In a previous research, we presented the Grid Method and confirmed it as a systematic and efficient problem formulation method for the task-oriented design of robot kinematics. However, our previous research was limited in two ways. First, it gave only a local optimum due to its use of a local optimization technique. Second, it used constant weights for a cost function chosen by the manual weights tuning algorithm, thereby showing low efficiency in finding an optimal solution. To overcome these two limitations, therefore, this paper presents a global optimization technique and an adaptive weights tuning algorithm to solve a formulated problem using the Grid Method. The efficiencies of the proposed algorithms have been confirmed through the kinematic design examples of various robot manipulators.