This paper investigates the leader-follower formation control problem of nonholonomic mobile robots based on backstepping technique composed with the bio-inspired neurodynamics while avoiding collision with obstacles. Kinematics model of robot and nonholonomic constraint are introduced and formation control scheme is formed based on backstepping technique. In order to solve velocity jump in backstepping kinematics model, the bio-inspired neurodynamic approach is used. In most of the previous studies, researches are used separation-bearing approach and also supposed that desired separation and bearing are constant. In this paper this assumption is relaxed and desired separation and bearing are considered to be time varying. Error dynamics equations are derived and a new controller is proposed. Also an auxiliary reference angular velocity control law is proposed to guarantee global asymptotic stability of the followers and local asymptotic stability of the entire formation according to direct method of Lyapunov. A common example of changing the formation is obstacle avoidance, when an obstacle is located within a follower path and is not in its leader path. Time varying functions for desired separation and bearing are chosen and the new controller is developed with its proof of stability. Simulations results reveal that each follower robot can track its real time leader employing the proposed kinematic controller while avoiding obstacles. Furthermore control inputs at the start moment and also while avoiding obstacles, do not contain impractical jumps and are reasonable thanks to integrating bio-inspired neurodynamic with backstepping technique.