Lagrangian dynamics of flexible manipulators using angular velocities instead of transformation matrices

Most recent computational algorithms for dynamic modeling of flexible manipulators are derived by using the 4×4 link-transformation matrix and Lagrange's equations. The resulting equations are computationally inefficient, even if special-purpose matrix multiplication routines are used to take advantage of the special structure of matrices and a recursive computational procedure is employed. A more efficient approach that uses angular velocities and 3×3 rotation matrices to represent link kinematics is presented. A truncated modal expansion is used to model link deflection. A recursive computational procedure is presented that is similar to the Newton-Euler dynamics formulation for rigid manipulators. The full nonlinear inverse dynamic equations are calculated in recursive form for manipulators with an arbitrary number of flexible links. Kinematics are computed recursively from the base to the tip, and torques are computed on the return recursion. Based on this formulation, a fast and accurate simulation algorithm is also presented. It is shown that this method offers significant improvement in computational speed without degrading numerical accuracy. The modeling accuracy of the method and some practical implementational problems are also discussed.