Rigid body dynamics in terms of quaternions: Hamiltonian formulation and conserving numerical integration

In the present paper unit quaternions are used to describe the rotational motion of a rigid body. The unit-length constraint is enforced explicitly by means of an algebraic constraint. Correspondingly, the equations of motion assume the form of differential-algebraic equations (DAEs). A new route to the derivation of the mass matrix associated with the quaternion formulation is presented. In contrast to previous works, the newly proposed approach yields a non-singular mass matrix. Consequently, the passage to the Hamiltonian framework is made possible without the need to introduce undetermined inertia terms. The Hamiltonian form of the DAEs along with the notion of a discrete derivative make possible the design of a new quaternion-based energy―momentum scheme. Two numerical examples demonstrate the performance of the newly developed method. In this connection, comparison is made with a quaternion-based variational integrator, a director-based energy―momentum scheme, and a momentum conserving scheme relying on the discretization of the classical Euler's equations.