We propose a nonlinear inverse kinematics formulation which solves for positions directly. Compared to various other popular methods that integrate velocities, this formulation can better handle fast, asymmetric and singular-postured balancing tasks for humanoid robots. We also introduce joint position and velocity boundaries as inequality constraints in the optimization to ensure feasibility. Such boundaries provide safety when approaching or getting away from joint limits or singularities. Besides, mixing positions and velocities in our proposed algorithm facilitates recovery from singularities, which is very difficult for conventional inverse kinematics methods. Extensive demonstrations on the real robot prove the applicability of the proposed algorithm while improving power consumption. Our formulation automatically handles different numerical and behavioral difficulties rising from singularities, which makes it a reliable low-level conversion block for different Cartesian planners.