Optimization and Systems Theory
For redundant robotic chains composed of simple one-degree of freedom joints or links, the forward kinematic map from joint space to the workspace of the end-effector is interpreted geometrically in terms of Riemannian submersions. Several properties of redundant robots then admit clear geometric characterizations, the most remarkable being that the Moore-Penrose pseudoinverse normally used in Robotics coincides with the horizontal lift of the Riemannian submersion. The end-effector of the robot leaves on the Special Euclidean group in 3 dimension. On SE(3), the dynamical equations of the robotic chain look like a set of controlled Euler-Lagrange equations, or Euler-Poincare' equations after reduction by group symmetry. Variational methods are used to generate a geometric spline for such equations and the extra complications of the corresponding reduction, due to the semidirect product structure of SE(3), are analyzed. The loop is closed in workspace using a PD controller which is then pulled back to joint space by means of the horizontal lift, all respecting the different geometric structures of the two underlying model spaces.