With wide applications of kinematically redundant manipulators in robotics, obstacle and singularity avoidance emerge as critical issues to be addressed. Correspondingly, three problems have to be considered, including the determination of critical points on a given manipulator, the computation of joint velocities using inverse kinematics, and the analysis of singularity caused by configurations of manipulators. In this paper, these tasks are formulated as a convex quadratic programming (QP) subject to equality and inequality constraints with time-varying parameters where physical constraints such as joint physical limits are also incorporated directly into the formulation. To solve the QP problem in real time, a recurrent neural network called the improved dual neural network is applied, which has lower structural complexity compared with existing neural networks for solving this particular problem. The effectiveness of the proposed approaches is demonstrated by simulation results based on the Mitsubishi PA10-7C manipulator.