Linear quaternion equations with application to spacecraft attitude propagation

The kinematic relationship between body rates and rate of change of Euler parameters is an instance of linear quaternion differential equations. This paper presents solutions for general linear quaternion equations in terms of state transition quaternions, in parallel to similar results for matrix/vector linear differential equations. Series expansions of these transition quaternions are presented. For constant coefficient quaternion differential equations, the transition quaternion is shown to be the quaternion exponential, which is described in this paper. These results are used to derive algorithms for the solution of the kinematic relationship between spacecraft body rates and Euler parameters. First, it is demonstrated mathematically, that the quaternion exponential gives the exact rotation quaternion for slews about inertially fixed rotation axis, with no coning. Secondly, a popular third-order attitude propagation algorithm is derived using this approach. Variants of this algorithm for accuracy enhancements of attitude propagation algorithms are discussed