  The direct kinematics analysis is the foundation of implementation of real
world application of parallel manipulators. For most parallel manipulators the
direct kinematics is challenging. In this paper, for the first time a fast and
efficient Homotopy Continuation Method, called the Ostrowski Homotopy
continuation method has been implemented to solve the direct and inverse
kinematics problem of the parallel manipulators. This method has advantage over
conventional numerical iteration methods, which is not rely on the initial
values and is more efficient than other continuation method and it can find all
solutions of equations without divergence just by changing auxiliary Homotopy
function. Numerical example and simulation was done to solve the direct
kinematic problem of the 3-UPU parallel manipulator that leads to 16 real
solutions. Results obviously reveal the fastness and effectiveness of this
method than the conventional Homotopy continuation methods such as Newton
Homotopy. The results shows that the Ostrowski-Homotopy reduces computation
time up to 80-97 % with more accuracy in solutions in comparison with the
Newton Homotopy.
