A conservative quaternion-based time integration algorithm for rigid body rotations with implicit constraints

A conservative time integration algorithm for rigid body rotations is presented in a purely algebraic form in terms of the four quaternions components and the four conjugate momentum variables via Hamilton’s equations. The introduction of an extended mass matrix leads to a symmetric set of eight state-space equations where constraints are embedded without explicit use of Lagrange multipliers.

The algorithm is developed by forming a finite increment of the Hamiltonian, which defines the proper selection of increments and mean values that leads to conservation of energy and momentum. The accuracy and conservation properties are illustrated by examples.