Explicit momentum-conserving integrator for dynamics of rigid bodies approximating the midpoint Lie algorithm

Abstract

SUMMARYWe reformulate the midpoint Lie algorithm, which is implicit in the torque calculation, to achieve explicitness in the torque evaluation. This is effected by approximating the impulse imparted over the time step with discrete impulses delivered at either the beginning of the time step or at the end of the time step. Thus, we obtain two related variants, both of which are explicit in the torque calculation, but only first order in the time step. Both variants are momentum conserving and both are symplectic. Consequently, drawing on the properties of the composition of maps, we introduce another algorithm that combines the two variants in a single time step. The resulting algorithm is explicit, momentum conserving, symplectic, and second order. Its accuracy is outstanding and consistently outperforms currently known implicit and explicit integrators.

Geometric methods and formulations in computational multibody system dynamics

“…In the field of MBS geometric integration, special attention is devoted to structure preserving methods that exploit rich geometric structure of rigid body rotational dynamics (see [7,16,25,58,67,73,84,85,87,89,90,97,104,135,146,147,152] and references cited therein). To this end, rigid body rotational dynamics is studied most conveniently as Lie-Poisson system that is defined on so * (3) (the dual space of so (3)).…”

“…Therefore, by following [58,84,89,152] a geometric scheme can be introduced that extends the coadjoint orbit-preserving integration method for SO(3).…”

“…As it is explained in [25,58,84,85,89,147,152], by starting from the geometric scheme that preserves coadjoint orbits, further algorithmic upgrades can be made to allow for numerical preservation of additional integrals of motion, such as kinetic energy of the free rotating body or heavy top constants of motion. Another route to derive algorithm that preserves rigid body integrals of motion is treating body as constrained mechanical system, which leads to Hamiltonian formalism in canonical form.…”

Lie group variational integrators for the full body problem

“…The idea of using the Lie group structure and the exponential map to numerically compute rigid body dynamics arises in the work of Simo et al [15], and in the work by Krysl [16]. A Lie group approach is explicitly adopted by Lee, Leok, and McClamroch in the context of a variational integrator for rigid body attitude dynamics with a potential dependent on the attitude, namely the 3D pendulum dynamics, in [17].…”

“…Equations of motion: Lagrangian form If we take variations of the Lagrangian using (16) and (17), we obtain the equations of motion from Hamilton's principle. We first take the variation of the kinetic energy of B i…”