By Fasano A., Marmi S.

ISBN-10: 0198508026

ISBN-13: 9780198508021

Robot manipulators have gotten more and more very important in study and undefined, and an realizing of statics and kinematics is key to fixing difficulties during this box. This ebook, written through an eminent researcher and practitioner, presents a radical advent to statics and primary order immediate kinematics with purposes to robotics. The emphasis is on serial and parallel planar manipulators and mechanisms. The textual content differs from others in that it truly is dependent exclusively at the ideas of classical geometry. it's the first to explain find out how to introduce linear springs into the connectors of parallel manipulators and to supply a formal geometric approach for controlling the strength and movement of a inflexible lamina. either scholars and training engineers will locate this publication effortless to persist with, with its transparent textual content, ample illustrations, routines, and real-world tasks Geometric and kinematic foundations of lagrangian mechanics -- Dynamics : basic legislation and the dynamics of some degree particle -- One-dimensional movement -- The dynamics of discrete structures : Lagrangian fomalism -- movement in a valuable box -- inflexible our bodies : geometry and kinematics -- The mechanics of inflexible our bodies : dynamics -- Analytical mechanics : Hamiltonian formalism -- Analytical mechanics : variational rules -- Analytical mechanics : canonical formalism -- Analytic mechanics : Hamilton-Jacobi idea and integrability -- Analytical mechanics : canonical perturbation concept -- Analytical mechanics : an advent to ergodic concept and the chaotic movement -- Statistical mechanics : kinetic idea -- Statistical mechanics : Gibbs units -- Lagrangian formalism in continuum mechanics

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The condition for a curve in the Euclidean space R3 to be a geodesic is satisﬁed by straight lines, for which d2 x/ds2 = 0. 23 It is easy to convince oneself that the maximal circles are geodesics on the sphere, while on a cylinder with circular normal section, the geodesics are the generating lines and helices (cf. 10), including the ones that degenerate to circles. 42) it is easy to derive a system of ordinary diﬀerential equations which the geodesics must satisfy: (u xu + (u )2 xuu + 2u v xuv + (v )2 xvv + v xv ) · xu = 0, (u xu + (u )2 xuu + 2u v xuv + (v )2 xvv + v xv ) · xv = 0.

We now turn to the problem of parametrising regular submanifolds. We have already remarked that for surfaces in R3 it is not possible in general to give a global parametric representation. For example, the sphere S2 is a regular submanifold of R3 , but the parametrisation given by the spherical coordinates x1 = (sin u1 cos u2 , sin u1 sin u2 , cos u1 ) is singular at the points (0, 0, 1) and (0, 0, −1). A regular parametrisation at those points is given instead by x2 = (cos u1 , sin u1 cos u2 , sin u1 sin u2 ), which however is singular at (1, 0, 0) and (−1, 0, 0).

44) for the geodesics can be written in the form 2 i j d2 uk k du du = 0, + Γ ij ds2 ds ds i,j=1 k = 1, 2. e. u = as + b, v = cs + d, with a, b, c, d arbitrary constants. 6 obtains the generating lines; a = 0 yields the normal sections; in all other cases v − d = c/a (u − b), and hence one ﬁnds helices. Since du/ds = a, the geodesics intersect the generating lines at a constant angle. 49) and hence the Christoﬀel symbols have the values Γ111 = f (u)f (u) , 1 + (f (u))2 Γ122 = − u , 1 + (f (u))2 Γ212 = Γ221 = 1 , u while Γ112 = Γ121 = Γ211 = Γ222 = 0.

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