Robust Real-Time Robot/Camera Calibration - Robotized Transcranial Magnetic Stimulation - page 61

Fig. 4.2 TMS session and
passive marker at link three:
With a subject sitting in front
of the articulated arm, the
marker at link three is always
visible for the tracking
system. In contrast, the
robot's base is occluded by
the patient during treatment

4.1 Hand-Eye Calibration

Whenever a camera or tracking system is used to detect objects for robot inter-
action, the spatial relationship between robot and tracking system must be known.
In this way, the spatial position (and orientation) of the tracked object can be
transformed into the robot coordinate frame. Commonly, Hand-Eye Calibration
(or tool/flange and robot/world calibration) is used to determine this spatial rela-
tionship. Typically, a marker M is attached to the robot's end effector E and
measured by the tracking system T. By moving the robot to multiple positions and
recording the marker positions, the unknown transforms

R T T , the transform from

E T M , the transform from end effector

the robot's base to the tracking system, and

to marker, can be estimated.
This problem can be generalized to a matrix equation of the type AX ¼ YB
[simultaneous tool/flange and robot/world calibration], where the matrices A and B
are known and the matrices X and Y are unknown. In our case, A is the pose matrix
of the robot ( R T E ) and B is the position and orientation of the attached marker with
respect to the tracking system ( T T M ). Consequently, the matrix X is the end
effector/ marker (flange/tool) matrix ( E T M ) and Y is the robot/world matrix rep-
resenting the spatial relation between robot and tracking system ( R T T ). Figure 4.3
illustrates this relationship. This problem is now solved by taking measurements at