It is known that nonholonomic systems on the plane cannot be stabilized on arbitrary leader's trajectories with piecewise continuous velocities via continuous controllers, even time-varying. In this article, we identify a wide class of smooth bounded reference trajectories (that includes set-points), and we propose a smooth time-varying controller that stabilizes such trajectories, not only for one nonholonomic mobile robot, but also for a network of mobile robots in a leader-follower configuration. We provide new constructions of strict Lyapunov functions with which we establish uniform global asymptotic stability and other strong robustness properties in the formation-error-coordinates space. Thus, our stability statements are unique in the literature and our tools of analysis are original.