This article proposes a tracking controller for servomechanisms with a continuous friction model. The parameters of this model are estimated through a proposed two-step offline identification methodology that uses a linear extended state observer (LESO) based on the integral of position measurements, termed as ILESO. In the first step, the viscous and Coulomb coefficients of this friction model are estimated by the ordinary least squares method. The second step employs these estimates to obtain the initial guess for the nonlinear least squares method that identifies all the parameters of the nonlinearly parameterized friction model. Finally, a backstepping controller is designed for reference tracking, which uses the estimated offline friction model and online estimations of the servo position, velocity, and system uncertainty. These online estimations are provided by a modified ILESO, termed as FILESO, that has a similar structure to that of the ILESO but incorporates the identified offline friction model into its dynamics. The convergence and stability of the proposed observer and controller are established using contraction analysis. Experimental results show that the proposed approach exhibits satisfactory performance.