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1
Content available Robot grasping and regrasping kinematics using Lie algebra, the geodesic, and Riemann curvature tensor
Sahin Haydar
Archives of Control Sciences
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2023
|
Vol. 33, No. 1
5--23
EN
Differential geometry is a strong and highly effective mathematical subject for robot gripper design when grasping within the predetermined trajectories of path planning. This study in grasping focuses on differential geometry analysis utilizing the Lie algebra, geodesic, and Riemann Curvature Tensors (RCT). The novelty of this article for 2RR robot mechanisms lies in the approach of the body coordinate with the geodesic and RCT. The importance of this research is significant especially in grasping and regrasping objects with varied shapes. In this article, the types of workspaces are clarified and classified for grasping and regrasping kinematics. The regrasp has not been sufficiently investigated of body coordinate systems in Lie algebra. The reason for this is the difficulty in understanding relative coordinates in Lie algebra via the body coordinate system. The complexity of the equations has not allowed many researchers to overcome this challenge. The symbolic mathematics toolbox in the Maxima, on the other hand, aided in the systematic formulation of the workspaces in Lie algebra with geodesic and RCT. The Lie algebra se(3) equations presented here have already been developed for robot kinematics from many references. These equations will be used to derive the following workspace types for grasping and regrasping. Body coordinate workspace, spatial coordinate workspace with constraints, body coordinate workspace with constraints, spatial coordinate workspace with constraints are the workspace types. The RCT and geodesic solutions exploit these four fundamental workspace equations derived using Lie algebra.
2
Content available remote On the Gauss-Bonnet for the quasi-Dirac operators on the sphere
Sitarz A.
Demonstratio Mathematica
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2017
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Vol. 50, nr 1
66--71
EN
We investigate examples of Gauss-Bonnet theorem and the scalar curvature for the two-dimensional commutative sphere with quasi-spectral triples obtained by modifying the order-one condition.
3
Content available remote The famous mathematician of Lithuanian University Otto Theodor Volk (1892‒1989)
Banionis J.
Czasopismo Techniczne. Nauki Podstawowe
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2014
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R. 111, z. 1-NP
5--12
EN
The article introduces a German mathematician Otto Theodor Volk (1892‒1989), who worked as a professor at Lithuanian University in 1922–1930, and sheds light on his merits in the science of mathematics in Lithuania.
PL
W artykule przedstawiono niemieckiego matematyka Otto Theodora Volka (1892‒1989), który był profesorem Uniwersytetu Litewskiego w latach 1922–1930. Zaprezentowano również jego zasługi dla rozwoju matematyki na Litwie.
4
Content available Friction compensation in nonlinear dynamical systems using fault-tolerant control methods
Bonfè M., Castaldi P., Preda N., Simani S.
Diagnostyka
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2013
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Vol. 14, No. 4
13--19
EN
This paper describes the application of differential geometry and nonlinear systems analysis to the estimation of friction effects in a class of mechanical systems. The proposed methodology relies on adaptive filters, designed with a nonlinear geometric approach to obtain the disturbance de-coupling property, for the estimation of the friction force. Thanks to accurate estimation, friction effects are compensated by injecting the on-line estimate of friction force to the control action calculated by a standard linear state feedback. The inverted pendulum on a cart is considered as an application example and the proposed approach is compared with a commonly used friction compensation strategy, based on an explicit model of the friction force.
5
Content available remote On semi-invariant submanifolds of a nearly trans-Sasakian manifold admitting a semi-symmetric semi-metric connection
Das L. S., Ahmad M., Danish Siddiqi M., Haseeb A.
Demonstratio Mathematica
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2013
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Vol. 46, nr 2
345--359
EN
We define a semi-symmetric semi-metric connection in a nearly trans-Sasakian manifold and we consider semi-invariant submanifolds of a nearly trans-Sasakian manifold endowed with a semi-symmetric semi-metric connection. Moreover, we also obtain integrability conditions of the distributions on semi-invariant submanifolds.
6
Content available remote Fractional-order derivative and time-dependent viscoelastic behaviour of rocks and minerals
Kawada Y., Yajima T., Nagahama H.
Acta Geophysica
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2013
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Vol. 61, no. 6
1690-1702
EN
A general constitutive equation for viscoelastic behaviour of rocks and minerals with fractional-order derivative is investigated. This constitutive law is derived based on differential geometry and thermodynamics of rheology, and the fractional order of derivative represents the degree of time delay. Analyzing some laboratory experimental data of high temperature deformation of rocks and minerals such as halite, marble and orthopyroxene, we propose how to determine the orders of fractional derivative for viscoelastic behaviours of rocks and minerals. The order is related to the exponents for the temporal scaling in the relaxation modulus and the stress power-law of strain rate, i.e., the non-Newtonian flow law, and considered as an indicator representing the macroscopic behaviour and microscopic dynamics of rocks.
7
Content available remote Sliding mode control of the DC drive with relative degree higher than one
Bisztyga B., Sieklucki G.
