On the determination of open-loop manipulator singularities subject to unilateral or non-unilateral constraints

This paper presents a general Jacobian row-rank deficiency method to obtain all singular curves/surfaces enveloping the workspace of n general n degree-of-freedom mechanism subject to unilateral or non-unilateral constraints. This method can determine the singular behaviour of manipulators with any joint profiles. For non-unilateral constraints, there are four types of singularities: Type I sets are position Jacobian singularities; Type II sets are instantaneous singularities that are due to a generalized joint that is reaching its apex; Type III sets are general domain boundary singularities, which are associated with the initial and final values of the time interval; and Type IV sets are also called coupled singularities, where the null space is reduced in one submatrix due to Type III singularity. For unilateral constraints there are three types of singularities: Type I sets are the same as non-unilateral constraints; Type II sets are a subdomnin of Type III for non-unilateral constraints, which are associated with the joint, limits; and Type III sets are coupled singularities, which are associated with a relative singular Jacobian, where the null space is reduced in one submatrix due to a Type II singularity. The complete mathematical formulation is presented and illustrated using a simple 3-DOF planar and a 4-DOF spatial manipulator.