When a robot is required to perform specific tasks defined in the world frame, there is a need for finding the coordinate transformation between the kinematic base frame of the robot and the world frame. The kinematic base frame used by the robot controller to define and evaluate the kinematics may deviate from the mechanical base frame constructed based on structural features. Besides, by using kinematic modeling rules such as the product of exponentials (POE) formula, the base frame can be arbitrarily located, and does not have to be related to any feature of the mechanical structure. As a result, the kinematic base frame cannot be measured directly. This paper proposes to find the kinematic base frame by solving a hand-eye calibration problem using 3D position measurements only, which avoids the inconvenience and inaccuracy of measuring orientations and thus significantly facilitates practical operations. A closed-form solution and an iterative solution are explicitly formulated and proved effective by simulations. Comprehensive analyses of the impact of key parameters to the accuracy of the solution are also carried out, providing four guidelines to better conduct practical operations. Finally, experiments on a 7-DOF industrial robot are performed with an optical tracking system to demonstrate the superiority of the proposed method using position data only over the method using full pose data.