In this paper, the kinematic structure of the geared robotic mechanism (GRM) is investigated with the aid of two different methods which are based on directed graphs and then these methods are compared, accordingly. One of the methods is Matroid method developed by Talpasanu and the other is Tsai-Tokad (T-T) graph method developed by Uyguroğlu and Demirel. The findings show that the kinematic structure of the geared robotic mechanism can be represented by directed graphs and angular velocity equations of the mechanisms can be systematically obtained from the graphs. The advantages and disadvantages of both methods are demonstrated in relation to each other.

The kinematics of parallel mechanisms are defined by means of a kinematic constraint map (KCM) that captures the constraints imposed on its links by the joints. The KCM incorporates both pose parameters describing the configuration of every link and the design parameters inherent in the mechanism architecture. This provides a coherent approach to determining C-space singularities and generalised Grashof conditions on the design parameters under which these can occur.

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A. ABSTRACTIn this paper, the kinematic structure of the geared robotic mechanism (GRM) is investigated with the aid of two different methods which are based on directed graphs and the methods are compared. One of the methods is Matroid Method developed by Talpasanu and the other method is Tsai-Tokad (T-T) Graph method developed by Uyguroglu and Demirel. It is shown that the kinematic structure of the geared robotic mechanism can be represented by directed graphs and angular velocity equations of the mechanisms can be systematically obtained from the graphs.The advantages and disadvantages of both methods are demonstrated relative to each other. B. INTRODUCTIONKinematic and dynamic analysis of mechanical systems have been well established by using graph theory in recent years. Non-oriented and oriented graphs were used for this purpose. Nonoriented graph technique is mainly used for the kinematic analysis of robotic bevel-gear trains [1]-[4]. The oriented graph technique has been used for electrical circuits and other types of lumped physical systems including mechanical systems in one-dimensional motion since the early sixties [5]-[8]. Chou et al. [9] used these techniques to three-dimensional systems. Recently, Tokad developed a systematic approach, the so called Network Model Approach, for the formulation of three dimensional mechanical systems [10], and Uyguroglu and Tokad extended this approach for the kinematic and dynamic analysis of spatial robotic bevel-gear trains [11]. Most recently the oriented and non-oriented graph techniques were compared by Uyguroglu and Demirel [12], and the advantages of the oriented graph over the non-oriented graph were demonstrated using the kinematic analysis of bevel-gear trains. In order to overcome the weaknesses of the methods developed by Tsai and Tokad, both method were combined and the so called T-T Graph method was introduced []. On the other hand, Talpasanu et al. [] developed Matroid Method for the kinematic analysis of geared mechanisms based on directed graph as well.In this paper the kinematic structures of the GRM is investigated with the aid of Matroid Method and T-T Graph method. Since both methods are used directed graph, the similarities and the differences are shown and the advantages of each system are indicated. C. Geared Robotic MechanismGRMs are closed-loop configurations which are used to reduce the mass and inertia of the actuators' loads. Gear trains in GRMs are employed such that actuators can be placed as closely as possible to the base. Figure 1 shows functional schematic of the GRM. It has 3 Degree of Freedom (i.e. it has 3 inputs) and the end-effector can have spatial motion (3 Dimensional) because this mechanism can generate two rotations about two intersected axes and one rotation of end-effector about its axis. In this mechanism 4, 5, and 6 are sun gears (input links), 1 and 2