Real Time Direct Kinematics Solution of 3-RPS Parallel Robot Using a Semi-Analytical Homotopy Method

In this paper, the forward kinematics of a parallel manipulator with three revolute-prismatic-spherical (3RPS), is analyzed using a combination of a numerical method with semi-analytical Homotopy Continuation Method (HCM) that due to its fast convergence, permits to solve forward kinematics of robots in real-time applications. The revolute joints of the proposed robot are actuated and direct kinematics equations of the manipulator leads to a system of three nonlinear equations with three unknowns that need to be solved. In this paper a fast and efficient Method, called the Ostrowski-HCM has been used to solve the direct kinematics equations of this parallel manipulator. This method has some advantages over conventional numerical iteration methods. Firstly, it is the independency in choosing the initial values and secondly, it can find all solutions of equations without divergence just by changing auxiliary Homotopy functions. Numerical example and simulation that has been done to solve the direct kinematic equations of the 3-RPS parallel manipulator leads to 7 real solutions. Results indicate that this method is more effective than other conventional Homotopy Continuation Methods such as Newton-HCM and reduces computation time by 77-97 % with more accuracy in solution in comparison with the Newton-HCM. Thus, it is appropriate for real-time applications.