The design for isotropy of a class of six-dof parallel-kinematics machines

The design for isotropy of a large class of six-dof parallel-kinematics machines (PKMs) whose six actuated wrenches intersect pairwise, is the subject of this paper. A PKM is called isotropic when it can achieve one or more postures under which the condition number of its Jacobian matrices becomes unity, thereby offering a high positioning accuracy. Based on a symbolic expression of the inverse of the forward Jacobian matrix, we analyze the isotropy condition for this class of PKMs. It is shown that isotropy can be achieved only when the moving platform (MP) bears an equilateral-triangular shape; however, the operation point need not be the centroid of this triangle. Moreover, for a MP with an acute-triangular shape, there exist postures that we call quasi-isotropic, under which the condition number is close to unity, while the six rows of the Jacobian matrix are mutually orthogonal. This greatly enriches the list of candidates for the MP shape and the location of the operation point, required, e.g., when a gripper or another tool is attached to the MP triangle.