Matroid Method versus Tsai-Tokad (T-T) Graph Method in the Kinematic Analysis of the Mechanical Systems consisting of Gears

In this thesis, Matroid and T-T Graph methods are compared. These are graphical methods which are used in kinematic analysis of mechanisms including gear trains. Both methods are based on Graph Theory. T-T graph method is developed by combining non-oriented graphs and oriented graphs. Whereas, incident matrix derived from oriented graph is used in Matroid. In order to perform kinematic analysis by using these two methods, a conventional Geared Robotic Mechanism (GRM) is considered as a sample mechanism. In Matroid, depending on the numbering links and joints, the digraph attached to kinematic chains is generated. Reduced incidence node-edge, spanning tree, path and cycle basis matrices are developed for this digraph. Equations for relative angular velocities of all turning and gear pairs are defined. Screw theory and plücker coordinates are defined to find the offset angles betwen z-axes of joints and z-axis of base. Twist intensities matrix is produced for turning and meshing pairs. Orthogonality condition for relative angular velocities is defined to acquire independent equations for relative velocities of turning pairs. Speed (teeth) ratio is used to express relative velocities of turning pairs as a function of input velocities. Finally, by using path and twist intensities matrices, link absolute angular velocities are determined in vectorial forms. In T-T, on the other hand, the graph associated to the mechanism is presented in terms of labeling of links, joints and axes of rotation. According to the graph, paths are stated. By investigating the level of axes of rotation, transfer vertices (carrier arms) are determined. By considering each link as a rigid body, terminal equations of turning and gear pairs are stated and for each terminal equation, gear ratio is obtained. Fundamental circuit equations are directly written from the graph and coaxial conditions are used for further kinematic analysis. Equations of output angular velocities in terms of input ones are developed in terms of gear ratios. Final results are obtained in vectorial forms by using Denavit-Hartenberg Convention. Finally, results of the relative and absolute angular velocities in both methods are identical. Differences are just related to how kinematic analysis is performed, how the final results are obtained and which definitions and techniques are used in both methods. Benefits and drawbacks of both methods are also specified. Keywords: Matroid Method, T-T Graph Method, Geared Robotic Mechanisms (GRMs), Kinematic analysis