Robust Real-Time Robot/Camera Calibration - Robotized Transcranial Magnetic Stimulation - page 63

orthogonal homogeneous matrices X and Y can be found which optimally solve the
relation AX ¼ YB. In reality, however, this is not necessarily the case: A typical
robot will not be calibrated perfectly. Also, for new industrial robots deviations up
to 2-3 mm can arise [ 2 ]. Neither will an arbitrary tracking device deliver results
which are exact. Optimal calibrated optical tracking systems will have a root mean
square (RMS) error of 0.2-0.3 mm [ 27 ]. For electromagnetic tracking systems
RMS errors of 1-1.5 mm can arise [ 9 ].
Therefore, another hand-eye calibration method has been used so far for the
robotized TMS system, called QR24 calibration algorithm,[ 8 ]. It is based on a
naïve least-squares solution of the equation system AX ¼ YB. Since we deal with
real-world tracking devices and imperfect robots, the calibration algorithm allows
the matrices X and Y to be non-orthogonal, i.e., to try to correct for system
inaccuracies in the tool/flange and robot/world calibration matrices. The QR24
algorithm computes simultaneously the rotational and translational parts of the
matrices X and Y . As calibration is not necessarily required for the full robot
workspace for the robotized TMS system (and many other medical applications),
the algorithm aims for high local accuracy. Furthermore, a variation of QR24
exists, which can deal with deficient tracking data, i.e., when the localization
device only provides translational data or does not provide full rotational data.
This variation is called QR15 calibration algorithm. For synthetic data, the QR24
algorithm performs as good as the standard hand-eye-calibration methods. For the
specific setup, however, where off-the-shelf tracking systems and robots are used,
the QR24 algorithm performs up to 50 % better than the standard algorithms [ 8 ].
As the QR24 algorithm has shown to be sufficient for the robotized TMS
system, we develop the online calibration method as an enhancement of QR24. In
particular, this real-time calibration algorithm is customized to the specific
requirements of the robotized TMS system. Note that generally any hand-eye
calibration method can be enhanced to the online calibration algorithm.
4.2 Online Calibration
In this section, we first describe the necessary prerequisites to enable us to use the
online calibration method. Subsequently, we describe how the robotized system is
adapted, how the constant transform between the marker M and the coordinate
system at S 3 can be determined (which needs to be done only once), and how we
validate the accuracy of the new calibration method in comparison to the standard
hand-eye calibration algorithms (i.e., full calibration before each use of the
system).