Constrained Multibody Kinematics and Dynamics in Absolute Coordinates: A Discussion of Three Approaches to Representing Rigid Body Rotation

Abstract

We compare three ways to pose the set of index 3 differential algebraic equations (DAEs) associated with the constrained multibody dynamics problem formulated in absolute coordinates. The first approach, named rA, works directly with the orientation matrix and therefore eschews the need for generalized coordinates used to produce the orientation matrix A. The approach is informed by the fact that rotation matrices belong to the SO(3) Lie matrix group. The second approach, referred to herein as rp, employs Euler parameters; the third, referred to as rε, uses Euler angles. In all cases, the index 3 DAE problem is solved via a first-order implicit numerical integrator. We note a roughly twofold speedup of rA over rε and a 1.2–1.3 times speedup of rε over rp. The tests were carried out in conjunction with four 3D mechanisms. The improvements in simulation speed of the rA approach are traced back to a simpler form of the equations of motion and more concise Jacobians that enter the numerical solution. The contributions made herein are twofold. First, we provide first-order variations of all the quantities that enter the rA formulation when used in the context of implicit integration; i.e., sensitivity of the kinematic constraints for all lower pair joints, as well as the sensitivity of the constraint reaction forces. Second, to the best of our knowledge, there is no other contribution that compares head to head the solution efficiency of rA, rp, and rε in the context of the multibody dynamics problem posed in absolute coordinates.