Robust Real-Time Robot/Camera Calibration - Robotized Transcranial Magnetic Stimulation - page 68

Fig. 4.6 Relationship
between the marker M and
the joints S 3 and S 2 for error
estimation of the marker
calibration. When we rotate
link three around joint two
(S 2 ) and track the marker M,
we can estimate the distance r
between M and S 2 . The same
distance can also be
calculated using the presented
marker calibration. For error
estimation, we compare both
of them

When we move the robot to its zero position (all joint angles equal to 0 ), we can
use the forward calculation to S 3 (joint 4) and S 2 (joint 3) to obtain R T S 3 and R T S 2 ,
respectively. Consequently, the transform between S 2 and S 3 is determined as

1 R T S 3 :

S 2 T S 3 ¼ R T S 2

ð 4 : 3 Þ

In addition, we get the transform from S 3 to marker M ( S 3 T M ) with the marker
calibration presented in Sect. 4.2.2 . Hence, we can calculate the transform from S 2
to the marker for this robot position (zero position):

S 2 T M ¼ S 2 T S 3 S 3 T M :

ð 4 : 4 Þ

The translational accuracy of the marker calibration can now be validated in part
because the following equation must hold:

q

S 2 T M

Þ 1 ; 4 þ S 2 T M Þ 2 ; 4

ð 4 : 5 Þ

r ¼ ð ð ;

i ; 4 denoting the i-th element of the translational part of the homoge-
neous transformation matrix S 2 T M . The relationship shown in Eq. 4.5 is visualized
in Fig. 4.6 .

with S 2 T M