In this article, an optimal controller for achieving constrained admittance control is proposed. This controller strictly adheres to the constraint boundaries while ensuring minimal variations in kinematic energy. The proposed method integrates admittance control for human-robot interaction with the Udwadia-Kalaba equations for constrained motion into a unified framework. The proposed architecture has been tested and validated both with simulations and real tests on a 6-DoF UR5e robot. The results demonstrate that the proposed architecture outperforms virtual fixtures, one of the most commonly used techniques to implement effective path-following control.