Добірка наукової літератури з теми "Homotopy technical"

AbstractWe describe the structure of the space of second order elliptic differential operators on a homogenous bundle over a compact Lie group. Subject to a technical condition, these operators are homotopic to the Laplacian. The technical condition is further investigated, with examples givenwhere it holds and others where it does not. Since many spectral invariants are also homotopy invariants, these results provide information about the invariants of these operators.

Homotopy classes of trajectories, arising due to the presence of obstacles, are defined as sets of trajectories that can be transformed into each other by gradual bending and stretching without colliding with obstacles. The problem of exploring/finding the different homotopy classes in an environment and the problem of finding least-cost paths restricted to a specific homotopy class (or not belonging to certain homotopy classes) arises frequently in such applications as predicting paths for unpredictable entities and deployment of multiple agents for efficient exploration of an environment. In [Bhattacharya, Kumar, Likhachev, AAAI 2010] we have shown how homotopy classes of trajectories on a two-dimensional plane with obstacles can be classified and identified using the Cauchy Integral Theorem and the Residue Theorem from Complex Analysis. In more recent work [Bhattacharya, Likhachev, Kumar, RSS 2011] we extended this representation to three-dimensional spaces by exploiting certain laws from the Theory of Electromagnetism (Biot-Savart law and Ampere's Law) for representing and identifying homotopy classes in three dimensions in an efficient way. Using such a representation, we showed that homotopy class constraints can be seamlessly weaved into graph search techniques for determining optimal path constrained to certain homotopy classes or forbidden from others, as well as for exploring different homotopy classes in an environment. (This is a condensed, non-technical overview of work previously published in the proceedings of Robotics: Science and Systems, 2011 conference [Bhattacharya, Likhachev, Kumar, RSS 2011].)

SynopsisFor an H-space with a generating subspace, we construct a space whose K-cohomology is a direct sum of a truncated polynomial algebra and an ideal, which enables technical restrictions to be removed from several known results in the homotopy theory of H-spaces.

Abstract We prove that any category of props in a symmetric monoidal model category inherits a model structure. We devote an appendix, about half the size of the paper, to the proof of the model category axioms in a general setting. We need the general argument to address the case of props in topological spaces and dg-modules over an arbitrary ring, but we give a less technical proof which applies to the category of props in simplicial sets, simplicial modules, and dg-modules over a ring of characteristic 0. We apply the model structure of props to the homotopical study of algebras over a prop. Our goal is to prove that an object 𝑋 homotopy equivalent to an algebra 𝐴 over a cofibrant prop P inherits a P-algebra structure so that 𝑋 defines a model of 𝐴 in the homotopy category of P-algebras. In the differential graded context, this result leads to a generalization of Kadeishvili's minimal model of 𝐴∞-algebras.

Editor-in-Chief of the journal Thermal Science request that it is necessary to retract paper SOLUTION OF BURGERS' EQUATION APPEARS IN FLUID MECHANICS BY MULTISTAGE OPTIMAL HOMOTOPY ASYMPTOTIC METHOD DOI: https://doi.org/10.2298/TSCI23S1087W by Fuzhang WANG, Niaz ALI SHAH, Imtiaz AHMAD Hijaz AHMAD, Muhammad Kamran ALAM, and Phatiphat THOUNTHONG published in the journal Thermal Science, Vol. 27, Year 2023, Special Issue 1, S87-S92 since by technical error of the Editorial staff, this paper has already been published in the journal Thermal Science Vol.26, Part 1B, pp. 815-821, 2022 https://doi.org/10.2298/TSCI210302343W <br><br><font color="red"><b> Link to the retracted article <u><a href="http://dx.doi.org/10.2298/TSCI23S1087W">10.2298/TSCI23S1087W</a></b></u>

Abstract Shifted symplectic Lie and $L_{\infty }$ algebroids model formal neighborhoods of manifolds in shifted symplectic stacks and serve as target spaces for twisted variants of the classical topological field theory defined by Alexandrov--Kontsevich--Schwarz--Zaboronsky. In this paper, we classify zero-, one-, and two-shifted symplectic algebroids and their higher gauge symmetries, in terms of classical geometric “higher structures”, such as Courant algebroids twisted by $\Omega ^{2}$-gerbes. As applications, we produce new examples of twisted Courant algebroids from codimension-two cycles, and we give symplectic interpretations for several well-known features of higher structures (such as twists, Pontryagin classes, and tensor products). The proofs are valid in the $C^{\infty }$, holomorphic, and algebraic settings and are based on a number of technical results on the homotopy theory of $L_{\infty }$ algebroids and their differential forms, which may be of independent interest.

