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1
Content available remote Explicit rational group law on hyperelliptic Jacobians of any genus
Urbanik David
Bulletin of the Polish Academy of Sciences. Mathematics
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2023
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Vol. 71, no 1
35--52
EN
It is well-known that abelian varieties are projective, and so there exist explicit polynomial and rational functions which define both the variety and its group law. It is however difficult to find any explicit polynomial and rational functions describing these varieties or their group laws in dimensions greater than two. One exception can be found in Mumford’s classic “Lectures on Theta”, where he describes how to obtain an explicit model for hyperelliptic Jacobians as the union of several affine pieces described as the vanishing locus of explicit polynomial equations. In this article, we extend this work to give explicit equations for the group law on a dense open set. One can view these equations as generalizations of the usual chord-based group law on elliptic curves.
2
Content available Modified linear-quadratic regulator used for controlling anti-tank guided missile in vertical plane
Nocoń Łukasz, Koruba Zbigniew
Journal of Theoretical and Applied Mechanics
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2020
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Vol. 58 nr 3
723--732
EN
The paper concerns the issue of optimum control of the strongly non-linear dynamic system, i.e. Anti-Tank Guided Missile (ATGM). The linear-quadratic regulator (LQR) was used to provide control capabilities. In order to use the classic LQR, the dynamics of the object must be presented in the form of a linear-stationary model. This is not possible in the case of the considered missile, mostly due to mass changing in time (intensive consumption of fuel) and varying aerodynamic conditions depending on the Mach number Ma. Thus, we are dealing with a non-stationary system. Moreover, state variables are frequently involved in complex functions, which do not allow one to separate coefficients related to state variables very easily. In order to linearize such a complex system, the paper uses Jacobian, as the matrix of state, calculated at each time instant. The automatic pilot of the ATGM, using the LQR method, determines the signals controlling the angles of flight control surfaces and the thrust vector using continuously calculated Jacobians. The paper presents the algorithm for the ATGM control.
3
Content available remote Jacobians of Hyperelliptic Curves over ℤn and Factorization of n
Dryło Robert, Pomykała Jacek
Fundamenta Informaticae
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2019
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Vol. 169, nr 4
275--283
EN
E. Bach showed that factorization of an integer n can be reduced in probabilistic polynomial time to the problem of computing exponents of elements in ℤn* (in particular the group order of ℤ*n). It is also known that factorization of square-free integer n can be reduced to the problem of computing the group order of an elliptic curve E/ℤn. In this paper we describe the analogous reduction for computing the orders of Jacobians over ℤn of hyperelliptic curves C over ℤn using the Mumford representation of divisor classes and Cantor’s algorithm for addition. These reductions are based on the group structure of the Jacobian. We also propose other reduction of factorization to the problem of determining the number of points |C(ℤn)|, which makes use of elementary properties of twists of hyperelliptic curves.
4
Content available Some remarks to the Jacobian conjecture
Lara-Dziembek S., Biernat G., Pawlak E.
Journal of Applied Mathematics and Computational Mechanics
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2017
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Vol. 16, nr 1
87--96
EN
This work is related to the Jacobian Conjecture. It contains the formulas concerning algebraic dependence of the polynomial mappings having two zeros at infinity and the constant Jacobian. These relations mean that such mappings are non-invertible. They reduce the Jacobian Conjecture only to the case of mappings having one zero at infinity. This case is already solved by Abhyankar. The formulas presented in the paper were illustrated by the large example.
5
Content available The non-Keller mapping with one zero at infinity
Biernat G., Ciekot A.
Journal of Applied Mathematics and Computational Mechanics
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2016
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Vol. 15, nr 4
5--10
EN
In this paper the polynomial mapping of two complex variables having one zero at infinity is considered. Unlike with Keller mapping, if determinant of the Jacobian of this mapping is constant then it must be zero.
6
Content available The Jacobians of non-maximal degree
Lara-Dziembek S., Biernat G., Pawlak E.
Journal of Applied Mathematics and Computational Mechanics
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2016
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Vol. 15, nr 4
99--104
EN
In the article the leading forms of the polynomial mapping having the Jacobians of non-maximal degree are considered. In particular, the mappings having two zeros at infinity are discussed.
7
Content available An example of non-Keller mapping
Pawlak E., Lara-Dziembek S., Biernat G., Woźniakowska M.
Journal of Applied Mathematics and Computational Mechanics
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2016
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Vol. 15, nr 1
115--121
EN
In the paper a nontrivial example of non-Keller mapping is considered. It is shown that the Jacobian of rare mapping, having one zero at infinity, being constant must vanish.
8
Content available A second example of non-Keller mapping
Lara-Dziembek S., Biernat G., Pawlak E., Woźniakowska M.
Journal of Applied Mathematics and Computational Mechanics
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2016
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Vol. 15, nr 2
65--70
EN
In the article the next nontrivial example of non-Keller mapping having two zeros at infinity is analyzed. The rare mapping of two complex variables having two zeros at infinity is considered. In the article it has been proved that if the Jacobian of the considered mapping is constant, then it is zero.
9
Content available Design of inverse kinematics algorithms: extended Jacobian approximation of the dynamically consistent Jacobian inverse
Ratajczak J.
