Eye-to-hand camera calibration, location of the calibration board

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There will always be an offset in translation and rotation between your end effector and the calibration board. But that offset is not important, as long as it does not change during the calibration process. That's why it is important to ensure the calibration board is "rigidly connected" to the end effector.

Generally, it is sufficient to make the calibration board reasonably rigid (no visible bending under its own load, or vibrations after robot movements) and attach it to the end effector with whatever method you have readily available. By grasping the board with your robot gripper or screwing it to the robot flange, for example.