<p style='text-indent:20px;'>The main purpose of this paper is to study the global propagation of singularities of the viscosity solution to discounted Hamilton-Jacobi equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE333"> \begin{document}$ \begin{align} \lambda v(x)+H( x, Dv(x) ) = 0 , \quad x\in \mathbb{R}^n. \quad\quad\quad (\mathrm{HJ}_{\lambda})\end{align} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with fixed constant <inline-formula><tex-math id="M1">\begin{document}$ \lambda\in \mathbb{R}^+ $\end{document}</tex-math></inline-formula>. We reduce the problem for equation <inline-formula><tex-math id="M2">\begin{document}$(\mathrm{HJ}_{\lambda})$\end{document}</tex-math></inline-formula> into that for a time-dependent evolutionary Hamilton-Jacobi equation. We prove that the singularities of the viscosity solution of <inline-formula><tex-math id="M3">\begin{document}$(\mathrm{HJ}_{\lambda})$\end{document}</tex-math></inline-formula> propagate along locally Lipschitz singular characteristics <inline-formula><tex-math id="M4">\begin{document}$ {{\bf{x}}}(s):[0,t]\to \mathbb{R}^n $\end{document}</tex-math></inline-formula> and time <inline-formula><tex-math id="M5">\begin{document}$ t $\end{document}</tex-math></inline-formula> can extend to <inline-formula><tex-math id="M6">\begin{document}$ +\infty $\end{document}</tex-math></inline-formula>. Essentially, we use <inline-formula><tex-math id="M7">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-compactness of the Euclidean space which is different from the original construction in [<xref ref-type="bibr" rid="b4">4</xref>]. The local Lipschitz issue is a key technical difficulty to study the global result. As a application, we also obtain the homotopy equivalence between the singular locus of <inline-formula><tex-math id="M8">\begin{document}$ u $\end{document}</tex-math></inline-formula> and the complement of Aubry set using the basic idea from [<xref ref-type="bibr" rid="b9">9</xref>].</p>

We settle several fundamental questions about the theory of universal deformation quantization of Lie bialgebras by giving their complete classification up to homotopy equivalence. Moreover, we settle these questions in a greater generality: we give a complete classification of the associated universal formality maps. An important new technical ingredient introduced in this paper is a polydifferential endofunctor ${\mathcal {D}}$ in the category of augmented props with the property that for any representation of a prop ${\mathcal {P}}$ in a vector space $V$ the associated prop ${\mathcal {D}}{\mathcal {P}}$ admits an induced representation on the graded commutative algebra $\odot ^\bullet V$ given in terms of polydifferential operators. Applying this functor to the minimal resolution $\widehat {\mathcal {L}\textit{ieb}}_\infty$ of the genus completed prop $\widehat {\mathcal {L}\textit{ieb}}$ of Lie bialgebras we show that universal formality maps for quantizations of Lie bialgebras are in one-to-one correspondence with morphisms of dg props \[F: \mathcal{A}\textit{ssb}_\infty \longrightarrow {\mathcal{D}}\widehat{\mathcal{L}\textit{ieb}}_\infty \] satisfying certain boundary conditions, where $\mathcal {A}\textit{ssb}_\infty$ is a minimal resolution of the prop of associative bialgebras. We prove that the set of such formality morphisms is non-empty. The latter result is used in turn to give a short proof of the formality theorem for universal quantizations of arbitrary Lie bialgebras which says that for any Drinfeld associator $\mathfrak{A}$ there is an associated ${\mathcal {L}} ie_\infty$ quasi-isomorphism between the ${\mathcal {L}} ie_\infty$ algebras $\mathsf {Def}({\mathcal {A}} ss{\mathcal {B}}_\infty \rightarrow {\mathcal {E}} nd_{\odot ^\bullet V})$ and $\mathsf {Def}({\mathcal {L}} ie{\mathcal {B}}\rightarrow {\mathcal {E}} nd_V)$ controlling, respectively, deformations of the standard bialgebra structure in $\odot V$ and deformations of any given Lie bialgebra structure in $V$. We study the deformation complex of an arbitrary universal formality morphism $\mathsf {Def}(\mathcal {A}\textit{ssb}_\infty \stackrel {F}{\rightarrow } {\mathcal {D}}\widehat {\mathcal {L}\textit{ieb}}_\infty )$ and prove that it is quasi-isomorphic to the full (i.e. not necessary connected) version of the graph complex introduced Maxim Kontsevich in the context of the theory of deformation quantizations of Poisson manifolds. This result gives a complete classification of the set $\{F_\mathfrak{A}\}$ of gauge equivalence classes of universal Lie connected formality maps: it is a torsor over the Grothendieck–Teichmüller group $GRT=GRT_1\rtimes {\mathbb {K}}^*$ and can hence can be identified with the set $\{\mathfrak{A}\}$ of Drinfeld associators.

