For a platform connected to its base through two chains forming a single loop, the instantaneous mobility may be expressed by a set of motion screws that is in the intersection of the sets of motion screws for each of the two chains. A recent work shows that the platform remains mobile after differential displacement along all mobile paths if the Lie closures of the screw sets of the two chains are each within the span of the union of screw sets of those chains. If this union span is one dimension short of containing the Lie closures of the two chains, a quadratic form determines whether the reference pose is at a constraint singularity and resolves the mobile paths at that singularity. Those results are now extended to a platform manipulator with more than two chains, using a recursive procedure for updating velocity, acceleration, and higher-order descriptions of platform mobility after adding successive chains. The new analytical technique characterizes the bifurcation of the mobility at constraint singularity of 3RSR, 3RER, and 3UPU platform mechanisms proposed for use in constant-velocity couplings, robotic wrists, and translational manipulators.