Analytical Determination of Principal Twists and Singular Directions in Robot Manipulators

The identification of principal twists of the end-effector of a manipulator undergoing multi-degree-of-freedom motion is considered to be one of the central problems in kinematics. In this paper, we use dual velocity vectors to parameterize se 3 , the space of twists, and define an inner product of two dual velocities as a dual number analog of a Riemannian metric on SE 3 . We show that the principal twists can be obtained from the solution of an eigenvalue problem associated with this dual metric. It is shown that the computation of principal twists for any degreeof-freedom of rigid-body motion, requires the solution of at most a cubic dual characteristic equation. Furthermore, the special nature of the coefficients yields simple analytical expressions for the roots of the dual cubic, and this is turn leads to compact analytical expressions for the principle twists. We also show that the method of computation allows us to separately identify the rotational and translational degrees-of-freedom lost or gained at singular configurations. The theory is applicable to serial, parallel, and hybrid manipulators, and is illustrated by obtaining the principal twists and singular directions for a three-degree-of-freedom parallel, and a hybrid six-degree-of-freedom manipulator.