Reducing the Constrained Multibody Dynamics Problem to the Solution of a System of ODEs via Velocity Partitioning and Lie Group Integration

Abstract

In multibody dynamics, formulating the equations of motion in absolute Cartesian coordinates results in a set of index-3 differential algebraic equations (DAEs). In this work, we present an approach that bypasses the DAE problem by partitioning the velocities in the system into dependent and independent coordinates, thereby reducing the task of producing the time evolution of the mechanical system to one of solving a set of ordinary differential equations (ODEs). In this approach, 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. This eliminates the need to choose generalized coordinates to capture the orientation of a body, as the matrix A is a byproduct of the solution algorithm. The approach is akin to the method presented in [1], which combines coordinate partitioning with an Euler parameter formulation. Herein, we outline the new approach and demonstrate it in conjunction with four mechanisms: a single pendulum, a double pendulum, a four-link mechanism, and a slider crank. We report on the convergence order behavior of the proposed method and a performance comparison with the solution introduced in [1]. The Python code developed to generate the reported results is open-source and available in a public repository for reproducibility studies [2].