation method is described which may be used to adjust the initial solution systematically to give agreement over the whole field. A descriptive discussion is appended.


Author(s):  
Guangfa Yao

Immersed boundary method has got increasing attention in modeling fluid-solid body interaction using computational fluid dynamics due to its robustness and simplicity. It usually simulates fluid-solid body interaction by adding a body force in the momentum equation. This eliminates the body conforming mesh generation that frequently requires a very labor-intensive and challenging task. But accurately tracking an arbitrary solid body is required to simulate most real world problems. In this paper, a few methods that are used to track a rigid solid body in a fluid domain are briefly reviewed. A new method is presented to track an arbitrary rigid solid body by solving a transformation matrix and identifying it using a level set function. Knowing level set function, the solid volume fraction can be derived if needed. A three-dimensional example is used to study a few methods used to represent and solve the transformation matrix, and demonstrate the presented new method.


Author(s):  
Suryanarayana R. Pakalapati ◽  
Hayri Sezer ◽  
Ismail B. Celik

Dual number arithmetic is a well-known strategy for automatic differentiation of computer codes which gives exact derivatives, to the machine accuracy, of the computed quantities with respect to any of the involved variables. A common application of this concept in Computational Fluid Dynamics, or numerical modeling in general, is to assess the sensitivity of mathematical models to the model parameters. However, dual number arithmetic, in theory, finds the derivatives of the actual mathematical expressions evaluated by the computer code. Thus the sensitivity to a model parameter found by dual number automatic differentiation is essentially that of the combination of the actual mathematical equations, the numerical scheme and the grid used to solve the equations not just that of the model equations alone as implied by some studies. This aspect of the sensitivity analysis of numerical simulations using dual number auto derivation is explored in the current study. A simple one-dimensional advection diffusion equation is discretized using different schemes of finite volume method and the resulting systems of equations are solved numerically. Derivatives of the numerical solutions with respect to parameters are evaluated automatically using dual number automatic differentiation. In addition the derivatives are also estimated using finite differencing for comparison. The analytical solution was also found for the original PDE and derivatives of this solution are also computed analytically. It is shown that a mathematical model could potentially show different sensitivity to a model parameter depending on the numerical method employed to solve the equations and the grid resolution used. This distinction is important since such inter-dependence needs to be carefully addressed to avoid confusion when reporting the sensitivity of predictions to a model parameter using a computer code. A systematic assessment of numerical uncertainty in the sensitivities computed using automatic differentiation is presented.


2004 ◽  
pp. 147-176
Author(s):  
Stanimir Rakic

In this paper I examine compound names of plants, animals, human beings and other things in which at least one nominal component designates a part of the body or clothes, or some basic elements of houshold in Serbian and English. The object of my analysis are complex derivatives of the type (adjective noun) + suffix in Serbian and componds of the type noun's + noun, noun + noun and adjective + noun in English. I try to show that there is a difference in metaphorical designation of human beings and other living creatures and things by such compound nouns. My thesis is that the metathorical designation of human beings by such compounds is based on the symbolic meaning of some words and expressions while the designation of other things and beings relies on noticed similarity. In Serbian language such designation is provided by comples derivatives praznoglavac 'empty-headed person', tupoglavac 'dullard' debolokoiac 'callos person', golobradac 'young, inexperienced person' zutokljunac 'tledling' (fig), in English chicken liver, beetle brain birdbrain, bonehead, butterfingers, bigwig, blackleg, blue blood bluestocking, eat's paw, deadhead,fat-guts,fathead, goldbrick (kol) hardhat, hardhead, greenhorn, redcoat (ist), redneck (sl), thickhead, etc. Polisemous compounds like eat's paw lend support for this thesis because their designation of human beings is based on symbolic meaning of some words or expressions. I hypothesize that the direction and extend of the possible metaphorization of names may be accounted for by the following hierarchy (11) people - animals - plants - meterial things. Such hierarchy is well supported by the observations of Lakoff (1987) and Taylor (1995) about the role of human body in early experience and perception ofthe reality. Different restrictions which may be imposed in the hierarchy (11) should be the matter of further study, some of which have been noted on this paper. The compounds of this type denoting people have metaphorically meaning conected with some pejorative uses. These compounds refer to some psychological or characteral features, and show that for the classification of people such features are much more important than physical properties. While the animals and plants are classified according to some charecteristics of their body parts, people are usually classified according to psychollogical characteristics or their social functions. I have also noted a difference in structure between compounds designation animals and those designating plants and other things. The designation of animals relies more on metonymy, and that of plants and other things on metaphor based on comparision of noticed similarities. In the compounds designating animals, the nominal component relatively seldom refers to the parts of plants or other things. I guess that the cause may be the fact that the anatomy of plants is very different from the anatomy of animals. As a consequence the structure adjective + noun is much more characteristic of the compounds designating animals in English than the structure noun's + noun, and the same holds, although in a lesser degree for the compounds designating humans. It is also noticeable that in English compounds whose second component a part of body or clothes the first component rarely designates animals. On the other hand the compounds (9), in which the nominal head refers to some superordinate species, the first component often designates animal species, but usually of a very different kind. These data seem to lend support for Goldvarg & Gluksberg's thesis (1998) that metaforical interpretation is favoured if the nominal constituents denote quite different entities.


2014 ◽  
Vol 66 (4) ◽  
pp. 783-825 ◽  
Author(s):  
Ivan Izmestiev

Abstract. The paper presents a new proof of the infinitesimal rigidity of convex polyhedra. The proof is based on studying derivatives of the discrete Hilbert–Einstein functional on the space of “warped polyhedra” with a fixed metric on the boundary.The situation is in a sense dual to using derivatives of the volume in order to prove the Gauss infinitesimal rigidity of convex polyhedra. This latter kind of rigidity is related to the Minkowski theorem on the existence and uniqueness of a polyhedron with prescribed face normals and face areas.In the spherical space and in the hyperbolic-de Sitter space, there is a perfect duality between the Hilbert–Einstein functional and the volume, as well as between both kinds of rigidity.We review some of the related work and discuss directions for future research.


Medicina ◽  
2018 ◽  
Vol 54 (4) ◽  
pp. 61 ◽  
Author(s):  
Agnieszka Chrustek ◽  
Iga Hołyńska-Iwan ◽  
Inga Dziembowska ◽  
Joanna Bogusiewicz ◽  
Marcin Wróblewski ◽  
...  

Pyrethroids are synthetic derivatives of natural pyrethrins extracted from Chrysanthemum cinerariaefolium. They are 2250 times more toxic to insects than to vertebrates due to insects’ smaller size, lower body temperature and more sensitive sodium channels. In particular, three pyrethroid compounds, namely deltamethrin, permethrin, and alpha-cypermethrin, are commonly used as insecticides and are recommended for in-home insect control because they are considered to be relatively non-toxic to humans in all stages of life. However, recent data show that they are not completely harmless to human health as they may enter the body through skin contact, by inhalation and food or water, and absorption level depending on the type of food. Permethrin seems to have an adverse effect on fertility, the immune system, cardiovascular and hepatic metabolism as well as enzymatic activity. Deltamethrin induces inflammation, nephro- and hepatotoxicity and influences the activity of antioxidant enzymes in tissues. Alpha-cypermethrin may impair immunity and act to increase glucose and lipid levels in blood. The aim of the review is to provide comprehensive information on potential hazards associated to human exposure to deltamethrin, permethrin and alpha-cypermethrin. The results of presented studies prove that the insecticides must be used with great caution.


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