Search

Aim: This work presents a method for developing a simplified but efficient model of the complex human hand kinematics with the aim of its implementation in rehabilitation robotics. Material and methods: The approach incorporates modularity by simplifying the available model comprising 24 degrees of freedom (DOFs) to 9 DOFs, with the introduction of additional joint coupling parameters specific to different grasp types. The effect of dependent joints to the ranges-of-motion (ROMs) of the model is investigated and compared to the anatomical one. The index, middle, ring and little finger solutions to forward and inverse kinematics problems are then acquired. The implementation of the model, based on the median male bones dimensions, is made available in the open-source Robot Operating System (ROS) framework. Results: By including additional four inclination angles per finger, the devised kinematic hand model encompasses also finger curvatures, resulting in significant positioning accuracy improvements compared to the conventional model. The used 3D spatial position improvement metrics are the mean absolute (MAE) and mean relative errors (MRE). The dependent joint position MAEs range from 0.22 to 0.34 cm, while MREs range from 2.8 and 3.5 %, whereas the highest absolute and relative errors during fingertip positioning can reach 0.5 cm and 10.5 %, respectively. Conclusion: The performed investigation allowed establishing that by modelling finger curvature and assuring the adaptability of the model to a variety of human hands and rehabilitation modalities through joint dependency, represents the best approach towards a relatively simple and applicable rehabilitation model with functional human-like hand movements., Cilj: U radu se prezentira metoda za razvoj pojednostavljenog ali učinkovitog kinematičkog modela ljudske ruke koji će se implementirati u rehabilitacijskoj robotici. Materijali i metode: Pristup se temelji na pojednostavljenju postojećeg modela koji ima 24 stupnja slobode gibanja (DOF) na 9 DOF-a, uz uvođenje dodatnih konstrukcijskih parametara, specifičnih za različite vrste hvata. Nakon što je istražen utjecaj ovisnih zglobova na raspon pokreta (ROM) modela, te je isti uspoređen s anatomskim modelom, dobivena su rješenja problema inverzne i direktne kinematike kažiprsta, srednjeg prsta, prstenjaka i malog prsta. Implementacija modela, temeljena na dimenzijama muških kostiju koje odgovaraju medijanu muške populacije, ostvarena je pomoću programskog okruženja otvorenog koda Robot Operating System (ROS). Rezultati: Uzimanjem u obzir četiriju dodatnih kutova nagiba po prstu, razvijeni model ruke obuhvaća i zakrivljenost prstiju, što omogućuje povećanje točnosti pozicioniranja u usporedbi s konvencionalnim modelom. Kao mjerila za utvrđivanje povećanja točnosti prostornog pozicioniranja, korištene su srednja apsolutna greška (MAE) i srednja relativna greška (MRE). MAE se, ovisno o položaju zgloba, kreće od 0,22 do 0,34 cm, dok se MRE kreće od 2,8 do 3,5 %. Najveće apsolutne i relativne greške tijekom pozicioniranja vrha prsta mogu, pak, doseći 0,5 cm, odnosno 10,5 %. Zaključak: Zaključuje se da modeliranje zakrivljenosti prsta i osiguranje prilagodljivosti modela različitim pacijentima i rehabilitacijskim modalitetima kroz međuovisnost zglobova, predstavlja najbolji pristup dobivanju relativno jednostavnog, primjenjivog i funkcionalnog modela šake.

U radu su provedene numeričke simulacije hidrauličkih tranzijenata u sustavu crpne hidroelektrane (CHE) Fužine pomoću nestacionarnog 1D modela strujanja pod tlakom u cijevi. Primjenom nekonzervativne formulacije modela nestacionarnog strujanja tekućine u cijevi pod tlakom i fluks-limitirane numeričke metode drugog reda postignuta je zadovoljavajuća točnost modela. Dobiveni numerički rezultati su pokazali dobro slaganje s izmjerenim podacima promatranog sustava., Hydraulic transients in pumping hydroelectric power plant (PHPP) Fužine were simulated with 1D unsteady pipe flow model. High accuracy of the model was accomplished with the use of nonconservative formulation of unsteady pipe flow model incorporating unsteady friction model and second order flux limited numerical scheme. Simulation results show very good agreement between the computed and measured piezometric heads, both in amplitude and frequency of the oscillation.

