Planning Rigid Body Motions and Optimal Control Problem on Lie Group SO(2, 1)
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 minimizes the integral of the Lorentz inner product 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.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1079938Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1461
 V. Jurdjevic, F. Monroy-Perez (2002), Variational problems on Lie groups and their homogeneous spaces: elastic curves, tops and constrained geodesic problems in nonlinear geometric control theory , World Scientific, Singapore.
 V. Jurdjevic, (1997), Geometric Control Theory, Advanced Studies in Mathematics, vol 52. Cambridge University Press, Cambridge.
 H.J. Sussmann, (1997), An introduction to the coordinate-free maximum principle , In: Jakubezyk B, Respondek W (eds) Geometry of feedback and optimal control. Marcel Dekker, New York, pp 463-557.
 J. Biggs, W. Holderbaum, (2008), Planning rigid body motions using elastic curves, Math. Control Signals Syst. 20: 351-367.
 A.Yucesan, A.C. Coken, N.Ayyildiz,(2004), On the darboux rotation axis of Lorentz space curve, Applied Mathematics and Computation, 155:345-351.
 R.Lopez, (2008), Differential geometry of curves and surfaces in Lorentz-Minkowski space, University of Granada.