Conserving integration of multibody systems with singular and non-constant mass matrix including quaternion-based rigid body dynamics

Mechanical systems with singular and/or configuration-dependent mass matrix can pose
difficulties to Hamiltonian formulations, which are the standard choice for the design
of energy-momentum conserving time integrators. In this work, we derive a structure-
preserving time integrator for constrained mechanical systems based on a mixed variational
approach. Livens’ principle (or sometimes called Hamilton–Pontryagin principle) features
independent velocity and momentum quantities and circumvents the need to invert the mass
matrix. In particular, we take up the description of rigid body rotations using unit quater-
nions. Using Livens’ principle, a new and comparatively easy approach to the simulation
of these problems is presented. The equations of motion are approximated by using (par-
titioned) midpoint discrete gradients, thus generating a new energy-momentum conserving
integration scheme for mechanical systems with singular and/or configuration-dependent
mass matrix. The derived method is second-order accurate and algorithmically preserves a
generalized energy function as well as the holonomic constraints and momentum maps cor-
... mehr
responding to symmetries of the system. We study the numerical performance of the newly
devised scheme in representative examples for multibody and rigid body dynamics.