Nowadays, the possibilities of predicting the behavior of different dynamical systems are in the focus of the research in many scientific disciplines. In this paper we consider the dynamical systems generated by chosen nonlinear and non-autonomous ordinary differential equations. These dynamical systems are approximated by using the Koopman operator based data-driven algorithms applied on the observable functions evaluated at some finite set of the system states. The learned model is then used as a prediction tool to envisage the values of the observable functions in the future system states. The aim of this work is to test and analyze the prediction capabilities of such data-driven algorithms on simple examples in order to make a foundation for understanding their behavior in the prediction task in more complex systems, even in those for which the associated differential equations modeling the system are not known.

The goal of this paper is to analyze various dynamical systems using algorithms based on the Koopman operator. The Koopman operator provides an alternative approach to describing the evolution of a dynamical system. It is a linear infinite-dimensional operator, so the analysis is based on the study of the spectral properties of this operator. The Koopman operator-based data-driven algorithms make it possible to identify the spectral elements of the Koopman operator from data. They can then be used to analyze the behavior of the system under consideration, to reconstruct or predict the system dynamics, and as a tool to reduce the dimensionality of the data provided. In this work, the algorithms are applied to data obtained from various dynamical systems, e.g., data from HVAC systems, fluid flow systems, etc. The corresponding discussions and analyzes were performed on the observed examples.

Razumijevanje transporta čestica fluida značajno je u mnogim prirodnim i industrijskim tokovima. U praksi se analiza miješanja fluida često temelji na aproksimaciji Lagrangeovih koherentnih struktura. Jedan od načina je mjerenje Lagrangeove deformacije pomoću mezohiperboličkih polja brzina. U ovom se radu promatra tok fluida definiran vektorskim poljem brzina uz prisustvo nesigurnosti. Trajektorije gibanja pod utjecajem nesigurnosti u vektorskom polju dobivaju se rješavanjem stohastičkih diferencijalnih jednadžbi, te se rezultati dobivenih simulacija koriste za računanje mezohiperboličkih polja brzina. Dobiveni rezultati uz prisustvo nesigurnosti uspoređeni su s rezultatima bez prisutnosti nesigurnosti te je provedena statistička analiza njezina utjecaja. Rezultati pokazuju da su za umjerene nesigurnosti u vektorskom polju koherentne strukture dosta stabilne. Kao što je i očekivano, utjecaj nesigurnosti veći je u nestacionarnom slučaju.

In practice, a complex dynamical system often include some time-dependent or random forcing so that it can not be modeled as autonomous system. Therefore, the extension of the Koopman operator theory to nonautonomous and stochastic systems is of great importance. It brings a new viewpoint to the existing theory of nonautonomous and random dynamical systems, particularly, with application of Koopman operator-based data-driven algorithms. We consider the nonautonomous Koopman operator family based on the skew-product definition of the nonautonomous dynamical system. Similarly, we introduce the stochastic Koopman operator family associated with the given random dynamical system. In both cases, we state basic properties of the operator and discuss the application of Koopman-operator based algorithms on the considered systems. The performance of the algorithms is illustrated in the context of discovering the spectral objects of the system, as well as, on the reconstruction and prediction of the system behaviour.

The problem of prediction of behavior of dynamical systems has undergone a change in the second half of the 20th century with the discovery of the possibility of chaotic dynamics in simple dynamical systems. However, that approach does not account for another type of unpredictability: the ``black swan" event. In our framework, the black-swan-type dynamics occurs when an underlying dynamical system becomes coupled to, or decoupled from, another one. Here we explore the problem of prediction in systems that exhibit such behavior. The mathematical theory and algorithms we use are based on an operator-theoretic approach in which the dynamics of the system are embedded into an infinite-dimensional space. We show that the framework correctly identifies a black swan event. Moreover, we show that the algorithms we developed enabled a successful prediction of the flu season, and prediction in other complex dynamics datasets such as physiology models.

A generalization of the Koopman operator framework, originally developed for deterministic dynamical systems, to discrete and continuous time random dynamical systems (RDS) results with the stochastic Koopman operators. We provide the results that characterize the spectrum and the eigenfunctions of the stochastic Koopman operator associated with different types of linear RDS. Then we consider the RDS for which the associated Koopman operator family is a semigroup, especially those for which the generator can be determined. We propose different approaches for using the data-driven DMD algorithms in the stochastic framework to approximate numerically the spectral objects (eigenvalues, eigenfunctions) of the stochastic Koopman operator. We prove that, under certain assumptions, the outputs of the stochastic Hankel DMD (sHankel-DMD) algorithm converge to the true stochastic Koopman eigenvalues and eigenfunctions. We apply the methodology to a variety of examples, revealing objects in spectral expansions of the stochastic Koopman operator.

