Efficient time-symmetric simulation of torqued rigid bodies using Jacobi elliptic functions

Abstract

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.

A novel Quaternion integration approach for describing the behaviour of non-spherical particles

“…In the literature, there are a number of papers describing the rotation of a rigid body in the framework of Hamilton dynamic systems (e.g [8,7,18,25,29]). Kosenko [25] reports a complex algorithm to represent the rotation of a free rigid body in the framework of Quaternions.…”

“…However, these improved DMV algorithms are limited to free body dynamics. To account for the full dynamic rigid body problem, thus including the application of a torque working on the body, Celledoni et al [8] propose a Stormer/Verlet splitting method to divide the rotation motion into two parts: the free rigid body kinetic part and the torque part, both in the Hamilton system. The free rigid body problem consists of Euler dynamic equations and a differential equation for updating the rotation matrix.…”

Detumbling and nutation canceling maneuvers with complete analytic reduction for axially symmetric spacecraft

“…the component on the plane perpendicular to the symmetry axis e. The torque direction is also equal to the direction of the transversal component of the angular velocity of the Virtual Sphere, because of Eq. (29), and also to the direction of the transversal component of the angular velocity of the body, because of Eq. (20).…”

“…For instance, as regards the kinematic problem, Iserles and Nørsett [27] study the solution in terms of series expansion for the more general problem of solving linear differential equation in Lie Groups, building upon the work of Magnus [28]. Celledoni and Saefstroem [29] propose ad hoc numerical integration algorithms. Finally, Livneh and Wie [30] introduce approximate results for the motion of a triaxial rigid body subjected to a constant torque.…”

Reducing round-off errors in rigid body dynamics

“…In several recent publications [3,2,13,14], it is proposed to integrate the equations of motion of the free rigid body analytically, using the Jacobi elliptic functions [10]. Although this approach yields the exact solution, a standard implementation yields an unexpected linear propagation (accumulation) of round-off errors (see Figure 1 and [2, Fig.…”