In this paper, a time-varying leader-follower formation control of nonholonomic mobile robots based on a trajectory tracking control strategy is considered. In the time-varying formation, the relative bearing and distance of each follower are variable parameters, and therefore, the followers can carry out various and complex behaviour even without changing the linear and angular velocities of the leader robot. After proposing the kinematic model of the time-varying leader-follower formation, the backstepping control method is exploited to keep the structure of the defined formation. The global stability of the formation is investigated using the Lyapunov theorem. Moreover, the designed nonlinear controller suffers from the ineffectual large input commands at the beginning of the formation. To rectify this problem, a fuzzy adaptive algorithm is proposed to improve the backstepping controller and the global stability of the resulting fuzzy adaptive backstepping controller is guaranteed. Considering the rate change of relative distance and bearing in the kinematic model of the leader-follower formation and controller design procedure, makes the formation more practical in dynamic and clutter environments, as well as capable of defining complicated behaviour for followers, and provides crash and obstacle avoidance without switching between different control strategies. Finally, the performance of the proposed kinematics model and designed controllers are investigated through simulations and experimental studies.