AbstractWe study left and right Bousfield localisations of stable model categories which preserve stability. This follows the lead of the two key examples: localisations of spectra with respect to a homology theory and A-torsion modules over a ring R with A a perfect R-algebra. We exploit stability to see that the resulting model structures are technically far better behaved than the general case. We can give explicit sets of generating cofibrations, show that these localisations preserve properness and give a complete characterisation of when they preserve monoidal structures. We apply these results to obtain convenient assumptions under which a stable model category is spectral. We then use Morita theory to gain an insight into the nature of right localisation and its homotopy category. We finish with a correspondence between left and right localisation.

Borel proved that, if a finite group F acts effectively and continuously on a closed aspherical manifold M with centerless fundamental group π1(M), then a natural homomorphism ψ from F to the outer automorphism group Out π1(M) of π1(M), called the associated abstract kernel, is a monomorphism. In this paper, we investigate to what extent Borel's theorem holds for a compact Lie group G acting effectively and smoothly on a particular orientable aspherical manifold N admitting a Riemannian metric g0 of non-positive curvature in case that π1(N) has a non-trivial center. It turns out that if G attains the maximal dimension equal to the rank of Center π1(N) and the metric g0 is real analytic, then any element of G defining a diffemorphism homotopic to the identity of N must be contained in the identity component G0 of G. Moreover, if the inner automorphism group of π1(N) is torsion free, then the associated abstract kernel ψ : G/G0 → Out π1(N) is a monomorphism. The same result holds for the non-orientable N's under certain technical assumptions. Our result is an application of a theorem by Schoen–Yau [12] on harmonic mappings.

