If the three moments of inertia are different from each other, the solution to the free rigid body (FRB) equations of motion is given in terms of Jacobi elliptic functions. Using the Arithmetic-Geometric mean algorithm, [1], these functions can be calculated efficiently and accurately. The overall approach yields a faster and more accurate numerical solution to the FRB equations compared to standard numerical ODE and symplectic solvers. This approach performs well also for mass asymmetric rigid bodies. In this paper we consider the case of rigid bodies subject to external forces. We consider a strategy similar to the symplectic splitting method proposed in [16]. The method here proposed is time-symmetric. We decompose the vector field of our problem in a FRB problem and another completely integrable vector field. In our experiments we observe that the overall numerical solution benefits greatly from the very accurate solution of the FRB problem. We apply the method to the simulation of artificial satellite attitude dynamics.
This article investigates the use of the computation of the exact free rigid body motion as a component of splitting methods for rigid bodies subject to external forces. We review various matrix and quaternion representations of the solution of the free rigid body equation which involve Jacobi ellipic functions and elliptic integrals and are amenable to numerical computations. We consider implementations which are exact (i.e., computed to machine precision) and semiexact (i.e., approximated via quadrature formulas). We perform a set of extensive numerical comparisons with state-of-the-art geometrical integrators for rigid bodies, such as the preprocessed discrete Moser-Veselov method. Our numerical simulations indicate that these techniques, combined with splitting methods, can be favorably applied to the numerical integration of torqued rigid bodies.
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