XVI CONGRESSO BRASILEIRO DE ENGENHARIA MECÂNICA 16th BRAZILIAN CONGRESS OF MECHANICAL ENGINEERING HIERARCHICALSINGULARITYANALYSISOFANARTICULATEDROBOT Daniel Martins Raul Guenther Henrique Simas Departamento de Engenharia Mecânica, Universidade Federal de Santa Catarina, Caixa Postal 476, 88040 900 - Florianópolis, SC, Brazil E-mails: dmartins,guenther,simas@lcmi.ufsc.br Abstract: Singularity avoidance is a current research topic. Singularities in general are analitically computed by equating the determinant of the Jacobian matrix to zero. However each singularity has a specific influence on the end-effector behaviour. This paper analyses the Jacobian matrix selected for an specific manipulator. The chosen manipulator is a non redundant six degree of freedom manipulator. It is a simplified version of the Roboturb manipulator that was specially designed for recovering turbine blades submitted to cavitation processes. The method is based on graph theory. Singularities are ranked according to the partial order of the associated graph. The results to the manipulator are discussed at the end of the paper. Keywords: Jacobian matrix, Singularity avoidance, Robot Analysis Screw Theory; Graph Theory. 1. Introduction Kinematic analysis derives the relative movements among various links of a given mechanism, for instance a robot manipulator. Typically the end-effector of a robot is programmed to follow a set of desired positions and orientations in space. The first derivatives (velocities) are often imposed as extra conditions on the path tracking. The inverse kinematics problem then is to find all sets of actuated joint variables and their corresponding time derivatives which will bring the end-effector to the desired motion characteristics. The Jacobian matrix, Jacobian for short, is probably the most important matrix related to the kinematics of serial manipulators. Inverse kinematics problem can be solved using closed form expressions (a global solution) for position inverse kinematics. This approach has several drawbacks (Leahy Jr et al., 1987) like computational time and uncer- tainties when a singularity is reached. The inverse kinematics problem is generally solved locally using the Jacobian. The Jacobian also maps the movement of the end-effector in the Cartesian space to the joint space where robot control takes place. Several attempts have been made to improve the velocity of the Jacobian computation via parallelisation (Zomaya et al., 1999) or via a favorable choice of auxiliary variables (Leahy Jr et al., 1987). The Jacobian J = J (q) is a map between velocities at the actuators joints and at the end-effector. Each element of this matrix depends upon position parameters, q, of the manipulator. A singularity occurs whenever the Jacobian drops rank. Singularities are normally detected by the determinant of the Jacobian matrix. This determinant (det J )is dependent upon geometrical and position parameters. Equation det J =0 is a necessary and sufficient condition for the Jacobian be rank deficient. Factoring det J as a product of minimal terms det J = m  i=1 Φ i =0 (1) Terms Φ i group variables responsible for the i-th singularity. The order of these terms is arbitrary. Terms Φ i are minimal in the sense that they cannot be splitted up as a product of smaller, non unity, terms. The manipulator has, in general, m different singularities. A common understanding is that all singularities equally affect, for worst, the manipulator response. This per- ception possibly comes out from eq. (1). There the order of the products Φ i is immaterial. Furthermore Φ i =0 → det J =0 no matter which i-th term is becoming zero. General singularity avoidance algorithms detect the proximity of a singularity but do not specify which particular singularity is being reached. Following developments presented in (Martins and Guenther, 2001), this paper emphasizes that singularities are NOT equallyimportanttothebehaviourofthemanipulator. Therobotkinematicanalysismethod,proposedin(Martins and Guenther, 2001) is applied to a new manipulator. This method is based on the structure of the Jacobian matrix. The sparsity of this matrix will be enhanced and geometrically explored using screw theory. Afterwards graph theory techniques are applied to extract qualitative relationships among input and output variables. The Jacobian matrix