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For a cylindrical, and a spherical robot, and a robot with a horizontal articulated arm with two links, time-optimal unconstrained trajectories for arbitrary fixed initial and final positions are calculated. The exact equations of motion are utilized. The controls (torques and forces) are limited. The general structure of the optimal solution is discussed and explained physically for each robot. The importance of such analyses during the mechanical design of a robot is pointed out. The reduction of the duration of an optimal motion, compared to more straightforward and natural ones, and hence the increase of the productivity of the robot in an assembly cycle can be considerable. The numerical examples include the "Automelec ACR" and the "IBM 7535 B 04" robots.