The article addresses the tracking control problem of differentially driven nonholonomic Wheeled Mobile Robots (WMRs) whose nonlinear kinematic-dynamic equations of motion subject to bounded inputs. To overcome the difficulties arising from the constraints in the system and to avoid singular states, the potential for their appearance is common in this system, we implement sigmoid functions in the feedback loop. Also, since the speed variables (linear and angular) that enter the kinematic subsystem are not independent variables and are the result of the action of bounded forces/torques on the dynamic system, we present a special way of reconstructing them. Using the flatness property of the system, the article presents a control algorithm for tracking a time-parameterized path (a geometric path with an associated timing-law). The proposed control scheme allows us to present conditions on the required trajectory that, if they are met, the input signals satisfy the given restrictions and ensures convergence to the reference trajectory.