Przegląd Elektrotechniczny
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2012
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R. 88, nr 11a
38-42
EN
Certain problems concerning the higher order sliding mode (SM) controller have been presented. SM methodology has been applied to DC drive system. The control system consists of two control feedbacks in a cascade structure - conventional PI current controller and SM position and velocity controller. Due to the relative degree of sliding function is higher than one, the second order sliding controller has been suggested. To verify the theoretical results, simulation studies have been carried out.
PL
W artykule przedstawiono zagadnienia syntezy regulatora ślizgowego wyższego rzędu. Metodę zastosowano dla układu napędowego prądu stałego o kaskadowej strukturze regulacji. Zaproponowano konwencjonalny PI regulator prądu, natomiast regulator ślizgowy został określony dla pętli regulacyjnej położenia i prędkości kątowej. Stopień relatywności dla funkcji ślizgowej jest większy od jeden, zastosowano więc regulator ślizgowy drugiego rzędu. Metodę zweryfikowano za pomocą badań symulacyjnych.
8
Content available remote Pseudogroups of transformations on antidiscrete topological spaces
Lipińska J.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2010
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Vol. 9, nr 2
131-133
EN
In this paper we show that groups of transformations defined on antidiscrete topological spaces are the only ones among pseudogroups. It means that if a topological structure of a pseudogroup is the weakest then an algebraic structure of this pseudogroup is a group.
9
Content available remote Some remarks on origami and its limitations
Maszczyk T., Świrszcz G.
Demonstratio Mathematica
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2008
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Vol. 41, nr 2
473-480
EN
From a mathematical point of view the Japanese art of Origami is an art of finding isometric injections of subsets of R2 into R3. Objects obtained in this manner are developable surfaces and they are considered to be fully understood. Nevertheless, until now it was not known whether or not the local shape of the Origami model determines the maximum size and shape of the sheet of paper it can be madę of. In the present paper we show that it does. We construct a set [...] containing the point (O, 1/2) and an isometry [...]such that for every neighborhood [...] restricted to u cannot be extended to an isometry of the set [...] into R3. We also prove that all the singularities of an Origami model are of the same type - there can appear only cones.
10
Content available remote Pseudo-Riemannien metric of the signature (n,n) associated with a symplectic form on a 2n-dimensional manifold
Grochulski J.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2007
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Vol. 6, nr 1
65-68
EN
In the paper the pseudo-Riemannien structure of the signature (n, n) has been applied a connection of this symplectic structure.
11
Content available remote Data probes, vertical trajectories and classification: A tentative study
Pearson D. W.
International Journal of Applied Mathematics and Computer Science
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2007
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Vol. 17, no 1
107-112
EN
In this paper we introduce a method of classification based on data probes. Data points are considered as point masses in space and a probe is simply a particle that is launched into the space. As the probe passes by data clusters, its trajectory will be influenced by the point masses. We use this information to help us to find vertical trajectories. These are trajectories in the input space that are mapped onto the same value in the output space and correspond to the data classes.
12
Content available remote Bundles of contact elements on fibered fibered manifolds and the flow operator
Mikulski W., Tomas J.
Demonstratio Mathematica
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2007
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Vol. 40, nr 2
485-493
EN
We define the concept of a fibered fibered (k1, k2, l1, l2)- contact element of order (r1, . . . , r8) for r8 > r4 < r5 > r3, r8 > r6 < r7 > r2 and r1 < ri for ri = 2, 3, . . . , 8. For k1 < m1, k2 < m2, l1 < n1, l2 < n2, we define a bundle func tor Kr1,...,r8/ k1,k2,l1,l2 defined on the category FM2 m1,m2,n1,n2 of (m1,m2, n1,n2)-fibere fibered manifolds. We prove that the only natural transformation on the bundle functor Kr1,...r8/ k1,k2,l1,l2 is the identity one. Moreover, we prove that any natural operator lifting projectable vector fields Y to Kr1,...r8/ k1,k2,l1,l2 Y is a constant multiple of the flow operator.
13
Content available remote On naturality of the formal Euler operator
Mikulski W.
Demonstratio Mathematica
|
2005
|
Vol. 38, nr 1
235--238
EN
That all natural operators of the type of formal Euler operator from the variational calculus are the constant multiples of the formal Euler operator is deduced.
14
Content available remote K-differential spaces as sheaves. Induced and coinduced structure
Piątkowski A.
Demonstratio Mathematica
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2005
|
Vol. 38, nr 2
495--506
EN
Any K-differential space determines the sheaf of germs of its elements. We consider the K-differential space induced and coinduced by a single mapping. In this special case, we compare the sheaf of germs of the induced K-differential space with the inverse image of the sheaf of germs of a given K-differential space, and the sheaf of germs of the coinduced K-differential space with the image of the sheaf of germs of a given K-differential space.
15
Tensor analysis of dislocation-stress relationship based on the extended deformation gradient
Yamasaki K.
