Robust Real-Time Robot/Camera Calibration - Robotized Transcranial Magnetic Stimulation - page 65

Note that the coordinate system S 3 belongs to the robot's third link and has its
origin in the robot's fourth joint. When using the forward calculation from the
robot's base R ( ¼ S 0 ) to the third link, we can calculate the transform R T S 3 . The
forward kinematics uses the robot specific DH-parameters to calculate the trans-
form from one joint to the following joint [ 6 ]. The free parameters are given by the
specific robot joint positions. In the DH-convention S i denotes to the coordinate
system which is associated to the robot's i þ 1-th joint, which also corresponds to
the robot's i-th link.
Now, according to standard hand-eye calibration methods, we move the robot
to a set of random positions within a sphere of 200 mm radius. At these positions
we track the marker with the tracking system and calculate the corresponding
position of S 3 . Based on this dataset, the QR24 algorithm for hand-eye calibration
calculates the transformation from the tracking system to the robot as well as the
transform from marker to robot link. To perform the marker calibration in such a
manner, at least three distinct measurements are theoretically required. As this
marker calibration must only be performed once (as long as the marker does not
shift), we use 500 random positions for measurement and computation. This
reduces the impact of noise and achieves an optimal calibration.

4.2.3 Robust Real-Time Calibration

Once the constant transform S 3 T M is estimated, online calibration can be per-
formed for any robot/tracking system position i: To do this, we track the marker M
with the tracking system T to obtain T T ð Þ i . Furthermore, we use the robot
forward calculation to the fourth joint to get R T S ð Þ i for this robot pose. Now we
can calculate the robot to tracking system calibration for position i as

i ¼ i S 3 T M 1

i

R T T R T S 3 T T M

ð 4 : 2 Þ

:

Note that T T ð Þ i and R T S ð Þ i can be obtained online. Thus, this calibration can
be performed while the application is running. We only have to make sure that
both measurements are synchronized.
Figure 4.5 illustrates the operation cycle of the algorithm. For the robotized
TMS application, online calibration is performed during the application start to
estimate the calibration from tracking system to robot. As a first step, the user has to
perform a registration of the patient's head to a virtual head contour. This is done
with a headband and a pointer, both measured by the tracking system (see
Sect. 1.2.1 ) . Before data is acquired from these two markers, the calibration is
checked with the online calibration method. This check just requires to track the
marker at link three, to compute a robot forward calculation to joint four and to
perform two matrix multiplications. It is thus available in less than 200 ms on any
standard desktop computer. Note that all computations are performed in the robot
coordinate frame. If an error occurs (shift of tracking system or robot), the user will