On propose par le biais de cette thèse un outil numérique pour l’étude statique et dynamique des structures viscoélastiques constituées de matériaux à gradient de propriétés (FGM) pour le contrôle des vibrations par amortissement passif. L’objectif est de mettre à la disposition des ingénieurs un code générique basé sur l’approche élément fini pour des calculs de dimensionnements sur des poutres sandwich FGM à âme viscoélastique destinées aux applications exigeant la légèreté et une bonne résistance thermique et mécanique comme le domaine de l’aérospatial, de l’automobile et du nucléaire. Pour atteindre cet objectif nous avons d’abord proposé un modèle numérique pour l’étude statique et des vibrations libres des poutres sandwich FGM à comportement élastique. Ce modèle élément fini est implémenté dans l’environnement du code Matlab. A l’aide de ce code nous comparons les différentes théories de poutre pour différentes configurations géométriques et différentes conditions aux limites. Ainsi, la limite de la théorie classique de poutre dans l’étude des structures courtes est soulignée. Aussi, avec ce modèle numérique, l’étude du couplage flexion membrane et rotation membrane est possible. De là, il est montré que les structures FGM sont très sensibles aux effets de couplages spatiaux et du gauchissement à cause de la répartition non symétrique de la matière dans leurs sections droites. Dans le code proposé, la résolution du problème de vibrations est possible grâce à des méthodes classiques de résolution des problèmes aux valeurs propres et vecteurs propres. Pour introduire de l’amortissement passif dans la structure sandwich FGM, nous avons proposé un modèle de poutre sandwich dont les faces sont en matériaux FGM et le cœur en matériaux viscoélastiques. Ce modèle est également implémenté dans le langage de programmation Matlab et proposé sous forme d’un outil générique. L’intérêt de cet outil numérique réside dans sa capacité à calculer les propriétés modales ainsi que le comportement de la structure sandwich FGM viscoélastique tout en prenant en compte la dépendance en fréquence du comportement viscoélastique, les conditions aux limites et le couplage membrane-flexion et membrane-rotation propres aux matériaux FGM. Le problème de vibrations libres est fortement non linéaire dans ce cas à cause de la non linéarité matériaux induite par la couche molle. Dans le code proposé, la résolution de ce problème est possible grâce au couplage de la technique d’homotopie, de la méthode asymptotique numérique et de la différentiation automatique. Par ce travail, l’apport des matériaux FGM dans l’amélioration du pouvoir amortissant des structures est démontré. Dans la suite du travail, nous proposons une formulation élément fini pour calculer l’amplitude des vibrations forcées des structures sandwich FGM viscoélastiques. La résolution du problème de vibration forcée est possible par utilisation de la méthode des bandes passantes. Une étude sur la contribution des matériaux FGM dans la réduction des amplitudes de vibrations est menée pour différentes lois viscoélastiques. Il est prouvé dans cette étude que par un contrôle direct du gradient de composition des matériaux FGM, il est possible d’optimiser le pouvoir amortissant des structures même pour les modes de basses fréquences pour lesquels les matériaux composites classiques présentent un pouvoir amortissant nécessitant des améliorations
This thesis proposes a numerical tool for the static and dynamic study of viscoelastic structures made of Functionally Graded Materials (FGM) for vibration control by passive damping. The objective is to make available to engineers a generic code based on the finite element approach for sizing calculations on FGM sandwich beam with viscoelastic core for applications requiring lightness and good thermal and mechanical resistance such as aerospace, automotive and nuclear. To reach this objective we first proposed a numerical model for the static and free vibration study of FGM sandwich beams with elastic behavior. This finite element model is implemented in the Matlab code environment. Using this code, we compare different beam theories for different geometric properties and boundary conditions. Thus, the limit of the classical beam theory in the study of short structures is highlighted. Also with this numerical model, the study of axial-bending and axial-rotation coupling is possible. From this, it is shown that FGM structures are very sensitive to spatial coupling and warping effects because of the non-symmetrical distribution of the material in their cross sections. In the proposed code, the resolution of the vibration problem is possible using classical eigenvalue and eigenvector problem solving methods. Then to introduce passive damping in the FGM sandwich structure, we proposed a sandwich beam model with FGM materials faces and viscoelastic materials core. This model is also implemented in the Matlab language and proposed as a generic tool. The interest of this numerical tool lies in its ability to compute the modal properties as well as the behavior of the viscoelastic FGM sandwich beam while taking into account the frequency dependence of the viscoelastic behavior, the boundary conditions and the axial-bending and axial-rotation coupling specific to FGM materials. The free vibration problem is non-linear in this case due to the material non-linearity induced by the soft layer. In the proposed code, the resolution of this problem is possible thanks to the coupling of the homotopy technical, the asymptotic numerical method and the automatic differentiation. Through this work, the contribution of FGM materials in the improvement of the damping power of structures is highlighted. In the continuation of the work, we propose a finite element formulation to compute the amplitude of forced vibrations of viscoelastic FGM sandwich structures. The resolution of the forced vibration problem is possible by using the bandwidths method. A study on the contribution of FGM materials in the reduction of vibration amplitudes is carried out for different viscoelastic laws. It is proved in this study that by a direct control of the composition gradient of FGM materials it is possible to optimize the damping power of structures even for low frequency modes for which classical composite materials have a damping power requiring improvement

