Robot manipulators are widely used in various areas of industrial factory automation. However, their base positioning is still achieved through trial-and-error methods based on the intuition and expertise of the engineer, even with the use of off-line programming software. Most previous studies do not provide on-line or on-site solutions suitable for practical applications because the nonlinearity and derivative complexity of the robot kinematics result in heavy computational burden and lengthy processing times. In this paper, we suggest a convex programming approach that uses time-efficient and reliable methods to solve the optimization problem in order to determine the base position of a six-degrees-of-freedom articulated robot with a spherical wrist. The proposed method uses convex optimization to accurately check the reachability of the given task without solving the inverse kinematics and to determine the feasible base position to satisfy singularity avoidance and spatial limitations. The feasibility of the proposed method is evaluated through various simulations, and the results show that not only the feasible base position but also the range of allowable base locations as an ellipsoidal volume can be provided within a few minutes without high computing performance or large resources.