A generalization of the Koopman operator framework, originally developed for de- terministic dynamical systems, to discrete and continuous time random dynamical systems results with the stochastic Koopman operators. The study of the spectral objects of these operators (Koopman eigenvalues, eigenfunctions and modes) could be useful in analysing the behaviour of the considered random dynamical system. We provide the results that characterize the spectrum and the eigenfunctions of the stochastic Koopman operators for the particular classes of random and stochastic dynamical systems. The numerical approximations of the eigenvalues and eigen- functions of the stochastic Koopman operator are computed by using DMD RRR algorithm. Its behaviour in the stochastic framework is explored on several test examples. Furthermore, we introduce the isostables and isochrones associated to the random dynamical systems and compute their numerical approximations on the chosen example.

U jeku napretka u nedavnim istraživanjima na polju teorije Koopmanovog operatora, područje dekompozicije dinamičkih sustava na Koopmanove modove postaje vrlo atraktivno i primjenjivo na mnoge realne probleme. Opstrujavanje cilindra, kako mirujućeg tako i rotirajućeg, je naizgled jednostavna tema koja, međutim, ima mnogo primjena u stvarnim inženjerskim problemima pa je zbog toga iscrpno istraživana u području mehanike fluida. U ovom radu je strujno polje oko rotirajućeg cilindra, dobiveno pomoću računalne simulacije strujanja, promatrano kao periodički dinamički sustav. Dekompozicijom takvog sustava na tzv. Koopmanove modove može se postići vrlo precizna redukcija modela i bolji uvid u dinamiku promatranog strujanja kroz frekvencijski spektar promatranog sustava. Prezentirani rezultati prikazuju spektar i prostorni utjecaj pojedinih modova, te daju ocjenu i usporedbu greške rekonstrukcije strujnog polja koristeći reducirani model za strujanje oko rotirajućeg cilindra.

We propose new methods for the evaluation of eigenvalues λ(t, t0)of the Koopman operator family K(t, t0), t>t0 of the non-autonomous dynamic systems. The first step in the development is a new data-driven method for very accurate evaluation of eigenvalues λ(t, t0), t>t0 in the hybrid linear non-autonomous case. Then, the approach is extended to continuous linear non-autonomous systems and non-autonomous systems in general. We also propose a relationship between eigenvalues λ(t, t0), t>t0 and eigenvalues computed by Arnoldi-like methods on large sets of snapshots. We apply the new approach to baroreflex physiology, i.e. to the resonant breathing which is used in PTSD treatment. For the resonant breathing we have both data and parameterized mathematical model. The model incorporates a delay and thus is infinite dimensional, hybrid, with a stochastic input. Its asymptotic dynamics is close to quasi-periodic. The applied new methods give an improved insight to the eigenvalues of the related Koopman operator family K(t, t0).

In this work we consider the nonstationary shear flow between two parallel solid and thermoinsulated horizontal plates, with the upper one moving irrotationally. The fluid is compressible, micropolar, viscous and heat- conducting, as well as is in thermodynamical sense perfect and polytropic. We assume that, given a Cartesian coordinate system x, y and z, solutions of corresponding problem are x- dependent only. Mathematical model is derived in the Lagrangian description. By using the Faedo-Galerkin method, as well as homogenization of boundary conditions, we derive an approximate system which we use to obtain a numerical solution to the given problem.

We consider the nonstationary 3D flow of a compressible viscous heat-conducting micropolar fluid in the domain to be a subset of R3, bounded with two concentric spheres. In the thermodynamical sense the fluid is perfect and polytropic. The homogeneous boundary conditions for velocity, microrotation, heat flux and spherical symmetry of the initial data are proposed. Due to the assumption of spherical symmetry, the problem can be considered as one– dimensional problem in Lagrangian description on the domain that is a segment. We define the approximate equations system by using the finite difference method and construct the sequence of approximate solutions for our problem. By analyzing the properties of these approximate solutions we prove their convergence to the generalized solution of our problem globally in time and establish the convergence of the defined numerical scheme, which is the main result of the paper. The practical application of the proposed numerical scheme is performed on the chosen test example.

