An Efficient Method for Solving the Direct Kinematics of Parallel Manipulators Following a Trajectory

This paper presents an efficient method for solving the direct kinematics of parallel manipulators that follow a defined singularity-free trajectory. Despite the main problem is to solve the inverse kinematics, calculating the errors between the desired path and the actual path requires solving the forward kinematics which is a challenging problem. The proposed method combines closed-loop Jacobian algorithm with Newton-Raphson method to efficiently identify the desired solution. Inverse kinematics is solved using the Jacobian algorithm whereas forward kinematics is solved using Newton-Raphson method. To avoid the numerical instabilities of the Newton-Raphson method, the current state of the manipulator is used as the initial guess for the next state. This makes the numerical solution converges to the correct and desired solution quickly with a few number of iterations. The proposed method is applied to a 3RRR planar parallel manipulator and the simulation results show the effectiveness of the method. The algorithms presented in this article can be applied to other parallel manipulators. 