The case of permanent singularity together with possible movement is corresponding to special mechanisms as Bricard mechanism. Then the jacobian analysis as related to first-order propoerties of movement will fail to predict the dof (or degree of mobilty). The jacobian of active joints still relate the active joint forces (or torques depending where you actuate) to the wrench applied on the platform. A zero determinant means that the 6 actuating forces are spanning a 5 dim subspace of the wrench space. So there must be a linear constraint between the actuating forces. Imposing independants actuating forces may cause troubles...

Jean-Pierre Merlet has done a reference work about singularities of parallel robots.

Thank for clarifying the concept. I will further refer the book by J P Merlet and concept of reciprocal screws.