This study was performed to evaluate the relations between robot pose and stiffness in a robot drilling system with the spindle motor mounted on the hand of an industrial robot. There is increasing demand for robotic machining of large and long workpieces. However, the challenge is that the robot has lower stiffness than the machine tools. Here, the stiffness of the robot was quantified with a dynamically manipulable ellipsoid (DME) for each robot pose. Although stiffness anisotropy was maximized at the singularity of the robot, the joints rotated rapidly when passing through the singularity. Therefore, an algorithm to pass through the singularity by changing the type of inverse kinematics solution was applied to achieve singularity machining. The results showed that the stiffness increased in a specific direction at the singularity.