Direct Kinematics of a Three Revolute-Prismatic-Spherical Parallel Robot Using a fast Homotopy Continuation Method

In this paper, the forward kinematics of a parallel manipulator with three revolute-prismatic- spherical (3RPS) is analyzed utilizing a fast Homotopy Continuation Method which contributes to solving the forward and inverse kinematic of this robot in real-time applications. Unlike the previous studies on this category of manipulators, the revolute joints of the proposed manipulator are actuated instead of the prismatic joints. Moreover, direct position kinematics of the manipulator leads to a system of three nonlinear equations with three unknowns that needs to be solved. In this paper, a fast and efficient Homotopy Continuation Method which is called the Ostrowski Homotopy Continuation Method has been used to solve the direct kinematics equations of this parallel manipulator. Fortunately, this method has some advantages over conventional numerical iteration methods. First, it is independent in choosing the initial values and second, it can find all solutions of equations without any divergence just by changing auxiliary Homotopy function. Numerical example and simulation that has been done to solve the direct kinematic equations of the 3RPS parallel manipulator lead to 7 real solutions. However, results indicate this method is more effective than other conventional Homotopy Continuation Methods such as Newton Homotopy. The results indicate that the Ostrowski Homotopy method reduces computation time by 77-97% with more accuracy in solution in comparison with the Newton Homotopy. Thus, it is suitable for real-time applications.