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 \(\boldsymbol{q}^{T}\boldsymbol{q} = 1\) 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.

中文翻译：

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基于四元数的刚体动力学的惯性表示的附加参数的数值影响

不同的惯性表示形式可能导致基于四元数的刚体动力学的微分-代数方程式的公式不同。在本文中，根据动能中的附加参数，惯性表示分为\（\ alpha \）型和\（\ gamma \）型。如果满足约束\（\ boldsymbol {q} ^ {T} \ boldsymbol {q} = 1 \），则这两种表示形式以及相应的参数\（\ alpha \）和\（\ gamma \）在理论上是等效的究竟。但是，误差估计表明它们可以在仿真中呈现完全不同的数值特征，并建议参数\（\ gamma \）可用于优化仿真中积分的数值性能。为了进一步验证\（\ alpha \）型和\（\ gamma \）型的惯性表示之间的数值差异，通过IMS（隐式中点方案），EMS（能量-动量）离散化了相应的修改后的汉密尔顿方程。保存方案）和Gauss-Lobatto SPARK方法。离散对称方案示例的数值性能显示是由于离散化方案的综合效果而产生的，这些方案包括离散点的分布和收敛顺序，惯性表示及其组合。数值结果进一步表明，\（\ gamma \） -型的积分优于\（\ alpha \） -类型和\（\ gamma \）的优化值可用于实现更好的数值精度，收敛速度和稳定性。