Passive Decomposition of Multiple Mechanical Systems under Motion Coordination Requirements

Consider a mechanical system whose Lagrangian dynamics evolves on a n-dimensional con guration manifold M with its kinetic energy as the Lagrangian and a sumbersion h : M→N from M to a m-dimensional manifold N (n ≥ m). In this paper, we reveal a fundamental property of the mechanical system in this setting s.t. its n-dimensional dynamics can be decomposed into 1) m-dimensional shape system describing the dynamics of the mapped point of h on N ; 2) locked system representing the (n−m)-dimensional dynamics on the level set of h; and 3) energetically conservative coupling between them. This coupling can be canceled out while preserving passivity of the open-loop mechanical system (i.e. passive with mechanical power and kinetic energy as supply rate and storage function). Moreover, the locked and shape systems also individually inherit passivity from the mechanical system. Due to these passivity properties, we name the presented decomposition passive decomposition. We exhibit and analyze intrinsic and geometric properties of the passive decomposition and the locked and shape systems. A control design example is also given for the decoupled locked and shape systems.