Dynamics and controller design of gyroscopic systems

The gyroscopic systems defined represent a large class of physical systems, including the spacecraft systems with rotational elements. The conglobation space of the spacecraft attitude should be modeled as SO(3), namely the rotation matrix group. How to suitably integrate the representations in terms of the rotation matrix and the design based on quaternions is the main issue as well as the contribution of the present work. By using the global representations of jet bundles of SO(3), we obtain a prominent result on Euler-Lagrange equation in terms of quasi-coordinates so that the dynamics of gyroscopic systems and its skew-symmetric type property can be derived and demonstrated. For practical implementation, using the least possible number of parameters (four) to represent the attitude, quaternion, is preferred. We thus propose an adaptive controller in terms of quaternion, where neither the inversion of the inertial matrix nor the body acceleration are needed.