Global Attitude Representation and its Lie Bracket

Global representation and various properties of the rigid body attitude, i.e., SO(3), are investigated in this paper. The representations, Euler-Lagrange equation, and Hamilton's equation on the second bundle of SO(3) are formally given. For its important role in nonlinear control theory, the global representation of Lie bracket on T ((SO(3))^m × Rⁿ), a general tangent bundle of the product space of SO(3) and Euclidean spaces, is also provided. As an application, the global controllability of spacecraft system with three gas jets is then easily guaranteed through this framework.