An Optimal Control Problem for Rigid Body Motions in Minkowski Space

In this paper smooth trajectories are computed in the Lie group SO(2, 1) as a motion planning problem by assigning a Frenet frame to the rigid body system to optimize the cost function of the elastic energy which is spent to track a spacelike curve in Minkowski space. In this case, the derivative of the tangent vector of the spacelike curve at a point s is taken as timelike. A method is proposed to solve a motion planning problem that minimizes the integral of the Lorentz inner product of Darboux vector of a spacelike curve. This method uses the coordinate free Maximum Principle of Optimal control and results in the theory of integrable Hamiltonian systems. The presence of several conversed quantities inherent in these Hamiltonian systems aids in the explicit computation of the rigid body motions.