The Numerical Influence of Additional Parameters of Inertia Representations for Quaternion-Based Rigid Body Dynamics

Different inertia representations can lead to different formulations of the differential-algebraic equations for the quaternion-based rigid body dynamics. In this paper, the inertia representations are classified into $alpha $-type and $gamma $-type, according to the additional parameters in the kinetic energy. These two types of representations and the corresponding parameters $alpha $ and $gamma $ are theoretically equivalent if the constraint $boldsymbolqtextasciicircumTboldsymbolq = 1$ is satisfied exactly. Nevertheless, the error estimation demonstrates that they can present entirely different numerical features in simulation and suggests that the parameter $gamma $ can be used to optimize the numerical performance of the integrations in simulation. To further verify the numerical difference between the inertia representations of $alpha $-type and $gamma $-type, the corresponding modified Hamilton’s equations are discretized by the IMS (implicit midpoint scheme), EMS (energy–momentum preserving scheme) and Gauss–Lobatto SPARK methods. Numerical performance for the examples of the spinning symmetrical top is shown to result from the comprehensive effect of the discretization schemes including the distribution of discretized points and the convergence order, the inertia representations and their combinations. Numerical results further suggest that the integrations of $gamma $-type are superior to those of $alpha $-type and the optimized values of $gamma $ can be used to achieve better numerical accuracy, convergence speed and stability.