This paper provides an algorithm for computing singularity-free paths on closed-chain manipulators. Given two nonsingular configurations of the manipulator, the method attempts to connect them through a path that maintains a minimum clearance with respect to the singularity locus at all points, which guarantees the controllability of the manipulator everywhere along the path. The method can be applied to nonredundant manipulators of general architecture, and it is resolution complete. It always returns a path whenever one exists at a given resolution or determines path nonexistence otherwise. The strategy relies on defining a smooth manifold that maintains a one-to-one correspondence with the singularity-free C-space of the manipulator, and on using a higher dimensional continuation technique to explore this manifold systematically from one configuration, until the second configuration is found. If desired, the method can also be used to compute an exhaustive atlas of the whole singularity-free component reachable from a given configuration, which is useful to rapidly resolve subsequent planning queries within such component, or to visualize the singularity-free workspace of any of the manipulator coordinates. Examples are included that demonstrate the performance of the method on illustrative situations.