Acta Geophysica Polonica
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2005
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Vol. 53, nr 1
1-12
EN
The dislocation density used to estimate the magnitude of paleostress in rocks has been expressed in terms of a scalar quantity. Dislocations are classified into two types: edge dislocations and screw dislocations. However, the scalar expression of dislocations does not contain information on the type of dislocations. Therefore, we cannot see the effect of stress on the type of dislocations. In other words, we can extract the information related to the magnitude but not the orientation from previous dislocation-stress relationship. Then, we attempted to derive the tensor equation for dislocation-stress field. For this analysis, we introduced the extended deformation gradient tensor, that is, a differential geometrical expression of the ordinary deformation gradient tensor. We assumed that: (1) the higher order terms and spatial derivatives of dislocation density can be ignored; (2) the material is isotropic. We found that our tensor equation for dislocation-stress field is the square root expression of the equation derived from the experimental data of aluminum under static tension. Moreover, we found that the type of dislocation affects the stress field through the difference in the value of coefficients of the dislocation-stress relation-ship.
16
Transformacja wejście-wyjście układu nieliniowego przez sprzężenie zwrotne metodą geometryczną
Zawadzki A.
Zeszyty Naukowe Politechniki Świętokrzyskiej. Elektryka
|
2004
|
Z. 41
163-172
PL
W pracy przedstawiono zastosowanie elementów geometrii różniczkowej do budowy transformacji linearyzującej układynieliniowe do postaci liniowej (bądź quasi-liniowej). Zaprezentowano metodę linearyzacji dotyczącą grupy zadań okreslonych jako transformacja wejście-wyjście układu i opartą na transformacji przestrzeni stanu oraz wprowadzeniu do układu sprzężenia zwrotnego. Metodę linearyzacji zbudowaną w oparciu o elementy geometrii różniczkowej zastosowano do modelu matamatycznego nieniowego obwodu elektrycznegon trzeciego rzędu z wyróżnionym wyjściem.
EN
In the paper, approaching of differential geometry elements to linearizing transformation of nonlinar system to linear (or quasi-linear) form has been shown. Presented method concerns group of tasks defined as input-output transformation. It is connected with the state space transformation and with introduction to feedback system. The paper presents approaching of linearization methods, supported on geometrical methods, to mathematic model of nonlinear, electrical, 3rd rank system with distinguished output.
17
Content available remote On the leaves of a prefoliation of a K-differential space
Piątkowski A.
Demonstratio Mathematica
|
2004
|
Vol. 37, nr 3
689--696
EN
The definition of a prefoliation (M,F) of a K-differential space and the theorem about regularity of the inclusion of a leaf of a prefoliation are reminded. An example of a pair (M, F) of SC-differential spaces with the same set of points, which shows that even if the identity of .M is an immersion F -> M and (top M, top F) is a topological foliation in the sense of Ehresmann then (M, F) has not to be a prefoliation, is given. In the end, we show that if L is a proper leaf of a prefoliation (M, F) then the both structures of a K-differential spaces coincide on L.
18
Content available remote Horizontal extension of connections into (2)-connections
Doupovec M., Mikulski W. M.
Demonstratio Mathematica
|
2004
|
Vol. 37, nr 4
963--975
EN
We discuss the prolongation of connections to to some non product preserving bundles. We introduce the concept of (r)-connection on a fibered manifold Y and for a given connection F on Y we construct its horizontal extension F(2). We also prove that F(2 ) is the unique (2)-connection on Y canonically dependent on F.
19
Content available remote A framed f(3,-1) structure on the tangent bundle of a lagrange space
Gîrţu M.
Demonstratio Mathematica
|
2004
|
Vol. 37, nr 4
955--961
EN
For a tangent bundle (TM,r,M), the kernel of the differential r* of the projection r defines the vertical subbundle VTM of the bundle (TTM, rTM , TM). A supplement HTM of it is called a horizontal subbundle or a nonlinear connection on M, (R. Miron and M. Anastasiei, [5]). The direct decomposition TTM = HTM VTM gives rise to a natural almost product structure P on the manifold TM. A general method to associate to P a framed f(3, -l)-structure of any corank is pointed out. When we endow M with a regular Lagrangian L and use as the nonlinear connection that canonically induced by L, a framed f(3, -l)-structure P2 of corank 2 naturally appears on TM. This reduces to that found by us in [3] when L = F2 , for F the fundamental function of a Finsler space Fn = (M,F). Then we show that on some conditions for L the restriction of P2 to the submanifold L = 1 of TO M is an almost paracontact structure on this submanifold. The conditions taken on L hold for the -Lagrangians introduced by P.L.Antonelli and D. Hrimiuc in [2] as well as for L = F2.
20
Modeling of material continua with defected and ordered microstructures
Rymarz Cz.
Journal of Technical Physics
|
2002
|
Vol. 43, no 3
277-285
EN
The paper contains a short presentation of some models of material continua with microstructure applying the methods of differential geometry. Two main kinds of microstructure are considered: dislocation, disclination, and unholonomic defects in plasticity, nematic liquid crystal as an example of oriented medium. The aim of this presentation is to illustrate the effectiveness of the methods of differential geometry in construction of the theory of microstructural media.
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