This paper provides an algorithm for computing singularity-free paths on non-redundant closed-chain manipulators. Given two non-singular configurations of the manipulator, the method attempts to connect them through a configuration space path that maintains a minimum clearance with respect to the singularity locus at all points. The method is resolutioncomplete, in the sense that it always returns a path if one exists at a given resolution, or returns “failure” otherwise. The path is computed by defining a new manifold that maintains a oneto- one correspondence with the singularity-free configuration space of the manipulator, and then using a higher-dimensional continuation technique to explore this manifold systematically from one configuration, until the second configuration is found. Examples are included that demonstrate the performance of the method on illustrative situations.