  Solving quaternion kinematical differential equations is one of the most
significant problems in the automation, navigation, aerospace and aeronautics
literatures. Most existing approaches for this problem neither preserve the
norm of quaternions nor avoid errors accumulated in the sense of long term
time. We present symplectic geometric algorithms to deal with the quaternion
kinematical differential equation by modeling its time-invariant and
time-varying versions with Hamiltonian systems by adopting a three-step
strategy. Firstly, a generalized Euler's formula for the autonomous quaternion
kinematical differential equation are proved and used to construct symplectic
single-step transition operators via the centered implicit Euler scheme for
autonomous Hamiltonian system. Secondly, the symplecitiy, orthogonality and
invertibility of the symplectic transition operators are proved rigorously.
Finally, the main results obtained are generalized to design symplectic
geometric algorithm for the time-varying quaternion kinematical differential
equation which is a non-autonomous and nonlinear Hamiltonian system
essentially. Our novel algorithms have simple algorithmic structures and low
time complexity of computation, which are easy to be implemented with real-time
techniques. The correctness and efficiencies of the proposed algorithms are
verified and validated via numerical simulations.
