Dynamic analysis and control design of kinematically-driven multibody mechanical systems

In this investigation, a method for solving the dynamics and control problems of multibody mechanical systems whose time evolution is induced by a kinematically-driven motion is presented. In particular, the motion of a double inverted pendulum is employed as a demonstrative example of the computational procedure developed in this work and the interaction between the pantograph and the catenary is considered as a case study. To this end, the dynamic analysis and the design of a control system are performed. The multibody approach is used for deriving a mechanical model of the double inverted pendulum as well as of the pantograph mechanism. The multibody mechanical model developed in this work is aimed at improving the interaction force between the catenary and the pantograph. The mathematical method devised in this work for constructing multibody models of constrained mechanical systems makes use of a recursive Lagrangian approach. The Lagrangian formulation used in this paper allows for effectively handling the redundancy of the generalized coordinates and leads to a straightforward determination of the nonlinear reaction force fields arising from the presence of the mechanical joints. While the pantograph mechanism is schematized as a multibody mechanism having a geometrically nonlinear structure, the interaction force between the pan-head and the suspended line is modeled in a simplified manner considering a linear elastic element. On the other hand, a force actuator is applied between the upper arm and the pan-head of the pantograph mechanism for controlling the coupled dynamics of the pantagraph system and the catenary cable. The control system is devised with a twofold objective, namely to attenuate the interaction forces generated by the pantograph/catenary contact and to reduce the nonlinear oscillations of the closed-loop mechanism. An optimal law for the control action is obtained by employing a numerical algorithm based on the adjoint method. The numerical results found by means of numerical experiments demonstrate the efficacy of the recursive Lagrangian approach proposed in this work.