Geometry of the motion of robot manipulators

Abstract

A p-parametric robot is a mapping g of R^p into the homogeneous space P=C₆×C₆/Diag(C₆×C₆) given by the formula g(u₁,...,u_p=exp u₁X₁..... exp u_pX_p, where C₆, is the Lie group of all congruences of E₃ and X₁,..., X_p are fixed vectors from the Lie algebra of C₆. We characterize the set g(R^p) locally by a system of PDE and give some geometrical properties of g as a p-dimensional motion for p<6. We also characterize the Frenet frame of g and show how to construct it for the robot manipulator given by its axes X₁,...,X_p.