We consider initial and homogeneous boundary problem for the model of nonstationary 3D flow of a compressible viscous and heat-conducting micropolar fluid with cylindrical symmetry in the Lagrangian setting. The fluid domain is the subset of R^3 bounded with two coaxial cylinders that present solid thermoinsulated walls. We assume that the fluid is in the thermodynamical sense perfect and polytropic, as well as that initial functions are radially dependent only. We apply the Faedo-Galerkin method to the problem with given initial functions. We construct and analyze the approximate numerical solution for this problem.

In this work the nonstationary flow of the compressible viscous and heat-conducting micropolar fluid is studied. We focus on the fluid flow between a static impermeable solid wall and a free boundary connected to a vacuum state. In the corresponding initial-boundary value problem we suppose that the homogeneous boundary conditions for velocity, microrotation and heat flux on the solid border are valid and that the normal stress, heat flux and microrotation are equal to zero on the free boundary. The aim of this work is to define the convergent numerical scheme for this type of problem and to analyze some numerical solutions. We use the finite difference approach on the staggered mesh and apply it to the model in Lagrangian description to avoid the difficulties associated with the moving boundary in Eulerian coordinate system. It has been shown that the defined scheme is convergent. The numerical solutions obtained by the proposed scheme on the chosen test examples are presented.

In this paper we consider the nonstationary 1D flow of the compressible viscous and heat-conducting micropolar fluid, assuming that it is in the thermodynamical sense perfect and polytropic. The homogeneous boundary conditions for velocity, microrotation and heat flux are introduced. In order to construct the sequence of approximate solution for our problem, we define the finite difference approximate equations system. We investigate the properties of these approximate solutions and establish their convergence to the strong solution of our problem globally in time. A numerical experiment is performed by solving the defined approximate ordinary differential equations system using strong-stability preserving (SSP) Runge-Kutta schemes for time discretization.

We define a finite difference method for the nonstationary 1D flow of the compressible viscous and heat-conducting micropolar fluid, assuming that it is in the thermodynamical sense perfect and polytropic. The homogeneous boundary conditions for velocity, microrotation and heat flux are proposed. The sequence of approximate solution for our problem is constructed by using the defined finite difference approximate equations system. We investigate the properties of these approximate solutions and establish their convergence to the strong solution of our problem globally in time. Numerical experiments are performed by solving the defined approximate ordinary differential equations system using strong-stability preserving (SSP) Runge-Kutta scheme for time discretization.

The irruption of gas and oil into the Gulf of Mexico during the Deepwater Horizon event fed a deep sea bacterial bloom that consumed hydrocarbons in the affected waters, formed a regional oxygen anomaly, and altered the microbiology of the region. In this work we describe the numerical results obtained from the proposed coupled physical-metabolic model to assess the impact of mixing processes on these deep ocean bacterial communities and their capacity for hydrocarbon and oxygen use.

Weighted essentially non-oscillatory (WENO) procedure is used for a non– oscillatory approximation of a function. The typical WENO procedure con- structs a rational approximation of a function v(x) based on known cell aver- ages of function v(x). This approximation is essentially non-oscillatory and high order accurate for smooth enough function v(x). However, the approximating rational function obtained by the classical WENO reconstruction contains poles. Therefore, if one needs to obtain a value of function v(x) in the interior of the numerical cell, the standard WENO procedure neads special treatment to over- come instabilities that can occur. We will present a new polynomial version of the WENO procedure that provides an elegant solution to this problem by constructing an approximating polynomial that is smooth, non-oscillatory and high order accurate not only at the cell boundaries but in the entire numerical cell.

Weighted essentially non-oscillatory (WENO) approximation has been mainly used in numerical schemes for solving hyperbolic partial differential equations. The typical WENO procedure constructs a rational function for approximating values of the function v(x) on cell boundaries from its known cell averages. This approximation is essentially non-oscillatory and high order accurate for smooth enough functions v(x). However, the approximating rational function obtained by the classical WENO reconstruction contains poles. Therefore, if one needs to reconstruct the value of function v(x) in the interior of the numerical cell, the standard WENO procedure fails. In this paper we have developed a polynomial version of the WENO procedure that provides an elegant solution to this problem by constructing an approximating polynomial that is non-oscillatory and high order accurate not only at the cell boundaries but in the entire numerical cell. The proposed new algorithm is computationally comparable to the standard WENO procedure and gives the same reconstruction values on the the cell boundaries. The obtained numerical results show that the newly proposed procedure performs very well on the considered test examples.