The algorithms of inverse kinematics based on optimality constraints have some problems because those are based only on necessary conditions for optimality. One of the problems is a switching problem, i.e., an undesirable configuration change from a maximum value of a performance measure to a minimum value may occur and cause an inverse kinematic solution to be unstable. In this paper, we derive sufficient conditions for the optimal solution of the kinematic control of a redundant manipulator. In particular, we obtain the explicit forms of the switching condition for the optimality constraints-based methods. We also show that the configuration at which switching occurs is equivalent to an algorithmic singularity in the extended Jacobian method. Through a numerical example of a cyclic task, we show the problems of the optimality constraints-based methods. To obtain good configurations without switching and kinematical singularities, we propose a simple algorithm of inverse kinematics.