A General Geometric Index for Solving the Forward Kinematics of Planar Parallel Manipulators

In this paper, a new geometrical method is proposed to solve the forward kinematics of planar parallel manipulators exhibiting two translational and one rotational DOF which leads to high-order system polynomial expressions. The main idea has arisen from a geometrical interpretation of the intersection among the vertex space of each kinematic chain. From the type synthesis performed for planar parallel manipulators with identical kinematic structure, it has been revealed that the number of every possible configuration for this type of manipulators is eighteen, which the solution of the forward kinematic problem can be expressed using a univariate polynomial expression. However, among these eighteen configuration only six of them lead to high order polynomials and therefore do not have closed-form solutions. The proposed method in this paper is based on an index representing the intersection of the circles which are extracted from the vertex space of each kinematic limb of the manipulator. In this approach, the orientation angle of the moving platform is selected from an interval; and based on the selected orientation angle, the circles obtained from the vertex space of each limb are determined. The forward kinematic solution happens where all the circles intersect at one common point where this geometrical phenomenon is expressed using the proposed index. In order to illustrate the performance of the proposed approach, three examples are solved. It should be noted that the foregoing method succeeded even in solving examples such as manipulators with six real solutions, manipulators with degenerate answers, which are reported in the literature as special cases.