Biomedical Engineering Reference
Note that r only describes the distance to the axis of rotation, whence the z-
3 ; 4 , cannot be taken into account. The z-axis of S 2 is in
accordance with the axis of rotation of joint three.
S 2 T M , i.e. S 2 T M
4.2.5 Error Calculation for Online Calibration
For calculating errors of the calibration methods, we compare two calibration
results. Therefore, the two general calibration results T 1 and T 2 are used. To
compare the difference between these two, we use
T e 1 ¼ T 1 T 1
and T e 2 ¼ T 2 T 1
ð 4 : 6 Þ
where T e i are homogeneous matrices with rotational parts R i and translational parts
t i :
R i t i
T e i ¼
ð 4 : 7 Þ
The translational error e rot is computed as
e trans ¼ 1
t kk 2 þ t kk 2
ð 4 : 8 Þ
and the rotational error e rot is computed as
e rot ¼ 1
ð 4 : 9 Þ
j h 1 jþj h 2 j
using the axis-angle (i.e., a i ; h ð Þ ) representation of the matrices R i .
Using both relationships T e 1 ¼ T 1 T 2 and T e 2 ¼ T 2 T 1 is necessary since
the matrices T 1 and T 2 may result from a calibration method which does not
produce orthogonal matrices. Consequently, since we do not wish to privilege one
frame of reference, the average of the errors is used. This, and the way of com-
puting the rotational error, is in line with the approaches proposed in [ 22 ].
4.2.6 Data Acquisition for Evaluation
Besides estimation of the translational error of marker calibration, we use three
different setups to evaluate the online calibration method and compare it with the
QR24 algorithm [ 8 ] and the hand-eye calibration method by Tsai and Lenz [ 25 ].
Therefore, we mount the tracking system to a KUKA KR 16 robot (Kuka AG,
Augsburg, Germany) such that it is about 2 m away from the Adept robot, as
shown in Fig. 4.7 . We can thus move the tracking system by fixed distances,
perform the calibration, and compare the results.