An algorithm is presented for the computation of path-constrained time-optimal motions of robotic manipulators exploring the nature of so-called critical points and critical arcs. At critical points the reflected inertia at one of the joints is zero, and the feasible acceleration range at the velocity limit is not unique. Time-optimal motions along critical arcs may be singular in the sense that the acceleration is neither maximum nor minimum. This is in contrast to existing motion optimization methods along specified paths which assume maximum acceleration or deceleration at all times. The consideration of critical arcs makes this algorithm robust near the switching points, which are potential points of failure in the other methods. Critical points can be anticipated along the path by mapping the locus of critical arcs to the position space. Examples are presented to demonstrate the algorithm and the existence of singular critical arcs.