solution of direct kinematic problem for stewart-gough platform with the use of analytical equation of plane

The paper concerns the solution of direct kinematic problem for the Stewart-Gough platform of the type 6-3. The article represents a detailed analysis of methods of direct kinematic problem solution for platform mechanisms based on the parallel structures. The complexity of the problem solution is estimated for the mechanisms of parallel kinematics in comparison with the classic manipulators, characterized by the open kinematic chain.

The method for the solution of this problem is suggested. It consists in setting up the correspondence between the functional dependence of Cartesian coordinates and the orientation of the moving platform centre on the values of generalized coordinates of the manipulator, which may be represented, in the case of platform manipulators, by the lengths of extensible arms to connect the foundation and the moving platform of the manipulator. The method is constructed in such a way that the solution of the direct kinematic problem reduces to solution of the analytical equation of plane where the moving platform is situated. The equation of the required plane is built according to three points which in this case are attachment points of moving platform joints. To define joints coordinates values it is necessary to generate a system of nine nonlinear equations. It ought to be noted that in generating a system of equation are used the equations with the same type of nonlinearity. The physical meaning of all nine equations of the system is Euclidean distance between the points of the manipulator. The location and orientation of the moving platform is represented as a homogeneous transformation matrix. The components of translation and rotation of this matrix can be defined through the required plane.

The obtained theoretical results are supposed to be used in the decision support system during the complex research of multi-sectional manipulators of parallel kinematics to describe the geometrically similar 3D-prototype of the manipulator.

In the process of work it is planned to make the optimal choice of numerical approach to the solution of the suggested equation system, considering the nonlinearity type etc., to develop the methodology for evaluation of received solutions, to adjust the suggested method for the solution of direct kinematic problem for the Stewart-Gough platform of the type 6-6.