Diagonalized Lagrangian robot dynamics

A diagonal equation _ + C( ; )= for robot dynamics is developed by combining recent mass matrix factorization results [1{7] with classical Lagrangian mechanics. Diagonalization implies that at each xed time instant the equation at each joint is decoupled from all of the other joint equations. The equation involves two important variables: a vector of total joint rotational rates and a corresponding vector of working joint moments. The nonlinear Coriolis term C( ; ) depends on the joint angles and the rates . The total joint rates are related to the relative joint-angle rates _ by a linear spatial operator m ( ) mechanized by a base-to-tip spatially recursive algorithm. The total rate (k) at a given joint k re ects, in a very unique sense the total rotational velocity about the joint, and includes the combined e ects from all the links between joint k and the manipulator base. This di ers from the more traditional joint-angle rates _ which only re ect the relative, as opposed to total, rotation about the joints. Similarly, the working moments = `T are related to the applied moments T by the spatial operator `( )=m ( ) mechanized by a tipto-base spatially recursive algorithm. The working moment (k) at a given joint k is that part of the applied moment T (k) which does actual mechanical work, while its other part a ects only the non-working internal constraint forces. The diagonal equations are obtained by using the recently developed [1] mass matrix factorization M( )=m( )m ( ) of the system Lagrangian. The diagonalization is achieved in velocity space. This means that only the velocity variables _ are replaced with the new variables , while the original con guration variables are retained. The new joint velocity variables can be viewed as time-derivatives of Lagrangian quasi-coordinates, similar to those of classical mechanics. The velocity transformations are shown to always exist for tree-like, articulated multibody systems, and they can be readily implemented using the spatially recursive ltering and smoothing methods [1,4,7] advanced by the authors in recent years. 1 Mass Matrix Factors Diagonalize Lagrange's Equations The main new result in this paper is a diagonalized equations of motion _ + C( ; )= , which embodies in a simple, elegant, diagonal equation the complete dynamical behavior of robotic manipulators, while simultaneously exploiting the computational e ciency of the spatially recursive ltering and smoothing algorithms of [4, 7] to conduct necessary velocity coordinate transformations. The diagonal equations of motion result by combining Lagrangian mechanics with the mass matrix factorization M = [I +H K]D[I +H K] (1.1)