Transverse-regularity is a point-wise manipulator property, ensuring that the singular set Σ is locally a smooth manifold. A generic manipulator is one which is transverse-regular in any configuration. The contribution of this paper is a necessary and sufficient condition for transverse-regularity. The condition is based on the manipulator’s joint screws and their screw products. An expression for the tangent space to Σ at transverse-regular singularities is derived. It is shown that a manipulator is non-generic if it can attain a pose where the rank of the manipulator’s screw system together with the screw products is not the maximal rank of the Jacobian. A necessary and sufficient criterion for degree-one singularities is given in terms of the mechanism’ joint screws. In particular, any non-redundant manipulator is transverse-regular in a degree-one singularity.

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