  Hand-eye calibration is the problem of solving the transformation from the
end-effector of a robot to the sensor attached to it. Commonly employed
techniques, such as AXXB or AXZB formulations, rely on regression methods that
require collecting pose data from different robot configurations, which can
produce low accuracy and repeatability. However, the derived transformation
should solely depend on the geometry of the end-effector and the sensor
attachment. We propose Geometry-Based End-Effector Calibration (GBEC) that
enhances the repeatability and accuracy of the derived transformation compared
to traditional hand-eye calibrations. To demonstrate improvements, we apply the
approach to two different robot-assisted procedures: Transcranial Magnetic
Stimulation (TMS) and femoroplasty. We also discuss the generalizability of
GBEC for camera-in-hand and marker-in-hand sensor mounting methods. In the
experiments, we perform GBEC between the robot end-effector and an optical
tracker's rigid body marker attached to the TMS coil or femoroplasty drill
guide. Previous research documents low repeatability and accuracy of the
conventional methods for robot-assisted TMS hand-eye calibration. When compared
to some existing methods, the proposed method relies solely on the geometry of
the flange and the pose of the rigid-body marker, making it independent of
workspace constraints or robot accuracy, without sacrificing the orthogonality
of the rotation matrix. Our results validate the accuracy and applicability of
the approach, providing a new and generalizable methodology for obtaining the
transformation from the end-effector to a sensor.
