This paper presents a computational method for generating time-optimal control (TOC) trajectories for robotic point-to-point motions. The algorithm is composed of two phases : initialization and refinement. In the initialization phase, a two-point boundary value problem (TPBVP) resulting from the perturbed TOC problem is solved for an appropriately chosen perturbation parameter. The solution obtained from the initialization phase is then refined by solving a set of initial value problems (IVPs) sequentially and/or in parallel until the desired solution is achieved. The proposed two-phase method is computationally efficient since the resulting TPBVP is solved only one time and the remaining problem becomes solutions to a set of IVP subproblems. The algorithm was used to investigate the effects of robot parameters on the TOC structure through computer simulations on an example robot system with different motion configurations, including both regular and singular trajectories.