Using Velocity Partitioning in the rA Formulation to Solve the Equations of Constrained Multibody Dynamics

Abstract

Abstract In mechanical systems, formulating the equations of motion in absolute coordinates results in a set of index-3 differential algebraic equations (DAEs). In this work, we present an approach to solving the DAE problem by partitioning the velocities in the system into dependent and independent coordinates, thereby reducing the problem to an ordinary differential equations (ODEs) problem. The independent coordinates are integrated directly, while the dependent coordinates are recovered through the kinematic constraint equations at the position and velocity levels. Notably, Lie group integration is employed to directly obtain the orientation matrix A at each time step of the simulation. The approach builds on a method presented in [1], which combines coordinate partitioning with an Euler parameter formulation. Herein, we outline the numerical method and demonstrate it in conjunction with three mechanisms: a single pendulum, a double pendulum, and a four-link mechanism. We report on the convergence order behavior of the proposed method. The open-source Python software developed to generate the reported results is available in a public repository for reproducibility studies [2].