By Dirk Aeyels (auth.), Bernard Bonnard, Bernard Bride, Jean-Paul Gauthier, Ivan Kupka (eds.)

The convention "Analysis of managed Dynamical structures" used to be held in July 1990 on the collage of LYON FRANCE. approximately hundred contributors attended this convention which lasted 4 days : there have been 50 audio system from departments of Engineering and arithmetic in east and west Europe, united states and USSR. the final topic of the convention was once procedure thought. the most issues have been optimum keep watch over, constitution and keep watch over of nonlinear platforms, stabilization and observers, differential algebra and structures thought, nonlinear features of Hoc idea, inflexible and versatile mechanical structures, nonlinear research of indications. we're indebted to the medical committee John BAILLIEUL, Michel FLIESS, Bronislaw JAKUBCZYCK, Hector SUSSMANN, Jan WILLEMS. We gratefully recognize the time and suggestion they gave to this activity. we might additionally prefer to thank Chris BYRNES for arranging for the e-book of those court cases in the course of the sequence "Progress in structures and keep watch over Theory"; BIRKHAUSER. ultimately, we're very thankful to the next associations who via their monetary aid contributed basically to the good fortune of this convention : CNRS, specified 12 months " Systemes Dynamiques", DRET, MEN-DAGIC, GRECO-AUTOMATIQUE, Claude Bernard Lyon I collage, Entreprise Rhone-Alpes foreign, Conseil basic du RhOne, the towns of LYON and VILLEURBANNE.

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**Extra info for Analysis of Controlled Dynamical Systems: Proceedings of a Conference held in Lyon, France, July 1990**

**Example text**

We assume also that any two points in the same connected component of U can be joined by a smooth curve lying entirely in int U (except possibly for endpoints). , there exists a Ck extension of f to an open neighbourhood of X x U in X x ]Rm). We call systems of class Coo smooth systems and those of class analytic systems. The most restrictive technical assumption to be made is that the system is invertible; this means that for each u E U the map fu = f(·,u): X ~ X is a global diffeomorphism of X.

We shall be especially interested in the case in which the forcing frequency is substantially larger than than the natural frequency of the unforced system. In this case, our objective is to describe the average behavior on a suitable time scale which is slower than that of the forcing. , qn). ) The generalized momentum is P = Mllq + vA. Applying the Legendre transformation H(q,PiV) = p. q - l(q,qiv), we have 1 - vA) T Mll -1( P - vA ) + Va(v,q). BAILUEUL In Hamiltonian form, the equations of motion are q= P= - aa [~(p q 2 Mii1(p - vA) (15) vAl Mil1(p - vA) + v;, (v, q).

While we assume that the actuator is capable of producing an arbitrary smooth 8(·)trajectory, all other degrees of freedom corresponding to the joints between the links in the chain are assumed to be uncontrolled. ) rotations depended crucially on the number of degrees-of-freedom of the joint or hinge. To pursue this dependence in the present section, we shall consider two types of kinematic chains: spatial chains in which there are universal or Hooke joints having two degrees of freedom connecting each pair of links and planar chains in which the revolute joint between each pair of links has one degree of freedom and in which all joint axes are parallel.