Stabilization and optimality results for the attitude control problem

We present some recent results on the description and control of the attitude motion of rotating rigid bodies. We derive a new class of globally asymptotically stabilizing feedback control laws for the complete (i.e., dynamics and kinematics) attitude motion. We show that the use of a Lyapunov function, which involves the sum of a quadratic term in the angular velocities and a logarithmic term in the kinematic parameters, leads to the design of linear controllers. We also show that the feedback control laws for the kinematics minimize a quadratic cost in the state and control variables for all initial conditions. For the complete system we construct a family of exponentially stabilizing control laws and we investigate their optimality characteristics. The proposed control laws are given in terms of the classical Cayley-Rodrigues parameters and the modified Rodrigues parameters.