Kinematics of single-loop mechanisms and serial robot arms: A systematic approach

This paper presents a systematic approach, based on displacement group properties, to the kinematic analysis of spatial linkages with one closed loop and to the solution of the inverse kinematic problem for robot manipulators. By using the proposed approach, a set of kinematic chains can be determined such that a first closure equation with one unknown can be derived directly and explicitly. Then the remaining closure equations are obtained: it is proved that they can be expressed in triangular form. The basic algorithms used to solve these equations in closed form are also presented. For each algorithm, the conditions of applicability, the initial information required, the results, the type and form of equations, and the maximum number of solutions are given. The proposed approach is well suited to the symbolic explicit solution of the inverse kinematic problem of a wide range of robut mechanisms. An example of its application is given.

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Sommario Il lavoro presenta un appreceio sistematico, basato sulle proprietà dei gruppi di spostamento, all'analisi cinematica di posizione di meccanismi spaziali ad una maglia e alla cinematica inversa di robot manipolatori seriali. L'approccio consente di determinare un insieme di catene cinematiche per le quali può essere scritta e risolta direttamente una prima equazione di chiusura in una sola incognita. Viene successivamente dimostrato esaustivamente che, per tali catene, le successive equazioni di chiusura possono essere espresse e risolte in forma triangolare. Inoltre sono presentati gli algoritmi di base utilizzabili per la soluzione del problema posto. Per ciascuno di essi sono dati: le condizioni di applicabilità, l'informazione iniziale richiesta, i risultati ottenuti, il tipo e la forma delle equazioni e il massimo numero di soluzioni possibili. L'approccio presentato è utilizzabile per la soluzione simbolica esplicita, manuale o automatica, di un esteso insieme di meccanismi per robot. Viene dato un esempio di uso del metodo.

     

Closed-form solution of the direct kinematics of the 6–3 type Stewart Platform using one extra sensor

A procedure for the determination of the actual configuration of the 6–3 type Stewart Platform is presented. The procedure makes use of one extra sensor in addition to the six that provide the mechanism leg lengths. The use of one extra sensor suitably located on the mechanism allows the direct position analysis to be solved in closed form and makes it possible to obtain a one-to-one correspondence between the sensor measurements and the configuration of the mechanism in almost the entire workspace. Particular configurations where the correspondence is not one-to-one are outlined and criteria to identify them given. Sensitivity to sensor measurement errors is investigated.Paper presented at the 12th Italian Congress on Theoretical and Applied Mechanics AIMETA 1995, Napoli, 3–6 October 1995     

The planar mechanism as a special case in the automatic computation of static equilibrium

Aspects of the planar mechanism are considered as a special case in the development of a scheme, where joint loads are calculated from equations of static equilibrium set up automatically by computer on the basis of input specifying the types of joints and link dimensions.     

A generalization of recursion operators of differential equations

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A generalization of the concept of equivalent linearization

A method is presented whereby a non-linear second order dynamical system is replaced by a linear system in such a way that an average of the difference between the two systems is minimized. Provided the averaging operator possesses certain properties, it is shown that the replacement is unique and can be accomplished in a straightforward manner. The parameters of the replacement linear system are expressed in terms of averages of functions of the linearized solution.     

A generalization of the Chapman-Enskog and Grad methods

By using the method of Chapman-Enskog as a base, we have determined successive approximations of any order to the transport coefficients of shear viscosity and thermal conductivity for a monatomic ideal gas. The expressions for the transport coefficients involve only integrals which can be evaluated once the law of interaction between the spherically symmetrical particles is known. Moreover, we have developed a (13 + 9N)-field theory based on the method of Grad and showed that the transition from this theory to a five-field theory leads to the same results as those obtained through the method of Chapman-Enskog.     

A generalization of Reiner's mathematical model for wet sand

In this paper we modify the constitutive relation derived by Reiner (1945), to describe dilatancy in wet sand, by suggesting that the shear viscosity would depend on the shear rate and the volume fraction. We then look at the flow of a saturated densely packed bed of particles (with liquid in the pores) between two horizontal flat plates. We obtain exact solutions for a very special case.     

