Local Dexterity Analysis for Open Kinematic Chains

Two methods for determining dexterity of manipulator arms are presented. A numerical method is illustrated for determining all possible orientations of the end-effector of an open kinematic chain based upon a row-rank deficiency of the sub-Jacobian of the kinematic constraint equations. This numerical method has been adapted from a theory for determining the accessible output set for planar and spatial manipulators. Boundaries of regions representing orientation solutions are mapped using continuation algorithms. One dimensional curves in three dimensional space are traced. Orientations of the manipulator end-effector at a point are studied (local dexterity). Only local dexterity solutions can be found. A second analytical method is also presented that determines the global dexterity of the system. This method relies upon determining all singularities of the system and identifying singular surfaces and singular curves. Regions of singular surfaces that are boundary are identified and intersected with a service sphere. This method, however, although analytical and accurate, has many limitations. The two methods are compared via implementation into a six degree of freedom manipulator. 1 . INTRODUCTION Workspaces of robotic manipulators are important characteristics associated with the kinematics of the underlying mechanism. Knowledge of the workspace allows a designer to better understand the manipulator’s functionality. The notion of workspaces has been studied by a number of researchers in the past [1-5]. Analytical conditions were used to obtain the workspace by Tsai and Soni [6], Yang and Lee [7], Lai and Menq [8], Freudenstein and Primrose [9], and Spanos and Kohli [10]. Numerical methods were also used by Haug et al. [11]. The general four degree of freedom revolute manipulator has been studied by Cecarelli and Vinciquerra [12]. Workspace of parallel manipulators has also been studied [13,4,14]. The dexterous workspace has been defined by Kumar and Waldron [15] as the subspace of the workspace within which a vector on the end-effector may assume all orientations. Because of joint limits and geometric constraints (e.g., at workspace boundaries), the dexterous workspace may not include all of the workspace. The subject of dexterity and orientability has also been studied by numerous researchers in the field [15-17]. This paper introduces two methods for determining dexterity. A numerical method based upon criteria for determining the accessible output set [11] was adapted to determine local dexterity. Local dexterity will be further clarified in the following section. Due to its inherent numerical formulation (using continuation methods), it will be shown that this method can only find local dexterity. It will also be shown that this numerical method is dependent upon the determination of a starting point on the onedimensional curve defining the region of dexterity (called the service region). Therefore, only individual regions can be traced. Another method for determining the workspace of manipulators presented by Abdel-Malek and Yeh [18] will be used to determine global dexterity. This method will be shown to be Abdel-Malek, K. and Yeh, H. J., (2000) "Local Dexterity Analysis for Open Kinematic Chains," Mechanism and Machine Theory, Vol. 35, pp. 131-154.