Influence of Generalized Coordinates on System Dynamics

We investigate the effect of the choice of a set of generalized coordinates (GCs) on the simulation of the behavior of the dynamical system using the single-link spherical pendulum as an example. Specifically, we focus our attention on numerical errors and the simulation time necessary to simulate system dynamics. The Lagrangian method is applied to obtain the equations of motion. The generalized Euler angles are used as GCs. The GCs depend on the direction of the axes along which they are defined. Therefore, by parameterizing the directions of these two axes, different sets of GCs with the corresponding system of nonlinear differential equations are obtained. For a spherical pendulum, we demonstrate that the optimal sets of GCs leading to the minimum simulation time are orthogonal sets. However, contrary to our expectations, orthogonal sets do not result in the minimum simulation error. Additionally, the intrinsic generalized Euler angles lead to faster simulations than the extrinsic ones. Therefore, different choices of GCs are not equivalent from a numerical point of view and further research is needed to develop a strategy for selecting an optimal set of GCs.