A generalization of Browder's fixed point theorem with applications

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A Simple Method for the Determination of Angular Velocity and Acceleration of a Spherical Motion Through Quaternions

Given a quaternion representation of a spherical motion of a rigid body with respect to another body, acting as a reference frame, this contribution presents a simple and straightforward method for determining both the angular velocity and angular acceleration of the moving body with respect to the reference frame. Instead of employing orthogonal matrices or their linear invariants, this contribution makes use of quaternions avoiding, in this way, the series of matrix identities or theorems that are required in a pair of previous approaches.     

A method of interface inversion in inhomogeneous media

First, an approximate solution of time domain interface scattering field is derived by extending the classical Born approximation in the problem of interface scattering. In accordance with the solution form, a projection density compensation (PDC) inversion method is developed according to the projection slice theorem, which is valid for the cases of inhomogeneous media and wave mode transformation. Finally, in the model of layered media, the calculation algorithm and the simulation inversion comparison results of point defect, crack, and crack on an interface, as well as the experiment method and results in the condition of acoustic wave, are given. The project supported by the National Natural Science Foundation of China (No.19232031) and Chinese Education Commission Science Foundation     

A generalization of wieghardt soil for two - dimensional foundation structures

Summary On the basis of a mechanical interpretation (see Reference[1]) the problem of generalization of Wieghardt soils for two-dimensional foundation structures is tackled. The results of experiments agree well with the schematization arrived at. The problem of the rigid circular plate on the thus schematized elastic soil is solved.

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Sommario Premesse alcune considerazioni di carattere generale sulla schematizzazione proposta dal Wieghardt per la legge carichi-spostamenti di un letto elastico di fondazione di un solido monodimensionale si affronta, al lume di un'interpretazione meccanica proposta dall'A. in un articolo precedente[1], il problema della generalizzazione di tale legge per i problemi di fondazione bidimensionali. Si perviene così in forma semplice alla formulazione della legge carichi-spostamenti per un generico piano di fondazione e si dimostra l'ottima rispondenza di tale schematizzazione con i risultati dedotti da prove sperimentali reali. Viene risolto in via completa il problema della piastra circolare rigida sul suolo elastico così schematizzato.

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First published in Italian in La ricerca scientifica, 35 (II-A), 1290–1304, Roma, 1965.     

Bounds on the Elastic Moduli of Completely Random Two-Dimensional Polycrystals

Explicit bounds on the elastic moduli of completely random planar polycrystals, the shape and crystalline orientations of the constituent grains of which are uncorrelated, are derived and calculated for a number of crystals of general two-dimensional anisotropy. The bounds on the elastic two-dimensional bulk modulus happen to coincide with the simple third order (in anisotropy contrast) bounds for the subclass of idealistic circular cell polycrystals. The bounds on the shear modulus are close to the much simpler bounds for circular cell polycrystals, which approximate aggregates of equiaxed grains.     

A wavelet multiscale method for inversion of Maxwell equations

This paper is concerned with estimation of electrical conductivity in Maxwell equations. The primary difficulty lies in the presence of numerous local minima in the objective functional. A wavelet multiscale method is introduced and applied to the inversion of Maxwell equations. The inverse problem is decomposed into multiple scales with wavelet transform, and hence the original problem is reformulated to a set of sub-inverse problems corresponding to different scales, which can be solved successively according to the size of scale from the shortest to the longest. The stable and fast regularized Gauss-Newton method is applied to each scale. Numerical results show that the proposed method is effective, especially in terms of wide convergence, computational efficiency and precision.     

ON A GENERALIZATION OF BERTRAND’S THEOREM

Bertrand’s theorem for the determination of the applied forces to a holonomicsystem from one of its first integrals,is extended to nonholonomic systems.Someinteresting applications of this new result are also given.     

A generalization of the method of averaging for systems with two time scales

In this paper we shall consider systems of the form x = ? f(t, ?t, x, y, ?), y = g(t,?t, x, y,?), where x and y are vectors of finite dimensions, f and g are assumed to be bounded for all t, and ? is a real parameter. Sufficient conditions are obtained for the existence of certain solutions which are bounded for all t. These solutions are shown to approach special solutions of a derived simpler averaged system of equations as ? → 0. Moreover, it is shown that there exists only one such bounded solution in the neighborhood of each special solution. In the special case when y is not present, it is shown that if a special solution is stable, solutions starting in nonlocal neighborhoods of this special solution approach the bounded solutions adjacent to it as t → ∞. These results generalize most of the existing work for systems of the type discussed here. Finally, we employ our results to study some problems of physical importance.