Archives of Control Sciences
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2015
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Vol. 25, no. 1
35--50
EN
The paper presents the approximation problem of the inverse kinematics algorithms for the redundant manipulators. We introduce the approximation of the dynamically consistent Jacobian by the extended Jacobian. In order to do that, we formulate the approximation problem and suitably defined approximation error. By the minimization of this error over a certain region we can design an extended Jacobian inverse which will be close to the dynamically consistent Jacobian inverse. To solve the approximation problem we use the Cholesky decomposition and the Ritz method. The computational example illustrates the theory.
10
Content available remote Analiza kinematyki manipulatora o pięciu stopniach swobody
Gierlak P.
Zeszyty Naukowe Politechniki Rzeszowskiej. Mechanika
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2014
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z. 86 [290], nr 4
491--500
PL
W artykule przestawiono analizę kinematyki manipulatora o pięciu stopniach swobody na przykładzie jednostki kinematycznej robota manipulacyjnego Scorbot-ER 4pc. Do opisu kinematyki układu zastosowano notację Denavita-Hartenberga. Przyjęto schemat kinematyki manipulatora i podano parametry opisujące układ. Zapisano odpowiednie macierze transformacji, które zastosowano w dalszej analizie. Wyznaczono jakobian analityczny manipulatora oraz jakobian geometryczny w ciele i przeprowadzono analizę osobliwości. Są to takie konfiguracje manipulatora, w których wyznaczenie rozwiązania zadania odwrotnego kinematyki jest znacznie utrudnione, a przy zastosowaniu klasycznych metod – niemożliwe. Dlatego znajomość konfiguracji osobliwych jest niezbędna w celu poprawnego planowania i generowania trajektorii manipulatora. Zaprezentowana metodyka jest uniwersalna i może być stosowana do analizy kinematyki manipulatorów o innej strukturze kinematycznej niż zaprezentowana w niniejszej pracy.
EN
In the paper the kinematics analysis of 5 degrees of freedom manipulator is presented. The analysis was realised for the Scorbot-ER 4pc robotic manipulator. To describe the kinematics of the manipulator the Denavit-Hartenberg notation is used. The kinematics scheme and parameters of the manipulator as well as appropriate transformation matrices, that were used in the further analysis, are given. The analytical Jacobian of the manipulator and the geometrical Jacobian in the body are determined and the analysis of singularities is realised. In singular manipulator’s configurations the solution of the inverse kinematics problem is very difficult, and using classical methods – impossible. Therefore, knowledge of the singular configurations is necessary for the proper planning and generating the trajectory of the manipulator. The presented methodology is universal and can be used to analyze the kinematics of manipulators with a other kinematic structure that is not presented in this paper.
11
Content available About some properties of jacobian for polynomial mappings of two complex variables
Gawroński B., Jankowska J., Moltzan R., Włodarczyk I., Biernat G.
Journal of Applied Mathematics and Computational Mechanics
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2013
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Vol. 12, nr 4
41--46
EN
In our article we consider jacobian Jac(f,h) of polynomial mapping f = Xk Yk +…+ f1, h = Xk–1 Yk–1 +…+ h1. We give conditions for coordinate h in which constant jacobian Jac(f,h) = Jac(f1,h1) vanishes.
12
Content available Loads of Lower Limb Joints During Bicycle Ride
Stępniewski A.
Acta Mechanica et Automatica
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2012
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Vol. 6, no. 2
84-87
EN
This work features a geometric and static analysis of the lower limb during bicycle ride. Basic dimensions are indicated, which are necessary for the description of movement geometry. The simplified flat model was adopted for the analysis. Using the transformation of Denavit-Hartenberg frames, vectors were developed for the position of the rotation axis of the joints in immovable frame. The inverse kinematics problem was solved. The course of displacement changes in the ankle joint was adopted as an angle function for crankset position, based on experimental research results, published in professional literature. The course was approximated with fifth grade polynomial. Joint displacements and loads were established. A sample calculation is presented, illustrating the subject computational algorithm.
13
Content available remote Kinematics of mobile manipulators : a control theoretic perspective
Tchoń K., Jakubiak J., Muszyński R.
Archives of Control Sciences
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2001
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Vol. 11, no. 3/4
195-221
EN
A mobile manipulator is a robotic system composed of a mobile platform and a manipulator mounted atop of the platform. From control theoretic viewpoint the kinematics of the mobile manipulator can be represented by means of a driftless control system with outputs. Assuming this kind of representation we define basic concepts concerned with the kinematics of mobile manipulators and develop a consistent theory involving these concepts in a way completely analogous to the existing theory of stationary manipulators. A key ingredient of our approach is a concept of endogenous configuration of a mobile manipulator that comprises a control function of the platform and a joint position of the manipulator. Relying on this concept we introduce the instantaneous kinematics, analytic Jacobian, regular and singular configurations, a Jacobian pseudoinverse, a dexterity matrix and a dexterity ellipsoid. Then we formulate the inverse kinematic problem (the motion planning problem), and derive two exemplary algorithms: the Jacobian pseudoinverse (Newton) algorithm and the Jacobian adjoint (Jacobian transpose) algorithm, that are applicable at regular configurations. In a vicinity of singular configurations a version of the singularity robust pseudoinverse is provided. Dexterity ellipsoids and inverse kinematics algorithms are illustrated with computer simulations.
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