The paper concerns the dynamics of curvilinear systems which are often met in mechanical systems (robots, artificial satellites and so on). We only suppose that each section is rigid. Using Lie group theory, a general curvilinear system is then equivalent to a differentiable distribution of displacements, elements of the Lie group of Euclidean displacements the algebra of which may be identified with the Lie algebra of screws. The kinematics is described by the lagrangian field of deformations and the lagrangian field of velocities elements of the Lie algebra and with standard hypotheses about the distribution of external forces, the intrinsic equations are obtained, the displacements or deformations being small or large. The non linearities (of inertia terms as for internal strenghts) appear by the adjoint mapping and its derivation: the Lie braket. Last, the elements to automatically obtain scalar equations and to come back to more classical models (beam, cable,) are given.
Intrinsic Optimal Control for Mechanical Systems on Lie Group
