  Methods for inverse kinematics computation and path planning of a three
degree-of-freedom (DOF) manipulator using the algorithm for quantifier
elimination based on Comprehensive Gr\"obner Systems (CGS), called CGS-QE
method, are proposed. The first method for solving the inverse kinematics
problem employs counting the real roots of a system of polynomial equations to
verify the solution's existence. In the second method for trajectory planning
of the manipulator, the use of CGS guarantees the existence of an inverse
kinematics solution. Moreover, it makes the algorithm more efficient by
preventing repeated computation of Gr\"obner basis. In the third method for
path planning of the manipulator, for a path of the motion given as a function
of a parameter, the CGS-QE method verifies the whole path's feasibility.
Computational examples and an experiment are provided to illustrate the
effectiveness of the proposed methods.
