We propose an efficient stochastic scheme for minimum-time trajectory planning of a nonholonomic unicycle mobile robot under constraints on path curvature, velocities, and torques. This problem, which is known to be complex, often requires important runtimes, particularly if obstacles are present and if full dynamics is considered. The proposed technique is a fast variant of the random-profile approach recently applied to wheeled-mobile robots. It incorporates a trapezoidal-velocity-profile constraint that helps reduce the number of unknown parameters and that speeds up the calculation steps. Results are presented for two- and three-wheel mobile robots in free/constrained workspaces. A comparison with reference solutions, which were obtained independently, shows that the proposed variant is able to achieve almost the same quality of calculated trajectories while reducing the runtime considerably.