Dynamics of multibody systems: Generalized mass metric in Riemannian velocity space and recursive momentum formulation

This paper describes two aspects of multibody system (MBS) dynamics on a generalized mass metric in Riemannian velocity space and recursive momentum formulation. Firstly, we present a detailed expression of the Riemannian metric and operator factorization of a generalized mass tensor for the dynamics of general-topology rigid MBS. The derived expression allows a clearly understanding the components of the generalized mass tensor, which also constitute a metric of the Riemannian velocity space. It is being the fact that there does exist a common metric in Lagrange and recursive Newton-Euler dynamic equation, we can determine, from the Riemannian geometric point of view, that there is the equivalent relationship between the two approaches to a given MBS. Next, from the generalized momentum definition in the derivation of the Riemannian velocity metrics, recursive momentum equations of MBS dynamics are developed for progressively more complex systems: serial chains, topological trees, and closed-loop systems. Through the principle of impulse and momentum, a new method is proposed for reorienting and locating the MBS form a given initial orientation and location to desired final ones without needing to solve the motion equations.