USE OF THE GROBNER BASIS IN THE STUDY OF MANIPULATORS TOPOLOGY

In the classic cases used in industry, manipulators need to pass through singularities of the joint space to change their posture: the end-effector must bump into the frontier of the workspace. A 3-DOF manipulator can execute a non singular change of posture if and only if there is at least one point in its workspace which has exactly three coincident solutions of the inverse kinematic model. Since it is difficult to express this condition directly from the kinematic model, it is proposed to eliminate two joint variables from the system in order to obtain a condition that depends only on the last joint variable. For this purpose, a powerful algebraic tool is used: the Grobner basis. With this approach, it is possible to obtain analytical expressions of the surfaces of the parameters space that separate the different types of manipulators. The determinant of Jacobian matrix of the direct kinematic model is considered equal to zero to obtain the other surfaces that separate the various regions for different topologies. In this paper, it will be presented the process of obtaining the surfaces of separation, as well as the corresponding curves due changes in the parameters of 3R orthogonal manipulators. Detailed knowledge of the various regions of the space of parameters is important for the optimal design of manipulators which obeys a topology specified by the designe r. In the study of manipulator robots is essential to know the topology of the singularity surfaces in the workspace. These singularities are defined as places where the determinant of the Jacobian matrix of direct kinematic model (DKM) is annulled, resulting in the equations of surfaces which divide the workspace in several regions that have manipulators with same properties (binary or quaternary, regions with the same numbers of cusps and node points). These regions are called domains. Since it is difficult to express them directly from the kinematic model, it is proposed to eliminate two joint variables from the system in order to obtain a condition that depends only on the last joint variable. For this purpose, a powerful algebraic tool is used: the Grobner basis. With this approach, it is possible to obtain analytical expressions of the surfaces of the parameters space that separate the different types of manipulators. The annulment of the determinant of Jacobian matrix of the inverse kinematic model (IKM) enables to obtain the other surfaces that separate the various regions for different topologies. Wenger and El Omri (1996a; 1996b) showed that for some choices of the parameters, manipulators with three rotational joints (3R) may be able to change posture without meeting a singularity in the joint space. They succeed in characterizing such manipulators (Wenger, 1998), but they needed general conditions on the design parameters. Corvez (2002) found important results about this issue. In 2004, Baili realized researches on the proprieties of 3R manipulators with orthogonal axes and made a classification in the parameters space. This article aims to show the achievement of the separation surfaces in parameters space of manipulators with three rotational joints with orthogonal axes as described in Fig. 1. It is described the process of obtaining the surfaces, as well as the corresponding curves due to changes in the parameters of 3R orthogonal manipulators. In addition, a brief classification of 3R manipulators is presented; this classification is based on two criteria to define a topology of the workspace: the number of cusp points and the number of nodes (intersection of two singularity branches). The study of this type of manipulator is done according to the Denavit-Hartenberg parameters: d2, d3, d4, r2, and r3. To reduce the number of parameters, will be considered d2 = 1 and r3 = 0. The joint variables are �1, �2 and �3 which represent the input angles of the actuators. For this type of manipulator, the direct kinematic model, obtained in Saramago et al. (2007), is given by: