The adaptive controller design of a large class of mechanical systems is addressed in this paper. The configuration space for the attitude of a vehicle is modeled as SO(3), namely the rotation matrix group. To avoid singularity problems, some studies adopt the representation of unit quaternions for modeling system dynamics and for the design of controllers. However, the complicated formulation of quaternions makes the application of this method limited to simple mechanical systems. Here, the strategy is to carry out the modeling and analysis in terms of the global representations of jet bundles of SO(3), while the unit quaternion is chosen to implement the controller for its least number of parameters (four) to represent SO(3) globally. Based on the transformation between the rotation matrix and the quaternion, the method explored in this paper gives rise to an adaptive controller in a clean and efficient way.