To address the degradation of tracking accuracy and insufficient control stability in robotic manipulator systems involving nonsmooth effects, such as viscous and sliding friction, a servo control strategy based on the Udwadia-Kalaba (U-K) method is proposed. Within an extended U-K dynamics framework that accounts for the coupling between normal and tangential constraint forces, the desired trajectory of the controlled system is formulated as a set of servo constraints. Analytical expressions for the active control forces and passive constraint forces are then explicitly derived, based on which a corresponding servo control method is developed. This method effectively mitigates system oscillations and tracking accuracy degradation caused by the coupling of constraint forces. To further enhance system stability, the Baumgarte stabilization method is introduced to correct constraint violations induced by initial perturbations. In addition, a smoothing Newton algorithm is employed to regularize the absolute-value term arising from the determination of the normal support-force direction, thereby improving numerical stability. Simulations of various complex trajectories are conducted in MATLAB. The results demonstrate that the proposed servo control strategy effectively suppresses friction-induced oscillations while ensuring high-precision and asymptotically stable trajectory tracking, exhibiting favorable robustness and generalizability.