It is known that the kinematic model of several nonholonomic systems can be converted into a {\it chained form} control system. Asymptotical stabilization of any equilibrium point of this system cannot be achieved by means of a continuous pure state feedback, but can be obtained by using a {\it time-varying } continuous feedback \cite{SA2}. In the present paper, a backstepping technique is used to derive explicit time-varying feedbacks that ensure {\it exponential} stability of the closed-loop system. Two classes of control laws are proposed, with one of them involving a dynamic extension of the original chained system. Like in other recent studies on the same topic, exponential convergence is obtained by using the properties associated with homogeneous systems. The control laws so obtained are continuous in both the state and time variables. A complementary and novel feature of the proposed control design technique lies in the estimation of a lowerbound of the asymptotical rate of convergence as a function of a reduced set of control parameters which is independent of the system's dimension. Moreover, the fact that this lowerbound may take any positive value indicates that any prespecified exponential rate of convergence can be achieved via a suitable choice of the control parameters.