Abazari N.Sager I.2023-06-162023-06-1620102010-376Xhttps://hdl.handle.net/20.500.14365/3939In 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.eninfo:eu-repo/semantics/closedAccessDarboux vectorHamiltonian vector fieldLie groupLorentz metricMaximum principleOptimal controlRigid body motionElastic energyFrenet frameHamiltonian systemsHamiltonian vector fieldsIntegrable Hamiltonian systemLorentzMinkowski spaceMotion planning problemsOptimal control problemOptimal controlsRigid body systemsRigid-body motionSmooth trajectoriesControlCurve fittingLie groupsMaximum principleMotion planningOptimal control systemsRigid structuresHamiltoniansArticle2-s2.0-84871273487