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Rigid body dynamics on the rotation group have typically been represented interms of rotation matrices, unit quaternions, or local coordinates, such asEuler angles. Due to the coordinate singularities associated with localcoordinate charts, it is common in engineering applications to adopt the unitquaternion representation, and the numerical simulations typically impose theunit length condition using constraints or by normalization at each step. Fromthe perspective of geometric structure-preserving, such approaches areundesirable as they are either computationally less efficient, or interferewith the preservation of other geometric properties of the dynamics.  In this paper, we adopt the approach used in constructing Lie groupvariational integrators for rigid body dynamics on the rotation group to therepresentation in terms of unit quaternions. In particular, the rigid bodydynamics is lifted to unit quaternions, and the Lie group structure of unitquaternions is used to represent tangent vector intrinsically, thereby avoidingthe use of a Lagrange multiplier. A Lie group variational integrator in theunit quaternion representation is derived, and numerical results are presented.

Lie group variational integrators for rigid body problems using quaternions