A new development of homotopy continuation method, applied in solving nonlinear kinematic system of equations of parallel mechanisms

Homotopy Continuation is known as a powerful systematic technique for solving kinematic equations of parallel or serial robots. This paper presents a new development of homotopy continuation which can provide more reliable solutions. The proposed approach, unlike former ones, finds the solutions of equations by direct investigation of collisions between continuation paths, which are known as singularity. In fact, in the proposed development, instead of escaping from singularity, it is investigated by using some hyper order terms of Taylor expansion of equations. Accordingly, the presented development of homotopy continuation is called as Collision-Based Homotopy Continuation technique. It is explained, that when there is no collision between continuation paths, the first term of Taylor Expansion is enough to find variation of roots in each iteration of the method. However, when two or more than two continuation paths are collided with each other, some hyper order terms of Taylor Expansion are taken into account in order to deal with singularity. To demonstrate the performance of the proposed methods, forward kinematics of a well known parallel mechanism robot is considered. The results reveal that Collision-Based Homotopy Continuation is able to find all roots (complex and real) of high-dimensional equations systems. Also, superiority of collision-based homotopy continuation in comparison solutions like Bertini solver package is shown in aspect of precision.