L’objectif de ce travail est de développer des outils numériques utilisables dans la détermination de manière exacte des propriétés modales des structures sandwichs viscoélastiques composites au vue de la conception des structures sandwichs viscoélastiques légères mais à haut pouvoir amortissant. Pour cela nous avons tout d’abord développé un outil générique implémenté en Matlab pour la détermination des propriétés modales en vibration libre des plaques sandwichs viscoélastiques dont les faces sont en stratifié de plusieurs couches orientées dans diverses directions. L’intérêt de cet outil, basé sur une formulation éléments finis, réside dans sa capacité à prendre en compte l’anisotropie des couches composites, la non linéarité matérielle de la couche viscoélastique traduit par diverses lois viscoélastiques dépendant de la fréquence ainsi que diverses conditions aux limites. La résolution du problème aux valeurs propres non linéaires complexes se fait par le couplage entre la technique d’homotopie, la méthode asymptotique numérique et la différentiation automatique. Ensuite pour permettre une étude continue des effets d’un paramètre de modélisation sur les propriétés modales des sandwichs viscoélastiques, nous avons proposé une méthode générique de résolution de problème résiduel non linéaire aux valeurs propres complexes possédant en plus de la dépendance en fréquence introduite par la couche viscoélastique du coeur, la dépendance du paramètre de modélisation qui décrit un intervalle d’étude bien spécifique. Cette résolution est basée sur la méthode asymptotique numérique, la différentiation automatique, la technique d’homotopie et la continuation et prend en compte diverses lois viscoélastiques. Nous proposons après cela, deux formulations distinctes pour étudier les effets, sur les propriétés amortissantes, de deux paramètres de modélisation qui sont importants dans la conception de sandwichs viscoélastiques à haut pouvoir amortissement. Le premier est l’orientation des fibres des composites dans la référence du sandwich et le second est l’épaisseur des couches qui lorsqu’elles sont bien définies permettent d’obtenir non seulement des structures sandwichs à haut pouvoir amortissant mais très légères. Les équations fortement non linéaires aux valeurs propres complexes obtenues dans ces formulations sont résolues par la nouvelle méthode de résolution d’équation résiduelle développée. Des comparaisons avec des résultats discrets sont faites ainsi que les temps de calcul pour montrer non seulement l’utilité de ces deux formulations mais également celle de la méthode de résolution d’équations résiduelles non linéaires aux valeurs propres complexes à double dépendance
Modeling and design of composite viscoelastic structures with high damping powerThe aim of this thesis is to develop numerical tools to determine accurately damping properties of composite sandwich structures for the design of lightweight viscoelastic sandwichs structures with high damping power. In a first step, we developed a generic tool implemented in Matlab for determining damping properties in free vibration of viscoelastic sandwich plates with laminate faces composed of multilayers. The advantage of this tool, which is based on a finite element formulation, is its ability to take into account the anisotropy of composite layers, the material non-linearity of the viscoelastic core induiced by the frequency-dependent viscoelastic laws and various boundary conditions . The nonlinear complex eigenvalues problem is solved by coupling homotopy technic, asymptotic numerical method and automatic differentiation. Then for the continuous study of a modeling parameter on damping properties of viscoelastic sandwichs, we proposed a generic method to solve the nonlinear residual complex eigenvalues problem which has in addition to the frequency dependence introduced by the viscoelastic core, a modeling parameter dependence that describes a very specific study interval. This resolution is based on asymptotic numerical method, automatic differentiation, homotopy technique and continuation technic and takes into account various viscoelastic laws. We propose after that, two separate formulations to study effects on the damping properties according to two modeling parameters which are important in the design of high viscoelastic sandwichs with high damping power. The first is laminate fibers orientation in the sandwich reference and the second is layers thickness which when they are well defined allow to obtain not only sandwich structures with high damping power but also very light. The highly nonlinear complex eigenvalues problems obtained in these formulations are solved by the new method of resolution of eigenvalue residual problem with two nonlinearity developed before. Comparisons with discrete results and computation time are made to show the usefulness of these two formulations and of the new method of solving nonlinear complex eigenvalues residual problem of two dependances

AbstractIn this chapterwe study extension conditions for dendroidal sets and lifting conditions for maps of dendroidal sets which parallel the conditions for simplicial sets of the previous chapter. The structure of this chapter follows the same plan to stress the analogy. The proofs of various pushout-product properties, which we develop in Section 6.3, become more technical in the case of dendroidal sets.

Chapter 18 describes a few systems where the classical action has an infinite number of degenerate minima but, in the quantum theory, this degeneracy is lifted by barrier penetration effects. The simplest example is the cosine periodic potential and leads to the band structure. Technically, this corresponds to the existence of instantons, solutions to classical equations in imaginary time. In all examples, we show that the classical solutions are constrained by Bogomolnyi’s inequalities, which involve topological charges associated to a winding number and defining homotopy classes. In the case of quantum chromodynamics, this leads to the famous strong CP violation problem.

This chapter proves the geometric picture of double resonance described in Chapter 4. There are two cases. In the simple critical homology case, the chapter shows the homoclinic orbit can be extended to periodic orbits both in positive and negative energy. The union of these periodic orbits forms a normally hyperbolic invariant manifold (which is homotopic to a cylinder with a puncture). In the non-simple homology case, the chapter demonstrates that for positive energy, there exist periodic orbits. The strategy is to prove the existence of these periodic orbits as hyperbolic fixed points of composition of local and global maps. A main technical tool to prove the existence and uniqueness of these fixed points is the Conley-McGehee isolation block.

Abstract A numerical method called “Homotopy Method” (or Continuation Method) is applied to the problem of four-bar coupler-point-curve synthesis. We have shown that: for five precision points, the “General Homotopy method” can be applied to find the link lengths of a number of four-bar linkages, and for nine precision points, a heuristic “Cheater’s Homotopy” can be applied to find some four-bar linkages. The nine-coupler-points synthesis problem is highly non-linear and highly singular. We have found that Newton-Raphson’s method and Powell’s method tend to converge to the singular solutions or do not converge at all, while the Cheater’s Homotopy always finds some non-singular solutions although sometimes the solutions may be complex.

