A new approach is proposed to construct the dynamic models of robots on the basis of Riemann geometry and tensor analysis, The suggested model belongs to the parametric type. The chosen parameter is a non-linear function of the entire kinetic energy of the robot mechanism. In such a way that the multidimensional complicated equations of motion are reduced to the elementary dynamics of a free material point. In the paper the concept, the geometrical interpretation and the metric coefficient computation method is outlined....The metric coefficient introduced in the paper makes possible to determine the correspondence between the robot' s end effector contour velocity and the mapping point parametric velocity in the Riemann space. A method of manipulator motion modelling along the given trajectory has been developed. The use of the results makes possible to develop optimal manipulator control according to energy criterion and ~ realize effective force control for contour operations. In the second part of the paper (Chapter 3) computing methods, modelling approach and the results of real experiments for PUMA-560 are given.