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 timelike curve in Minkowski space. A method is proposed to solve a motion planning problem that minimize the integral of the square norm of Darboux vector of a timelike 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.<\/p>\r\n","references":"[1] V. Jurdjevic, F. Monroy-Perez (2002), Variational problems on Lie\r\ngroups and their homogeneous spaces: elastic curves, tops and constrained\r\ngeodesic problems in nonlinear geometric control theory ,\r\nWorld Scientific, Singapore.\r\n[2] V. Jurdjevic, (1997), Geometric Control Theory, Advanced Studies in\r\nMathematics, vol 52. Cambridge University Press, Cambridge.\r\n[3] H.J. Sussmann, (1997), An introduction to the coordinate-free maximum\r\nprinciple , In: Jakubezyk B, Respondek W (eds) Geometry of feedback\r\nand optimal control. Marcel Dekker, New York, pp 463-557.\r\n[4] J. Biggs, W. Holderbaum, (2008), Planning rigid body motions using\r\nelastic curves, Math. Control Signals Syst. 20: 351-367.\r\n[5] A.Yucesan, A.C. Coken, N.Ayyildiz,(2004), On the darboux rotation\r\naxis of Lorentz space curve, Applied Mathematics and Computation,\r\n155:345-351.\r\n[6] R.Lopez, (2008), Differential geometry of curves and surfaces in\r\nLorentz-Minkowski space, University of Granada.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 42, 2010"}