This paper provides analytical solutions to the closed loop system kinematics for a class of almost globally asymptotically stable feedback laws on SO(n). The resulting closed loop kinematics are solved for the respective matrices as functions of time, the initial conditions and the gain parameters of the control laws. The analytical solutions provide insight into the transient dynamics of the system and can be used to prove almost global attractiveness of the identity matrix. We consider an application of these results towards model predictive control where the transient phase of the system is utilized to attempt to complete a task of secondary importance by choosing the gain parameters as functions of time and the initial conditions.