  The goal of this paper is to present a rigorous and intrinsic formulation of
a Riemannian PD-regulator of the robot's tool, The first one is based upon the
Lasalle's invariance principle, we use it to control the tool's position in the
workspace under the assumption of absence of singularities in configuration
space, The second method deals with geometrical constraints on the trajectory
of the robot's tool with the same assumption, we construct a unique orthogonal
force that is viewed as a gravitational force that keeps the tool constrained,
We also present a variation of the first method in the case of double pendulum
based on the Lyapunov stability theorem. With this modification, we control the
tool and the difference between the two angles, we did simulations on a
two-link manipulator that shows the efficiency of the presented methods.
