This paper proposes a navigation control law based on a Lyapunov function for the construction of a generalized Voronoi diagram (GVD), applicable for mobile robots with ultrasonic sensors operating in indoor environments. The proposed control law is a modification of the one in [H. Choset and K. Nagatani, “Topological simultaneous localization and mapping (SLAM): Toward exact localization without explicit localization,” IEEE Trans. Robot. Autom., vol. 17, no. 2, pp. 125-137, Apr. 2001.]. A new definition is proposed for the distance function that renders the proposed modified control law well defined for all points in free space. To follow the trajectory closely, an architecture for the control law based on simultaneous handling of sensors and the motors is proposed. An efficient implementation of the control law using merely a low-end field-programmable gate array and no external memory is presented. Experimental results validate the efficacy of the proposed modified control law. The paper also proves that the proposed modified control law traces the GVD without any approximation. The results are compared with two earlier studies.