  Trajectory planning is a critical step while programming the parallel
manipulators in a robotic cell. The main problem arises when there exists a
singular configuration between the two poses of the end-effectors while
discretizing the path with a classical approach. This paper presents an
algebraic method to check the feasibility of any given trajectories in the
workspace. The solutions of the polynomial equations associated with the
tra-jectories are projected in the joint space using Gr{\"o}bner based
elimination methods and the remaining equations are expressed in a parametric
form where the articular variables are functions of time t unlike any numerical
or discretization method. These formal computations allow to write the Jacobian
of the manip-ulator as a function of time and to check if its determinant can
vanish between two poses. Another benefit of this approach is to use a largest
workspace with a more complex shape than a cube, cylinder or sphere. For the
Orthoglide, a three degrees of freedom parallel robot, three different
trajectories are used to illustrate this method.
