By Fasano A., Marmi S.

Robot manipulators have gotten more and more vital in examine and undefined, and an figuring out of statics and kinematics is vital to fixing difficulties during this box. This e-book, written by means of an eminent researcher and practitioner, presents an intensive creation 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 really is dependent exclusively at the techniques 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 process for controlling the strength and movement of a inflexible lamina. either scholars and practising engineers will locate this publication effortless to stick with, with its transparent textual content, considerable illustrations, routines, and real-world initiatives Geometric and kinematic foundations of lagrangian mechanics -- Dynamics : common legislation and the dynamics of some extent particle -- One-dimensional movement -- The dynamics of discrete structures : Lagrangian fomalism -- movement in a relevant box -- inflexible our bodies : geometry and kinematics -- The mechanics of inflexible our bodies : dynamics -- Analytical mechanics : Hamiltonian formalism -- Analytical mechanics : variational ideas -- 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 conception -- Statistical mechanics : Gibbs units -- Lagrangian formalism in continuum mechanics

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5 Geometric and kinematic foundations of Lagrangian mechanics 15 x3 n t x2 w A x1 Fig. 10 yielding for the torsion χ=− R2 λ . + λ2 Curvature and torsion are the only two geometric invariants of a curve in space. Namely we have the following. 4 Let k(s) > 0 and χ(s) be two given regular functions. There exists a unique curve in space, up to congruences (rotations and translations), which has s as natural parameter, and k and χ as curvature and torsion, respectively. 2 and is based on the fact that t(s) solves the diﬀerential equation d2 t k dt dt + k 2 t + χt × = 0.

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. 50) = 0. 51) and hence, if c = / 0, ds = 1 2 u dv. 52) this leads to the elimination of ds and one can hence consider v as a function of u.

61) is simply u˙ i (t) = αi (u1 (t), . . , u (t)), i = 1, . . , . 30 Let M be the unit sphere; consider the parametrisation x = (sin u1 cos u2 , sin u1 sin u2 , cos u1 ), with the tangent vectors ∂x = (cos u1 cos u2 , cos u1 sin u2 , − sin u1 ), ∂u1 ∂x = (− sin u1 sin u2 , sin u1 cos u2 , 0). ∂u2 A vector ﬁeld tangent over M takes the form α1 (u1 , u2 ) ∂x ∂x + α2 (u1 , u2 ) . ∂u1 ∂u2 For example, if α1 = constant, α2 = constant the integral curves are given by (0) (0) u1 (t) = α1 t + u1 , u2 (t) = α2 t + u2 .