Optimal attitude control of a rigid body on special orthogonal group using LGVI and FD-GMRES

A new optimal attitude control method is proposed for controlling the rigid body on the special orthogonal group. First, the discrete kinematics and dynamics of attitude motion are derived based on discrete d´ Alembert-Pontryagin variational principle, which are identical with Lie group variational integrator. Second, a constrained optimal control problem on Lie group is described and its first-order necessary optimality condition is derived using Lagrange multipliers and Lie group operators. Then, a forward difference-generalized minimum residual based algorithm is proposed for solving the first-order necessary optimality condition in receding manner. Finally, through the numerical tests and simulation examples for controlling a 3D pendulum rotating on the special orthogonal group, it is concluded that the proposed method is efficient and can converge well.