Analysis Of Singularities Condition Of Robots Manipulators Based On Grassmann-Cayley Algebra

Robotics research and development depend on progresses of Mechanical Engineering, Electrical Engineering, Control Systems, and Computer Sciences as well as Mathematics. In recent years, researches show Grassmann-Cayley Algebra (GCA) will be a promising framework for applications in Robotics. In this research, the motion of robots manipulators is investigated based on Grassmann-Cayley Algebra in order to perform its specified tasks which would be benefit for the designer as well as the user.In this dissertation, an extensive mathematics tools and software are studied to analyze the kinematics of instantaneous motion of robots manipulators.One of the key problems of Mechanic of robots manipulators which make trouble in engineering practice is the rapidly increase of the difficulty to control a manipulator when the links augmented. Since a robot manipulator is classified by several criteria in order to determine which robot is suitable and adequate for a given task, in this work we devote to the study of a configuration that provides a description of the instantaneous motion of robot using above mentioned mathematics tools.The Jacobian matrix is one of the most important quantities in the analysis of robot motion.In practice calculus, the difficulty to solve the nonlinear equation det (J)=0increases when the architecture of the robot becomes complex. To overcome this difficulty, we use the Grassmann-Cayley Algebra to determine the singular critical poses in coordinate-free manners. The principal advantage of this method also called Double Algebra is that it can simultaneously investigate the singularity conditions and the pure conditions of the rigidity of the framework for robots manipulators.The first portion of this thesis is devoted to the calculus of the Global Wrench System and the Twist System respectively for the parallel and serial manipulators, using both Geometric approach of Screw Theory and its reciprocate for parallel manipulators while only Screw Theory is applied to determine the Twist System for serial robot. Additionally the symbolic approach of Pliicker coordinates lines using meet and join operators is developed to represent the Jacobian matrix of the robot motion when the Superbracket which represented the symbolic approach of the determinant of this matrix is described. The Graphic User Interface (GUI) of Pr Stephane Caro is applied here.The second portion of this thesis is focused on the implementation of the above mentioned method developed in the first portion on the3-PRS Parallel Manipulators (PMs) and the Wrist-Partitioned6-R Serial Manipulator (SM):the singularities conditions are obtained in Grassmann-Cayley Algebra language.Finally the third portion of this thesis presented the interpretation of all obtained results and the verification of our hypothesis in this work in Grassmann-Cayley Algebra language when the parallel or the serial robot is in singular configuration.Since one of the advantage of this Double Algebra is to determine simultaneously singularity condition and pure condition of the rigidity framework of robot manipulator, the stiffness and the rigidity of the3PRS Parallel Manipulators and Wrist-Partitioned6-R Serial Manipulator are also investigated in coordinate free expression and give respectively double, single and undetermined singularities condition for3-PRS,3-PRS and3-PRS Parallel Manipulators while a single singularity condition is obtained for the Wrist-Partitioned6-R Serial Manipulator. The dependence between the position of the actuated joint and the rigidity of the framework is discussed in detail through the3-PRS Parallel Manipulators with variable actuated joints.