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This paper proposes a new inequality-based criterion/constraint with its algorithmic and computational details for obstacle avoidance of redundant robot manipulators. By incorporating such a dynamically updated inequality constraint and the joint physical constraints (such as joint-angle limits and joint-velocity limits), a novel minimum-velocity-norm (MVN) scheme is presented and investigated for robotic redundancy resolution. The resultant obstacle-avoidance MVN scheme resolved at the joint-velocity level is further reformulated as a general quadratic program (QP). Two QP solvers, i.e., a simplified primal-dual neural network based on linear variational inequalities (LVI) and an LVI-based numerical algorithm, are developed and applied for online solution of the QP problem as well as the inequality-based obstacle-avoidance MVN scheme. Simulative results that are based on PA10 robot manipulator and a six-link planar robot manipulator in the presence of window-shaped and point obstacles demonstrate the efficacy and superiority of the proposed obstacle-avoidance MVN scheme. Moreover, experimental results of the proposed MVN scheme implemented on the practical six-link planar robot manipulator substantiate the physical realizability and effectiveness of such a scheme for obstacle avoidance of redundant robot manipulator.