Abstract This paper presents complete solutions to the function generation problem of six-link Watt and Stephenson mechanisms, with multiply separated precision positions (PP), using homotopy methods with m-homogenization. It is seen that using the matrix method for synthesis, applying m-homogeneous group theory and by defining auxiliary equations in addition to the synthesis equations, the number of homotopy paths to be tracked in obtaining all possible solutions to the synthesis problem can be drastically reduced. Numerical work dealing with the synthesis of Watt and Stephenson mechanisms for 6 and 9 multiply separated precision points is presented. For both mechanisms, it is seen that complete solutions for 6 and 9 precision points can be obtained by tracking 640 and 286,720 paths, respectively. A parallel implementation of homotopy methods on the Connection Machine on which several thousand homotopy paths can be tracked concurrently is also discussed.

Abstract Precision position synthesis of an RSSR-SS mechanism is considered. The RSSR-SS mechanism is broken up into two RS and one SS dyads. These components are designed independently using a continuation method. This treatment is possible for a maximum of four precision positions. A parameter homotopy is implemented for the RS dyads and a 3-homogeneous traditional homotopy is used for the SS dyad design. Two example problems are considered.

Abstract This paper presents complete solutions to the function, motion and path generation problems of Watt’s and Stephenson six-link, slider-crank and four-link mechanisms using homotopy methods with m-homogenization. It is shown that using the matrix method for synthesis, applying m-homogeneous group theory, and by defining compatibility equations in addition to the synthesis equations, the number of homotopy paths to be tracked can be drastically reduced. For Watt’s six-link function generators with 6 thru 11 precision positions, the number of homotopy paths to be tracked in obtaining all possible solutions range from 640 to 55,050,240. For Stephenson-II and -III mechanisms these numbers vary from 640 to 412,876,800. For 6, 7 and 8 point slider-crank path generation problems, the number of paths to be tracked are 320, 3840 and 17,920, respectively, whereas for four-link path generators with 6 thru 8 positions these numbers range from 640 to 71,680. It is also shown that for body guidance problems of slider-crank and four-link mechanisms, the number of homotopy paths to be tracked is exactly same as the maximum number of possible solutions given by the Burmester-Ball theories. Numerical results of synthesis of slider-crank path generators for 8 precision positions and six-link Watt and Stephenson-III function generators for 9 prescribed positions are also presented.

Abstract A formulation for solving the direct kinematics of the general Stewart platform consisting of six moving and six grounded spheric joints is presented. The homotopy method is used for solving the direct kinematics of the platform, and it is shown that there exist a maximum of 40 possible solutions to the direct kinematics problem. These 40 solutions can be obtained by tracking only 64 homotopy paths. It is also shown that there are a maximum of 24 solutions for the Stewart platform with four spheric joints, and there exist a maximum of 16 solutions to the direct kinematics of the Stewart platform with three moving and three grounded spheric joints thus confirming the correctness of 24th and 16th degree direct kinematics polynomials obtained by other researchers.

In reality, the behavior and nature of nonlinear dynamical systems are ubiquitous in many practical engineering problems. The mathematical models of such problems are often governed by a set of coupled second-order differential equations to form multi-degree-of-freedom (MDOF) nonlinear dynamical systems. It is extremely difficult to find the exact and analytical solutions in general. In this paper, the homotopy analysis method is presented to derive the analytical approximation solutions for MDOF dynamical systems. Four illustrative examples are used to show the validity and accuracy of the homotopy analysis and modified homotopy analysis methods in solving MDOF dynamical systems. Comparisons are conducted between the analytical approximation and exact solutions. The results demonstrate that the HAM is an effective and robust technique for linear and nonlinear MDOF dynamical systems. The proof of convergence theorems for the present method is elucidated as well.

This paper presents a robustness study of 3R manipulators and aims at answering the following question: are generic manipulators more robust than non-generic manipulators? We exploit several properties specific to 3R manipulators such as singularities, cuspidality, homotopy classes, and path feasibility, in order to find some correlations between genericity and robustness concepts. For instance, we show that generic manipulators, close to non-generic ones in the space of geometric parameters, are not robust with respect to their homotopy class and to the feasibility of paths. Moreover, we notice that the dexterity and the accuracy of 3R manipulator do not depend on genericity.