Optimization problem of the rigid body motion with the geodesic frame

This study tries to solve the motion of a rigid body, its optimal control problem on the Lie group SE(3) with respect to geodesic frame of curves on the surface in Euclidian 3-space. In this case, optimal control problem is solved on the Lie group SE(3). The motion planning problem is formulated as an optimal control problem in which the cost function to be minimized is equivalent to integrate the conjugated square norm of Darboux vector with respect to the geodesic frame of the curve. The coordinate free Maximum Principle is applied to the theory of integrable Hamiltonian systems to solve this problem.