Tabulation of the Symbolic Midframe Jacobian of a Robot Manipulator

Abstract

The symbolic midframe Jacobian for a general 6-DOF ma nipulator, a 12-DOF, 6-cylindric-jointed manipulator (6C) and a 24-DOF manipulator in which all four kinematic parameters of each link transform vary, admits to direct, manageable tabulation. For 6-link manipulators, the mid frame Jacobians³J_αand³J_αassociated with rates of change in the link lengths a and twists a do not depend upon the kinematic parameters of the first link frame ( d₁, θ₁, a,, α₁). Similarly, the Jacobians³J_dand³J_θassociated with the rates of change in link offsets d and rotations θ do not depend on the kinematic parameters of the sixth link frame ( d₆, θ₆, a₆, α₆). A general property of a manipulator Jacobian, ex pressed in any frame, is that the column associated with the rate of change of a kinematic parameter is independent of that kinematic parameter. Furthermore, the determinant of the manipulator Jacobian is invariant with respect to its frame of representation, hence, the manipulator singularities may be calculated with relative ease in midframe. Finally, the complexity of the general form of³J_aversus³J_dand³J_αversus³J_θ, as measured in types of symbols (+, -, ., sines, cosines, parameters) and quantity of symbols of each type, is the same.